Advanced Algebra II Activities and Homework 2009
Advanced Algebra II Activities and Homework 2009
Advanced Algebra II Activities and Homework 2009
CONNEXIONS
Rice University, Houston, Texas
This selection and arrangement of content as a collection is copyrighted by Kenny M. Felder. It is licensed under the Creative Commons Attribution 2.0 license (http://creativecommons.org/licenses/by/2.0/). Collection structure revised: September 15, 2009 PDF generated: August 8, 2011 For copyright and attribution information for the modules contained in this collection, see p. 245.
Table of Contents
The Philosophical Introduction No One Reads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 How to Use Advanced Algebra II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Functions 1.1 The Function Game: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2 The Function Game: Leader's Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3 The Function Game: Answer Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 Homework: The Function Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 Homework: Functions in the Real World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.6 Algebraic Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.7 Homework: Algebraic Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.8 Homework: Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.9 Horizontal and Vertical Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.10 Homework: Horizontal and Vertical Permutations I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.11 Homework: Horizontal and Vertical Permutations II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.12 Sample Test: Function I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.13 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.14 Homework: Graphing Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.15 Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.16 Homework: Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1.17 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.18 Homework: Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1.19 TAPPS Exercise: How Do I Solve That For y? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.20 Sample Test: Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2 Inequalities and Absolute Values 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 4.3 4.4 4.5
Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Homework: Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Inequality Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Absolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Homework: Absolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Homework: Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Graphing Inequalities and Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Homework: Graphing Inequalities and Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Sample Test: Inequalities and Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Simultaneous Equations
Distance, Rate, and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Homework: Simultaneous Equations by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Homework: Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 The Generic Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Sample Test: 2 Equations and 2 Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Quadratics
Multiplying Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Homework: Multiplying Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Homework: Factoring Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Introduction to Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
iv
4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 7.1 7.2 7.3 7.4 7.5 7.6 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8
Homework: Introduction to Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Homework: Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 The Generic Quadratic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Homework: Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Sample Test: Quadratic Equations I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Graphing Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Graphing Quadratic Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Homework: Graphing Quadratic Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Solving Problems by Graphing Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Homework: Solving Problems by Graphing Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Homework: Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Sample Test: Quadratics II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 105
5 Exponents
Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Homework: Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 108 Extending the Idea of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Homework: Extending the Idea of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Fractional Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Homework: Fractional Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Real Life Exponential Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Homework: Real life exponential curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Sample Test: Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Logarithms
Introduction to Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Homework: Logs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Homework: Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Using the Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 So What Are Logarithms Good For, Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Homework: What Are Logarithms Good For, Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Sample Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Rational Expressions
Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Homework: Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Rational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Homework: Rational Expressions and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Dividing Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Sample Test: Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8 Radicals
Radicals (aka* Roots) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Radicals and Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Some Very Important Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Simplifying Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Homework: Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A Bunch of Other Stu About Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Homework: A Bunch of Other Stu About Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.9 Homework: Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.10 Sample Test: Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 149 9 Imaginary Numbers 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8
Imaginary Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Homework: Imaginary Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Homework: Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Me, Myself, and the Square Root of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 The Many Merry Cube Roots of -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Homework: Quadratic Equations and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 161 Sample Test: Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10 Matrices 10.1 Introduction to Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.2 Homework: Introduction to Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.3 Multiplying Matrices I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 10.4 Homework: Multiplying Matrices I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 10.5 Multiplying Matrices II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 10.6 Homework: Multiplying Matrices II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 10.7 The Identity and Inverse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 10.8 Homework: The Identity and Inverse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 10.9 The Inverse of the Generic 2x2 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10.10 Using Matrices for Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.11 Homework: Using Matrices for Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 10.12 Sample Test : Matrices I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 10.13 Homework: Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.14 Homework: Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 10.15 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.16 Homework: Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 10.17 Sample Test: Matrices II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
11 Modeling Data with Functions 11.1 11.2 11.3 11.4 11.5 11.6 11.7 12 Conics 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11
Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Homework: Inverse Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Homework: Direct and Inverse Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 From Data Points to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Homework: From Data Points to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 197 Homework: Calculator Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Sample Test: Modeling Data with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Homework: Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 All the Points Equidistant from a Given Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Homework: Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 All the Points Equidistant from a Point and a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Homework: Vertical and Horizontal Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Parabolas: From Denition to Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Sample Test: Distance, Circles, and Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Distance to this point plus distance to that point is constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Homework: Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 The Ellipse: From Denition to Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
vi
12.12 12.13 12.14 13.1 13.2 13.3 13.4 13.5 13.6 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8
Distance to this point minus distance to that point is constant . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Homework: Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Sample Test: Conics 2 (Ellipses and Hyperbolas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
14 Probability
How Many Groups? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Homework: Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Introduction to Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Homework: The Multiplication Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Homework: Trickier Probability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Homework: Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Homework: Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Sample Test: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
(*but it's real short so please read it anyway) Welcome to Advanced Algebra II at Raleigh Charter High School! There are three keys to succeeding in this math class. 1. Do the homework 2. Ask questions in class if you don't understand anything. 3. Focus on
understanding, not just doing the problem right. (Hint: you understand something
when you say Gosh, that makes sense! I should have thought of that myself !) Here's how it works. The teacher gets up and explains something, and you listen, and it makes sense, and you get it. You work a few problems in class. Then you go home, stare at a problem that looks exactly like the one the teacher put up on the board, and realize you have no idea how to do it. How did that happen? It looked so simple when the teacher did it! Hmm. . .. So, you dig through your notes, or the book, or you call your friend, or you just try something, and you try something else, and eventually. . .ta-da! You get the answer! Hooray! Now, you have learned the concept. You didn't learn it in class, you learned it when you gured out how to do it. Or, let's rewind time a bit. You dig through your notes, you just try something, and eventually. . .nothing. You still can't get it. work! Either way, you win. But if you don't do the homework, then even if the teacher explains the exact same thing in class the next day, it won't help. . .any more than it helped the previous day. The materials in this course-pack were originally developed for Mr. Felder's Advanced Algebra II classes in the 2001-2002 school year. Every single student in those classes got an A or a B on the North Carolina End of Course test at the end of the year. You can too! Do your homework, ask questions in class, and always keep your focus on real understanding. The rest will take care of itself. teacher explains how to do it, you will have that Aha!
That's OK! Come in the next day and say I couldn't get it. This time, when the experience: So that's why I couldn't get it to
1 This
Over a period of time, I have developed a set of in-class assignments, homeworks, and lesson plans, that work for me and for other people who have tried them. The complete set comprises three separate books
The Homework and Activities Book the students day-by-day. The Concepts Book
4
provides conceptual explanations, and is intended as a reference or review guide provides lesson plans; it is your guide to how I envisioned these materials being
for students; it is not used when teaching the class. The Teacher's Guide
5
used when I created them (and how I use them myself ). Instructors should note that this book probably contains more information than you will be able to cover in a single school year. I myself do not teach from every chapter in my own classes, but have chosen to include these additional materials to assist you in meeting your own needs. As you will likely need to cut some sections from the book, I strongly recommend that you spend time early on to determine which modules are most important for your state requirements and personal teaching style. One more warning is important: these materials were designed for an
such a course, I hope this will provide you with ready-to-use textbook and lesson plans. If you are teaching a Standard or Remedial-level course, these materials will still be useful, but you will probably have to cut or reduce some of the most conceptual material, and supplement it with more drill-and-practice than I provide. The following table of contents provides a list of topics covered in this course with links to each module. You can use these links to move between the books or to jump ahead to any topic.
[m19335]
646
2 This content is available online at <http://cnx.org/content/m19435/1.6/>. 3 Advanced Algebra II: Activities and Homework <http://cnx.org/content/col10686/latest/> 4 Advanced Algebra II: Conceptual Explanations <http://cnx.org/content/col10624/latest/> 5 Advanced Algebra II: Teacher's Guide <http://cnx.org/content/col10687/latest/>
3
The Rule of Consistency Four Ways to Represent a Function Domain and Range The Function Game The Function Game: Introduction The Function Game:
[m19190] [m18195]
650 651
[m18191]
[m19121] [m19125]
[m19342]
[m19126]
656
in
the
Real
[m18193]
658
[m19331]
659
in the Real World Function Notation Algebraic tions Homework: Algebraic [m19108]
665
[m18188] [m18186]
Generaliza-
[m19114]
[m19332]
Generalizations Graphing Homework: Graphing Horizontal and Vertical Permutations Homework: and Vertical tions I Homework: and Vertical tions II Sample Test: Functions I Lines [m18197]
675
[m18196]
667
[m19339]
670
Horizontal Permuta-
[m19119]
671
Horizontal Permuta-
[m31952]
672
[m19122]
673
[m19340]
674
[m19113]
676
[m19337]
677
Homework: Lines
Graphing
[m19118]
678
Composite Functions Homework: Functions Inverse Functions Homework: Functions TAPPS Exercise: How Inverse Composite
[m18187]
679
[m19109] [m19107]
680 682
[m19333]
681
[m18198]
683
[m19112] [m19120]
684 686
[m19336]
685
[m19123]
687
[m19432] [m18205]
690
689 692
[m19430]
[m19428]
695
Value
Equa-
[m18201]
696
[m19148]
697
[m19426]
698
[m18207]
700
[m19151]
701
[m19431]
702
Value Inequalities Graphing Absolute Values Graphing Inequalities Graphing Inequalities [m18208]
705 706 707
[m18199]
704
[m19150]
[m19433]
"Piecewise
Functions"
[m18200]
708
and Absolute Value Homework: Inequalities lute Values Sample Test: Inequali[m19166]
710
[m19153]
709
Simultaneous tions
Equa[m19497]
711
Introduction to Simultaneous Equations Distance, Time Simultaneous Equations by Graphing Homework: neous Graphing Substitution Elimination Special Cases Word Problems Using Letters as Numbers Simultaneous Equations Homework: Simultane[m19293] [m19289]
722 724
Rate,
and
[m18211]
712
[m19288]
713
[m18209]
714
[m19291]
715
Simultaby
[m19291]
716
Equations
[m19498]
723
[m19294]
725
[m19499]
726
Quadratics
Introduction [m19469]
728
[m18224]
729
[m19247] [m19253]
730 732
[m19472]
731
[m18227] [m18222]
733 736
[m19243]
734
[m19466]
735
Equations by Factoring Homework: Expressions Introduction Quadratic Equations Homework: tions Solving Quadratic [m18217]
741
Factoring
[m19248]
737
to
[m19246]
738
[m19470]
739
Introduc-
[m19251]
740
Equations by Completing the Square Completing the Square Homework: Completing the Square The Quadratic Formula The "Generic" [m18231]
745 746 747
[m19242] [m19249]
742 744
[m19465]
743
[m19262]
[m19480]
Quadratic Equations Sample Test: Quadratic Equations I Dierent Types of Solutions to Quadratic Equations Graphing Functions Graphing Functions II Quadratic [m19244]
754
[m19259]
749
[m18216]
750
Quadratic
[m18228]
751
[m19245]
752
[m19468]
753
[m19467]
755
Homework:
Graphing
[m19250]
756
Problems
by
[m18220]
757
[m19260]
758
[m19479]
759
Quadratic
Problems by Graphing Quadratic Equations Quadratic Inequalities Homework: Inequalities Sample Test: ics II Quadrat[m19258]
765
[m18230]
761
[m19257] [m19254]
Quadratic
[m19473]
Exponents
Introduction Exponent Concepts Laws of Exponents Zero, bers, Negative Num[m18232] [m18235] [m18234]
767 768 769
[m19325]
766
and Fractions as
Exponents Exponential Curves Rules of Exponents Homework: Exponents Extending the Idea of Exponents Homework: Extending [m19098]
776
[m18233]
[m19104] of [m19101]
[m19327]
Rules
[m19096]
774
[m19328]
775
the Idea of Exponents Fractional Exponents Homework: Exponents "Real Life" Exponential Curves [m19103]
780
[m19097] [m19100]
777 779
[m19322]
778
Fractional
[m19329]
781
[m19102]
782
[m19105]
783
Logarithms
Logarithm Concepts Logarithms Explained [m18242] [m18236]
784 785
by Analogy to Roots Rewriting Equations The Logarithm Dened as an Inverse Function Introduction Homework: Logs Properties rithms Homework: Properties [m19177]
794
Logarithm
[m18241]
786
Equations as Exponent
[m18240]
787
[m19436]
789
of
[m19269]
[m19438]
793
of Logarithms Using the Laws of Logarithms Common Logarithms Graphing Functions So What Are Loga[m19181]
799
[m19184]
795
[m19440]
796
[m18237] [m18238]
797 798
Logarithmic
rithms Good For, Anyway? Homework: Are For, Anyway? Sample Test [m19180]
802
So
What Good
[m19268]
800
[m19439]
801
Logarithms
Rational Expressions
Introduction [m19486]
803
10
Expressions
[m18304]
804
Rational
[m18296]
805
Rational
[m18301]
806
Adding and Subtracting Rational Expressions Rational Expressions Homework: Expressions Rational Equations Homework: tions Dividing Polynomials Sample Test: Expressions Rational Rational Rational
[m18303]
807
[m19278] [m19275]
808 810
[m19488]
809
[m18302]
811
[m19279] [m19277]
812 814
[m19489]
813
[m18299]
815
[m19276] [m19274]
816 818
[m19487]
817
Radicals
Radical Concepts Radicals (*aka Roots) Properties of Radicals Radicals and Exponents Some Very Important [m18271]
821 822 823
[m18244]
819 820
[m19420]
[m19419] [m19422]
Generalizations Simplifying Radicals Introduction Homework: Radicals A Bunch of Other Stu About Radicals Homework: A Bunch of Other Stu About Radicals [m19264]
830
[m18274]
824
[m19421]
825 826
[m19483]
829
11
[m18273]
831
[m19272] [m19271]
832 834
[m19485]
833
[m19273]
835
Imaginary Numbers
Introduction Imaginary Concepts Playing with i Introduction to Imaginary Numbers Imaginary Numbers Homework: Numbers Complex Numbers Equality and Inequality in Complex Numbers Homework: Numbers Quadratic Equations [m18288]
847
836
[m18286]
838 839
[m21990]
[m19129] [m19130]
840 841
Imaginary
[m18282] [m18283]
842 845
[m19128]
843
[m19423]
844
Complex
[m19132]
846
[m19425]
849
Square Root of i The Many Merry Cube Roots of -1 Homework: Numbers A Few "Extras Thoughts For on [m18284]
852
[m19131]
850
Quadratic
[m19127]
851
Experts"
12
Matrices
Matrices Introduction to Matrices Homework: Introduc[m19205]
857
[m18311]
[m19206]
[m19445]
tion to Matrices Multiplying Matrices Multiplying Matrices I Homework: Multiplying Matrices I Multiplying Matrices II Homework: Multiplying Matrices II The Identity Matrix The Inverse Matrix The Identity and In[m18293] [m18294]
865 866 867 868
[m18291]
[m19207] [m19196]
[m19448]
[m19208] [m19201]
862 864
[m19449]
863
[m19213]
[m19443]
The Iden-
[m19194]
869
[m19446]
871
[m19451]
873
Transformations Homework: trices tions Sample Test: Matrices I Matrices on a TI-83 or TI-84 Calculator Matrices on the Calculator [m19447]
877
Using MaTransforma-
[m19190]
874
for
[m19210] [m18290]
876
875
13
Homework: Calculators Determinants Homework: ers Solving tions Homework: Solving Linear Equa[m18292]
882
[m19188] [m18289]
879
878 880
[m19442] [m19193]
881
Determin-
[m19212]
883
[m19450]
884
[m19204]
885
[m19454] Con[m18277]
888
887
Modeling
[m18281]
889
[m19452]
890
891 892
Inverse Variation Finding a Linear Function For Any Two Points Finding Function a Parabolic For Any [m18279]
895
[m18278]
894
[m19453] [m19231]
899
14
Sample Test:
Modeling
[m19222]
901
Conics
Introduction Conic Concepts A Mathematical Look [m18265] [m18246]
903 904
[m19307]
902
at Distance Distance Homework: Distance Circles All the Points Equidistant from a Given Point Homework: Circles Parabolas All the Points Equidistant from a Point and a Line Parabolas: Day 1 Homework: las Parabolas: Day 2 Parabolas: From Denition to Equation Sample Test: Distance, [m19094]
919
905 907
[m19299]
906
[m19298] [m19078]
910
909
[m19084] [m18268]
912
911
[m19079]
913
[m19315] [m19086]
915
914
Vertical
[m19313] [m19092]
917
916 918
[m19311]
Circles, and Parabolas Ellipses Distance from this point plus distance to that point is Constant Homework: Ellipses Ellipses: From Deni[m19088] [m19095]
923 924 925
[m18247]
920 922
[m19303] [m19083]
921
[m19305]
tion to Equation
15
Hyperbolas Distance from this point minus distance from that point is constant Homework: Hyperbolas A Brief Recap: How Do You Tell What Shape It Is? Sample Test: las) Conics 2
[m18249]
926 928
[m19306] [m19082]
927
[m19089] [m18270]
930
929
[m19093]
931
[m19495]
932
[m19285]
[m19490]
metric Sequences Homework: metic and Sequences Series Series and Series Notation Homework: Series and [m19280]
939
ArithGeometric
[m19284]
936
[m19074]
937 938
[m19491]
and
Geo-
[m19494]
940
and Geometric Series Proof by Induction Homework: Proof by Induction Extra Credit Sample Test: Sequences and Series [m19283]
946
[m19075]
942 944
[m19492] [m19281]
943
[m19493]
945
16
Probability
How Many Groups? Tree Diagrams Homework: grams Probability Concepts Introduction to Probability Homework: The Multiplication Rule Trickier Problems Homework: Trickier [m19235]
955
[m19236]
947 948
[m19463] Dia[m19234]
949
Tree
[m19073]
[m19237]
[m19461]
[m19233]
953
Probability
[m19464]
954
[m19072] Permuta-
956 958
[m19462] [m19241]
957
[m19460] [m19240]
960
Probabil-
[m19238]
962
17
644 Advanced Algebra II: Conceptual Explanations <http://cnx.org/content/col10624/latest/> 645 Advanced Algebra II: Activities and Homework <http://cnx.org/content/col10686/latest/> 646 http://cnx.org/content/m19335/latest/?collection=col10687 647 http://cnx.org/content/m18192/latest/?collection=col10624 648 http://cnx.org/content/m18194/latest/?collection=col10624 649 http://cnx.org/content/m18189/latest/?collection=col10624 650 http://cnx.org/content/m18190/latest/?collection=col10624 651 http://cnx.org/content/m18195/latest/?collection=col10624 652 http://cnx.org/content/m18191/latest/?collection=col10624 653 http://cnx.org/content/m19121/latest/?collection=col10686 654 http://cnx.org/content/m19342/latest/?collection=col10687 655 http://cnx.org/content/m19125/latest/?collection=col10686 656 http://cnx.org/content/m19126/latest/?collection=col10686 657 http://cnx.org/content/m19124/latest/?collection=col10686 658 http://cnx.org/content/m18193/latest/?collection=col10624 659 http://cnx.org/content/m19331/latest/?collection=col10687 660 http://cnx.org/content/m19115/latest/?collection=col10686 661 http://cnx.org/content/m18188/latest/?collection=col10624 662 http://cnx.org/content/m18186/latest/?collection=col10624 663 http://cnx.org/content/m19114/latest/?collection=col10686 664 http://cnx.org/content/m19332/latest/?collection=col10687 665 http://cnx.org/content/m19108/latest/?collection=col10686 666 http://cnx.org/content/m18196/latest/?collection=col10624 667 http://cnx.org/content/m19334/latest/?collection=col10687 668 http://cnx.org/content/m19116/latest/?collection=col10686 669 http://cnx.org/content/m19110/latest/?collection=col10686 670 http://cnx.org/content/m19339/latest/?collection=col10687 671 http://cnx.org/content/m19119/latest/?collection=col10686 672 http://cnx.org/content/m31952/latest/?collection=col10686 673 http://cnx.org/content/m19122/latest/?collection=col10686 674 http://cnx.org/content/m19340/latest/?collection=col10687 675 http://cnx.org/content/m18197/latest/?collection=col10624 676 http://cnx.org/content/m19113/latest/?collection=col10686 677 http://cnx.org/content/m19337/latest/?collection=col10687 678 http://cnx.org/content/m19118/latest/?collection=col10686 679 http://cnx.org/content/m18187/latest/?collection=col10624 680 http://cnx.org/content/m19109/latest/?collection=col10686 681 http://cnx.org/content/m19333/latest/?collection=col10687 682 http://cnx.org/content/m19107/latest/?collection=col10686 683 http://cnx.org/content/m18198/latest/?collection=col10624 684 http://cnx.org/content/m19112/latest/?collection=col10686 685 http://cnx.org/content/m19336/latest/?collection=col10687 686 http://cnx.org/content/m19120/latest/?collection=col10686 687 http://cnx.org/content/m19123/latest/?collection=col10686 688 http://cnx.org/content/m19117/latest/?collection=col10686 689 http://cnx.org/content/m19432/latest/?collection=col10687 690 http://cnx.org/content/m18205/latest/?collection=col10624 691 http://cnx.org/content/m19158/latest/?collection=col10686 692 http://cnx.org/content/m19430/latest/?collection=col10687 693 http://cnx.org/content/m19154/latest/?collection=col10686 694 http://cnx.org/content/m19163/latest/?collection=col10686 695 http://cnx.org/content/m19428/latest/?collection=col10687 696 http://cnx.org/content/m18201/latest/?collection=col10624 697 http://cnx.org/content/m19148/latest/?collection=col10686 698 http://cnx.org/content/m19426/latest/?collection=col10687 699 http://cnx.org/content/m19151/latest/?collection=col10686 700 http://cnx.org/content/m18207/latest/?collection=col10624 701 http://cnx.org/content/m19149/latest/?collection=col10686 702 http://cnx.org/content/m19431/latest/?collection=col10687 703 http://cnx.org/content/m19155/latest/?collection=col10686 704 http://cnx.org/content/m18199/latest/?collection=col10624
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Chapter 1
Functions
1.1 The Function Game: Introduction
1
Each group has three people. Designate one person as the Leader and one person as the Recorder. (These roles will rotate through all three people.) At any given time, the Leader is looking at a sheet with a list of functions, or formulas; the Recorder is looking at the answer sheet. Here's how it works.
The Leader does the formula (silently), comes up with another number, and says it. The Recorder writes down both numbers, in parentheses, separated by a comma. (Like a point.) Keep doing this until someone guesses the formula. (If someone guesses incorrectly, just keep going.) The Recorder now writes down the formulanot in words, but as an algebraic function. Then, move on to the next function.
Sound confusing? It's actually pretty easy. Suppose the rst formula was Add ve. One player says 4 and the Leader says 9. One player says -2 and the Leader says 3. One player says 0 and the Leader says 5. One player says You're adding ve and the Leader says Correct. At this point, the Recorder has written down the following: 1.
Points:
Answer:
square root and someone gives you 4. Well, you can't take the square root of a negative number: 4 is not in your domain, meaning the set of numbers you are allowed to work on. So you respond that 4 is not in my domain. Leader, help.
do not ever give away the answer!!! But everyone, feel free to ask the teacher if you need
Only the leader should look at this sheet. Leader, use a separate sheet to cover up all the functions below the one you are doing right now. That way, when the roles rotate, you will only have seen the ones you've done.
1. Double the number, then add six.
1 This 2 This
23
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CHAPTER 1. FUNCTIONS
2. Add three to the number, then double. 3. Multiply the number by 1, then add three. 4. Subtract one from the number. Then, compute one 5. Divide the number by two. 6. No matter what number you are given, always answer 3. 7. Square the number, then subtract four. 8. Cube the number. 9. Add two to the number. Also, subtract two from the original number.
10. Take the square root of the number. Round up to the nearest integer. 11. Add one to the number, then square. 12. Square the number, then add 1. 13. Give back the same number you were given. 14. Cube the number. Then subtract the original number from that answer. 15. Give back the
16. If you are given an odd number, respond 1. If you are given an even number, respond 2. (Fractions are not in the domain of this function.)
Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer content is available online at <http://cnx.org/content/m19124/1.1/>.
3 This
25
16.
Points Answer -
There are seven functions below (numbered #2-8). For each function,
Write the same function in algebraic notation. Generate three points from that function.
For instance, if the function were Add ve the algebraic notation would be x + 5. The three points might be
and
(5, 0).
Exercise 1.2
Triple the number, then subtract six.
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CHAPTER 1. FUNCTIONS
Exercise 1.8
Add three. Then, multiply by four. Then, subtract twelve. Then, divide by the original number.
x+35=x2
Note that this is not an equation you can solve for
all
values. It is a way of indicating that if you do the calculation on the left, and the calculation
on the right, they will always give you the same answer. In the functions #2-8 above, there are three such pairs of equal functions. Which ones are they? Write the algebraic equations that state their equalities (like my
x+35 = x2
equation).
Exercise 1.10
Of the following sets of numbers, there is one that could not possibly have been generated by any function whatsoever. Which set it is, and why? (No credit unless you explain why!)
a. b. c. d. e.
(3, 6) (4, 8) (2, 4) (6, 9) (2, 9) (3, 9) (1, 112) (2, 4) (3, 3) (3, 4) (3, 9) (4, 10) (2, 4) (1, 1) (0, 0) (1, 1) (2, 4)
each.
wants, and then brings the box to Laura to pay for them. Let n represent the number of doughnuts
a. b. c. d.
If the box has 3 doughnuts, how much does the box cost? If
c = 245
, how much does the box cost? How many doughnuts does it have?
c (n) that gives the cost of a box, as a function of the number of doughnuts
27
b. If this pattern keeps up, how many grati marks will there be on day 10? c. If this pattern keeps up, on what day will there be 40 grati marks? d. Write a function g (d)) that gives the number of grati marks as a function of the day. Exercise 1.13
Each of the following is a set of points. Next to each one, write yes if that set of points could have been generated by a function, and no if it could not have been generated by a function. (You do not have to gure out what the function is. But you may want to try for funI didn't just make up numbers randomly. . .)
a. b. c. d. e.
(1, 1) (3, 3) (1, 1) (3, 3) ________ (1, ) (3, ) (9, ) (, ) ________ (1, 1) (1, 1) (2, 4) (2, 4) (3, 9) (3, 9) ________ (1, 1) (1, 1) (4, 2) (4, 2) (9, 3) (9, 3) ________ (1, 1) (2, 3) (3, 6) (4, 10) ________
Exercise 1.14
f (x) = x2 + 2x + 1
a. f (2) = b. f (1) = c. f 3 = 2 d. f (y) = e. f (spaghetti) f. f ( x) g. f (f (x)) Exercise 1.15
Make up a function that has something to do with
movies.
a. Think of a scenario where there are two numbers, one of which depends on the other. Describe the scenario, clearly identifying the independent variable and the dependent variable. b. Write the function that shows how the dependent variable depends on the independent variable. c. Now, plug in an example number to show how it works.
a. Pick a number:_____ b. Add three:_____ c. Subtract three from your answer in part (b):_____
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content is available online at <http://cnx.org/content/m19108/1.1/>.
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CHAPTER 1. FUNCTIONS
d. What happened?_________________________________ e. Write an algebraic generalization to represent this rule._____ f. Is there any number for which this rule will not work?_____ Exercise 1.17 a. Pick a number:_____ b. Subtract ve:_____ c. Double your answer in part (b):_____ d. Add ten to your answer in part (c):_____ e. Divide your answer in part (d) by your original number (a):_____ f. Now, repeat that process for three dierent numbers. Record the number you started with
(a) and the number you ended up with (e).
Started With:_____ Ended With:_____ Started With:_____ Ended With:_____ Started With:_____ Ended With:_____ g. What happened? h. Write an algebraic generalization to represent this rule. i. Is there any number for which this rule will not work? Exercise 1.18
Here are the rst six powers of two.
21 = 2 22 = 4 23 = 8 24 = 16 25 = 32 26 = 64
a. If I asked you for 27 (without a calculator), how would you get it? More generally, how do
you always get from one term in this list to the next term?________________
8 8 = 64 7 9 = 63 5 5 = 25 4 6 = 24
29
30 30 = 900,
29 31
is?________________
b. Express this rulethe pattern in these numbersin words. c. Whew! That was ugly, wasn't it? Good thing we have math. Write the algebraic generalization for this rule.________________
d. Try out this generalization with negative numbers, with zero, and with fractions. (Show your work below, trying all three of these cases separately.) Does it always
work, or are there cases where it doesn't?
If you multiply
210
by 2, you get
211 . 23
Now, we're going to make that rule even more general. Suppose I want to multiply
25
times
means
22222
, and
222
Figure 1.1
25
23 = 28
a. Using a similar drawing, demonstrate what 103 104 must be. b. Now, write an algebraic generalization for this rule.________________ Exercise 1.21
The following statements are true.
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CHAPTER 1. FUNCTIONS
Write an algebraic generalization for this rule.________________
Exercise 1.22
In class, we talked about the following four pairs of statements.
10 10
thing.
10 10 =
one away from 10; these numbers are, of course, 9 and 11. As we saw, 9 11 is 99. It is one less than 100. b. Now, suppose we look at the two numbers that are two away from 10? Or three away? Or four away? We get a sequence like this (ll in all the missing numbers):
100
1 away from 10, the product is 1 less than 100 2 away from 10, the product is ____ less than 100 3 away from 10, the product is ____ less than 100 __ away from 10, the product is ____ less than 100 __ away from 10, the product is ____ less than 100
Table 1.2
c. Do you see the pattern? What would you expect to be the next sentence in this sequence? d. Write the algebraic generalization for this rule. e. Does that generalization work when the ___away from 10 is 0? Is a fraction? Is a negative
number? Test all three cases. (Show your work!)
The following graph shows the temperature throughout the month of March. pretending, OK?
graph upthe numbers do not actually reect the temperature throughout the month of March. We're just
8 This
31
Figure 1.2
Exercise 1.23
Give a weather report for the month of March, in words.
Exercise 1.24
On what days was the temperature exactly 0 C?
Exercise 1.25
On what days was the temperature below freezing?
Exercise 1.26
On what days was the temperature above freezing?
Exercise 1.27
What is the domain of this graph?
Exercise 1.28
During what time periods was the temperature going up?
Exercise 1.29
During what time periods was the temperature going down?
Exercise 1.30
Mary started a company selling French Fries over the Internet. For the rst 3 days, while she worked on the technology, she lost $100 per day. Then she opened for business. People went wild over her French Fries! She made $200 in one day, $300 the day after that, and $400 the day after that. The following day she was sued by an angry customer who discovered that Mary had been using genetically engineered potatoes. She lost $500 in the lawsuit that day, and closed up her business. Draw a graph showing Mary's prots as a function of days.
Exercise 1.31
Fill in the following table. Then draw graphs of the functions
y = (x + 3)
y = x2 , y = x2 + 2, y = x2 1,
y = 2x
, and
y = x
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CHAPTER 1. FUNCTIONS
x
-3 -2 -1 0 1 2 3
x2
x2 + 2
x2 1
(x + 3)
2x2
x2
Table 1.3
a. How did adding 2 to the function change the graph? b. How did subtracting 1 from the function change the graph? c. How did adding three to x change the graph? d. How did doubling the function change the graph? e. How did multiplying the graph by 1 change the graph? f. By looking at your graphs, estimate the point of intersection of the graphs
y = x2
(x + 3)
and
y =
Standing at the edge of the Bottomless Pit of Despair, you kick a rock o the ledge and it falls into the pit. The height of the rock is given by the function you dropped the rock, and
h (t) = 16t2 ,
where
h (t) = 16t
Table 1.4
b. h (0) = 0. What does that tell us about the rock? c. All the other heights are negative: what does that tell us about the rock? d. Graph the function h (t). Be sure to carefully label your axes! Exercise 1.33
Another rock was dropped at the exact same time as the rst rock; but instead of being kicked from the ground, it was dropped from your hand, 3 feet up. So, as they fall, the second rock is always three feet higher than the rst rock.
9 This
33
Table 1.5
b. Graph the function h (t) for the new rock. Be sure to carefully label your axes! c. How does this new function h (t) compare to the old one? That is, if you put them side by
side, what change would you see?
d. The original function was h (t) = 16t2 . What is the new function? . h (t) = . (*make sure the function you write actually generates the points in your table!) e. Does this represent a horizontal permutation or a verticalpermutation? f. Write a generalization based on this example, of the form: when you do such-and-such to a function, the graph changes in such-and-such a way. Exercise 1.34
A third rock was dropped from the exact same place as the rst rock (kicked o the ledge), but it was dropped
h = 0)
at that time.
0 0
1 0
1 0
Table 1.6
b. Graph the function h (t) for the new rock. Be sure to carefully label your axes! c. How does this new function h (t) compare to the original one? That is, if you put them
side by side, what change would you see?
d. The original function was h (t) = 16t2 . What is the new function? . h (t) = . (*make sure the function you write actually generates the points in your table!) e. Does this represent a horizontal permutation or a vertical permutation? f. Write a generalization based on this example, of the form: when you do such-and-such to a function, the graph changes in such-and-such a way.
10
table.
0 1
f (x).
1 2
2 4
3 8
f (x)
10 This
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CHAPTER 1. FUNCTIONS
Table 1.7
a. b. c.
f (2) = f (3) = f (4) = g (x) = f (x) 2. Think of this as a set of instructions, Whatever number you are given, plug that number into f (x), and then subtract two from the
answer.
d. e. f.
g (2) = g (3) = g (4) = h (x) = f (x 2). Think of this as a set of instructions, Whatever number you are given, subtract two. Then, plug that number into f (x).
as
g. h(2)= h. h(3)= i. h(4)= j. Graph all three functions below. Label them clearly so I can tell which is which!
2. Standing at the edge of the Bottomless Pit of Despair, you kick a rock o the ledge and it falls into the pit. The height of the rock is given by the function dropped the rock, and
h (t) = 16t2 ,
time (seconds)
height (feet)
Table 1.8
b.
h (0) = 0.
c. All the other heights are negative: what does that tell us about the rock? d. Graph the function
h (t).
35
Figure 1.3:
h (t) = 16t2
3. Another rock was dropped at the exact same time as the rst rock; but instead of being kicked from the ground, it was dropped from your hand, 3 feet up. So, as they fall, the second rock is always three feet higher than the rst rock.
second
rock.
time (seconds)
height (feet)
Table 1.9
h (t)
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CHAPTER 1. FUNCTIONS
Figure 1.4
c. How does this new function change would you see? d. The original function was
h (t)
compare to the old one? That is, if you put them side by side, what What is the new function?
h (t) = 16t2 .
h (t) =
(*make sure the function you write actually generates the points in your table!) e. Does this represent a the graph changes in
or a
to a function,
4. A third rock was dropped from the exact same place as the rst rock (kicked o the ledge), but it was dropped
1 seconds later,
h = 0)
at that time.
third
0 0
rock.
time (seconds)
1 0
1 0
height (feet)
37
Table 1.10
h (t)
Figure 1.5
c. How does this new function what change would you see? d. The original function was
h (t)
compare to the
original
h (t) = 16t2 .
h (t) =
(*make sure the function you write actually generates the points in your table!) e. Does this represent a
horizontal permutation
a way.
or a
Write a generalization based on this example, of the form: when you graph changes in
such-and-such
11
M = 2t ,
where
is the time (in years) your money has been in the bank,
11 This
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CHAPTER 1. FUNCTIONS
and M is the amount of money (in dollars) you have. Don puts $1 into the bank at the very beginning (t Susan
= 0).
also
t = 0.
mattress at home.
Of course, her $2 stash doesn't grow: so at any given time t, she has the same
amount of money that Don has, plus $2 more. Cheryl, like Don, starts with $1. But during the rst year, she hides it under year (t
her
mattress. After a
= 1)
a. Fill in the following table to show how much money each person has.
t=1
t=2
t=3
1
Table 1.11
Figure 1.6
c. Below each graph, write the function that gives this person's money as a function of time. Be sure your function correctly generates the points you gave above! (*For Cheryl, your function will not accurately represent her money between
t=0
and
t = 1,
39
2. The function
y = f (x)
Figure 1.7
(That is, what does this function give you if you give it a -2?)
The function g (x) is dened by the equation: g (x) = f (x) 1. That is to say, for any x-value you put into g (x) , it rst puts that value into f (x) , and then it subtracts 1 from the answer.
e. What is f. What is g. h.
to the
f (x)
drawing above.
The function h (x) is dened by the equation: h (x) = f (x + 1) . That is to say, for any x-value you put into h (x), it rst adds 1 to that value, and then it puts the new x-value into f (x).
i. What is j. What is k. l.
to the
f (x)
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CHAPTER 1. FUNCTIONS
Figure 1.8
of the function?
Y 1 = x3 13x 12.
x = 3, x = 1,
and
x = 4.
Figure 1.9:
y = x3 13x 12
41
up 10.
on the calculator (as Y2, so you can see the two functions together). When you have a function that
When
right.
1 unit to the
When you have a function that works, write your new function below.
12
Let
represent
a. If Chris is fteen years old, how old is David?______ b. Write a function to show how to nd David's age, given Chris's age. Exercise 1.36
D (C) =______
Sally slips into a broom closet, waves her magic wand, and emerges as. . .the candy bar fairy! Flying through the window of the classroom, she gives every student two candy bars. gone to do more good in the world. Let Then ve candy bars oat through the air and land on the teacher's desk. And, as quickly as she appeared, Sally is
bars distributed. Two for each student, and ve for the teacher.
a. Write a function to show how many candy bars Sally gave out, as a function of the number
of students.
c (s) =______
b. Use that function to answer the question: if there were 20 students in the classroom, how many candy bars were distributed? First represent the question in functional notationthen answer it. ______ c. Now use the same function to answer the question: if Sally distributed 35 candy bars, how many students were in the class? First represent the question in functional notationthen answer it. ______ Exercise 1.37
The function
f (x) =
a. Express this function algebraically, instead of in words: f (x) =______ b. Give any three points that could be generated by this function:______ c. What x-values are in the domain of this function?______ Exercise 1.38
The function
y (x)
a. Express this function algebraically, instead of in words: y (x) =______ b. Give any three points that could be generated by this function:______ c. What x-values are in the domain of this function?______
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content is available online at <http://cnx.org/content/m19122/1.1/>.
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CHAPTER 1. FUNCTIONS
Exercise 1.39
z (x) = x2 6x + 9
a. z (1) =______ b. z (0) = ______ c. z (1) =______ d. z (3) =______ e. z (x + 2) =______ f. z (z (x)) =______ Exercise 1.40
Of the following sets of numbers, indicate which ones could possibly have been generated by a function. All I need is a Yes or Noyou don't have to tell me the function! (But go ahead and do, if you want to. . .)
a. b. c. d.
(2, 4) (1, 1) (0, 0) (1, 1) (2, 4) (4, 2) (1, 1) (0, 0) (1, 1) (4, 2) (2, ) (3, ) (4, ) (5, 1) (, 2) (, 3) (, 4) (1, 5)
Exercise 1.41
Make up a function involving
music.
depends on another.
b. Name, and clearly describe, two variables. Indicate which is dependent and which is independent. c. Write a function showing how the dependent variable depends on the independent variable. If you were explicit enough in parts (a) and (b), I should be able to predict your answer to part (c) before I read it. d. Choose a sample number to show how your function works. Explain what the result means. Exercise 1.42
Here is an algebraic generalization: for any number
x , x2 25 = (x + 5) (x 5).
a. Plug x = 3 into that generalization, and see if it works. b. 20 20 is 400. Given that, and the generalization, can you nd 15 25 without a calculator?
(Don't just give me the answer, show how you got it!)
Exercise 1.43
Amy has started a company selling candy bars. Each day, she buys candy bars from the corner store and sells them to students during lunch. The following graph shows her March.
43
Figure 1.10
a. On what days did she break even? b. On what days did she lose money? Exercise 1.44
The picture below shows the graph of right forever.
y=
x.
(0, 0)
Figure 1.11
a. What is the domain of this graph? b. Write a function that looks exactly the same, except that it starts at the point
moves up-and-right from there.
(3, 1) and
Exercise 1.45
The following graph represents the graph
y = f (x).
44
CHAPTER 1. FUNCTIONS
Figure 1.12
a. b. c. d. e.
Is it a function? Why or why not? What are the zeros? For what Below is For what
y = f (x) 2.
Figure 1.13
f (x).
y = f (x).
45
Figure 1.14
Extra credit:
Here is a cool trick for squaring a dicult number, if the number immediately below it is easy to square. Suppose I want to nd 31 . That's hard. But it's easy to nd 30 , that's 900. Now, here comes the trick: add 30, and then add 31. 900
+ 30 + 31 = 961.
= 961.
a. Use this trick to nd 412 . (Don't just show me the answer, show me the work!) b. Write the algebraic generalization that represents this trick.
1.13 Lines
13
Exercise 1.46
You have $150 at the beginning of the year. (Call that day 0.) Every day you make $3.
a. b. c. d.
How much money do you have on day 1? How much money do you have on day 4? How much money do you have on day 10? How much money do you have on day money you have on any given day.
n?
e. How much is that function going up every day? This is the slope of the line. f. Graph the line. Exercise 1.47
Your parachute opens when you are 2,000 feet above the ground. (Call this time probably not realistic!)
t = 0.)
Thereafter,
you fall 30 feet every second. (Note: I don't know anything about skydiving, so these numbers are
a. How high are you after one second? b. How high are you after ten seconds? c. How high are you after fty seconds? d. How high are you after t seconds? This gives you a general formula for your height. e. How long does it take you to hit the ground? f. How much altitude are you gaining every second? This is the slope of the line. Because
you are falling, you are actually gaining negative altitude, so the slope is negative.
13 This
46
CHAPTER 1. FUNCTIONS
g. Graph the line. Exercise 1.48
Make up a word problem like exercises #1 and #2. dependent variables, as always. general equation and the slope of the line. Be very clear about the independent and Give the Make sure the relationship between them is linear!
Exercise 1.49
Compute the slope of a line that goes from
(1, 3)
to
(6, 18).
Exercise 1.50
For each of the following diagrams, indicate roughly what the slope is.
Figure 1.15:
a.
Figure 1.16:
b.
47
Figure 1.17:
c.
Figure 1.18:
d.
48
CHAPTER 1. FUNCTIONS
Figure 1.19:
e.
Figure 1.20:
f.
Exercise 1.51
Now, for each of the following graphs, draw a line with roughly the slope indicated. For instance, on the rst little graph, draw a line with slope 2.
49
Figure 1.21:
m=
1 2
Figure 1.22:
m=
1 2
50
CHAPTER 1. FUNCTIONS
Figure 1.23:
Solve for
y,
y = mx + b (. . .if
y -intercept,
and graph it
Use the slope to nd one point other than the Graph the line
y -intercept
on the line
Exercise 1.52
y = 3x 2
Slope:___________
y -intercept:___________
Other point:___________
Exercise 1.53
2y x = 4
Equation in
y = mx + b
Slope:___________
n-intercept:___________
Other point:___________
14
2y + 7x + 3 = 0
a. b. c. d. e.
Put this equation into the slope-intercept form slope = ___________ y-intercept = ___________ x-intercept = ___________ Graph it.
y = mx + b
Exercise 1.55
The points
(5, 2)
and
(7, 8)
lie on a line.
14 This
51
a. Find the slope of this line b. Find another point on this line Exercise 1.56
When you're building a roof, you often talk about the pitch of the roofwhich is a fancy word that means its slope. You are building a roof shaped like the following. The roof is perfectly symmetrical. The slope of the left-hand side is . In the drawing below, the roof is the two thick black linesthe ceiling of the house is the dotted line 60' long.
Figure 1.24
a. What is the slope of the right-hand side of the roof ? b. How high is the roof ? That is, what is the distance from the ceiling of the house, straight up to
the point at the top of the roof ?
c. How long is the roof ? That is, what is the combined length of the two thick black lines in the
drawing above?
Exercise 1.57
In the equation
y=
mx
class
y = 3x, explain why 3 is the slope. (Don't just say because it's the m + b. Explain why y will be 3 for any two points on this line, just like we explained x why b is the y-intercept.)
in in
Exercise 1.58
How do you measure the height of a very tall mountain? You can't just sink a ruler down from the top to the bottom of the mountain! So here's one way you could do it. You stand behind a tree, and you move back until you can look straight over the top of the tree, to the top of the mountain. Then you measure the height of the tree, the distance from you to the mountain, and the distance from you to the tree. So you might get results like this.
52
CHAPTER 1. FUNCTIONS
Figure 1.25
Exercise 1.59
The following table (a relation, remember those?) shows how much money Scrooge McDuck has been worth every year since 1999.
1999
$3 Trillion
2000
$4.5 Trillion
2001
$6 Trillion
Table 1.12
2002
$7.5 Trillion
2003
$9 Trillion
2004
$10.5 Trillion
a. b. c. d.
How much is a trillion, anyway? Graph this relation. What is the slope of the graph? How much money can Mr. McDuck earn in 20 years at this rate?
Exercise 1.60
Make up and solve your own word problem using slope.
15
covered with big plastic numbers for our customers. Your two best employees are Katie and Nicolas. Both of them stand at their stations by the conveyor belt. Nicolas's job is: whatever number comes to your station, next on the line. Her job is: whatever number comes to you, down the line to Sesame Street.
add 2 and then multiply by 5, and send out the resulting number. Katie is subtract 10, and send the result
15 This
53
This number comes down the line Nicolas comes up with this number, and sends it down the line to Katie Katie then spits out this number
-5
-3 -1 2 4 6 10
x 2x
Table 1.13
b. In a massive downsizing eort, you are going to re Nicolas. Katie is going to take over both functions (Nicolas's and her own). So you want to give Katie a number, and she
rst does Nicolas's function, and then her own. But now Katie is overworked, so she comes up with a shortcut: one function she can do, that covers both Nicolas's job and her own. What does Katie do to each number you give her? (Answer in words.)
Exercise 1.62
Taylor is driving a motorcycle across the country. Each day he covers 500 miles. A policeman started the same place Taylor did, waited a while, and then took o, hoping to catch some illegal activity. The policeman stops each day exactly ve miles behind Taylor. Let the number of miles Taylor has driven. Let
d equal the number of days they have been driving. (So after the rst day, d = 1.) Let T be p equal the number of miles the policeman has driven.
a. After three days, how far has Taylor gone? ______________ b. How far has the policeman gone? ______________ c. Write a function T (d) that gives the number of miles Taylor has traveled, as a function
of how many days he has been traveling. ______________
d. Write a function
p (T )
e. Now write the composite function p (T (d)) that gives the number of miles the policeman has traveled, as a function of the number of days he has been traveling. Exercise 1.63
Rashmi is a honor student by day; but by night, she works as a hit man for the mob. Each month she gets paid $1000 base, cashall $20 bills. Let
plus an extra $100 for each person she kills. Of course, she gets paid in
equal the number of people Rashmi kills in a given month. Let m be the amount of money
m (k) b (m)
that tells how much money Rashmi makes, in a given month, as a that tells how many bills Rashmi gets, in a given month, as a
function of the number of people she kills. ______________ function of the number of dollars she makes. ______________
b (m (k))b
d. If Rashmi kills 5 men in a month, how many $20 bills does she earn? First, translate this question into function notationthen solve it for a number. e. If Rashmi earns 100 $20 bills in a month, how many men did she kill? First, translate this question into function notationthen solve it for a number.
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CHAPTER 1. FUNCTIONS
Exercise 1.64
Make up a problem like exercises #2 and #3. Be sure to take all the right steps: dene the scenario, dene your variables clearly, and then show the functions that relate the variables. This is just like the problems we did last week, except that you have to use three variables, related by a composite function.
Exercise 1.65
f (x) =
x+2
g (x) = x2 + x
a. f (7) = ______________ b. g (7) = ______________ c. f (g (x)) =______________ d. f (f (x)) = ______________ e. g (f (x)) = ______________ f. g (g (x)) = ______________ g. f (g (3)) =______________ Exercise 1.66
h (x) = x 5. h (i (x)) = x.
i (x)
16
An inchworm (exactly one inch long, of course) is crawling up a yardstick (guess how long that is?). After the rst day, the inchworm's head (let's just assume that's at the front) is at the 3" mark. After the second day, the inchworm's head is at the 6" mark. After the third day, the inchworm's head is at the 9" mark. Let
equal the number of days the worm has been crawling. (So after the rst day,
d = 1.)
Let
a. After 10 days, where is the inchworm's head? ______________ b. Its tail? ______________ c. Write a function h (d) that gives the number of inches the head has traveled, as a function
of how many days the worm has been traveling. ______________
d. Write a function
t (h)
t (h (d))
Exercise 1.68
The price of gas started out at 100 /gallon on the 1st of the month. Every day since then, it has gone up 2 /gallon. My car takes 10 gallons of gas. (As you might have guessed, these numbers are all ctional.) Let
equal the date (so the 1st of the month is 1, and so on). Let
g (d)
that gives the price of gas on any given day of the month.
______________
c (g)
55
c (g (d))
d. How much money does it take to ll up my car on the 11th of the month? First, translate
this question into function notationthen solve it for a number.
First,
Exercise 1.69
Make up a problem like numbers 1 and 2. Be sure to take all the right steps: dene the scenario, dene your variables clearly, and then show the (composite) functions that relate the variables.
Exercise 1.70
f (x) =
x x2 +3x+4 . Find
f (g (x)) if. . .
a. g (x) = 3 b. g (x) = y c. g (x) = oatmeal d. g (x) = x e. g (x) = (x + 2) x f. g (x) = x2 +3x+4 Exercise 1.71
i (x)
17
We are playing the function game. Every time you give Christian a number, he doubles it and subtracts six.
a. b. c. d. e.
If you give Christian a ten, what will he give you back? If you give Christian an
x,
What number would you give Christian, that would make him give you a 0? What number would you give Christian, that would make him give you a ten? What number would you give Christian, that would make him give you an
x?
note: Try to follow the process you used to answer part (d).
Exercise 1.73
A television set dropped from the top of a 300' building falls according to the equation: 300
h (t) =
of time that has passed since it was dropped (measured in seconds), and
a. Where is the television set after 0 seconds have elapsed? b. Where is the television set after 2 seconds have elapsed? c. A man is watching out of the window of the rst oor, 20' above ground. At what time
does the television set go ying by?
56
CHAPTER 1. FUNCTIONS
e. Find a general formula
like (c) and (d).
t (h)
Find the inverse of each function. For each one, check your answer by plugging in two dierent numbers to see if they work.
Exercise 1.74
y =x+5
Inverse function: Test: Test:
Exercise 1.75
y =x+6
Inverse function: Test: Test:
Exercise 1.76
y = 3x
Inverse function: Test: Test:
Exercise 1.77
y=
x 4
Inverse function: Test: Test:
Exercise 1.78
y = 3x + 12
Inverse function: Test: Test:
Exercise 1.79
y=
100
x
Inverse function: Test: Test:
Exercise 1.80
y=
2x+3 7
Inverse function: Test: Test:
57
Exercise 1.81
y = x2
Inverse function: Test: Test:
Exercise 1.82
y = 2x
Inverse function: Test: Test:
18
On our last Sample Test, we did a scenario where Sally distributed two candy bars to each student and ve to the teacher. We found a function
c (s)that
a. b. c. d.
What was that function again? How many candy bars would Sally distribute if there were 20 students in the room? Find the inverse function. Nowthis is the key partexplain what that inverse function actually represents. Ask a wordproblem question that I can answer by using the inverse function.
Exercise 1.84
Make up a problem like #1. That is, make up a scenario, and show the function that represents that scenario. Then, give a word problem that is answered by the inverse function, and show the inverse function. For each function, nd the domain, the range, and the inverse function.
Exercise 1.85
y =2+
2x+3 7
1 x
2 (x + 3)
Exercise 1.88
x2
Exercise 1.89
x3
Exercise 1.90
(x +10)
3 3
Exercise 1.91
y=
18 This
2x+1 x
58
CHAPTER 1. FUNCTIONS
Exercise 1.92
y=
x 2x[]1
Exercise 1.93
The functions
f (x)
and
g (x)
19
y 2y+1 . Now you have
y=
x=
y,
in it anywhere. So
x,
y
y
1. The biggest problem we have is the fraction. To get rid of it, we multiply both sides by 2. Now, we distribute through. 3. Remember that our goal is to isolate we get all the things the things 4.
x (2y + 1) = y 2xy + x = y
2y + 1. y.
So now
x = y 2xy
with
without
factor out a
x = y (1 2x)
This is the
distributive property (like we did in step 2) done in reverse, and you should check it by distributing through. 5. Finally, we divide both sides by what is left in the parentheses!
Table 1.14
x 12x
=y
x x 12x is the inverse function of 2x+1 . Not convinced? Try two tests.
Now, you try it! Follow the above steps one at a time. You should switch roles at this point: the previous
student should do the work, explaining each step to the previous teacher. Your job: nd the inverse
function of
x+1 x1 .
20
cookies of dierent radii (*the plural of radius). Unknown to Lisa, Joe is very competitive about
19 This 20 This
59
his baking. He sneaks in to measure the radius of Lisa's cookies, and then makes his have a 2" bigger radius. Let
own cookies
be the
area
of Joe's cookies.
a. Write a function
Lisa's cookies.
J (L) a (J)
that shows the radius of Joe's cookies as a function of the radius of that shows the area of Joe's cookies as a function of their radius.
b. Write a function
(If you don't know the area of a circle, ask methis information will cost you 1 point.)
c. Now, put them together into the function a (J (L)) that gives the area of Joe's cookies, as a direct function of the radius of Lisa's. d. Using that function, answer the question: if Lisa settles on a 3" radius, what will be the area of Joe's cookies? First, write the question in function notationthen solve it. e. Using the same function, answer the question: if Joe's cookies end up 49 square inches in area, what was the radius of Lisa's cookies? First, write the question in function notationthen solve it. Exercise 1.95
Make up a word problem involving composite functions, and having something to do with drug use. (*I will assume, without being told so, that your scenario is entirely ctional!)
a. Describe the scenario. Remember that it must have something that depends on something else that depends on still another thing. If you have described the scenario
carefully, I should be able to guess what your variables will be and all the functions that relate them.
b. Carefully name and describe all three variables. c. Write two functions. One relates the rst variable to the second, and the other relates the
second variable to the third.
d. Put them together into a composite function that shows me how to get directly from the
third variable to the rst variable.
e. Using a sample number, write a (word problem!) question and use your composite function
to nd the answer.
Exercise 1.96
Here is the algorithm for converting the temperature from Celsius to Fahrenheit. First, multiply the Celsius temperature by
a. Write this algorithm as a mathematical function: Celsius temperature (C) goes in, Fahrenheit temperature (F ) comes out. F = (C)______ b. Write the inverse of that function. c. Write a real-world word problem that you can solve by using that inverse function. (This
does not have to be elaborate, but it has to show that you know what the inverse function
does.) d. Use the inverse function that you found in part (b) to answer the question you asked in
part (c).
Exercise 1.97
f (x) =
should
1 x + 1. g (x) = x . For (a)-(e), I am not looking for not have a g or an f in them, just a bunch of " x"'s.
answers like
[g (x)]
. Your answers
a.
f (g (x)) =
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CHAPTER 1. FUNCTIONS
b. g (f (x)) = c. f (f (x)) = d. g (g (x)) = e. g (f (g (x))) = f. What is the domain of f (x)? g. What is the domain of g (x)? Exercise 1.98
f (x) = 20 x
a. What is the domain? b. What is the inverse function? c. Test your inverse function. (No credit for just the words it worksI have to see your test.) Exercise 1.99
f (x) = 3 +
x 7
a. What is the domain? b. What is the inverse function? c. Test your inverse function. (Same note as above.) Exercise 1.100
f (x) =
2x 3x4
a. What is the domain? b. What is the inverse function? c. Test your inverse function. (Same note as above.) Exercise 1.101
For each of the following diagrams, indicate roughly what the slope is.
Figure 1.26:
a.
61
Figure 1.27:
b.
Figure 1.28:
c.
Exercise 1.102
6x + 3y = 10 y = mx + b
format:___________
Slope:___________
y -intercept:___________
Graph it!
Extra credit:
Two numbers have the peculiar property that when you the same answer.
62
CHAPTER 1. FUNCTIONS
b. If one of the numbers is x, what is the other number? (Your answer will be a function of x.) c. What number could x be that would not have any possible other number to go with it?
Chapter 2
4<6
a. Add 4 to both sides of the equation. ___________ Is it still true? b. Add 4 to both sides of the (original) equation. ___________ Is it still true? c. Subtract 10 from both sides of the (original) equation. ___________ Is it still true? d. Multiply both sides of the (original) equation by 4. ___________ Is it still true? e. Divide both sides of the (original) equation by 2. ___________ Is it still true? f. Multiply both sides of the (original) equation by 3. ___________ Is it still true? g. Divide both sides of the (original) equation by 2. ___________ Is it still true? h. In general: what operations, when performed on an inequality, reverse the inequality? Exercise 2.2
2x + 3 < 7
a. Solve for x. b. Draw a number line below, and show where the solution set to this problem is. c. Pick an x value which, according to your drawing, is inside the solution set. Plug it into
the original inequality the original inequality
2x + 3 < 7. 2x + 3 < 7.
Does the inequality hold true? Does the inequality hold true?
d. Pick an x-value which, according to your drawing, is outside the solution set. Plug it into
Exercise 2.3
10
x4 x. x
Your rst step should be adding
a. Solve for b. c. d. e.
Solve for
is on the right side. again from the original equation. This time, leave
Did your two answers come out the same? Draw a number line, and show where the solution set to this problem is. Pick an
x-value x-value
x 4. x 4.
f. Pick an
1 This
63
64
x = 4
a. Rewrite this statement as two dierent statements, joined by and or or. b. Draw a number line, and show where the solution set to this problem is. Exercise 2.5
3 < x 6
a. Rewrite this statement as two dierent statements, joined by and or or. b. Draw a number line, and show where the solution set to this problem is. Exercise 2.6 x > 7 or x < Exercise 2.7 x > 7 and x < Exercise 2.8 x < 7 or x > Exercise 2.9
Draw a number line, and show where the solution set to this problem is.
Draw a number line, and show where the solution set to this problem is.
Draw a number line, and show where the solution set to this problem is.
x> 4
a. Rewrite this statement as two dierent statements, joined by and or or. b. Draw a number line below, and show where the solution set to this problem is.
2x + 7 4x + 4
a. Solve for x. b. Draw a number line, and show where the solution set to this problem is. c. Pick an x-value which, according to your drawing, is inside the solution set. Plug it into the
original inequality
2x + 7 4x + 4. 2x + 7 4x + 4.
d. Pick an
x-value
original inequality
Exercise 2.11
14
2x < 20
a. Solve for x. b. Draw a number line, and show where the solution set to this problem is. c. Pick an x-value which, according to your drawing, is inside the solution set. Plug it into the
original inequality 14
d. Pick an
2 This
x-value
original inequality 14
65
Exercise 2.12
10 < 3x + 2 5
a. Solve for x. b. Draw a number line, and show where the solution set to this problem is. c. Pick an x-value which, according to your drawing, is inside the solution set. Plug it into the
original inequality
10 < 3x + 2 5. 10 < 3x + 2 5.
d. Pick an
x-value
original inequality
Exercise 2.13 x < 3 and x < 7. Draw a number line, and show where the solution set to this problem is. Exercise 2.14 x < 3 or x < 7. Draw a number line, and show where the solution set to this problem is. Exercise 2.15
x 2y 4
a. Solve for y . b. Nowfor the momentlet's pretend that your equation said equals instead of greater than
or less than. Then it would be the equation for a line. Find the slope and the y-intercept of that line, and graph it. Slope: _________ y-intercept__________
(x, y) (x, y)
that is that is
above that line. Plug the below that line. Plug the
x x
and and
y y
inequality from part (a). Does this point t the inequality? (Show your work. . .) inequality from part (a). Does this point t the inequality? (Show your work. . .)
e. So, is the solution to the inequality the points below or above the line? Shade the appropriate
region on your graph.
Exercise 2.16
Using a similar technique, draw the graph of
y x2 .
yx
Jacob is giving a party. 20 people showed up, but he only ordered 4 pizzas! Fortunately, Jacob hasn't
there are enough slices for everyone at the party to get at least one. Write an inequality or set that describes what n has to be.
Exercise 2.18
Whitney wants to drive to Seattle. She needs 100 gallons of gas to make the trip, but she has only $80 allocated for gas. Her strategy is to wait until the price of gas is low enough that she can make the trip. Write an inequality or set that describes what the price of gas be able to reach Seattle. Be sure to clearly dene your variable(s)!
3 This
66
to bring home an A on your report card. (A means 93 or above.) Be sure to clearly dene your
Exercise 2.20
Laura L is going to build a movie theater with
for the audience to watch that movie. (So maximum capacity is 200 audience members per screen.) In addition to audience members, there are 20 employees on the premises at any given time (selling tickets and popcorn and so on). According to code (which I am making up), she must have at least one bathroom for each 100 people in the building. (Of course, it's ne to build more bathrooms than that, if she wants!)
(n)
to the total
(p).
b. Write an inequality relating the total number of people who can possibly be in the
building
(p)
(b).
c. Now write a composite inequality (I just made that word up) that tells Laura: if you
build this many screens, here is how many bathrooms you need.
Exercise 2.21
Make up your own word problem for which the solution is an inequality, and solve it. The topic should be breakfast.
| 4 |=
Exercise 2.23
| 5 |=
Exercise 2.24
| 0 |=
Exercise 2.25
OK, now, I'm thinking of a number. All I will tell you is that the
is 7. a. Rewrite my question as a math equation instead of a word problem. b. What can my number be? Exercise 2.26
I'm thinking of a dierent number. This time, the
a. Rewrite my question as a math equation instead of a word problem. b. What can my number be?
4 This
content is available online at <http://cnx.org/content/m19148/1.1/>.
67
Exercise 2.27
I'm thinking of a dierent number. This time, the
a. Rewrite my question as a math equation instead of a word problem. b. What can my number be? Exercise 2.28
I'm thinking of a dierent number. This time, the 7.
a. Rewrite my question as a math inequality instead of a word problem. b. Does 8 work? c. Does 6 work? d. Does 8 work? e. Does 6 work? f. Write an inequality that describes all possible values for my number. Exercise 2.29
I'm thinking of a dierent number. This time, the
than 4. a. Rewrite my question as a math inequality instead of a word problem. b. Write an inequality that describes all possible values for my number. (Try a few numbers,
as we did in #7.)
Exercise 2.30
I'm thinking of a dierent number. This time, the
than
4.
a. Rewrite my question as a math inequality instead of a word problem. b. Write an inequality that describes all possible values for my number.
Stop at this point and check your answers with me before going on to the next questions.
Exercise 2.31
| x + 3 |= 7
a. First, forget that it says x + 3 and just think of it as a number. The absolute value of this number is 7. So what can this number be? b. Now, remember that this number is x + 3. So write an equation that says that x + 3 can be <your answer(s) in part (a)>. c. Solve the equation(s) to nd what x can be. d. Plug your answer(s) back into the original equation | x + 3 |= 7 and see if they work. Exercise 2.32
4 | 3x 2 | +5 = 17
a. This time, because the absolute value is not alone, we're going to start with some algebra. Leave | 3x 2 | alone, but get rid of everything around it, so you end up with | 3x 2 |
alone on the left side, and some other number on the right.
b. Now, remember that some number is 3x 2. So write an equation that says that can be <your answer(s) in part a>. c. Solve the equation(s) to nd what x can be.
3x 2
68
4 | 3x 2 | +5 =
Exercise 2.33
| 3x 3 | +5 = 4
a. Solve, by analogy to the way you solved the last two problems. b. Plug your answer(s) back into the original equation | 3x 3 | +5 Exercise 2.34
=4
| x 2 |= 2x 10.
a. Solve, by analogy to the way you solved the last two problems. b. Plug your answer(s) back into the original equation | x 2 |= 2x 10 and see if they work.
| x |= 5
a. Solve for x b. Check your answer(s) in the original equation. Exercise 2.36
| x |= 0
a. Solve for x b. Check your answer(s) in the original equation. Exercise 2.37
| x |= 2
a. Solve for x b. Check your answer(s) in the original equation. Exercise 2.38
10
| x |= 5
a. Solve for x b. Check your answer(s) in the original equation. Exercise 2.39
| x + 3 |= 1
a. Solve for x b. Check your answer(s) in the original equation. Exercise 2.40
4|x2| 3
=2
5 This
69
a. Solve for x b. Check your answer(s) in the original equation. Exercise 2.41
7 | 2x + 3 | 4 = 4
a. Solve for x b. Check your answer(s) in the original equation. Exercise 2.42
| 2x 3 |= x
a. Solve for x b. Check your answer(s) in the original equation. Exercise 2.43
| 2x + 2 |= x
a. Solve for x b. Check your answer(s) in the original equation. Exercise 2.44
| x 5 |= 2x 7
a. Solve for x b. Check your answer(s) in the original equation.
Check-yourself hint: For exercises 8, 9, and 10, one of them has no valid solutions, one has
one valid solution, and one has two valid solutions.
| x | 7
a. Solve. b. Graph your solution on a number line c. Choose a point that is in your solution set, and test it in the original inequality. Does it
work?
d. Choose a point that is not in your solution set, and test it in the original inequality. Does
it work?
Exercise 2.46
| 2x + 3 | 7
a. Write down the solution for what b. Now, solve that inequality for
6 This
2x + 3 has (2x + 3)
to be.
instead of an
(x).
x.
70
e. Choose a point that is not in your solution set, and test it in the original inequality. Does
it work?
Exercise 2.47
4 | 3x 6 | +7 > 19
a. Solve for b. c. d. e.
| 3x 6 |.
| 3x 6 |
around it.) Write down the inequality for what Now, solve that inequality for Choose a point that work?
(3x 6)
has to be.
x.
f. Choose a point that is not in your solution set, and test it in the original inequality. Does
it work?
Exercise 2.48
|3x4| 2
+6<3
a. Solve for x. (You know the drill by now!) b. Graph your solution on a number line c. Choose a point that is in your solution set, and test it in the original inequality. Does it
work?
d. Choose a point that is not in your solution set, and test it in the original inequality. Does
it work?
Exercise 2.49
6 | 2x2 17x 85 | +5 3
a. Solve for x. b. Graph your solution on a number line c. Choose a point that is in your solution set, and test it in the original inequality. Does it
work?
d. Choose a point that is not in your solution set, and test it in the original inequality. Does
it work?
| 4 + 3x |= 2 + 5x (. . .OK,
at these)
this isn't an
Exercise 2.51
| x |= x 1
Exercise 2.52
4 | 2x 3 | 5 3
7 This
content is available online at <http://cnx.org/content/m19155/1.2/>.
71
a. b. c. d.
Solve for
x.
is in your solution set, and test it in the original inequality. Does it work? not in your solution set, and test it in the original inequality. Does it
Graph your solution on a number line Choose a point that work? Choose a point that is
Exercise 2.53
3 | x 5 | +2 < 17
a. b. c. d.
Solve for
x.
is in your solution set, and test it in the original inequality. Does it work? not in your solution set, and test it in the original inequality. Does it
Graph your solution on a number line Choose a point that work? Choose a point that is
Exercise 2.54
3 | x 5 | +2 < 17
a. b. c. d.
Solve for
x.
is in your solution set, and test it in the original inequality. Does it work? not in your solution set, and test it in the original inequality. Does it
Graph your solution on a number line Choose a point that work? Choose a point that is
Exercise 2.55
2 | x + 2 | +6 < 6
a. b. c. d.
Solve for
x.
is in your solution set, and test it in the original inequality. Does it work?
Graph your solution on a number line Choose a point that work? Choose a point that is not in your solution set, and test it in the original inequality. Does it
9x + 3y 6
a. Put into a sort of y = mx + b format, except that it will be an inequality. b. Now, ignore the fact that it is an inequalitypretend it is a line, and graph that line. c. Now, to graph the inequality, shade in the area either above the line, or below the line,
as appropriate.
note: Does
have to be
d. Test your answer. Choose a point (any point) in the region you shaded, and test it in the
inequality. Does the inequality work? (Show your work.)
e. Choose a point (any point) in the region you did not shade, and test it in the inequality.
Does the inequality work? (Show your work.)
8 This
72
4x 2y + 5
a. Graph the inequality, using the same steps as above. b. Test your answer by choosing one point in the shaded region, and one point that is not in
the shaded region. Do they give you the answers they should? (Show your work.)
Exercise 2.58
y =| x |
a. Create a table of points. Your table should include at least two positive
negative
x-values,
two
x-values,
and
x = 0.
b. Graph those points, and then draw the function. Exercise 2.59
y =| x | +3.
Graph this without a table of points, by remembering what adding 3 does to any
y -values
y -values
in #3?)
Exercise 2.60
y = | x |.
Graph this without a table of points, by remembering what multiplying by 1 does
y -values
y -values
in #3?)
Exercise 2.61
Now, let's put it all together!!!
a. Graph b. Graph
answer will
or on a number line.
c. Test your answer by choosing one point in the shaded region, and one point that is not in
the shaded region. Do they give you the answers they should? (Show your work.)
d. Graph |x|+2
<
or on a number line.
e. Test your answer by choosing one point in the shaded region, and one point that is not in
the shaded region. Do they give you the answers they should? (Show your work.)
y 3|x+4|
a. Graph it. Think hard about what that +4 and that 3 will do. Generate a few points if it
will help you!
b. Test your answer by choosing one point in the shaded region, and one point that is not in
the shaded region. Do they give you the answers they should? (Show your work.)
The famous detectives Guy Noir and Nick Danger are having a contest to see who is better at catching bad guys. At 8:00 in the evening, they start prowling the streets of the city. They have twelve hours. Each of them gets 10 points for every mugger he catches, and 15 points for every underage drinker. At 8:00 the next morning, they meet in a seedy bar to compare notes. I got 100 points, brags Nick. If Guy gets enough muggers and drinkers, he will win the contest.
9 This
73
a. Label and clearly describe the relevant variables. b. Write an inequality relating the variables you listed in part (a). I should be able to read it as
If this inequality is true, then Guy wins the contest.
y = f (x).
Figure 2.1
a. Graph b. Graph
a number line. Your answer will either be a shaded region on a 2-dimensional graph, number line.
Exercise 2.65
x 2y > 4
a. Graph. b. Pick a point in your shaded region, and plug it back into our original equation
the inequality work? (Show your work!)
x 2y > 4.
Does
c. Pick a point which is not in your shaded region, and plug it into our original equation
Does the inequality work? (Show your work!)
x2y > 4.
Exercise 2.66
y > x3
a. Graph. (Plot points to get the shape.) b. Pick a point in your shaded region, and plug it back into our original equation
the inequality work? (Show your work!)
y > x3 .
Does
c. Pick a point which is not in your shaded region, and plug it into our original equation
Does the inequality work? (Show your work!)
y > x3 .
Exercise 2.67
Graph:
y+ | x |< | x |.
74
1 < 4 3x 10
a. Solve for x. b. Draw a number line below, and show where the solution set to this problem is. c. Pick an x-value which, according to your drawing, is inside the solution set. Plug it into
the original inequality work!)
1 < 4 3x
10.
(Show your
d. Pick an
x-value
1 < 4 3x 10.
Exercise 2.69
Find the
x x
4 | 2x + 5 | 3 = 17 | 5x 23 |= 21 6x
Exercise 2.70
Find the value(s) that make this equation true:
Exercise 2.71
|2x3| 3
+7>9
a. Solve for x. b. Show graphically where the solution set to this problem is. Exercise 2.72
3 | x + 4 | +7 7
a. Solve for x. b. Show graphically where the solution set to this problem is. Exercise 2.73
Make up and solve an inequality word problem, having to do with
hair.
a. Describe the scenario in words. b. Label and clearly describe the variable or variables. c. Write the inequality. (Your answer here should be completely determined by your
answers to (a) and (b)I should know exactly what you're going to write. If it is not, you probably did not give enough information in your scenario.)
Exercise 2.74
Graph
y | x | +2.
Exercise 2.75
| 2y | | x |> 6
a. Rewrite this as an inequality with no absolute values, for the fourth quadrant (lowerright-hand corner of the graph).
b. Graph what this looks like, in the fourth quadrant only. Exercise 2.76
Graph
y = x | x |.
10 This
Chapter 3
Simultaneous Equations
3.1 Distance, Rate, and Time
Exercise 3.1
You set o walking from your house at 2 miles per hour.
(t) (d)
hour
1 hour 2 hours
3 hours
10 hours
Table 3.1
d (t).
Exercise 3.2 You set o driving from your house at 60 miles per hour. a. Fill in the following table.
(t) (d)
hour
1 hour 2 hours
3 hours
10 hours
Table 3.2
d (t).
(t) (d)
hour
1 hour 2 hours
3 hours
10 hours
75
76
d (t).
(d),
rate
(r),
and time
(t).
Exercise 3.5
You start o for school at 55 mph. of an hour later, your mother realizes you forgot your lunch. She dashes o after you, at 70 mph. Somewhere on the road, she catches up with you, throws your lunch from her car into yours, and vanishes out of sight. Let
equal the distance from your house that your mother catches up with you. Let
equal
you took to reach that distance. (Note that you and your mother traveled the same distance, but in dierent times.) d should be measured in miles, and t in hours (not minutes).
the time that
a. Write the distance-rate-time relationship for you, from the time when you leave the house
until your mother catches up with you.
b. Write the distance-rate-time relationship for your mother, from the time when she leaves
the house until she catches up with you.
c. Based on those two equations, can you gure out how far you were from the house when
your mother caught you?
and
Exercise 3.6
2y = 6x + 10
and
3y = 12x + 9
a. Put both equations into y = mx + b format. Then graph them. b. List all points of intersection. c. Check these points to make sure they satisfy both equations. Exercise 3.7
y = 2x 3
and
3y = 6x + 3
a. Put both equations into y = mx + b format. Then graph them. b. List all points of intersection. c. Check these points to make sure they satisfy both equations. Exercise 3.8
y =x3
and
2y = 2x 6
a. Put both equations into y = mx + b format. Then graph them. b. List all points of intersection. c. Check these points to make sure they satisfy both equations. Exercise 3.9
y=x
2 This
and
y = x2 1
77
a. Graph them both on the back. b. List all points of intersection. c. Check these points to make sure they satisfy both equations. Exercise 3.10
y = x2 + 2
and
y=x
a. Graph them both on the back. b. List all points of intersection. c. Check these points to make sure they satisfy both equations. Exercise 3.11
y = x2 + 4
and
y = 2x + 3
a. Put the second equation into y = mx + b format. Then graph them both on the back. b. List all points of intersection. c. Check these points to make sure they satisfy both equations. Exercise 3.12
Time for some generalizations. . .
a. When graphing two lines, is it possible for them to never meet? _______
To meet exactly once? _________ To meet exactly twice? _________ To meet more than twice? ___________
b. When graphing a line and a parabola, is it possible for them to never meet? _______
To meet exactly once? _________ To meet exactly twice? _________ To meet more than twice? ___________ This last problem does not involve two lines, or a line and a parabola: it's a bit weirder than that. It is the only problem on this sheet that should require a calculator.
Exercise 3.13
y=
6x x2 +1 and
y =4 x5
a. Graph them both on your calculator and nd the point of intersection as accurately as you can. b. Check this point to make sure it satises both equations.
First
(because of the re codes) the total number of people attending (teachers and students combined) must be 56. Second (for obvious reasons) there must be one teacher for every seven students. How many students and how many teachers are invited to the party?
a. Name and clearly identify the variables. b. Write the equations that relate these variables.
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Exercise 3.15
A group of 75 civic-minded students and teachers are out in the eld, picking sweet potatoes for the needy. Working in the eld, Kasey picks three times as many sweet potatoes as Davisand then, on the way back to the car, she picks up ve sweet potatoes did Kasey and Davis each pick?
her newly increased pile, Davis remarks Wow, you've got 29 more potatoes than I do! How many
a. Name and clearly identify the variables. b. Write the equations that relate these variables. c. Solve. Your nal answers should be complete English sentences. Exercise 3.16
A hundred ants are marching into an anthill at a slow, even pace of 2 miles per hour. Every ant is carrying either one bread crumb,
a. Name and clearly identify the variables. b. Write the equations that relate these variables. c. Solve. Your nal answers should be complete English sentences. Exercise 3.17
Donald is 14 years older than Alice. 22 years ago, she was only half as old as he was. How old are they today?
a. Name and clearly identify the variables. b. Write the equations that relate these variables. c. Solve. Your nal answers should be complete English sentences. Exercise 3.18
Make up your own word problem like the ones above, and solve it.
Exercise 3.19
3x 2y = 16 7x y = 30
a. Solve by substitution b. Solve by elimination c. Check your answer Exercise 3.20
3x + 2y = 26 2x + 4y = 32
a. Solve by substitution b. Solve by elimination c. Check your answer Exercise 3.21
Under what circumstances is substitution easiest?
Exercise 3.22
Under what circumstances is elimination easiest?
79
Years after their famous race, the tortoise and the hare agree on a re-match. As they begin the race, the tortoise plods along at
21 2
81 2
mph, determined
not to repeat his original mistake... and he doesn't! The hare never slows down, and reaches the nish line 45 minutes (that is,
a. Did they run the same amount of time as each other? b. Did hey run the same distance? c. Clearly dene and label the two variables in this problem. Note that your answers to (a) and
(b) will have a lot to do with how you do this?
d. Based on your variables, write the equation d = rt for the tortoise. e. Based on your variables, write the equation d = rt for the hare. f. Now answer the question: how long did the tortoise run? Exercise 3.24
6x + 2y = 6 xy =5
a. Solve by substitution. b. Check your answers. Exercise 3.25
3x + 2y = 26 2x 4y = 4
a. Solve by substitution. b. Check your answers. Exercise 3.26
2y x = 4 2x + 20y = 4
a. Solve by substitution. b. Check your answers. Exercise 3.27
7x 3 = y 14x = 2y + 6
a. Solve by substitution. b. Check your answers. Exercise 3.28
3x + 4y = 12 5x 3y = 20
a. Solve any way you like. b. Check your answers.
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+y =6 2x + y = 4
a. Solve any way you like. You are solving for x and y ; a is just a constant. (So your nal answer will say x=blah-blah, y =blah-blah. The blah-blah will both have a in them.) b. Check your answers.
ax cx
+ by = e + dy = f a, b, c, d, e, f.
I call them generic because every possible pair of simultaneous equations looks exactly like that, except with numbers instead of and We are going to solve these equations.
Very important!!!
When I say solve it I mean nd a formula no
x =blah-blah where the blah-blah has only a, b, c, d, e, and f : dierent formula with only a, b, c, d, e, and f . If we can do that, we will be able to use these formulas to immediately solve any pair of simultaneous equations, just by plugging x
or
y.
And, of course,
y=
some
in the numbers. We can solve this by elimination or by substitution. I am going to solve for all the exact same steps we have always used in class.
a.
terms go away.
acx + bcy = ec -
bcy - ady = ec - af
y;
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3x + 4y = 18 5x + 2y = 16
We now have a new way of solving this equation: just plug into
y =
ecaf bcad
y=
(18)(5)(3)(16) (4)(5)(3)(2)
9048 206
42 14
=3
instead
y. x
in the two equations I did at the bottom (under So what did we do?)
and my
y=3
Evan digs into his pocket to see how much pizza he can aord. He has $3.00, exactly enough for two slices. But it is
all in dimes and nickels! Counting carefully, Evan discovers that he has twice
a. Identify and clearly label the variables. b. Write two equations that represent the two statements in the question. c. Solve these equations to nd how many nickels and dimes Evan has. Exercise 3.31
Black Bart and the Sheri are having a gunght at high noon. They stand back to back, and start walking away from each other: Bart at 4 feet per second, the Sheri at 6 feet per second. When they turn around to shoot, they nd that they are 55 feet away from each other.
Figure 3.1
a. Write the equation d = rt for Bart. b. Write the equation d = rt for the Sheri. c. Solve, to answer the question: for how long did they walk away from each other?
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a. Identify and clearly label the variables. b. Write two equations that represent the two conditions that Mrs. V imposed. c. Solve these equations to nd the number of works she will be assigning. Exercise 3.33 Solve by graphing. Answers may be approximate. (But use the graph paper and get as close as
you can.)
y = x2 3 y = | x | +2
Exercise 3.34 Solve, using substitution.
3x + y = 2 6x 2y = 12
Exercise 3.35 Solve, using elimination.
2x + 3y = 11 3x 6y = 4
Exercise 3.36
Solve any way you like.
2x = 6y + 12 x 9 = 3y
Exercise 3.37
Solve any way you like.
2y + 3x = 20 y+x=6
Exercise 3.38 a. Solve for
ax
x.
+ by = e cx + dy = f
b. Use the formula you just derived to nd
in these equations.
3x + 4y = 7 2x + 3y = 11
Extra credit:
Redo #9. If you used elimination before, use substitution. If you used substitution, use elimination.
Chapter 4
Quadratics
4.1 Multiplying Binomials
Exercise 4.1
Multiply:
(x + 2) (x + 2) x=3
into both my original function, and your resultant function.
Exercise 4.2
Multiply:
Exercise 4.3
Multiply:
(x + 5) (x + 5) x=
into both my original function, and your resultant function.
Exercise 4.4
Multiply: Now, leave
(x + a) (x + a) x as it is, but plug a = 3 into both my original function, and your resultant function.
Do you get two functions that are equal? Do they look familiar?
Exercise 4.5 Do not answer these questions by multiplying them out explicitly. Instead, plug these numbers
into the general formula for
(x + a)
a.
(x + 4) (x + 4) 2 (y + 7) 2 c. (z + ) 2 d. m + 2 2 e. (x 3) (*so in 2 f. (x 1) 2 g. (x a)
b. Exercise 4.6
this case,
(a
is 3.)
(x + a) (x a) = x2 a2 .
Just to refresh
your memory on how we found that, test this generalization for the following cases.
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84
CHAPTER 4. QUADRATICS
a. b. c. d.
Exercise 4.7
Test the same generalization by multiplying out
Exercise 4.8
Now, use that dierence between two squares generalization. As in #5, do not solve these by multiplying them out, but by plugging appropriate values into the generalization in #6.
(x + a) = x2 + 2ax + a2 2 (x a) = x2 2ax + a2 2 2 x a = (x + a) (x a)
Exercise 4.9
In the following drawing, one large square is divided into four regions. The four small regions are labeled with Roman numerals because I like to show o.
Figure 4.1
2 This
85
a. How long is the left side of the entire gure? _______________ b. How long is the bottom of the entire gure? _______________ c. One way to compute the area of the entire gure is to multiply these two numbers (total height
times total width). Write this product down here: Area = _______________
d. Now: what is the area of the small region labeled I? _______________ e. What is the area of the small region labeled II? _______________ f. What is the area of the small region labeled III? _______________ g. What is the area of the small region labeled IV? _______________ h. The other way to compute the area of the entire gure is to add up these small regions. Write
this sum down here: Area =_________________
i. Obviously, the answer to (c) and the answer to (h) have to be the same, since they are both the
area of the entire gure. So write down the equation setting these two equal to each other here: ____________________________________
a. (x + 3) (x + 4) b. (x + 3) (x 4) c. (x 3) (x 4) d. (2x + 3) (3x + 2) e. (x 2) x2 + 2x + 4 f. Check your answer to part (e) by substituting the number 3 into both my original function, and
your answer. Do they come out the same?
Exercise 4.11
Multiply these out using the formulae above.
(x + 3/2) 2 b. (x 3/2) 2 c. (x + 3) 2 d. (3 + x) 2 e. (x 3) 2 f. (3 x)
g. Hey, why did (e) and (f ) come out the same? ( x 3 isn't the same as 3 x, is it?) h. (x + 1/2) (x + 1/2) i. (x + 1/2) (x 1/2) 5 3 5+ 3 j. k. Check your answer to part by running through the whole calculation on your calculator: (j)
a.
3 =_______,
5+
3=
Exercise 4.12
Now, let's try going
(x + something)
, or as
(x something)
a. b. c.
, or as
(x + something) (x something).
x2 8x + 16 = x2 25 =
____________
Check by multiplying back: _______________ ____________ ____________ Check by multiplying back: _______________
x2 + 2x + 1 =
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CHAPTER 4. QUADRATICS
Check by multiplying back: _______________
d. e.
____________
Exercise 4.13
Enough squaring: let's go one higher, and see what
(x + a)
is!
a.
(x + a) (x + a) (x + a). You already know what (x + a) (x + a) is. So multiply (x + a) to nd the cubed formula. (Multiply term-by-term, then collect like terms.) 3 b. Use the formula you just found to nd (y + a) . 3 c. Use the same formula to nd (y 3) .
means (
(x + a)
that by
4.3 Factoring
Exercise 4.14
3x 4x2 + 5x + 2 =
Exercise 4.15
____________________
backward.
a. Pull out the common term of 4y from the following expression. . 16y 3 + 4y + 8y = 4x (_________) b. Check yourself, by multiplying 4y by the term you put in parentheses. c. Did it work? _______________
For each of the following expressions, pull out the highest common factor you can nd.
Exercise 4.16
+ 9y 2 =____________________
3
Exercise 4.18
100x
+ 25x2 =____________________
Exercise 4.19
4x2 y + 3y 2 x =____________________
Next, look to apply our three formulae... Factor the following by using our three formulae for
(x + y)
(x y)
, and
x2 y 2 .
Exercise 4.20
x2 9 =______________
Exercise 4.21
x2 10x + 25 =______________
Exercise 4.22
x2 + 8x + 16 =______________
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Exercise 4.23
x2 + 9 =______________
Exercise 4.24
3x2 27 =______________
note: Start by pulling out the common factor!
If all else fails, factor the "old-fashioned way"...
Exercise 4.25
Exercise 4.27
x2 6x + 5 =______________
Exercise 4.28
x2 + 8x + 6 =______________
Exercise 4.29
x2 x 12 =______________
Exercise 4.30
x2 + x 12 =______________
Exercise 4.31
x2 + 4x 12 =______________
Exercise 4.32
2x2 + 7x + 12 =______________
Exercise 4.33
Exercise 4.34
x2 12x + 32
Check your answer by plugging the number 4 into both my original expression, and your factored expression. Did they come out the same? Is there any number
Exercise 4.35
x2 4x 32
Exercise 4.36
x82 18y 4
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CHAPTER 4. QUADRATICS
Exercise 4.37
x2 + 18x + 32
Exercise 4.38
x2 18x + 32
Exercise 4.39
2x2 + 12x + 10
Exercise 4.40
100x
+ 8000x + 1000
Exercise 4.41
x2 + 4x 32
Exercise 4.42
x4 y 4
Exercise 4.43
x2 + 13x + 42
Exercise 4.44
x2 + 16
Exercise 4.45
4x2 9
Exercise 4.46
3x2 48
Exercise 4.47
2x2 + 10x + 12
Exercise 4.48
x2 9
Exercise 4.49
2x2 + 11x + 12
and
y.
and
y if. . .
Exercise 4.50
x+y =0
Exercise 4.51
xy
=0 =1 >0 <0
Exercise 4.52
xy
Exercise 4.53
xy
Exercise 4.54
xy
Exercise 4.55
OK, here's a dierent sort of problem. A swimming pool is going to be built, 3 yards long by 5 yards wide. Right outside the swimming pool will be a tiled area, which will be the same width all around. The total area of the swimming pool plus tiled area must be 35 yards.
5 This
89
a. Draw the situation. This doesn't have to be a fancy drawing, just a little sketch that shows
the 3, the 5, and the unknown width of the tiled area.
b. Write an algebra equation that gives the unknown width of the tiled area. c. Solve that equation to nd the width. d. Check your answerdoes the whole area come to 35 yards?
Solve for
by factoring.
Exercise 4.56
x2 + 5x + 6 = 0
Check your answers by plugging them into the original equation. Do they both work?
Exercise 4.57
2x2 16x + 15 = 0
Exercise 4.58
x3 + 4x2 21x = 0
Exercise 4.59
3x2 27 = 0
Solve for
x.
You may be able to do all these without factoring. Each problem is based on the previous
Exercise 4.60
x2 = 9
Exercise 4.61
(x 4) = 9
Exercise 4.62
x2 8x + 16 = 9
Exercise 4.63
x2 8x = 7
h (t) = ho + vo t 16t2
where. . .
h is the heightgiven as a function of time, of coursemeasured in feet. t is the time, measured in seconds. ho is the initial height that it had when it was thrownor, to put it another way, ho is t = 0. vo is the initial velocity that it had when it was thrown, measured in feet per secondor, it another way, vo is the velocity when t = 0.
This is sometimes called the (its height) at any given time. Use that equation to solve the questions below.
equation of motion for an object, since it tells you where the object is
Exercise 4.64
I throw a ball up from my hand. It leaves my hand 3 feet above the ground, with a velocity of 35 feet per second. (So these are the
ho
and
vo .)
a.: Write the equation of motion for this ball. You get this by taking the general equation I gave
you above, and plugging in the specic
ho
and
vo
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CHAPTER 4. QUADRATICS
b.: How high is the ball after two seconds? (In other words, what
plug in
t = 2?) t=0
into the equation? Explain in words what this
c.: What
result means.
Exercise 4.65
I throw a dierent ball, much more gently. This one also leaves my hand 3 feet above the ground, but with a velocity of only 2 feet per second.
a.: Write the equation of motion for this ball. You get this by taking the general equation I gave
you above, and plugging in the specic plug in
ho
and
vo
b.: How high is the ball after two seconds? (In other words, what
t = 2?) t=0
into the equation? Explain in words what this
c.: What
result means.
Exercise 4.66
A spring leaps up from the ground, and hits the ground again after 3 seconds. velocity of the spring as it left the ground? What was the
Exercise 4.67
I drop a ball from a 100 ft building. How long does it take to reach the ground?
Exercise 4.68
Finally, one straight equation to solve for
x:
(x 2) (x 1) = 12
x.
Exercise 4.69
x2 = 18
Exercise 4.70
x2 = 0
Exercise 4.71
x2 = 60
Exercise 4.72
x2 + 8x + 12 = 0
a. Solve by factoring. b. Now, we're going to solve it a dierent way. Start by adding four to both sides. 2 c. Now, the left side can be written as (x + something) . Rewrite it that way, and then solve
from there.
d. Did you get the same answers this way that you got by factoring?
Fill in the blanks.
Exercise 4.73
(x 3) = x2 6x + ___
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Exercise 4.74
3 2 2
= x2 ___x + ___
2
Exercise 4.75
Exercise 4.77
x2 20x + 90 = 26
Exercise 4.78
3x2 + 2x 4 = 0
note: Start by dividing by 3. The
the square.
x2
A pizza (a perfect circle) has a 3" radius for the real pizza part (the part with cheese). But they advertise it as having an area of 25 the crust?
Exercise 4.80
According to NBA rules, a basketball court must be precisely 94 feet long and 50 feet wide. (That part is truethe rest I'm making up.) I want to build a court, and of course, bleachers around it. The bleachers will be the same depth (*by depth I mean the length from the court to the back of the bleachers) on all four sides. I want the total area of the room to be 8,000 square feet. How deep must the bleachers be?
Exercise 4.81
Recall that the height of a ball thrown up into the air is given by the formula:
h (t) = ho + vo t 16t2
I am standing on the roof of my house, 20 feet up in the air. I throw a ball
up with an initial
velocity of 64 feet/sec. You are standing on the ground below me, with your hands 4 feet above the ground. The ball travels up, then falls down, and then you catch it. How long did it spend in the air? Solve by completing the square.
Exercise 4.82
x2 + 6x + 8 = 0
Exercise 4.83
x2 10x + 3 = 5
Exercise 4.84
x2 + 8x + 20 = 0
Exercise 4.85
x2 + x = 0
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CHAPTER 4. QUADRATICS
Exercise 4.86
3x2 18x + 12 = 0
Exercise 4.87
Consider the equation equation have. . .
x2 + 4x + 4 = c
where
is some constant.
will this
a. Two real answers? b. One real answer? c. No real answers? Exercise 4.88
Solve by completing the square:
x2 + 6x + a = 0. (a
is a constant.)
OK, let's say I wanted to solve a quadratic equation by completing the square. Here are the steps I would take, illustrated on an example problem. (These steps are exactly the same for any problem that you want to solve by completing the square.) Note that as I go along, I simplify thingsfor instance, rewriting
31 + 9 2
as 12 , or
1 2
12
5 1 . It is 2 as 2
Step
The problem itself Put all the the other Divide both sides by the coecient of
Example
2x2 3x 7 = 9x
and the number on
x2
(The coe-
x, squared.
both sides.) Rewrite the left side as a perfect square Square rootbut with a plus or minus! (*Remember, if
1 (x 3) = 12 2
x3= x=3
5 2
12
1 2
25
is 25,
may be 5 or -5
5 = 2
( .5, 6.5)
Table 4.1
Now, you're going to go through that same process, only you're going to start with the generic quadratic equation: ax
+ bx + c = 0
(4.1)
As you know, once we solve this equation, we will have a formula that can be used to solve equationsince every quadratic equation is just a specic case of that one! Walk through each step. Remember to simplify things as you go along!
any quadratic
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Exercise 4.89
Put all the
Exercise 4.90
Divide both sides by the coecient of
x2 .
Half the coecient of
Exercise 4.91
Add the same number to both sides. What number?
x, squared.
of that?
x?
Exercise 4.92
>This
brings us to a rational expressions momenton the right side of the equation you will be
Exercise 4.93
Rewrite the left side as a perfect square.
Exercise 4.94
Square rootbut with a plus or minus! (*Remember, if
x2 = 25, x
may be 5 or 5!)
Exercise 4.95
Finally, add or subtract the number next to the
x.
Did you get the good old quadratic formula? If not, go back and see what's wrong. If you did, give it a try on these problems! (Don't solve these by factoring or completing the square, solve them using the quadratic formula that you just derived!)
Exercise 4.96
4x2 + 5x + 1 = 0
Exercise 4.97
9x2 + 12x + 4 = 0
Exercise 4.98
2x2 + 2x + 1 = 0
Exercise 4.99
In general, a quadratic equation may have may have
two real roots, or it may have one real root, or it no real roots. Based on the quadratic formula, and your experience with the previous
2
three problems, how can you look at a quadratic equation ax roots it will have?
+ bx + c = 0
10
2x2 5x 3 = 0
a. Solve by factoring b. Solve by completing the square
b c. Solve by using the quadratic formula x = b 2a 4ac d. Which way was easiest? Which way was hardest? e. Check your answers by plugging back into the original equation.
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CHAPTER 4. QUADRATICS
Exercise 4.101
x2 5x + 30 = 5 (x + 1)
Exercise 4.102
3x2 + 24x + 60 = 0
Exercise 4.103
2 2 3x
+ 8.5x = x
Exercise 4.104
x2 x = 0
Exercise 4.105
9x2 = 16
Exercise 4.106
Consider the equation equation have. . .
x2 + 8x + c = 0
where
is some constant.
will this
a. Two real answers? b. One real answer? c. No real answers? Exercise 4.107
Starting with the generic quadratic equation ax
+ bx + c = 0,
11
a.
b. x 3 c. (x 7) (x + 7) d. (x 2) x2 4x + 4 e. (x + 3) (2x 5) f. Check your answer to part (e) by substituting in the number 1 for
here for just saying yesI have to be able to see your work!)
3 2 2
expression, and your resultant expression. Do they come out the same? (No
credit
Exercise 4.109
Here is a formula you probably never saw, but it is true: for any
and
a, (x + a) = x4 + 4x3 a +
6x a + 4xa + a
a.
2 2
(x + 2) = 4 b. (x 1) =
Exercise 4.110
Factor:
11 This
95
a. x2 36 b. 2x2 y 72y c. Check your answer to part (b) by multiplying back. (*I have to see your work!) d. x3 6x2 + 9x e. 3x2 27x + 24 f. x2 + 5x + 5 g. 2x2 + 5x + 2 Exercise 4.111
Geo has a rectangular yard which is 55' by 75'. He is designing his yard as a big grassy rectangle, surrounded by a border of mulch and bushes. The border will be the same width all the way around. The area of his entire yard is 4125 square feet. The grassy area will have a smaller area, of courseGeo needs it to come out exactly 3264 square feet. How wide is the mulch border?
Exercise 4.112
Standing outside the school, David throws a ball up into the air. The ball leaves David's hand 4' above the ground, traveling at 30 feet/sec. Raven is looking out the window 10' above ground, bored by her class as usual, and sees the ball go by. How much time elapsed between when David threw the ball, and when Raven saw it go by?
h (t) = h0 vo t 16t2 .
Exercise 4.113
Solve by factoring:
Exercise 4.114
Solve by completing the square:
Exercise 4.115
Solve by using the quadratic formula:
Exercise 4.116
Solve. (*No credit unless I see your work!) ax Solve any way you want to:
+ bx + c = 0
Exercise 4.117
2x2 + 4x + 10 = 0
Exercise 4.118
1 2
x2 x + 2 1 = 0 2
Exercise 4.119
x3 = x
Exercise 4.120
Consider the equation this equation have. . .
3x2 bx + 2 = 0,
where
is some constant.
will
a. b. c. d.
No real answers: Exactly one answer: Two real answers: Can you nd a value of b for which this equation will have two rational answersthat is, answers that can be expressed with no square root? (Unlike (a)-(c), I'm not asking for all such solutions, just one.)
Extra Credit:
(5 points) Make up a word problem involving throwing a ball up into the air. The problem should have one negative answer and one positive answer. Give your problem in wordsthen show the equation that represents your problemthen solve the equationthen answer the original problem in words.
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CHAPTER 4. QUADRATICS
12
Graph by plotting points. Make sure to include positive and negative values of
x!
y = x2 x y
Table 4.2
Note that there is a little point at the bottom of the graph. This point is called the vertex. Graph each of the following by drawing these as variations of #1that is, by seeing how the various numbers transform the graph of the vertex.
y = x2 .
Exercise 4.122
y = x2 + 3
Vertex:
Exercise 4.123
y = x2 3
Vertex:
Exercise 4.124
y = (x 5)
Vertex:
Exercise 4.125
Plot a few points to verify that your graph of #4 is correct.
Exercise 4.126
y = (x + 5)
Vertex:
Exercise 4.127
y = 2x2
Vertex:
Exercise 4.128
y = 1 x2 2
Vertex:
Exercise 4.129
y = x2
Vertex: In these graphs, each problem transforms the graph in several dierent ways.
Exercise 4.130
y = (x 5) 3
Vertex:
Exercise 4.131
Make a graph on the calculator to verify that your graph of #10 is correct.
Exercise 4.132
y = 2 (x 5) 3
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Vertex:
Exercise 4.133
y = 2 (x 5) 3
Vertex:
Exercise 4.134
y=
1 2 (x + Vertex:
5) + 3
Exercise 4.135
Where is the vertex of the general graph
y = a (x h) + k ? y! x = y2
Exercise 4.136
Graph by plotting points. Make sure to include positive and negative values of
y x
Table 4.3
Graph by drawing these as variations of #16that is, by seeing how the various numbers transform the graph of
x = y2 .
Exercise 4.137
x = y2 + 4
Exercise 4.138
x = (y 2)
Exercise 4.139
Plot a few points to verify that your graph of #18 is correct.
Exercise 4.140
x = y 2
Exercise 4.141
x = 2 (y 2) + 4
13
Yesterday we played a bunch with quadratic functions, by seeing how they took the equation permuted it. Today we're going to start by making some generalizations about all that.
y = x2
and
Exercise 4.142
y = x2
a. Where is the vertex? b. Which way does it open (up, down, left, or right?) c. Draw a quick sketch of the graph. Exercise 4.143
y = 2 (x 5) + 7
a. Where is the vertex?
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CHAPTER 4. QUADRATICS
b. Which way does it open (up, down, left, or right?) c. Draw a quick sketch of the graph. Exercise 4.144
y = (x + 3) 8
a. Where is the vertex? b. Which way does it open (up, down, left, or right?) c. Draw a quick sketch of the graph. Exercise 4.145
y = (x 6)
a. Where is the vertex? b. Which way does it open (up, down, left, or right?) c. Draw a quick sketch of the graph. Exercise 4.146
y = x2 + 10
a. Where is the vertex? b. Which way does it open (up, down, left, or right?) c. Draw a quick sketch of the graph. Exercise 4.147
Write a set of rules for looking at any quadratic function in the form where the vertex is, and which way it opens.
y = a (x h) + k
and telling
Exercise 4.148
Now, all of those (as you probably noticed) were quadratic function in the form opens.
the same thing for their cousins, the horizontal parabolas. Write a set of rules for looking at any
x = a (y k) + h
After you complete #7, stop and let me check your rules before you go on any further.
OK, so far, so good! But you may have noticed a problem already, which is that most quadratic functions that we've dealt with in the past did not look like know,
y = a (x k) + h.
x 2x 8
Answer: we put it
into the forms we now know how to graph. that? is like, and (more
Answer: Completing the square! The process is almostbut not entirelylike the one we used before to solve equations. Allow me to demonstrate. Pay careful attention to the ways in which is importantly)
Step
Example
99
The function itself We used to start by putting the number (8 in this case) on the other side. In this case, we don't have another side. But I still want to set that 8 apart. So I'm going to put the rest in parenthesesthat's where we're going to complete the square.
x2 6x 8 x2 6x 8
x2 6x+9 9 8
inside the parentheses, and at the same time subtract it outside the parentheses, so the function is,
in total, unchanged. Inside the parentheses, you now have a perfect square and can rewrite it as such. Outside the parentheses, you just have two numbers to combine. Voila! You can now graph it! Vertex
Table 4.4
(x 3) 17
(3, 17)opens
up
Your turn!
Exercise 4.149
y = x2 + 2x + 5
a. Complete the square, using the process I used above, to make it y = a (x h) b. Find the vertex and the direction of opening, and draw a quick sketch. Exercise 4.150
2
+ k.
x = y 2 10y + 15
a. Complete the square, using the process I used above, to make it x = a (y b. Find the vertex and the direction of opening, and draw a quick sketch.
k) + h.
14
y = a (x h) + k
or
x = a (y k) + h,
and graph.
Exercise 4.151
y = x2
Exercise 4.152
y = x2 + 6x + 5
Exercise 4.153
Plot at least three points to verify your answer to #2.
Exercise 4.154
y = x2 8x + 16
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CHAPTER 4. QUADRATICS
Exercise 4.155
y = x2 7
Exercise 4.156
y + x2 = 6x + 3
Exercise 4.157
Use a graph on the calculator to verify your answer to #6.
Exercise 4.158
y + x2 = x2 + 6x + 9
Exercise 4.159
y = 2x2 + 12x + 4
Exercise 4.160
x = 3y 2 + 6y
Exercise 4.161
x2 + y 2 = 9
Exercise 4.162
Explain in words how you can look at any equation, in any form, and tell if it will graph as a parabola or not.
15
h (t) = ho + vo t 16t2
Exercise 4.163
A ball is thrown upward from the ground with an initial velocity of 64 ft/sec.
a. Write the equation of motion for the ball. b. Put that equation into standard form for graphing. c. Now draw the graph. h (the height, and also the dependent variable) should be on the y -axis, and t (the time, and also the independent variable) should be on the x-axis. d. Use your graph to answer the following questions: at what time(s) is the ball on the
ground?
e. At what time does the ball reach its maximum height? f. What is that maximum height? Exercise 4.164
Another ball is thrown upward, this time from the roof, 30' above ground, with an initial velocity of 200 ft/sec.
a. b. c. d. e.
15 This
Write the equation of motion for the ball. Put that equation into standard form for graphing, and draw the graph as before. At what time(s) is the ball on the ground? At what time does the ball reach its maximum height? What is that maximum height?
101
OK, we're done with the height equation for now. The following problem is taken from a Calculus book. Just so you know.
Exercise 4.165
A farmer has 2400 feet of fencing, and wants to fence o a rectangular eld that borders a straight river. He needs no fence along the river. What are the dimensions of the eld that has the largest area?
a. We're going to start by getting a feeling for this problem, by doing a few drawings. First
of all, draw the river, and the fence around the eld by the river, assuming that the farmer makes his eld 2200 feet long. What is the total area of the eld? How far out from the river does the eld go?
After you do part (a), please stop and check with me, so we can make sure you have the right idea, before going on to part (b). b. Now, do another drawing where the farmer makes his eld only 400 feet long. How far out
from the river does the eld go? What is the total area of the eld?
c. Now, do another drawing where the farmer makes his eld 1000 feet long. How far out
from the river does the eld go? What is the total area of the eld? The purpose of all that was to make the point that if the eld is too short or too long then the area will be small; somewhere in between is the length that will give the biggest eld area. For instance, 1000 works better than 2200 or 400. But what length works best? Now we're going to nd it.
d. Do a nal drawing, but this time, label the length of the eld simply
the river does the eld go?
x.
e. What is the area of the eld, as a function of x? f. Rewrite A (x) in a form where you can graph it, and do a quick sketch. (Graph paper not
necessary, but you do need to label the vertex.)
g. Based on your graph, how long should the eld be to maximize the area? What is that
maximum area?
note: Make sure the area comes out bigger than all the other three you already did, or
something is wrong!
4.16
Homework:
16
Solving
Problems
by
Graphing
Quadratic
Functions
h (t) = ho + vo 16t2 .
Exercise 4.166
Michael Jordan jumps into the air at a spectacular 24 feet/second.
a. b. c. d. e.
16 This
Write the equation of motion for the ying Wizard. Put that equation into standard form for graphing, and draw the graph as before. How long does it take him to get back to the ground? At what time does His Airness reach his maximum height? What is that maximum height?
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CHAPTER 4. QUADRATICS
Exercise 4.167
Time to generalize! A ball is thrown upward from the ground with an initial velocity of what time does it reach its maximum height, and what is that maximum height?
vo .
At
Exercise 4.168
Find the dimensions of a rectangle with perimeter 100 ft whose area is as large as possible. (Of course this is similar to the one we did in class, but without the river.)
Exercise 4.169
There are lots of
8 + 2,
or
1 92 +
1 2 . Find the
Exercise 4.170
A pharmaceutical company makes a liquid form of penicillin. If they manufacture sell them for 200 manufacturing
units, they
dollars (in other words, they charge $200 per unit). However, the total
cost of
+ 80x + 0 + 0.003x2 .
17
Exercise 4.171
The following graph shows the temperature throughout the month of March. of March. We're just pretending, OK? Actually, I just made this graph upthe numbers do not actually reect the temperature throughout the month
Figure 4.2
17 This
103
a. On what days was the temperature exactly 0 C? b. On what days was the temperature below freezing? c. On what days was the temperature above freezing? d. What is the domain of this graph? e. During what time periods was the temperature going up? f. During what time periods was the temperature going down? Exercise 4.172
The following graph represents the graph
y = f (x).
Figure 4.3
a. Is it a function? Why or why not? b. What are the zeros? c. For what x-values is it positive? d. For what x-values is it negative? e. Draw the graph y = f (x) 2. f. Draw the graph y = f (x)
OK, your memory is now ocially refreshed, right? You remember how to look at a graph and see when it is zero, when it is below zero, and when it is above zero. Now we get to the actual quadratic inequalities part. But the good news is, there is nothing new here! First you will graph the function (you already know how to do that). Then you will identify the region(s) where the graph is positive, or negative (you already know how to do that).
Exercise 4.173
x2 + 8x + 15 > 0
a. Draw a quick sketch of the graph by nding the zeros, and noting whether the function opens
up or down.
>0that
x-values
c. Based on your answer to part (b), choose one x-value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should. Exercise 4.174
A ying sh jumps from the surface of the water with an initial speed of 4 feet/sec.
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CHAPTER 4. QUADRATICS
a. Write the equation of motion for this sh. b. Find the zeros of the graph, and graph it. c. Based on your graph, answer the question: during what time interval was the sh above the
water?
d. During what time interval was the sh below the water? e. At what time(s) was the sh exactly at the level of the water? f. What is the maximum height the sh reached in its jump?
18
x2 + 8x + 7 > 2x + 3
a. Collect all the terms on one side, so the other side of the inequality is zero. Then graph the
function by nding the zeros.
b. Based on your graph, for what x-values is this inequality true? c. Based on your answer, choose one x-value for which the inequality should hold, and one for
which it should not. Check to make sure they both do what they should.
Exercise 4.176
2x2 + 8x + 8 0
a. Graph b. Based on your graph, for what x-values is this inequality true? c. Based on your answer, choose one x-value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should. Exercise 4.177
2x2 + 8x > 9
a. Collect terms and graph b. Based on your graph, for what x-values is this inequality true? c. Based on your answer, choose one x-value for which the inequality should hold, and one for
which it should not. Check to make sure they both do what they should.
Exercise 4.178
x2 + 4x + 3 > 0
a. Graph the function b. Based on your graph, for what x-values is this inequality true? c. Based on your answer, choose one x-value for which the inequality should hold, and one for
which it should not. Check to make sure they both do what they should.
Exercise 4.179
x2 > x
a. Collect terms and graph b. Based on your graph, for what
18 This
x-values
105
c. Now, let's solve the original equation a dierent waydivide both sides by
wrong with the other one?
x.
same answer this way? If not, which one is correct? (Answer by trying points.) What went
Exercise 4.180
x2 + 6x + c < 0
a. For what values of c will this inequality be true in some range? b. For what values of c will this inequality never be true? c. For what values of c will this inequality always be true?
19
x = 3y 2 + 5
a. Opens (up / down / left / right) b. Vertex: ___________ c. Sketch a quick graph on the graph paper Exercise 4.182
y + 9 = 2x2 + 8x + 8
a. b. c. d.
Put into the standard form of a parabola. Opens (up / down / left / right) Vertex: ___________ Sketch a quick graph on the graph paper
Exercise 4.183
For what
x-values x-values
Exercise 4.184
For what is this inequality true?
Exercise 4.185
A rock is thrown up from an initial height of 4' with an initial velocity of 32 ft/sec. As I'm sure you recall,
h (t) = ho + vo t 16t2 .
a. Write the equation of motion for the rock. b. At what time (how many seconds after it is thrown) does the rock reach its peak? How
high is that peak? (Don't forget to answer both questions. . .)
c. During what time period is the rock above ground? Exercise 4.186
A hot dog maker sells hot dogs for $2 each. (So if he sells for manufacturing
2x.)
His cost
1 + 2x +
x2 . 1000
P (x)
106
CHAPTER 4. QUADRATICS
b. How many hot dogs should he make in order to maximize his prots? What is the maximum
prot?
c. How many hot dogs does he need to make, in order to make any prot at all? (The answer will be in the form as long as he makes more than this and less than that or, in other words, between this and that, he will make a prot.) Extra Credit:
a b -
y = ax2 + bx + c.
y = 3x2 + 5x + 6
Chapter 5
Exponents
5.1 Rules of Exponents
Exercise 5.1
Here are the rst six powers of two.
21 22 23 24 25 26
=2 =4 =8 = 16 = 32 = 64
a. If I asked you for 27 (without a calculator), how would you get it? More generally, how do you always get from one term in this list to the next term? b. IWrite an algebraic generalization to represent this rule. Exercise 5.2
Suppose I want to multiply
25
times
23 .
Well,
25
23
means 222. So
a. b. c. d.
5 3
2 = 2[ ] 103 104
must be.
Now, write an algebraic generalization for this rule. Show how your answer to 1b (the getting from one power of two, to the next in line) is a special case of the more general rule you came up with in 2c (multiplying two exponents).
Exercise 5.3
Now we turn our attention to division. What is
312 310 ?
1 This
107
108
CHAPTER 5. EXPONENTS
a. Write it out explicitly. (Like earlier I wrote out explicitly what
exponents into a big long fraction.)
25 23
was:
expand the
b. Now, cancel all the like terms on the top and the bottom. (That is, divide the top and
bottom by all the 3s they have in common.)
c. What you are left with is the answer. So ll this in: 310 3[ ] . 3 d. Write a generalization that represents this rule. e. Suppose we turn it upside-down. Now, we end up
bottom. Write it out explicitly and cancel 3s, as ___________________________ = applies, as opposed to the one in part (d)!
12
with you
some did
3s
on
before:
1 3[] f. Write a generalization for the rule in part (e). Be sure to mention when that generalization
the 310 12 = 3
Exercise 5.4
Use all those generalizations to simplify
x3 y 3 x7 x5 y 5
Exercise 5.5
Now we're going to raise exponents,
to exponents. What is
23
? Well,
23
when you raise anything to the fourth power, you multiply it by itself, four times. So we'll multiply
23
23
= 2[ ] .
in a similar way, and show what power of 10 it equals. b. Expand out 10 c. Find the algebraic generalization that represents this rule.
Exercise 5.6
310 35
Exercise 5.7
310 35
Exercise 5.8
35 310
Exercise 5.9
35 310
10
Exercise 5.10
5
Exercise 5.11
310 + 35
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Exercise 5.12
310 35
Exercise 5.13
6x3 y 2 z 5 4x10 yz2
Exercise 5.14
6x2 y 3 +3x3 y xy+x2 y
Exercise 5.15
(3x2 y)
+xy (xy)
Exercise 5.16
3x2 + 4xy
x
4 3 2 1
3x 3 3 3 3 = 81
Table 5.1
Exercise 5.18
In this table, every time you go to the next row, what happens to the left-hand number (x)?
Exercise 5.19
What happens to the right-hand number (3 )?
Exercise 5.20
Now, let's assume that pattern continues, and ll in the next few rows.
x
1
3x
Table 5.2
3 This
110
CHAPTER 5. EXPONENTS
Exercise 5.21
Based on this table,
30
Exercise 5.22
31
Exercise 5.23
32
Exercise 5.24
What would you expect
34
to be?
Exercise 5.25
Now check
34
Exercise 5.26
50
Exercise 5.27
52
Exercise 5.28
= (2) = (2) = 63
Exercise 5.29
3
x0 =
Exercise 5.32
(x a) =
In those last two problems, of course, you have created the general rules for zero and negative exponents. So hey, what happens to our trusty rules of exponents? Let's try. . .
Exercise 5.33
Let's look at the problem
60 6x
a. What is 60 ? Based on that, what is 60 6x ? b. What do our rules of exponents tell us about Exercise 5.34
Let's look at the problem
60 6x ?
a. What is
60 ?
Exercise 5.35
Let's look at the problem
64 63
4 This
111
a. What does 64 mean? Based on that, what is 64 63 ? b. What do our rules of exponents tell us about 64 63 ? Exercise 5.36
What would you Simplify:
x36 ?
Exercise 5.37
1 x5
Exercise 5.38
8x3 y 7
12x3 y 4
Exercise 5.39
Solve for
If the bases are the same, the exponents must be the same!)
Exercise 5.40
Solve for
Exercise 5.41
Solve for
Exercise 5.42
Solve for
x: 7x 7x+2 = 1 x: (7x )
x+2
Exercise 5.43
Solve for
=1
On the homework, we demonstrated the rule of negative exponents by building a table. Now, we're going to demonstrate it another wayby using the rules of exponents.
a. According to the rules of exponents, 75 7[ ] . 7 73 b. But if you write it out and cancel the excess 7s, then 75 = . 73 c. Therefore, since 75 can only be one thing, we conclude that these two things must be equal:
write that equation!
Exercise 5.45
Now, we're going to approach
nents,
1 2
2
=
Exercise 5.46
So, what does that tell us about
92 ?
Well, it is
92
Exercise 5.47
Using the same logic, what is
16 2 ?
5 This
112
CHAPTER 5. EXPONENTS
Exercise 5.48
What is
25 2 ? x2 ? 8 2 = 2.
1 1
Exercise 5.49
What is
Exercise 5.50
Construct a similar argument to show that
Exercise 5.51
What is
27 3 ? (1) 3 ? x3 ? x5
1 1 1
Exercise 5.52
What is
Exercise 5.53
What is
Exercise 5.54
What would you expect
to be?
Exercise 5.55
What is
25
1 2
Exercise 5.56
Check your answer to #12 on your calculator. Did it come out the way you expected? OK, we've done negative exponents, and fractional exponentsbut always with a 1 in the numerator. What if the numerator is not 1?
Exercise 5.57
Using the rules of exponents,
83
= 8[ ] .
1
2
is, so now we know what 82/3 is.
82
Exercise 5.58
83 =
Exercise 5.59
Construct a similar argument to show what
16 4
should be.
Exercise 5.60
Check
16 4
Now let's combine all our rules! number it is. (For instance, for
For each of the following, say what it means and then say what actual
92
so it is 3.)
Exercise 5.61
8 8
1 2
= = 32
1
Exercise 5.62
2 3
means
3,
end of story.)
Exercise 5.63
104
3 4
Exercise 5.64
Exercise 5.65
xb x
Exercise 5.66
a b
113
x0 = 1 1 xa = xa a b x b = xa
Let's get a bit of practice using these denitions.
Exercise 5.67
100 2
Exercise 5.68
1002 100
1 2
100 2
Exercise 5.71
100
3 2
Exercise 5.72
Check all of your answers above on your calculator. If any of them did not come out right, gure out what went wrong, and x it!
Exercise 5.73
Solve for
x:
1 x 2 x 17 1
17 2
Exercise 5.74
Solve for
x: x 2 = 9
x
Exercise 5.75
x Simplify:
Exercise 5.76
Simplify:
x+ x 1 x+ x
x2 . x
and the
y.
1
Find the inverse of each of the following functions. To do this, in some cases, you will have to rewrite the things. For instance, in #9, you will start by writing Now what? Well, remember what that means: it
y = x 2 . Switch the x and the y , and you get x = y 2 . means x = y . Once you've done that, you can solve for
y,
right?
Exercise 5.77
x3
a. Find the inverse function. b. Test it.
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CHAPTER 5. EXPONENTS
Exercise 5.78
x2
a. Find the inverse function. b. Test it. Exercise 5.79
x0
a. Find the inverse function. b. Test it. Exercise 5.80
Can you nd a generalization about the inverse function of an exponent?
Exercise 5.81
Graph
y = 2x
values.
Exercise 5.82
Graph
y = 2 2x y = 2x + 1 y=
y = 2x .
Exercise 5.83
Graph by taking the graph
y = 2x
Exercise 5.84
Graph
values.
This is a famous ancient story that I am not making up, except that I am changing some of the details. A man did a great service for the king. The king oered to reward the man every day for a month. So the man said: Your Majesty, on the rst day, I want only a penny. On the second day, I want twice that: 2 pennies. On the third day, I want twice that much again: 4 pennies. On the fourth day, I want 8 pennies, and so on. On the thirtieth day, you will give me the last sum of money, and I will consider the debt paid o. The king thought he was getting a great deal. . .but was he? Before you do the math, take a guess: how much do you think the king will pay the man on the 30th day? Now, let's do the math. For each day, indicate how much money (in pennies) the king paid the man. Do this without a calculator, it's good practice and should be quick.
Day 1: _1 penny_
Day ________
7:
Day _______
13:
Day _______
19:
Day _______
25:
7 This
115
Day ___2____ Day ___4____ Day ___8____ Day ________ Day ________
2:
Day ________
8:
Day _______
14:
Day _______
20:
Day _______
26:
3:
Day ________
9:
Day _______
15:
Day _______
21:
Day _______
27:
4:
Day _______
10:
Day _______
16:
Day _______
22:
Day _______
28:
5:
Day _______
11:
Day _______
17:
Day _______
23:
Day _______
29:
6:
Day _______
12:
Day _______
Table 5.3
18:
Day _______
24:
Day _______
30:
How was your guess? Now let's get mathematical. On the nth day, how many pennies did the king give the man? Use your calculator, and the formula you just wrote down, to answer the question: what did the king pay the man on the 30th day? _______ Does it match what you put under Day 30 above? (If not, something's wrong somewherend it and x it!) Finally, do a graph of this function, where the day is on the x-axis and the pennies is on the y axis (so you are graphing pennies as a function of day). Obviously, your graph won't get past the fth or sixth day or so, but try to get an idea for what the shape looks like.
Year
0 1 2 3 4 5
Table 5.4
Now, let's start generalizing. Suppose at the end of one year, you have x dollars. How much does the bank give you that year? And when you add that, how much do you have at the end of the next year? (Simplify as much as possible.) So, now you know what is happening to your money each year. money do you have? Give me an equation. Test that equation to see if it gives you the same result you gave above for the end of year 5. Once again, graph that. The x-axis should be year. The y-axis should be the total amount of money you end up with after each year. So after year n, how much
116
CHAPTER 5. EXPONENTS
How is this graph like, and how is it unlike, the previous graph? If you withdraw all your money after the equation you found above!) If you withdraw all your money after 2
a year, how much money will the bank give you? years, how much money will the bank give you?
(Use
Suppose that, instead of starting with $1,000, I just tell you that you had $1,000 at year 0. How much money did you have ve years before that (year 5)? How many years will it take for your money to triple? That is to say, in what year will you have $3,000?
Radioactive substances decay according to a half-life. The half-life is the period of time that it takes for For instance, if the half-life is 20 minutes, then every 20 minutes, half the
As you can see, this is the sort of exponential curve that goes down instead of up: at each step (or half-life) the total amount
Time
0 1 minute 2 minutes 3 minutes 4 minutes 5 minutes
Substance remaining
1 gram
gram
Table 5.5
a. After n minutes, how many grams are there? Give me an equation. b. Use that equation to answer the question: after 5 minutes, how many grams of substance
are there? Does your answer agree with what you put under 5 minutes above? (If not, something's wrong somewherend it and x it!)
c. How much substance will be left after 4 minutes? d. How much substance will be left after half an hour? e. How long will it be before only one one-millionth of a gram remains? f. Finally, on the attached graph paper, do a graph of this function, where the minute is on the
x-axis and the amount of stu left is on the y-axis (so you are graphing grams as a function of minutes). Obviously, your graph won't get past the fth or sixth minute or so, but try to get an idea for what the shape looks like.
8 This
117
Time
0 20 minutes 40 minutes 60 minutes 80 minutes 100 minutes
Half-Lives
0 1
Substance remaining
1000 grams 500 grams
Table 5.6
a. After n half-lives, how many grams are there? Give me an equation. b. After n half-lives, how many minutes have gone by? Give me an equation. c. Now, let's look at that equation the other way. After t minutes (for instance, after 60 minutes, or 80 minutes, etc), how many half-lives have gone by? Give me an equation. d. Now we need to put it all together. After t minutes, how many grams are there? This equation
should take you directly from the rst column to the third: for instance, it should turn 0 into 1000, and 20 into 500. (*Note: you can build this as a two of your previous answers!)
e. Test that equation to see if it gives you the same result you gave above after 100 minutes. f. Once again, graph that do a graph on the graph paper. The x-axis should be minutes. The
y-axis should be the total amount of substance. In the space below, answer the question: how is it like, and how is it unlike, the previous graph?
g. How much substance will be left after 70 minutes? h. How much substance will be left after two hours? (*Not two minutes, two hours!) i. How long will it be before only one gram of the original substance remains? Exercise 5.89 Compound Interest
Finally, a bit more about compound interest If you invest $A into a bank with i% interest compounded n times per year, after t years your bank account is worth an amount M given by:
i nt n For instance, suppose you invest $1,000 in a bank that gives 10% interest, compounded semi-
M =A 1+
A,
i,
n,
the number of times compounded per year, is 2. So after 30 years, you would have:
0.10 230 =$18,679. (Not bad for a $1,000 investment!) 2 Now, suppose you invest $1.00 in a bank that gives 100% interest (nice bank!). How much do
$1,000
1+
a. Compounded annually (once per year)? b. Compounded quarterly (four times per year)? c. Compounded daily?
118
CHAPTER 5. EXPONENTS
d. Compounded every second?
Simplify. Your answer should not contain any negative or fractional exponents.
x6
Exercise 5.91
x0
Exercise 5.92
x8
Exercise 5.93
x3
Exercise 5.94
1 a3
Exercise 5.95
2 2 3
Exercise 5.96
1 x 2 2
Exercise 5.97
Exercise 5.98
3
Exercise 5.99
1
Exercise 5.100
Exercise 5.101
1
Exercise 5.104
3
x2 x2
1
Exercise 5.105
2
Exercise 5.106
x3
Exercise 5.107
3 4
9 This
119
Exercise 5.108
(4 9) 2
Exercise 5.109
42 92
Exercise 5.110
Give an algebraic formula that gives the generalization for #18-19. Solve for
x:
Exercise 5.111
8x = 64
Exercise 5.112
8x = 8
Exercise 5.113
8x = 1
Exercise 5.114
8x = 2
Exercise 5.115
8x = 8x = 8x =
1 8 1 64 1 2
8x = 0
Exercise 5.119
1 Rewrite 3
Solve for
x2
as xsomething.
x:
Exercise 5.120
2x+3 2x+4 = 2
Exercise 5.121
2 3(x )
1 3x 9
Exercise 5.122
A friend of yours is arguing that or division, or something.
x1/3
Exercise 5.123
On October 1, I place 3 sheets of paper on the ground. Each day thereafter, I count the number of sheets on the ground, and add that many again. (So if there are 5 sheets, I add 5 more.) After I add my last pile on Halloween (October 31), how many sheets are there total?
a. Give me the answer as a formula. b. Plug that formula into your calculator to get a number. 1 c. If one sheet of paper is 250 inches thick, how thick is the nal pile?
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CHAPTER 5. EXPONENTS
Exercise 5.124 Depreciation
The Web site www.bankrate.com denes
course of its useful life (and also as something new-car buyers dread). The site goes on to say: Let's start with some basics. Here's a standard rule of thumb about used cars. A car loses 15 percent to 20 percent of its value each year.
For the purposes of this problem, let's suppose you buy a new car for exactly $10,000, and it loses only 15% of its value every year.
a. b. c. d.
How much is your car worth after the rst year? How much is your car worth after the second year? How much is your car worth after the nth year? How much is your car worth after ten years? buyers dread depreciation.) (This helps you understand why new-car
Exercise 5.125
Draw a graph of
y = 2 3x . y=
1 x 3
x. x.
Exercise 5.126
Draw a graph of
3.
Exercise 5.127 What are the domain and range of the function you graphed in number 36? Extra Credit:
We know that they the same?
(a + b)
a2 + b2 .
are
Chapter 6
Logarithms
6.1 Introduction to Logarithms
Exercise 6.1
On day 0, you have 1 penny. Every day, you double.
a. How many pennies do you have on day 10? b. How many pennies do you have on day n? c. On what day do you have 32 pennies? Before you answer, express this question as an equation, where x is the variable you want to solve for. d. Now, what is x? Exercise 6.2
A radioactive substance is decaying. There is currently 100
of the substance.
a. How much substance will there be after 3 half-lives? b. How much substance will there be after n half-lives? c. After how many half-lives will there be 1 g of the substance left? Before you answer, express this question as an equation, where x is the variable you want to solve for. d. Now, what is x? (Your answer will be approximate.)
In both of the problems above, part (d) required you to going doesit goes the other wayremember?) So let's go ahead and talk formally about an inverse exponential function. Remember that an inverse function goes
invert the normal exponential function. Instead of from time to amount, it asked you to go from amount to time. (This is what an inverse function
backward. If
f (x) = 2x
So, ll in the following table (on the left) with a bunch of
inverse
function of
2x .
Pick
inverse function of
1 This
121
122
CHAPTER 6. LOGARITHMS
Inverse of
2x
Inverse of
10x
x
8
y
3
Table 6.1
Now, let's see if we can get a bit of a handle on this type of function. In some ways, it's like a square root. Now, we have the inverse of
is the inverse of
x2 .
x? x2
of course). But this new function is also
2x
a question: see if you can gure out what it is. That is, try to write a question that will reliably get me from the left-hand column to the right-hand column in the rst table above. Do the same for the second table above. Now, come up with a word problem of your own, similar to the rst two in this exercise, but related to compound interest.
2 to what power is 8? Based on that, you can answer the following questions:
Exercise 6.3
log2 8 log3 9
= = = = = =
Exercise 6.6
log10 100
Exercise 6.7
log10 1000
Exercise 6.8
log10 1, 000, 000
Exercise 6.9
Looking at your answers to exercises #3-6, what does the log10 tell you about a number?
Exercise 6.10
Multiple choice: which of the following is closest to log10 500 ?
A. 1 1 B. 1 2 C. 2 1 D. 2 2
2 This
content is available online at <http://cnx.org/content/m19176/1.2/>.
123
E. 3 Exercise 6.11
log10 1
= = = =
Exercise 6.12
log10
1 10
Exercise 6.13
log10
1 100
Exercise 6.14
log2 (0.01) log10 0
Exercise 6.15
= =
Exercise 6.16
log10 (1) log9 81
Exercise 6.17
Exercise 6.18
log9
1 9
= = =
Exercise 6.19
log9 3
Exercise 6.20
log9
1 81 1 3
Exercise 6.21
log9
= =
Exercise 6.22
log5 (54)
Exercise 6.23
5log5 4 =
OK. When I say
36
= 6,
that's the
What squared equals 36? So the equation 32? And I'm answering: two to the
62 = 36.
Why? Because
36 asks a question:
25 = 32.
ments.
You can look at logs in a similar way. If I say log2 32 = 5 I'm asking a question: 2 to what power is fth power is 32. So saying log2 32 = 5 is the same thing as saying Based on this kind of reasoning,
Exercise 6.24
log2 8
=3 = 1 =0
Exercise 6.25
log3
1 3
Exercise 6.26
logx (1) loga x
Exercise 6.27
=y
rewrite the following exponent statements as logarithm state-
43 = 64
124
CHAPTER 6. LOGARITHMS
Exercise 6.29
2 3
1 4
Exercise 6.30
ab = c
Finally...you don't understand a function until you graph it...
Exercise 6.31 a. Draw a graph of y = log2 x . Plot at least 5 points to draw the graph. b. What are the domain and range of the graph? What does that tell you about this function?
(2) = (2 2) = (2 2 2) = (2 2 2 2) = (2 2 2 2 2) =
Exercise 6.33
log2
Exercise 6.34
log2
Exercise 6.35
log2
Exercise 6.36
log2
Exercise 6.37
Based on exercises #1-5, nish this sentence in words: when you take log2 of a number, you nd:
Exercise 6.38
log2
Exercise 6.39
log2
Exercise 6.40
log2
Exercise 6.41
log3
Exercise 6.42
log3
Exercise 6.43
log3
Exercise 6.44
Based on exercises #7-12, write an algebraic generalization about logs.
Exercise 6.45
Now, let's dig more deeply into that one. Rewrite exercises #7-9 so they look like exercises #1-5: that is, so the thing you are taking the log of is written as a power of 2.
a. #7: b. #8:
3 This
content is available online at <http://cnx.org/content/m19269/1.1/>.
125
c. #9: d. Based on this rewriting, can you explain why your generalization from #13 works? Exercise 6.46
log5
(25) =
1 25
Exercise 6.47
log5
Exercise 6.48
log2
(32) =
1 32
Exercise 6.49
log2
Exercise 6.50
Based on exercises #15-18, write an algebraic generalization about logs.
Exercise 6.51
log3
Exercise 6.52
log3
Exercise 6.53
log3 812
Exercise 6.54
log3
(81 81 81) =
Exercise 6.55
log3 813
Exercise 6.56
log3
(81 81 81 81) =
Exercise 6.57
log3 814
Exercise 6.58
Based on exercises #20-26, write an algebraic generalization about logs.
logx
logx
Exercise 6.59
In class, we demonstrated the log2 8
add.
Now, you come up with a similar demonstration of the second rule of logs, that shows why when
4 This
126
CHAPTER 6. LOGARITHMS
applying those three rules. Take my word for these two facts. (You don't
have to memorize them, but you will be using them for this homework.)
log5 8
log5 60
= 1.29 = 2.54
Exercise 6.60
log5 480
=
So this is log5
Hint:
480 = 8 60.
(8 60).
Exercise 6.61
How can you use your calculator to test your answer to #2? (I'm assuming here that you can't nd log5 480 on your calculator, but you can do exponents.) Run the testdid it work?
Exercise 6.62
log5
2
15
= =
Exercise 6.63
log5
15
Exercise 6.64
log5 64
=
23
Exercise 6.65
log5
(5)
Exercise 6.66
5(log5 23) =
Simplify, using the log (xy) property:
Exercise 6.67
loga
(x x x x) (x 1)
x y
property:
Exercise 6.68
loga
Exercise 6.69
loga
1 x x 1 x x b
property:
Exercise 6.70
loga
Exercise 6.71
loga Simplify, using the log (x)
Exercise 6.72
loga
Exercise 6.73
loga
Exercise 6.74
loga
y = log 1 x. 2
127
b. What are the domain and range of the graph? What does that tell you about this function?
Exercise 6.76
Simplify: log3
Exercise 6.77
Simplify: log3
log
Exercise 6.78
Simplify:
(x2 )
log(x)
Exercise 6.79
Solve for log (2x
Exercise 6.80
Solve for log (3)
Exercise 6.81
Solve for
Exercise 6.82
Solve for log (x)
Andy invests $1,000 in a bank that pays out 7% interest, compounded annually. Note that your answers to parts (a) and (c) will be numbers, but your answers to parts (b) and (d) will be formulae.
a. b. c. d.
m (t) =
After how many years does Andy have exactly $14,198.57? After how many years
t (m) =
5 This 6 This
128
CHAPTER 6. LOGARITHMS
is approximately 1000; the intensity of a normal conversation is approximately 1,000,000; a rock concert (and the threshold of pain) has an intensity around 1,000,000,000,000. Place these points on a number line, and label them. Then answer the question: what's wrong with this number line? That was pretty ugly, wasn't it? It's almost impossible to graph or visualize something going from a hundred to a trillion: the range is too big. Fortunately, sound volume is usually not measured in intensity, but in dened by the formula: the intensity.
L = 10
log10 I , where
is
a. What is the loudness, in decibels, of a whisper? b. What is the loudness, in decibels, of a rock concert? c. Now do the number line again, labeling all the soundsbut this time, graph loudness instead of
intensity.
d. That was a heck of a lot nicer, wasn't it? (This one is sort of rhetorical.) e. The quietest sound a human being can hear is intensity 1. What is the loudness of that sound? f. Sound intensity can never be negative, but it can be less than 1. What is the loudness of such
inaudible sounds?
g. The formula I gave above gives loudness as a function of intensity. Write the opposite function,
that gives intensity as a function of loudness.
h. If sound
is
is it?
r = log10 a).
a. A microearthquake is dened as 2.0 or less on the Richter scale. Microearthquakes are not
felt by people, and are detectable only by local seismic detectors. If a is the amplitude of an earthquake, write an inequality that must be true for it to be classied as a microearthquake. b. A great earthquake has amplitude of 100,000,000 or more. There is generally one great earthquake somewhere in the world each year. If r is the measurement of an earthquake on the Richter scale, write an quake.
c. Imagine trying to show, on a graph, the amplitudes of a bunch of earthquakes, ranging from
microearthquakes to great earthquakes. (Go on, just imagine itI'm not going to make you do it.) A lot easier with the Richter scale, ain't it?
d. Two Earthquakes are measuredthe second one has 1000 times the amplitude of the rst. What
is the dierence in their measurements on the Richter scale?
Exercise 6.86 pH
In Chemistry, a very important quantity is the
[H + ]this is related to the acidity of a liquid. In a normal pond, the concentration of Hydrogen ions 6 6 1 is around 10 moles/liter. (In other words, every liter of water has about 10 , or 1,000,000 moles
of Hydrogen ions.) Now, acid rain begins to fall on that pond, and the concentration of Hydrogen ions begins to go up, until the concentration is How much did the concentration go up by?
104
1
10,000
moles of
H + ).
a. Acidity is usually not measured as concentration (because the numbers are very ugly, as you can
see), but as pH, which is dened as
log10 [H + ].
129
b. What is the pH of the pond after the acid rain? Exercise 6.87
Based on exercises #24, write a brief description of scientists to want to use a logarithmic scale.
300 (1.05)
years, I have
dollars in the bank. When I come back, I nd that my account is worth $1000. How
1 a. If the Hydrogen concentration is 10000 , what is the pH? 1 b. If the Hydrogen concentration is 1000000 , what is the pH? c. What happens to the pH every time the Hydrogen concentration divides by 10?
You may have noticed that all our logarithmic functions use the base 10. Because this is so common, it is given a special name: the
common log. When you see something like log (x) with no base written at all,
that means the log is 10. (So log (x) is a shorthand way of writing log10 (x), just like is a shorthand way of writing . With roots, if you don't see a little number there, you assume a 2. With logs, you assume a 10.)
Exercise 6.90
In the space below, write the question that log (x) asks.
Exercise 6.91
log100
Exercise 6.92
log1000
Exercise 6.93
log10000
Exercise 6.94
log( 1 with n 0s after it )
Exercise 6.95
log500(use the log button on your calculator) OK, so the
log
button on your calculator does common logs, that is, logs base 10.
There is one other log button on your calculator. It is called the "natural log," and it is written sort of stands for "natural log" only backward personally, I blame the French).
ln (which
What is
e?
on forever and you can only approximate it, but it is somewhere around 2.7. Answer the following question about the natural log.
Exercise 6.96
ln (e)
7 This
130
CHAPTER 6. LOGARITHMS
Exercise 6.97
ln (1)
= =
Exercise 6.98
ln (0)
Exercise 6.99
ln
e5 =
Exercise 6.100 ln (3) = (*this is the only one that requires the
ln
= = = =
Exercise 6.104
(approximately)
Exercise 6.105
log3 1
=
1 3
Exercise 6.106
log3
Exercise 6.107
log3
1 9
Exercise 6.108
log3
(3) = =
Exercise 6.109
log9 3
Exercise 6.110
3log3 8 =
Exercise 6.111
log3 9
= =
Exercise 6.112
log100, 000
Exercise 6.113
log
1
100,000
Exercise 6.114
lne3 =
Exercise 6.115
ln
4= qz = p
Exercise 6.116
Rewrite as a logarithm equation (no exponents):
8 This
131
Exercise 6.117
Rewrite as an exponent equation (no logs): logw g For exercises #18-22, assume that. . .
=j
log5 12 log5 20
= 1.544 = 1.861
Exercise 6.118
log5 240
= =
Exercise 6.119
log5 log5 1
3 5 2 3
Exercise 6.120
= =
Exercise 6.121
log5
3 2 5
Exercise 6.122
log5 400
Exercise 6.123
Graph
y = log2 x + 2.
Exercise 6.124
What are the domain and range of the graph you drew in exercise #23?
Exercise 6.125
I invest $200 in a bank that pays 4% interest, compounded annually.
200 (1.04)
So after
years, I have
dollars in the bank. When I come back, I nd that my account is worth $1000. How
Exercise 6.126
The loudness of a sound is given by the formula in decibels), and
L = 10
logI , where
a. b. c. d.
If the sound wave intensity is 10, what is the loudness? If the sound wave intensity is 10,000, what is the loudness? If the sound wave intensity is 10,000,000, what is the loudness? What happens to the loudness every time the sound wave intensity
multiplies by 1,000?
Exercise 6.127
Solve for
Exercise 6.128
Solve for log2
Extra Credit:
Solve for
x. ex = (cabin)
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CHAPTER 6. LOGARITHMS
Chapter 7
Rational Expressions
7.1 Rational Expressions
Exercise 7.1
1 x
1 y
x=2
and
y =4
Exercise 7.2
1 x 1 y = 1 x 1 y
Exercise 7.3
Exercise 7.4 1
x 1 y
Exercise 7.5
x2 +2x+1 x2 1
c. Are there any x-values for which the new expression does not equal the old expression? Exercise 7.6
2 x2 9
4 x2 +2x15
Exercise 7.7
4x2 25 x2 +2x+1
x = 2
simplied expression. (If they don't come out the same, you did something wrong!)
c. Are there any x-values for which the new expression does not equal the old expression?
1 This
133
134
x2 +8x+16 x2 7x+10
a. Simplify b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.9
2x2 +7x+3 x3 +4x x2 +x6 3x
a. Simplify b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.10
1 x1
1 x+1
a. b. c. d.
x are not allowed in the original expression? x are not allowed in your simplied expression? answer by choosing an x value and plugging it into the
Exercise 7.11
x3 x2 +9x+20
x4 x2 +8x+15
a. Simplify b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.12
x+1 4x2 9
4x 6x2 9x
x
36
15 36
OK, that was easy, wasn't it? So the moral of that story is: rational equations are easy to solve, if they have a common denominator. If they don't, of course, you just get one!
2 This 3 This
135
Exercise 7.14
Now suppose I tell you that true?
x
18
15 36
Hey,
Exercise 7.15
x2 +2
21
9 7
Did you get only one answer? Then look againthis one has two! Once you are that far, you've got the general ideaget a common denominator, and then set the numerators equal. So let's really get into it now. . .
Exercise 7.16
x+2 x+3
Exercise 7.17
2x+6 x+3 x+3 2x3
= =
Exercise 7.18
a. Solve. You should end up with two answers. b. Check both answers by plugging them into the original equation.
4x3 +x 3x3 3x
a. Simplify b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.20
2 4x3 x
3 2x3 +3x2 +x
a. Simplify b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.21
x2 +9 x2 4 x2 +7x+12 x2 +2x8
a. b. c. d.
4 This
x are not allowed in the original expression? x are not allowed in your simplied expression? answer by choosing an x value and plugging it into the
136
x2 +1 x3 9x2 +20x
a. What is the something? b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.23
x6 x3
x+18 2x+7
a. Solve for x. You should get two answers. b. Check by plugging one of your answers back into the original equation. Exercise 7.24
If
f (x) =
f (x + h),
is a composite function!
p42kp2 +56kp3
14kp
Exercise 7.26
x2 12x45 x+3
Exercise 7.27 a. = y3 b. Test your answer by choosing a number for Exercise 7.28 Exercise 7.29
2x3 4x2 +6x15 x2 +3 x3 4x2 x4 2y 2 +y16
Exercise 7.30
Exercise 7.31
3 27 a. xx3 = b. Test your answer by multiplying back.
Exercise 7.32
After dividing two polynomials, I get the answer I divide?
1 r2 6r + 9 r3 .
5 This
137
x4 x2 +8x+15
a. Simplify b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.34
2 x2 1
x x2 2x+1
a. Simplify b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.35
4x3 9x x2 3x10
a. Simplify b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.36 1
x x1 x2
a. Simplify b. What values of x are not allowed in the original expression? c. What values of x are not allowed in your simplied expression? Exercise 7.37
6x3 5x2 5x+34 2x+3
a. Solve by long division. b. Check your answer (show your work!!!). Exercise 7.38
If
f (x) = x2 , =
x+7 7x+4
nd
Exercise 7.39
x1 2x1
a. Solve for x. b. Test one of your answers and show that it works in the original expression. (No credit
unless you show your work!)
Extra credit:
I am thinking of two numbers,
and
y,
product. (Sum means add them; product means multiply them.) a. Can you nd any such pairs? b. To generalize: if one of my numbers is
the other one?
x,
to work with?
138
Chapter 8
Radicals
8.1 Radicals (aka* Roots)
1
As a student of mine once said, In real life, no one ever says `Here's 100 dollars, what's the square root of it? ' She's right, of courseas far as I know, no one ever takes the square root of money. And she is asking exactly the right question, which is: why do we need roots anyway?
Exercise 8.1
If a square is
49ft2
Q. You call that real life? A. OK, you asked for it. . . Exercise 8.2
A real estate developer is putting houses down on a plot of land that is 50 acres large. He wants
1 2 -acre lot. (1 acre is 43,560 square feet.) If each house sits on a square lot, how long are the sides of each lot?
to put down 100 houses, so each house will sit on a
Exercise 8.3
A piano is dropped from a building 100 ft high. (Dropped implies that someone just let go of it, instead of throwing itso it has no initial velocity.)
a. Write the equation of motion for this piano, recalling as always that h (t) = ho +vo t16t2 b. According to the equation, how high is the piano when t = 0? Explain in words what this
answer means.
c. After 2 seconds, how high is the piano? d. How many seconds does it take the piano to reach the ground? e. Find the inverse function that will enable to me nd the time
any given height
h.
Convinced? Square roots come up all the time in real life, because squaring things comes up all the time in real life, and the square root is how you get back. So we're going to have a unit on square roots.
25 = 5,
52 = 25.
1 This 2 This
139
140
CHAPTER 8. RADICALS
Exercise 8.4
Exercise 8.5
3
Exercise 8.6
a
Exercise 8.7
Now, rewrite all three as
9= 4=
4=
Exercise 8.11
9 4
Exercise 8.12
Based on #1-4, write an algebraic generalization.
Exercise 8.13
Now, give me a completely dierent example of that same generalization: four dierent statements, like #1-4, that could be used to generate that same generalization.
Exercise 8.14
9 4
Exercise 8.15
9 4
a. What do you think is the answer? b. Test by squaring back. (To square anything, multiply it by itself. So this just requires
multiplying fractions!) If it doesn't work, try something else, until you are convinced that you have a good [xxx].
Exercise 8.16
Based on exercises #7-8, write an algebraic generalization.
Exercise 8.17
9+
4=
Exercise 8.18
9+4
Exercise 8.19
Based on exercises #10-11, write an algebraic generalization that is not true.
3 This
141
Based on the generalization you wrote in #5 in the module "Some Very Important Generalizations", and
4 2 = 8, 8=
we can simplify
as follows:
42 42= 4 2 4 2=2 2
4 2)
Table 8.1
2 2.
Exercise 8.20
Find
8 2
on your calculator.
Exercise 8.21
Find
on your calculator.
Exercise 8.22
Double your answer to #2 on your calculator.
Exercise 8.23
So, did it work?
Exercise 8.24
Good! Then let's try another one. Simplify
72,
8.
Exercise 8.25
Check your answer on the calculator. Did it work?
Exercise 8.26
Oh yeah, one nal question: what is
x16 ?
For the following problems, I am not looking for big long ugly decimalsjust simplied expressions. You should not have to use your calculator at all, except possibly to check your answers. Remember, you can always check your answer by squaring back!
Exercise 8.27
64
Exercise 8.28
3
64
Exercise 8.29
3 64
Exercise 8.30
64 64
Exercise 8.31
4 This 5 This
142
CHAPTER 8. RADICALS
Exercise 8.32
64 n x
has a real answer, what can you say about
Exercise 8.33
If
n?
Exercise 8.34
Exercise 8.35
18 48 70 72
100
1 4
Exercise 8.42
8 12
Exercise 8.43
16x2
a. Simplify as much as possible (just like all the other problems) b. Check your answer with x = 3. Did it work? Exercise 8.44
75x3 y 6 z 5
Exercise 8.48
x2 y 2
Exercise 8.49
x2 + y 2
Exercise 8.50
x2 + 2xy + y 2
Exercise 8.51
x2 + 9
Exercise 8.52
x2 + 6x + 9
143
Exercise 8.53
f (x) = x2
a. Find the inverse of the function. b. Test it. Exercise 8.54
f (x) = x3
a. Find the inverse of the function. b. Test it. Exercise 8.55
f (x) = 3x
a. Find the inverse of the function. b. Test it.
Exercise 8.56
Albert Einstein's Special Theory of Relativity tells us that matter and energy are dierent forms of the same thing. (Previously, they were thought of as two completely dierent things.) If you have some matter, you can convert it to energy; if you have some energy, you can convert it to matter. This is expressed mathematically in the famous equation amount of energy,
E = mc2 , E
where
is the
me the equation I could use that would help me gure out, from these two numbers, what the speed
Exercise 8.57
The following gure is an Aerobie, or a washer, or whatever you want to call itit's the shaded area, a ring with inner thickness
r1
r2 .
6 This
144
CHAPTER 8. RADICALS
Aerobie
Figure 8.1
a. What is the area of this shaded region, in terms of r1 and r2 ? b. Suppose I told you that the area of the shaded region is 32 , and that the inner radius r1
is 7. What is the outer radius
r2 ? A,
and that the outer radius is
r2 .
Exercise 8.58
100y
x 25
x + 16 =
Exercise 8.61
50 2
5 2+2 33 2= 27 48 =
Let's try some that are a bit trickiersort of like rational expressions. Don't forget to start by getting a common denominator!
Exercise 8.64
1 2
2 2
145
a. Simplify. (Don't use your calculator, it won't help.) b. Now, check your answer by plugging the original formula into your calculator. What do
you get? Did it work?
Exercise 8.65
1 2
1 3
a. Simplify. (Don't use your calculator, it won't help.) b. Now, check your answer by plugging the original formula into your calculator. What do
you get? Did it work?
Exercise 8.66
1 3+1
3 2
a. Simplify. (Don't use your calculator, it won't help.) b. Now, check your answer by plugging the original formula into your calculator. What do
you get? Did it work? And now, the question you knew I would ask. . .
Exercise 8.67
Graph
y=
a. Plot a whole mess of points. (Choose x-values that will give you pretty easy-to-graph
y-values!)
b. What is the domain? What is the range? c. Draw the graph. Exercise 8.68
Graph
y=
x3
a. Plug in a couple of points to make sure your shift was correct. Fix it if it wasn't. b. What is the domain? What is the range? Exercise 8.69
Graph
y=
x3
a. Plug in a couple of points to make sure your shift was correct. Fix it if it wasn't. b. What is the domain? What is the range?
Several hundred years before Einstein, Isaac Newton proposed a theory of gravity. According to Newton's theory, any two bodies exert a force on each other, pulling them closer together. force is given by the equation
1 F = G mrm2 2
The
, where
is a constant,
m1
7 This
and
m2
146
CHAPTER 8. RADICALS
Exercise 8.71
In the following drawing,
is the
length of the ski lift (the diagonal line); and lift to the bottom of the mountain.
Ski Lift
Figure 8.2
a. Label these three numbers on the diagram. Note that they make a right triangle. b. Write the relationship between the three. (Pythagorean Theorem.) c. If you build the ski lift starting 1,200 feet from the bottom of the mountain, and the
mountain is 800 feet high, how long is the ski lift?
d. If the ski lift is s feet long, and you build it starting x feet from the bottom of the mountain,
how high is the mountain?
Exercise 8.72
Simplify
Exercise 8.73
1 Simplify 5
Exercise 8.74
Graph
y= y=
x 3. x 3.
Exercise 8.75
Graph
Before I get into the radical equations, there is something very important I have to get out of the way.
Exercise 8.76
2+
=
2
Exercise 8.77
3+
=
stop! Go back and look again,
How'd it go?
If you got six for the rst answer and ve for the second,
because those answers are not right. (If you don't believe me, try it on your calculator.) When you've got those correctly simplied (feel free to askor, again, check on your calculator) then go on.
8 This
147
Exercise 8.78
2x + 3 = 7
I call this an easy radical equation because there is no just subtract 3, then divide by
Sure, there's a ,
but that's just a number. So you can solve it pretty much the same way you would solve
4x + 3 = 7;
2.
a. Solve for x b. Check your answer by plugging it into the original equation. Does it work?
This next one is denitely trickier, but it is still in the category that I call easy because there is still no x under the square root.
Exercise 8.79
2x + 3x = 7
a. Solve for x b. Check your answer by plugging it into the original equation. Does it work? (Feel free to
use your calculator, but show me what you did and how it came out.) Now, what if there
is an
under the square root? Let's try a basic one like that.
x=9
What did you get? If you said the answer is three: shame, shame. The square root of 3 isn't 9, is it? Try OK, that's better. You probably guessed your way to the answer. But if you had to be systematic about it, you could say I got to the answer by squaring both sides. The rule is:
a radical, you will have to square both sides. If there is no both sides. Exercise 8.81
It worked out this time, but squaring both sides is fraught with peril. Here are a few examples.
x = 9
a. Solve for x, by squaring both sides. b. Check your answer by plugging it into the original equation.
Hey, what happened? When you square both sides, you get wrong answer:
x = 81,
81
is not
9.
The moral of the story is that when you square both sides, you
can introduce false answers. So whenever you square both sides, you have to check your answers to see
if they work. (We will see that rule come up again in some much less obvious places, so it's a good idea to get it under your belt now:
whenever you square both sides, you can introduce false answers!)
But that isn't the only danger of squaring both sides. Check this out. . .
2+
x=5
Hey, what happened there? When you square the left side, you got (I hope)
x + 4 x + 4.
you have to get the square root by itself before you can square
2+
x=5 x
by rst getting the square root by itself, and then squaring both sides
a. Solve for
148
CHAPTER 8. RADICALS
b. Check your answer in the original equation.
Whew! Much better! Some of you may have never fallen into the trapyou may have just subtracted the two to begin with. But you will nd you need the same technique for harder problems, such as this one:
Exercise 8.84
x=6 x
by rst getting the square root by itself, and then squaring both sides, and
a. Solve for
Exercise 8.85
x2=
What do you do now? You're going to have to square both sides. . .that will simplify the left, but the right will still be ugly. But if you look closely, you will see that you have changed an equation with
you can solve it the way you did above: get the square root by itself and square both sides. Before you are done, you will have squared both sides twice!
For each of the following, you will rst identify it as one of three types of problem:
No x under a radical, so don't square both sides. x under a radical, so you will have to isolate it and square both sides. More than one x under a radical, so you will have to isolate-and-square
Then you will solve it; and nally, you will check your answers (often on a calculator). Remember that if you squared both sides, you may get false answers even if you did the problem correctly! If you did not square both sides, a false answer means you must have made a mistake somewhere.
Exercise 8.86
x = 3
a. Which type of problem is it? b. Solve for x. c. Check your answer(s).
Exercise 8.87
2x 3 =
3x
149
Exercise 8.88
2x = 4
a. Which type of problem is it? b. Solve for x. c. Check your answer(s). Exercise 8.89
4x + 2
2x = 1
a. Which type of problem is it? b. Solve for x. c. Check your answer(s). Exercise 8.90
x 2 = 4
a. Which type of problem is it? b. Solve for x. c. Check your answer(s). Exercise 8.91
x + 2 x = 15
a. Which type of problem is it? b. Solve for x. c. Check your answer(s). Exercise 8.92
x+4+
x=2
10
Punch this into your calculator and give the answer rounded to three decimal places. This is the only question on the quiz where I want an answer in the form of an ugly decimal:
69 = 3 10.
Exercise 8.94
Give me an approximate answer for
Simplify the following problems in #3-#15. Give answers using radicals, not decimals or approximations.
Exercise 8.95
400
10 This
150
CHAPTER 8. RADICALS
Exercise 8.96
27
Exercise 8.97
3 27 108
20
300 64
45
x16
Exercise 8.102
5
x38
Exercise 8.103
98x20 y 5 z
Exercise 8.104
4x2 + 9y 4
Exercise 8.105
Exercise 8.106
54 2+ 2
Exercise 8.107
something
= x + 2.
What is the
something?
Exercise 8.108
Rewrite as an exponent equation:
x=
Exercise 8.109
Rewrite as a radical equation:
ab = c ab = c
x =5 x=1 8=0
Exercise 8.111
3+ x 2
Exercise 8.112
2x + 1
Exercise 8.113
3x +
3 (x) 4 + 2 (x)
Exercise 8.114
x+
2=0
Exercise 8.115
For an object moving in a circle around the origin, whenever it is at the point (x,y ), its distance to the center of the circle is given by:
r=
x2 + y 2
x?
151
Exercise 8.116
Graph
y =
x + 3.
Exercise 8.117
What are the domain and range of the graph you drew in #24?
Extra Credit:
Draw a graph of
y=
x.
152
CHAPTER 8. RADICALS
Chapter 9
Imaginary Numbers
9.1 Imaginary Numbers
Exercise 9.1
Explain, using words and equations, why the equation does. OK, so now we are going to use our imaginations. (Didn't think we were allowed to do that in math class, did you?) Suppose there
x2 = 1
x3 = 1
were an answer to
x2 = 1?
).
i,
because it's
imaginary. What
Denition of i
The denition of the imaginary number
i=
i is 1 or, equivalently, i2 = 1
1:
Based on that denition, answer the following questions. In each case, don't just guessgive a good mathematical reason why the answer should be what you say it is!
Exercise 9.2
What is
i ( i)? ( i) (3i)
2
?
(*Remember that
means
1 i.)
Exercise 9.3
What is
Exercise 9.4
What is
Exercise 9.5
What is
( 3i) 2i 2i
2
?
Exercise 9.6
What is
Exercise 9.7
What is
Exercise 9.8
What is
25? 3? 8?
Exercise 9.9
What is
Exercise 9.10
What is
1 This
153
154
Exercise 9.12
Fill in the following table.
Exercise 9.13
(3 + 4i) =
Exercise 9.14
(3 + 4i) (3 4i) =
Exercise 9.15
1 2 i
=
1 i.
Exercise 9.16
Simplify the fraction
Exercise 9.17
Square your answer to #16. Did you get the same answer you got to #15? Why or why not?
Exercise 9.18
Simplify the fraction
1 3+2i .
155
3 2i.
not. Let's talk about the same thing graphically. Exercise 9.19
On the graph below, do a quick sketch of ppp
y = x3 .
a. Draw, on your graph, all the points on the curve where y = 1. How many are there? b. Draw, on your graph, all the points on the curve where y = 0. How many are there? c. Draw, on your graph, all the points on the curve where y = 1. How many are there? Exercise 9.20
On the graph below, do a quick sketch of ppp
y = x2 .
a. Draw, on your graph, all the points on the curve where y = 1. How many are there? b. Draw, on your graph, all the points on the curve where y = 0. How many are there? c. Draw, on your graph, all the points on the curve where y = 1. How many are there? Exercise 9.21
Based on your sketch in exercise #2. . .
a. b. c. d.
a is some number such that a > 0, how many solutions are there to the equation x2 = a? 2 If a is some number such that a = 0, how many solutions are there to the equation x = a? 2 If a is some number such that a < 0, how many solutions are there to the equation x = a? 2 If i is dened by the equation i = 1, where the heck is it on the graph?
If
i.
Exercise 9.22
In class, we made a table of powers of i, and found that there was a repeating pattern. Make that table again quickly below, to see the pattern.
2 This
156
Exercise 9.23
Now let's walk that table
backward. Assuming the pattern keeps up as you back up, ll in the
i4 i3 i2 i1 i0
Table 9.4
Exercise 9.24
Did it work? Let's gure it out. What should
i0
Exercise 9.25
What should
i1
be, according to our general rules of exponents? Can you simplify it to look
Exercise 9.26
What should
i2
be, according to our general rules of exponents? Can you simplify it to look
Exercise 9.27
What should
i3
be, according to our general rules of exponents? Can you simplify it to look
Exercise 9.28
Simplify the fraction
i 43i .
157
a + bi
where
and
bi
3 + 4i (a
is 3,
is 4) and
3 4i (a a a a
and
is 3,
is 4).
Exercise 9.29
Is 4 a complex number? If so, what are
b? b? b?
Exercise 9.30
Is
i 0
and
Exercise 9.31
Is a complex number? If so, what are and If not, why not? All four operationsaddition, subtraction, multiplication, and divisioncan be done to complex numbers, and the answer is always another complex number. So, for the following problems, let X = 3 + 4i Y = 5 12i. In each case, your answer should be a complex number, in the form a + bi. and
Exercise 9.32
Add:
X +Y. X Y . XY
(*To get the answer in
Exercise 9.33
Subtract:
Exercise 9.34
Multiply:
Exercise 9.35
Divide:
X/Y . X2
Exercise 9.36
Square: The
a + bi
is dened as
a bi.
(5 12i)? (5 12i)
by its complex conjugate?
Exercise 9.38
What do you get when you multiply
Exercise 9.39
Where have we used complex conjugates before? For two complex numbers to be or to
equal, there are two requirements: the real parts must be the same, and
2 + 3i
is only equal to
2 + 3i.
It is not equal to
2 + 3i
3 + 2i
or to anything else. So it is very easy to see if two complex numbers are the same, as long as
a + bi form:
(If they are not written in that form, it can be very tricky to tell: for instance, we saw earlier that
i!)
and
Exercise 9.40
If
2 + 3i = m + ni,
and
and
n?
Exercise 9.41
Solve for the real numbers
and
y : (x 6y) + (x + 2y) i = 1 3i
Finally, remember. . .rational expressions? We can have some of those with complex numbers as well!
Exercise 9.42
4+2i ( 3+2i ) 53i ( 7i )
a + bi.
3 This
158
53i 7i Simplify.
(3 + 7i) (4 + 7i) =
Exercise 9.45
(5 3i) + (5 3i) =
Exercise 9.46
2 (5 3i) =
Exercise 9.47
(5 3i) (2 + 0i) =
Exercise 9.48
What is the complex conjugate of
(5 3i)? (5 3i)
by its complex conjugate?
Exercise 9.49
What do you get when you multiply
Exercise 9.50
What is the complex conjugate of 7?
Exercise 9.51
What do you get when you multiply 7 by its complex conjugate?
Exercise 9.52
What is the complex conjugate of
2i? 2i
by its complex conjugate?
Exercise 9.53
What do you get when you multiply
Exercise 9.54
What is the complex conjugate of
(a + bi)? (a + bi)
by its complex conjugate?
Exercise 9.55
What do you get when you multiply
Exercise 9.56
I'm thinking of a complex number
z.
z)
a. What might z be? b. Test it, and make sure it worksthat is, that Exercise 9.57
I'm thinking of a answer is
z.
When I multiply it by
a. What might z be? b. Test it, and make sure it worksthat is, that
4 This
159
Exercise 9.58
Solve for
and
y : x2 + 2x2 i + 4y + 40yi = 7 2i
Exercise 9.59
Finally, a bit more exercise with rational expressions. We're going to take one problem and solve it two dierent ways. The problem is
3 2+i
a + bi.
a. Here is one way to solve it: the common denominator is
over the common denominator and combine them. and simplify it into
(2 + i) (3 + 4i).
a + bi
form.
b. Here is a completely dierent way to solve the same problem. Take the two fractions we are subtracting and simplify them both into a + bi form, and then subtract. c. Did you get the same answer? (If not, something went wrong. . .) Which way was easier?
3 2i and rewrite it in a + bi format. There are many and log (i)that do not look like a + bi, but all of them can be turned into
a + bi
ithat
so that we can
clearly see its real part and its imaginary part. How do we do that? Well, we want to nd some number terms of its real and imaginary partsthat is, in the form equation:
z such that z 2 = i. And we want to express z in a + bi. So what we want to solve is the following
(a + bi) = i
You are going to solve that equation now. When you nd
and
b,
Stop now and make sure you understand how I have set up this problem, before you go on to solve it.
Exercise 9.60
What is
(a + bi)
? Multiply it out.
Exercise 9.61
Now, rearrange your answer so that you have collected all the real terms together and all the imaginary terms together. Now, we are trying to solve the equation and set it equal to on the right.
(a + bi) = i.
i.
This will give you two equations: one where you set the real part on the left equal to
the real part on the right, and one where you set the imaginary part on the left equal to the imaginary part
Exercise 9.62
Write down both equations.
Exercise 9.63
Solve the two equations for you should have two
a and b. (Back to simultaneous equations, (a ,b pairs that both work in both equations. a
and
Exercise 9.64
So. . . now that you know
b,
write down
x2 = i.
Exercise 9.65
Did all that work? Well, let's nd out. Take your answers in #5 and test them: that is, square it, and see if you get
i.
5 This
160
i:
z 2 = i.
Check them!
x2 = 1
But
x2 = 1.
In fact,
x2 = n
Exercise 9.67
What is the one exception? You might suspect that
x3 = n
should have
x3 = 1
Exercise 9.68
What is the one solution? But if we allow for other two. How do we do that? Well, we know that every complex number can be written as are real numbers. So if there is some complex number that solves the
complex answers,
x3 = 1
(a + bi),
where
a and b
x3 = 1 ,
You are going to solve that equation now. When you nd
and
b,
Stop now and make sure you understand how I have set up this problem, before you go on to solve it.
Exercise 9.69
What is
(a + bi)
? Multiply it out.
Exercise 9.70
Now, rearrange your answer so that you have collected all the real terms together and all the imaginary terms together. Now, we are trying to solve the equation number 4, and set it equal to
(a + bi) = 1.
1.
This will give you two equations: one where you set the real part on
the left equal to the real part on the right, and one where you set the imaginary part on the left equal to the imaginary part on the right.
Exercise 9.71
Write down both equations. OK. If you did everything right, one of your two equations factors as equations doesn't factor that way, go backsomething went wrong! If it did, then let's move on from there. As you know, we now have two things being multiplied to give 0, which means one of them must be 0. One possibility is that possibility is that
b 3a2 b2 = 0.
b = 0:
3a2 b2 = 0,
which means
3a2 = b2 . 3a2 = b2
into the other equation
Exercise 9.72
Solve the two equations for
and
by substituting
Exercise 9.73
So. . .now that you know
and
b,
write down
x3 = 1 .
If
6 This
161
Exercise 9.74
But wait. . .shouldn't there be a third answer? Oh, yeah. . .what about that to pick that one up. If to
b=0
business? Time
b = 0,
what is
a?
Based on this
and
b,
x3 = 1?
Exercise 9.75
Did all that work? Well, let's nd out. Take either of your answers in #7 and test it: that is, cube it, and see if you get 1. If you don't, something went wrong!
b b2 4ac . Back when we were doing quadratic 2a equations, if we wound up with a negative number under that square root, we just gave up. But now we can
x =
Exercise 9.76
Use the quadratic formula to solve:
2x2 + 6x + 5 = 0. x2 2x + 5 = 0.
Exercise 9.77
Use the quadratic formula to solve:
Exercise 9.78
Check one of your answers to #2.
2x2 + 10x + 17 = 0.
a. In general, what has to be true for a quadratic equation to have two non-real roots? b. What is the relationship between the two non-real roots? c. Is it possible to have a quadratic equation with one non-real root?
i1 i0 i1 i2 i3 i4 i5 i6 i7
7 This 8 This
content is available online at <http://cnx.org/content/m19127/1.1/>. content is available online at <http://cnx.org/content/m19133/1.1/>.
162
Simplify.
Exercise 9.82
(i)
85
Exercise 9.83
(5i) =
Exercise 9.84
(ni)
103
Exercise 9.85 a. 20 = b. Other than your answer to part (a), is there any other number that you can squareto
get 20? If so, what is it?
Exercise 9.86
(3w zi) =
Exercise 9.87 a. Complex conjugate of 4 + i = b. What do you get when you multiply
If the following are simplied to the form
4+i
a + bi,
what are
and
in each case?
Exercise 9.88
i
a. b.
a= b=
Exercise 9.89
n i
a. b.
a= b=
Exercise 9.90
4x 16ix
2i 3i
a. b.
a= b=
Exercise 9.91
If
2x + 3xi + 2y = 28 + 9i,
what are
and
y?
Exercise 9.92
Make up a quadratic equation (using all real numbers) that has two non-real roots, and solve it.
Exercise 9.93 a. Find the two complex numbers (of course in the form
z = a + bi
z 2 = 2i.
b. Check one of your answers to part (a), by squaring it to make sure you get
2i.
163
Extra credit:
Complex numbers cannot be graphed on a number line. But they can be graphed on a 2-dimensional graph: you graph the point
x + iy
at (x,
y ).
1. If you graph the point 2. If you graph the point 3. What do you get if
5 + 12i, how far is that point from the origin (0,0)? x + iy , how far is that point from the origin (0,0)? you multiply the point x + iy by its complex conjugate?
164
Chapter 10
Matrices
10.1 Introduction to Matrices
1
The following matrix, stolen from a rusted lockbox in the back of a large, dark lecture hall in a school called Hogwart's, is the gradebook for Professor Severus Snape's class in potions.
Poison
Granger, H Longbottom, N Malfoy, D Potter, H Weasley, R 100 80 95 70 85
Cure
105 90 90 75 90
Love philter
99 85 0 70 95
Invulnerability
100 85 85 75 90
Table 10.1
When I say this is a matrix I'm referring to the numbers in boxes. The labels (such as Granger, H or Poison) are labels that help you understand the numbers in the matrix, but they are not the matrix itself. Each student is designated by a
Exercise 10.1
Below, copy the row that represents all the grades for Malfoy, D. Each assignment is designated by a
if you picture columns in Greek architecture, which are big and tall and. . .well, you know. . .vertical.)
Exercise 10.2
Below, copy the column that represents all the grades on the Love philter assignment. I know what you're thinking, this is so easy it seems pointless. Well, it's going to stay easy until tomorrow. So bear with me. The
dimensions of a matrix are just the number of rows, and the number of columns. . .in that order.
So a 10
Exercise 10.3
What are the dimensions of Dr. Snape's gradebook matrix? For two matrices to be
equal, they must be exactly the same in every way: same dimensions, and every cell
the same. If everything is not precisely the same, the two matrices are not equal.
1 This
165
166
and
be, in order to make the following matrix equal to Dr. Snape's gradebook
100 80 95 70 85
105
99 85 0
100 85 85 75 90
x+y
90 75 90
xy
95
Table 10.2
Finally, it is possible to
add or subtract matrices. But you can only do this when the matrices have the
same dimensions!!! If two matrices do not have exactly the same dimensions, you cannot add or subtract them. If they do have the same dimensions, you add and subtract them just by adding or subtracting each individual cell.
Exercise 10.5
As an example: Dr. Snape has decided that his grades are too high, and he needs to curve them downward. So he plans to subtract the following matrix.
5 5 5 10 5
0 0 0 5 0
10 10 10 15 10
0 0 0 5 0
Table 10.3
Exercise 10.6
In the grade-curving matrix, all rows except the fourth one are identical. What is the eect of the dierent fourth row on the nal grades?
3 3 5 9 7
7 7 0 5 4
4 0 7 4 2
9 8 3 3 9
6 8 8 6
2 This
1 2 0 1
167
a. b. c. d.
What are the dimensions? __ Copy the second column here: Copy the third row here:
__
Exercise 10.8
Add the following two matrices.
2 9
6 n
4 8
5 9
7 n
1 3n
Exercise 10.9
Add the following two matrices.
2 9
6 n
4 8
5 9
7 n
Exercise 10.10
Subtract the following two matrices.
2 9
6 n
4 8
5 9
7 n
1 3n x
Exercise 10.11
Solve the following equation for be true.) and
y.
and
2x 5y
x+y 6x
6 2
Exercise 10.12
Solve the following equation for be true.)
and
y. 5 9
and
x+y 3x 2y
4x y x + 5y
3 7
Just to limber up your matrix muscles, let's try doing the following matrix addition.
2 3
5 7
2y
2 3
5 7
x 2y
2 3
5 7
x 2y
Exercise 10.14
How many times did you add that matrix to itself ?
Exercise 10.15
Rewrite #1 as a
3 This
168
with a matrix that has the same dimensions as the original, but all the individual cells have been multiplied Let's do another example. I'm sure you remember Professor Snape's grade matrix.
Poison
Granger, H Longbottom, N Malfoy, D Potter, H Weasley, R 100 80 95 70 85
Cure
105 90 90 75 90
Love philter
99 85 0 70 95
Invulnerability
100 85 85 75 90
Table 10.4
Now, we saw how Professor Snape could lower his grades (which he loves to do) by subtracting a curve matrix. But there is another way he can lower his grades, which is by multiplying the entire matrix by a number. In this case, he is going to multiply his grade matrix by then the resulting matrix could be written as
9
10
[S]
[S]. 10 [S]
is just the grades, not the names.
Exercise 10.16
Write down the matrix
9
10
[S].
Finally, it's time to Prof. Snape to calculate nal grades. He does this according to the following formula: Poison counts 30%, Cure counts 20%, Love philter counts 15%, and the big nal project on Invulnerability counts 35%. calculation: For instance, to calculate the nal grade for Granger, H he does the following
To make the calculations easier to keep track of, the Professor represents the various weights in his
multiplying a row matrix by a column
matrix, as follows.
.3 .2 .15 .35
= [100.85]
A row matrix means a matrix that is just one row. A column matrix means. . .well, you get the idea. When a row matrix and a column matrix have the same number of items, you can multiply the two matrices. What you do is, you multiply both of the rst numbers, and you multiply both of the second numbers, and so on. . .and you add all those numbers to get one big number. The nal answer is not just a numberit is a 11 matrix, with that one big number inside it.
Exercise 10.17
Below, write the matrix multiplication that Prof. Snape would do to nd the grade for Potter, H. Show both the problem (the two matrices being multiplied) and the answer (the 11 matrix that contains the nal grade).
169
1 2
2 9
6 n
4 8
Exercise 10.19
Multiply.
3 2 3 4
6 7
Exercise 10.20
Multiply.
3 6 7
y z
Exercise 10.21
Solve for
x. x 3
x = [6] 5
Just for a change, we're going to start with. . .Professor Snape's grade matrix!
Poison
Granger, H Longbottom, N Malfoy, D Potter, H Weasley, R 100 80 95 70 85
Cure
105 90 90 75 90
Love philter
99 85 0 70 95
Invulnerability
100 85 85 75 90
Table 10.5
As you doubtless recall, the good Professor calculated nal grades by the following computation: Poison counts 30%, Cure counts 20%, Love philter counts 15%, and the big nal project on Invulnerability
4 This 5 This
170
counts 35%.
Exercise 10.22
Just to make sure you remember, write the matrix multiplication that Dr. Snape would use to nd the grade for Malfoy, D. Make sure to include both the two matrices being multiplied, and the nal result! I'm sure you can see the problem with this, which is that you have to write a separate matrix multiplication problem for every student. To get around that problem, we're going to extend our denition of matrix multiplication so that the rst matrix no longer has to be a rowit may be many rows. Each row of the rst matrix becomes a new row in the answer. So, Professor Snape can now multiply his entire student matrix by his weighting matrix, and out will come a matrix with all his grades!
Exercise 10.23
Let's try it. Do the following matrix multiplication. The answer will be a 31 matrix with the nal grades for Malfoy, D, Potter, H, and Weasley, R.
95 70 85 90 75 90 0 70 95 85
.3
.2 = 75 .15 90 .35
OK, let's step back and review where we are. Yesterday, we learned how to multiply a row matrix times a column matrix. Now we have learned that you can add more rows to the rst matrix, and they just become extra rows in the answer. For full generality of matrix multiplication, you just need to know this: if you
second matrix, they become additional columns in the answer! As an example, suppose Dr. Snape wants to try out a dierent weighting scheme, to see if he likes the new grades better. So he adds the new column to his weighting matrix. The rst column represents the original weighting scheme, and the second column represents the new weighting scheme. The result will be a 3x2 matrix where each row is a dierent and each column is a dierent
student
Exercise 10.24
85
95 70 85
90 75 90
0 70 95
.3
.4
.2 .2 = 75 .15 .3 90 .35 .1
[A]
is
1 3
2 4
.
Matrix
[B]
is
5 7
6 8
AB. BA.
6 This
171
2 9
6 5
5 4 4
4 6 2
3 8 8
9 0
Exercise 10.27
Multiply.
5 4 4 9
4 6 2
3 8 5 3 5 7
2 9 9 0 x
6 5
4 8
Exercise 10.28
7 2
3 y z 5
a. Multiply.
b. Now, multiply
Instead,
plug
7 5 3 10 but not by manually multiplying it out! 2 7 5 5 x = 2, y = 10, and z = 5 into the formula you came up with in part (a).
Exercise 10.29
Multiply.
2 5 8 2 3
0 1 0
4 7 3
6 0 9 0 x y
0 1 9 3
Exercise 10.30
a. Find the x and y values that will make this matrix equation true. b. Test your answer by doing the multiplication to make sure it works out. Exercise 10.31
1 3
2 4
Matrix
Some
1 3
2 4
a. Find the some matrix that will make this matrix equation true. b. Test your answer by doing the multiplication to make sure it works out.
172
This assignment is brought to you by one of my favorite numbers, and I'm sure it's one of yours. . .the number 1. Some people say that 1 is the loneliest number that you'll ever do. (*Bonus: who said that?) But I say, 1 is the multiplicative identity. Allow me to demonstrate.
Exercise 10.32
51=
Exercise 10.33
2 3
Exercise 10.34
1=
Exercise 10.35
1x=
You get the idea? 1 is called the multiplicative identity because it has this lovely property that whenever you multiply it by anything, you get that same thing back. But that's not all! Observe. . .
Exercise 10.36
1 2
= =
Exercise 10.37
2 3 3 x 2
The fun never ends! The point of all that was that every number has an inverse. The inverse is dened by the fact that, when you multiply a number by its inverse, you get 1.
Exercise 10.38
Write the equation that denes two numbers
and
Exercise 10.39
Find the inverse of
4 5.
Exercise 10.40
Find the inverse of 3.
Exercise 10.41
Find the inverse of
x.
not have an inverse, according to your denition in #7?
(I hear you cry.) Well, we've
Exercise 10.42
Is there any number that does
1 0
0 1 8 12
= = 1 0 0
by another 22 matrix, you get that other matrix back. the multiplicative identity.
4 1 0 0 1
12
Exercise 10.44
1 as [I])
7 This
173
Remember that matrix multiplication does not, in general, commute: that is, for any two matrices and
[A]
[B],
the product
AB
is not necessarily the same as the product BA. But in this case, it is:
[I]
times
another matrix gives you that other matrix back no matter which order you do the multiplication in. This is a key part of the denition of
I,
which is. . .
Denition of [I]
The matrix
Which, of course, is just a fancy way of saying what I said before. If you multiply either order, you get that other matrix back.
AI = IA = A I by any
matrix, in
Exercise 10.45
We have just seen that
1 0
0 1
acts as the multiplicative identify for a 22 matrix.
a. b. c. d. e.
What is the multiplicative identity for a 33 matrix? Test this identity to make sure it works. What is the multiplicative identity for a 55 matrix? (I won't make you test this one. . .) What is the multiplicative identity for a 23 matrix? Trick question! There isn't one. You could write a matrix that satises would not
AI = A,
but it
also satisfy
IA = Athat
requirement. Don't take my word for it, try it! The point is that only square matrices (*same number of rows as columns) have an identity matrix. So what about those inverses? Well, remember that two numbers might guess, we're going to dene two matrices
and
and
as inverses if
As you
Exercise 10.46
Multiply:
21 2 1 1 2 5 4
3 2 2
5 4 21 2 1 1 2
[I].
We
1 3
Exercise 10.47
Multiply:
inverses: no matter which order you multiply them in, you get
A1
f 1
Denition of A-1
The matrix (*where
A1
A1 A = AA1 = I I!
Now we
Of course, only a square matrix can have an inverse, since only a square matrix can have an know what an inverse matrix
Exercise 10.48
Find the inverse of the matrix
3 5
2 4
a. Since we don't know the inverse yet, we will designate it as a bunch of unknowns:
a c
b d
will be our inverse matrix. Write down the equation that denes this unknown matrix as our inverse matrix.
b. Now, in your equation, you had a matrix multiplication. Go ahead and do that multiplication, and write a new equation which just sets two matrices equal to each other.
174
d. Solve for a, b, c, and d. e. So, write the inverse matrix A1 . f. Test this inverse matrix to make sure it works!
is
4 2
10
. I
for Matrix
a. b. c. d.
Write the identity matrix Show that it works. Find the inverse matrix Show that it works.
A.
A1 .
Exercise 10.50
Matrix
is
3 5
4 . 6
a. Can you nd a matrix that satises the equation BI = B ? b. Is this an identity matrix for B ? If so, demonstrate. If not, why not? Exercise 10.51
2 6
10 14
3 7
11 15
Matrix
is
5 9
13
8 . 12
16
C.
Exercise 10.52
Matrix
is
1 3
2 n
. D1
8 This
175
generic 22 matrix. Once you have done that, you will any 22 matrix.
[A] =
a c
b d w y
w
in terms of our original variables
Since its inverse is unknown, we will designate the inverse like this:
A1 =
x z
a, b, c,
and
d.
w , x, y ,
or
in it, since those are unknowns! Just the original four variables in our original
[A].
x, y ,
and
Our approach will be the same approach we have been using to nd an inverse matrix. proceeding to the next.
you through the stepsafter each step, you may want to check to make sure you've gotten it right before
Exercise 10.53
Write the matrix equation that denes
A1
as an inverse of
A.
Exercise 10.54
Now, do the multiplication, so you are setting two matrices equal to each other.
Exercise 10.55
Now, we have two 22 matrices set equal to each other. That means every cell must be identical, so we get four dierent equations. Write down the four equations.
Exercise 10.56
Solve. Remember that your goal is to nd four equationsone for for zwhere each equation has only the four original constants
one for
y,
and one
Exercise 10.57
Now that you have solved for all four variables, write the inverse matrix
Exercise 10.58
As the nal step, to put this in the form that it is most commonly seen in, note that all four terms have an
ad bc
bc ad
of it after the next test. In the mean time, note that we can write our answer much more simply
1 pull out the common factor of adbc . (This is similar to pulling out a common term from a polynomial. Remember how we multiply a matrix by a constant? This is the same thing in
A1 =
You're done! You have found the generic formula for the inverse of any 2x2 matrix. Once you get the hang of it, you can use this formula to nd the inverse of any 2x2 matrix very quickly. Let's try a few!
Exercise 10.59
The matrix
2 4
3 5
a. Find the inversenot the long way, but just by plugging into the formula you found above.
9 This
content is available online at <http://cnx.org/content/m19214/1.1/>.
176
3 9
2 5
a. Find the inversenot the long way, but just by plugging into the formula you found above. b. Test the inverse to make sure it works. Exercise 10.61
Can you write a 22 matrix that
has no inverse?
10
You are an animator for the famous company Copycat Studios. Your job is to take the diagram of the sh below (whose name is Harpoona) and animate a particular scene for your soon-to-be-released movie. In this particular scene, the audience is looking down from above on Harpoona who begins the scene happily oating on the surface of the water. Here is a picture of Harpoona as she is happily oating on the surface.
10 This
177
Figure 10.1
[H] =
0 0
10 0
10 5
0 0
Exercise 10.62
Explain, in words, how this matrix represents her position. instructions to a computer on exactly how to draw Harpoona? That is, how can this matrix give
Exercise 10.63
The transformation
1 2
[H]
is applied to Harpoona.
a. Write down the resulting matrix. b. Draw Harpoona after this transformation. c. Then answer this question in words: in general, what does the transformation 1 2
to a picture?
[H]
do
Exercise 10.64
Now, Harpoona is going to swim three units to the left. Write below a general transformation that can be applied to any 24 matrix to move a drawing three units to the left.
178
the transformation
0 1
1 0
[H].
a. Write the new matrix that represents Harpoona. b. Draw Harpoona after this transformation. c. In the space below, answer this question in words: in general, what does the transforma
tion
1 0
[H]
do to a picture?
Exercise 10.66
Now: in the movie's key scene, the audience is looking down from above on Harpoona who begins the scene happily oating on the surface of the water. As the scene progresses, our heroine spins around and around in a whirlpool as she is slowly being sucked down to the bottom of the sea. Being sucked down is represented visually, of course, by shrinking.
a. Write a single transformation that will rotate Harpoona by 900 and shrink her. b. Apply this transformation four times to Harpoona's original state, and compute the resulting matrices that represent her next four states.
11
Harpoona's best friend is a sh named Sam, whose initial position is represented by the matrix:
[S1 ] =
0 0
4 0
4 3
0 3
0 0
4 3
Draw Sam:
11 This
179
Exercise 10.68
When the matrix
3 T = 1 2 1
1 3
transfor-
S2 = T S1 .
3.)
Exercise 10.69
Draw Sam's resulting condition,
S2 . T.
Find
Exercise 10.70
The matrix
T 1
T 1 .
the inverse matrix that we derived in class, instead of starting from rst principles. But make sure to rst multiply the
1 2 into
T,
Exercise 10.71
Sam now undergoes this transformation, so his new state is given by graph his new position.
S3 = T 1 S2 .
Find
S3
and
Exercise 10.72
Finally, Sam goes through nal position.
T 1
S4 = T 1 S3 .
Exercise 10.73
Describe in words: what do the transformations
and
T 1
180
2 5 8
b e h
1 1 1
What are
4 7
6 2 d g 9
f = 1 1 i
1 . 1
a, b, c, d, e, f , g , h,
and
i?
Exercise 10.75
(9 points)
Matrix
[A]
is
4 6
2 8
0
10
.
What is
A + A + 1 A? 2
Exercise 10.76
(9 points) Using the same Matrix
[A],
what is
2 1 A? 2
Exercise 10.77
(9 points)
4 5
6 3
x y
0
22
.
What are
and
y?
Exercise 10.78
(9 points)
1 3 4 n
5 6 7
Exercise 10.79
(9 points)
4 0 n 2
6 9 1 4
2 3 1 1 0
0 = 4 2
Exercise 10.80
(9 points)
6 9 1 4
2 1 0
4 2 0 0 3 = 4 n 1 2
Exercise 10.81
(9 points)
12 This
181
b e h
some matrix
b e h
d g
f i
= d g
f i
Exercise 10.82
(5 points)
b e h
some matrix
b e h
d g
f i
= f i
d g
Exercise 10.83
(8 points)
a. b. c. d.
Write two matrices that can be added and can be multiplied. Write two matrices that
cannot be added or multiplied. can be added but cannot be multiplied. Write two matrices that can be multiplied but cannot be added.
Write two matrices that
Exercise 10.84
(15 points)
4 1
x 2
by using the denition of an inverse matrix.
note: If you are absolutely at stuck on part (a), ask for the answer. You will receive no
credit for part (a) but you may then be able to go on to parts (b) and (c).
b. Test it, by showing that it fullls the denition of an inverse matrix. c. Find the inverse of the matrix
4 1
by plugging
x=3
Extra Credit:
(5 points) Use the generic formula for 4 x . Does it agree with your answer 1 2
13
1 4 1 4
2 5 2 5
3 6 3 6
+
7
10
8
11
9
12
Exercise 10.86
Solve on a calculator:
7
10
8
11
9
12
13 This
182
1 4
2 5
3 6
7
10
8
11
9
12
Exercise 10.88
Find the inverse of the matrix
2 5 8
4 7
6 . 9
Exercise 10.89
Multiply the matrix
2 5 8 .
4 7
6 9
Exercise 10.90
Matrix
is
6 0
1 9
2 5
Matrix
is
10
2 7
3 0
.
Matrix
C
is
12
Use your
9 3
7 . 2
calculator to nd. . .
a. b. c. d.
AC CA (A + B) C 2A + 1 B 2
14
3 5
4 6
a. Find the determinant by hand. b. Find the determinant by using your calculator. c. Finally, use your calculator to nd the determinant of the inverse of this matrix.
note: You should not have to type in the inverse by hand.
What did you get? Can you make a generalization about the "determinant of an inverse matrix"?
Exercise 10.92
1 3
2 n
|=2
183
Exercise 10.93
2 | 1 5 2 | 1 5
4 3 7 4 3 7
6 n |= 9 6 8 | 9
Exercise 10.94
a. Find the determinant by hand. b. Find the determinant by using the formula you found in #3. c. Find the determinant by using your calculator. Exercise 10.95
A triangle has vertices at (-1,-2), (1,5), and (3,4).
a. b. c. d.
Do a quick sketch of the triangle. Set up a determinant that will nd its area. Evaluate that determinant (use your calculator) to nd the area. Can you think of any
Exercise 10.96
1
Use your calculator to nd the determinant
3 5 5 1
4 3 9 2
2 8 7 0
not use all zeros!) Try to nd its inverse
6 9 4
|.
Exercise 10.97
Make up a 33 matrix whose determinant is zero. (Do on your calculator. What happens?
15
I'm sure you remember our whole unit on solving linear equations. . .by graphing, by substitution, and by elimination. Well, now we're going to nd a new way of solving those equations. . .by using matrices!
Oh, come on. . .why do we need another way when we've already got three?
Glad you asked! There are two reasons. First, this new method can be done entirely on a calculator.
Um. . .it would take a while. How about Please don't do that. With matrices and your calculator, all of these Wow! Do those really come up in real life? Yes, all the time. Actually, this is just Do you have an example?
Oh, look at the
about the only real-life application I can give you for matrices, although there are also a lot of other ones. But solving many simultaneous equations is incredibly useful. time! I have to explain how to do this method.
15 This
184
equation:
9 18
6 9
x y
9 3
Exercise 10.98
The rst thing you had to do was to rewrite this as two equations with two unknowns. Do that now. (Don't bother solving for The point is that
and
y,
one matrix equation is the same, in this case, as two simultaneous equations. What reverse: I give you simultaneous equations, and you turn
them into a matrix equation that represents the same thing. Let's try a few.
Exercise 10.99
Write a single matrix equation that represents the two equations:
3x + y = 2 6x 2y = 12
Exercise 10.100
Now, let's look at three equations:
7a + b + 2c = 1 8a 3b = 12 a b + 6c = 0
a. Write a single matrix equation that represents these three equations. b. Just to make sure it worked, multiply it out and see what three equations you end up with
OK, by now you are convinced that we can take simultaneous linear equations and rewrite them as a single matrix equation. In each case, the matrix equation looks like this:
AX = B where A
X a
and
solve forthat is, it has all our variables in it, so if we nd what
was
b .) c
X?
Time for
some matrix algebra! We can't divide both sides by can do the next best thing.
A,
Exercise 10.101
Take the equation inverse of
AX = B , A1
where
A, X ,
and
A1
(the
A)
in front. (Why did I say in front? Remember that order matters when multiplying in front of both sides, we have done the same thing to both sides.)
matrices. If we put
Exercise 10.102
Now, we have
A1 Agee,
didn't that equal something? Oh, yeah. . .rewrite the equation simpli-
Exercise 10.103
Now, we're multiplying again a bit simpler. We're done! We have now solved for the matrix
by something. . .what does that do again? Oh, yeah. . .rewrite the equation
X.
So, what good is all that again? Oh, yeah. . .let's go back to the beginning. 3x + y = 2 6x 2y = 12
185
You showed in #2 how to rewrite this as one matrix equation solve such an equation for
AX = B .
X.
and
Exercise 10.104
Solve those two equations for
and
Did it work? We nd out the same way we always haveplug our
original equations and make sure they work.
and
Exercise 10.105
Check your answer to #7.
Exercise 10.106
Now, solve the three simultaneous equations from #3 on your calculator, and check the answers.
16
4x + 2y = 3 3y 8x = 8
a. b. c. d.
Solve these two equations by either substitution or elimination. Now, rewrite those two equations as a matrix equation. Solve the matrix equation. Your answer should be in the form of a matrix equation: with the answers you found in part (a)?
[x] =
Now, using your calculator, nd the numbers for your equation in part (c). Do they agree
Exercise 10.108
6x 8y = 2 9x 12y = 5
a. Solve by using matrices on your calculator. b. Hey, what happened? Why did it happen, and what does it tell you about these two
equations?
Exercise 10.109
3x + 4y 2z = 1 8x + 3y + 3z = 4 xy+z =7
a. Solve. b. Check your answers. Exercise 10.110
2x 5y + z = 1 6x y + 2z = 4 4x 10y + 2z = 2
a. Solve b. Check your answer
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186
3x + 3y 2z = 4 x 7y + 3z = 9 5x + 2z = 6
a. Solve. ("Hey there's no b. Check your answers. Exercise 10.112
is 0.)
3w + 4x 8y + 2z = 4 7w 9x 3y + 4z = 2 2w + 5x + 2y 10z = 7 8w + 3x 6y z = 6
a. Solve. b. Check your answers.
17
(Solution on p. 188.)
and
A, B , C , D , ABC = DE
C.
(Solution on p. 188.)
Exercise 10.114
Here are two equations and two unknowns.
6m + 2n = 2 3 n = 1
a. Rewrite this problem as a matrix equation. b. Solve. What are m and n? Exercise 10.115
Solve the following equations for
(Solution on p. 188.)
a, b, c,
and
d.
2a + 3b 5c + 7d = 8 3a 4b + 6c + 8d = 10 10a + c + 6d = 3 a b c d = 69
Exercise 10.116
(Solution on p. 189.)
2 5
2n 4
2 5 2 5
6 4 6 4
187
Exercise 10.117
(Solution on p. 189.)
0 | 2 2
3 4 8
6 x |
1 2
a. Find the determinant. b. Check your answer by nding the determinant of that same matrix when
x = 10
on your
calculator. Does it come out the way your equation predicted? Show your work!
Exercise 10.118
(Solution on p. 189.)
4
Find the determinant
2 3 42 3
5 9
1 3
6 13 0 2
(Solution on p. 189.)
2 23 14
35
Exercise 10.119
Write a 22 matrix that has no inverse. No two of the four numbers should be the same.
188
A1 in front: A1 ABC = A1 DE But A A = I : IBC = A1 DE 1 But IB = B : BC = A DE 1 1 Multiply both sides by B in front: B BC = B 1 A1 DE 1 1 1 But B B = I : IC = B A DE 1 1 But IC = C : C = B A DE
Multiply both sides by
That is the solution. Note that solving this uses both the denition of an inverse matrix (A denition of the identity matrix (IB would not be correct. Incidentally, it may help to think of this in analogy to numerical equations. equation:
= B ).
3x = 12
You might say I would divide both sides by 3. But what if I told you there is no such thing as division, only multiplication? Hopefully you would say No problem, I will multiply both sides by
1 both sides by the inverse of 3 because 3 times 3 is 1, and 1 times 1 Multiplying by A to get rid of A is exactly like that.
Solution to Exercise 10.114 (p. 186) a. Rewrite this problem as a matrix equation.
m n
2 1
(*I urge you to conrm this for yourself. Multiply the two matrices on the left, then set the resulting matrix equal to the matrix on the right, and conrm that you get the two equations we started with.)
and
n? AX = A,
then it solves out as
If you think of that previous equation as the rst matrix into your calculator as First, matrix
B,
then type
can type
an error! Singular matrix! What happened? I can answer that question on two levels.
A,
thus dened, has a determinant of 0. (You can conrm this easily, with or without the
calculator.) Hence, it has no inverse. Second, these two equations are actually the same equationas you can see if you multiply the bottom equation by 2. They cannot be solved, because they have an innite number of solutions.
A = B,
where...
a 8 b 10 6 8 , X = , and B = c 3 1 6 1 1 d 69 1 Then the solution is X = A B , which comes out on the calculator: 3.345 39.564 X= 25.220 0.871 Since this equals the X I dened earlier, that means a = 3.345, b = 39.564,c = 25.220, d = 0.871.
2 3 3 4 A= 10 0 1 1
and
189
It's that easy...and it's also very, very dangerous. Because if you make one tiny little mistake (such as not noticing the 0b in the third equation, or mistyping one little number on the calculator), you get a completely wrong answer, and no credit. So what can you do about this? Here are a few tips.
Even on a problem like this, you can show me your work. Show me your you typed matrix
and your
and tell me
A1 B
into your calculator. Then I can see exactly what went wrong.
After you type in the matrices, always check them: just ask the calculator to dump out matrix
and
and match them against the original equation. Type: to put that number into memory
If you have time after you're done with everything else, come back and check the answers!
3.345 STO A
it right!
B, C ,
and
D.
Then type:
2A+3B5D+7D
you get approximately 8; and so on for the other three equations. If they all work, you know you got
2 5
6 4
2 5
6 4
0 (2 8x) 3 ( 1 2x) + 6 ( 16 8) = 3 + 6x + 6 ( 24) = 6x 141 b. Check your answer by nding the determinant of that same matrix when x = 10 on
Does it come out the way your equation predicted? Show your work! Our solution above predicts an answer of you don't get that, nd your mistake!
your calculator.
60 141 = 81.
ad bc,
1 3
2 6
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Chapter 11
much money I make in that particular hourit's how much total money I have made, after working that many hours.)
time (t)
money (m)
Table 11.1
b. c. d. e.
Which is the dependent variable, and which is the independent variable? Write the function. Sketch a quick graph of what the function looks like. In general: if I double the number of hours, what happens to the amount of money?
Exercise 11.2
I am stacking bricks to make a wall. Each brick is 4" high. Let and
1 This
191
192
b. c. d. e.
Which is the dependent variable, and which is the independent variable? Write the function. Sketch a quick graph of what the function looks like. In general: if I triple the number of bricks, what happens to the height?
Exercise 11.3
The above two scenarios are examples of
x,
direct variation. If a variable y varies directly with y = kx, where k is called the constant of variation. (We proportional to x, where k is called the constant of proportionality. if
Why do we say it two dierent ways? Because, as you've always suspected, we enjoy making your life dicult. Not students in general but just you personally.) So,
y if x doubles? (Hint: You can nd and prove the answer y = kx.) b. What happens to y if x is cut in half ? c. What does the graph y (x) look like? What does k represent in this graph?
equation
a. What happens to
Exercise 11.4
Make up a word problem like Exercises 1 and 2 above, on the subject of should
fast food. Your problem not involve getting paid or stacking bricks. It should involve two variables that vary directly
with each other. Make up the scenario, dene the variables, and then do parts (a) (e) exactly like my two problems.
dierent sizes but they have the same amount of air in them.
balloons that are very small experience a great deal of air pressure (the air inside pushing out on the balloon); the balloons that are very large, experience very little air pressure. He measures the volumes and pressures, and comes up with the following chart.
Volume (V )
5 10 15 20
Pressure (P )
270 135 90 67
Table 11.3
a. Which is the dependent variable, and which is the independent variable? b. When the volume doubles, what happens to the pressure? c. When the volume triples, what happens to the pressure?
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d. Based on your answers to parts (a) (c), what would you expect the pressure to be for
a balloon with a volume of 30?
e. On the right of the table add a third column that represents the quantity
PV:
pressure
times volume. Fill in all four values for this quantity. What do you notice about them?
f. Plot all four points on the graph paper, and ll in a sketch of what the graph looks like. g. Write the function P (V ). Make sure that it accurately gets you from the rst column to
the second in all four instances! (Part (e) is a clue to this.)
P (V )
on your calculator, and copy the graph onto the graph paper.
Exercise 11.6
The three little pigs have built three housesmade from straw, Lincoln Logs 1" thick; the bricks are 4" thick. Let
and bricks,
respectively. Each house is 20' high. The pieces of straw are 1/10" thick; the Lincoln Logs
are
be the
note: There are 12" in 1'. But you probably knew that. . .
values.
Building Blocks
Straw Lincoln Logs Bricks
thickness (t)
number (n)
Table 11.4
a. Which is the dependent variable, and which is the independent variable? b. When the thickness of the building blocks doubles, what happens to the number required?
(*Not sure? Pretend that the pig's cousin used 8" logs, and his uncle used 16" logs. See what happens to the number required as you go up in this sequence. . .)
c. When the thickness of the building blocks is halved, what happens to the number required? d. On the right of the table add a fourth column that represents the quantity tn: thickness
times number. Fill in all three values for this quantity. What do you notice about them? What do they actually represent, in our problem?
e. Plot all three points on the graph paper, and ll in a sketch of what the graph looks like. f. Write the function n (t). g. Graph your function n (t) on your calculator, and copy the graph onto the graph paper.
Does it match your graph in part (f )?
Exercise 11.7
The above two scenarios are examples of
inverse variation. If a variable y varies inversely with k x , where k is called the constant of variation. So, if
y y
if
x x
doubles?
note: You can nd and prove the answer from the equation
y=
k x.
b. What happens to
if
is cut in half ?
194
may want to try a few dierent ones on your calculator to see the eect
k increases? k has.)
(*You
Exercise 11.8
Make up a word problem like #1 and #2 above. Your problem should volume, or building a house. It problems.
not involve pressure and should involve two variables that vary inversely with each other.
Make up the scenario, dene the variables, and then do problems (a) - (h) exactly like my two
note: For #13, please note that these numbers are meant to simulate real world datathat is to
Exercise 11.9
For the following set of data. . .
x
3 6 21
y
5 11 34
Table 11.5
a. b. c. d.
Does it represent direct variation, inverse variation, or neither? If it is direct or inverse, what is the constant of variation? If
x = 30,
what would
be?
Exercise 11.10
For the following set of data. . .
x
3 4 10
y
18 32 200
Table 11.6
a. b. c. d.
3 This
Does it represent direct variation, inverse variation, or neither? If it is direct or inverse, what is the constant of variation? If
x = 30,
what would
be?
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Exercise 11.11
For the following set of data. . .
x
3 6 21
y
20 10 3
Table 11.7
a. b. c. d.
Does it represent direct variation, inverse variation, or neither? If it is direct or inverse, what is the constant of variation? If
x = 30,
what would
be?
Exercise 11.12
In #2 above, as you (hopefully) saw, the relationship is neither direct nor inverse. However, the relationship can be expressed this way:
y is directly proportional to x2 . k?
note: Once you have written the relationship, you can use it to generate more points that
(c).
b. What happens to y when you double x ? c. What happens to y when you double x? d. What happens to y when you triple x? Exercise 11.13
In June, 2007, Poland argued for a change to the voting system in the European Union Council The Polish suggestion: each member's voting strength should be directly proportional to the square root of his country's population. This idea, traditionally known of Ministers. as Pensore's Rule, is almost sacred among people versed in the game theory of voting according to one economist. I swear I am not making this up. Also in the category of things I am not making up, the following table of European Populations comes from Wikipedia.
Table 11.8
a. Write an equation that represents Pensore's Rule. Be sure to clearly label your variables. b. Suppose that Pensore's Rule was followed, and suppose that Poland voting strength was
exactly 100 (which I did actually make up, but of course it doesn't matter). What would the voting strength of Germany, Italy, and Luxembourg be?
196
x?
What happens
if you multiply
by 9?
d. Now, suppose a dierent country proposed the rule: each member's voting strength should be directly proportional to his country's population. Compared to
Pensore's Rule, how would that change things? Would it make things better for smaller countries, better for larger countries, or would it not make a dierence at all?
Exercise 11.14
Write a real world word problem involving an inverse relationship, on the topic of Identify the constant of variation. depends inversely upon the independent variable. your function to answer that question.
movies.
Write the function that shows how the dependent variable Create a specic numerical question, and use
a. If the number of people at the table doubles, does Al's expected tip double? b. If the average cost per meal doubles, does Al's expected tip double? c. Write the function that expresses the dependent variable, T , as a function of the two
independent variables, will have an unknown
N k
and
C. x
and
y .
Your function
Andrea is in charge of a new chemical plant. It's a technological miracleyou put in toxic waste products (which everyone is trying to get rid of anyway) and out comes Andreum, the new wonder chemical that everyone, everyone, everyone needs! (*What am I quoting?) Andrea's rst task is to nd a model to predict, based on the amount of toxic waste that goes
Exercise 11.16
Andrea makes two initial measurements. If 10 lb of toxic waste go in, 4 lb of Andreum come out. If 16 lb of toxic waste go in, 7 lb of Andreum come out.
a. Create a model based on these two experiments. This model should be a function
that comes out;
A (t)
that correctly predicts both of Andrea's data points so far. (A = amount of Andreum
b. Test your model, to conrm that it does correctly predict both data points. c. Now use your model to make a new predictionhow much Andreum should be produced
if 22 lb of toxic waste are put in?
Exercise 11.17
Being a good scientist, Andrea now runs an experiment to validate her model. She pumps in 22 lb of toxic waste and waits eagerly to see what comes out. Unfortunately, the reality does not match her prediction (the prediction you made in part (c) above). Instead, she only gets 4 lb of Andreum out this time.
4 This
197
a. Create a new model based on the two previous experiments, plus this new one. This new
function
A (t)
b. Test your model, to conrm that it does correctly predict all three data points. c. Now use your model to make a new predictionhow much Andreum should be produced
if 23 lb of toxic waste are put in?
Exercise 11.18
The new model works much better than the old one. When Andrea experiments with 23 lb of toxic waste, she gets exactly what she predicted. Now, her next job is (of course) optimization. Based on the model you previously calculated, how much toxic waste should she put in to get the
most
Andreum possible?
5 This
198
(a)
(b)
199
Exercise 11.20
A certain function contains the points (3,5) and (5,2).
a. Find the function. b. Verify that it contains both points. c. Sketch it on the graph paper, noting the two points on it. Exercise 11.21
A certain function contains the points (3,5), (5,2), and (1,4).
a. Find the function. b. Verify that it contains all three points. c. Sketch it on the graph paper, noting the three points on it. Exercise 11.22
A certain function contains the points (3,5), (5,2), (1,4), and (8,2).
a. Find the function. b. Verify that it contains the point (8,2). c. Sketch it on the graph paper (by graphing it on your calculator and then copying the sketch
onto the graph paper). Note the four points on it.
The following table shows the percentage of Canadian voters who voted in the 1996 federal election.
Age % voted
20 59
30 86
40 87
50 91
60 94
Table 11.9
a. Enter these points on your calculator lists. b. Set the Window on your calculator so that the
x-values
y -values
go from 0 to 100. Then view a graph of the points on your calculator. Do they increase steadily (like a line), or increase slower and slower (like a log), or increase more and more quickly (like a parabola or an exponent)?
c. Use the
STAT
d. Graph the function on your calculator. Does it match the points well? Are any of the
points outliers?
6 This
200
Height Weight
68 180
74 185
66 150
68 150
72 200
69 160
65 125
71 220
69 220
72 180
71 190
64 120
65 110
Table 11.10
a. Enter these points on your calculator lists. b. Set the Window on your calculator appropriately, and then view a graph of the points on
your calculator. Do they increase steadily (like a line), or increase slower and slower (like a log), or increase more and more quickly (like a parabola or an exponent)?
c. Use the
STAT
d. Graph the function on your calculator. Does it match the points well? Are any of the
points outliers?
Weight Mileage
29 31
35 27
28 29
44 25
25 31
34 29
30 28
33 28
28 28
24 33
Table 11.11
a. Enter these points on your calculator lists. b. Set the Window on your calculator appropriately, and then view a graph of the points on
your calculator. Do they decrease steadily (like a line), or decrease slower and slower (like a log), or decrease more and more quickly (like a parabola or an exponent)?
c. Use the
STAT
d. Graph the function on your calculator. Does it match the points well? Are any of the
points outliers?
or sitcomsand found that it made very little dierence. Quantity, not quality, mattered.) In a study that you can read all about at www.iusb.edu/journal/2002/hershberger/hershberger.html , Angela found that her data could best be modeled by the linear function Assuming that this line is a good t for the data...
x = 0.0288x+3.4397.
a. What does the number 3.4397 tell you? (Don't tell me about lines and points: tell me
about students, TV, and grades.)
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201
8 This
202
(a)
(b)
203
Exercise 11.28
Three cars and an airplane are traveling to New York City. But they all go at dierent speeds, so they all take dierent amounts of time to make the 500-mile trip. Fill in the following chart.
Time (t)hours
a. Is this an example of direct variation, inverse variation, or neither of the above? b. Write the function s (t). c. If this is one of our two types, what is the constant of variation? Exercise 11.29
There are a bunch of squares on the board, of dierent sizes.
a. Is this an example of direct variation, inverse variation, or neither of the above? b. Write the function A (s). c. If this is one of our two types, what is the constant of variation? Exercise 11.30
Anna is planning a party. Of course, as at any good party, there will be a lot of ppp on hand! 50 Coke cans t into one recycling bin. So, based on the amount of Coke she buys, Anna needs to make sure there are enough recycling bins.
Table 11.14
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A typical
beer is 10-proof. With beer in the tank, my car goes up to 30 mph. The maximum proof of wine (excluding dessert wines such as port) is 28-proof. With wine
b. Test your model to make sure it correctly predicts that a 28-proof drink will get me up to
75 mph.
c. Use your model to predict how fast I can get with a 151-proof drink. Exercise 11.32
Oops! I gave it a try. I poured Bacardi 151 into the car (so-called because it is indeed 151-proof ). And instead of speeding up, the car only went up to 70 mph. Your job is to create a
new quadratic
model
s (p) = ap2 + bp + c
a. Write simultaneous equations that you can solve to nd the coecients a, b, and c. b. Solve, and write the new s (p) function. c. Test your model to make sure it correctly predicts that a 151-proof drink will get me up
to 70 mph.
Exercise 11.33
Make up a word problem involving inverse variation, on the topic of
skateboarding.
a. Write the scenario. b. Label and identify the independent and dependent variables. c. Show the function that relates the dependent to the independent variable. This function
should (of course) be an inverse relationship, and it should be obvious from your scenario!
Exercise 11.34
I found a Web site (this is true, really) that contains the following sentence:
[This process] introduces an additional truncation error [directly] proportional to the original error and inversely proportional to the gain (g ) and the truncation parameter (q ).
I don't know what most of that stu means any more than you do. But if we use additional truncation error and relationship.
for the
Exercise 11.35
Which of the following correctly expresses, in words, the relationship of the area of a circle to the radius?
205
B. The area is directly proportional to the square of the radius C. The area is inversely proportional to the radius D. The area is inversely proportional to the square of the radius Exercise 11.36
Now, suppose we were to write the The radius of a
function of the area. Then we would write: circle _____________________ the area.
E: C:
10 102
30 65
50 34
70 27
90 22
100 17
350 8
Table 11.15
a. Use your calculator to create the following models, and write the appropriate functions
C (E)
Linear:
in the blanks.
C= C= Logarithmic: C = Exponential: C =
Quadratic:
b. Which model do you think is the best? Why? c. Based on his very strong correlation, Farr concluded that bad air had settled into low-lying
areas, causing outbreaks of cholera. We now know that air quality has nothing to do with causing cholera: the water-borne bacterial
What
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Chapter 12
Conics
12.1 Distance
Exercise 12.1
Draw the points (2,5), (10,5), and (10,1), and the triangle they all form.
Exercise 12.2
Find the distance from (2,5) to (10,5) (just by looking at it).
Exercise 12.3
Find the distance from (10,5) to (10,1) (just by looking at it).
Exercise 12.4
Find the distance from (2,5) to (10,1), using your answers to #2 and #3 and the Pythagorean Theorem.
Exercise 12.5
I start at Raleigh Charter High School. I drive 5 miles West, and then 12 miles North. How far am I now from RCHS?
note:
Exercise 12.6
Draw a point anywhere in the rst quadrant. Instead of labeling the specic coordinates of that point, just label it (x,y ).
x-axis?
note: think about specic pointssuch as (4,2) or (1,10)until you can see the pattern, and
answer the question for the general point (x,y ).
Exercise 12.8 How far across is it from your point to the Exercise 12.9
Find the distance
y -axis?
Exercise 12.10
Find the distance from the point (3,7) to the line
y = 2.
1 This
207
208
y = 2.
Exercise 12.12
Find the distance from the point (3,7) to the line
y = 2. y = 2. y -coordinate of my
Exercise 12.13
Find the distance from the generic point (x,y ) to the line
Exercise 12.14
I'm thinking of a point which is exactly 5 units away from the point (0,0). The point is 0. What is the
x-coordinate?
Exercise 12.15
I'm thinking of two points which are exactly 5 units away from (0,0). The points is 4. What are the
x-coordinates
of both
y -coordinates?
Draw these points on the same graph that you did #14.
Exercise 12.16
I'm thinking of two points which are exactly 5 units away from (0,0). The points is 4. What are the
x-coordinates
of both
y -coordinates?
Draw these points on the same graph that you did #14.
Exercise 12.18
Draw another point somewhere else. Label it (x2 ,y2 ). To make life simple, make this point higher and to the right of the rst point.
Exercise 12.19
Draw the line going from (x1 ,y1 ) to (x2 ,y2 ). Then ll in the other two sides of the triangle
Exercise 12.20 How far up is it from the rst point to the second? (As always, start by thinking about specic
numbersthen see if you can generalize.)
Exercise 12.21 How far across is it from the rst point to the second? Exercise 12.22
Find the distance
from (x1 , y1 ) to (x2 , y2 ), using the Pythagorean Theorem. This will give you
Exercise 12.23
Plug in
x2 = 0
and
y2 = 0
into your formula. You should get the same formula you got on the
previous assignment, for the distance between any point and the origin. Do you?
Exercise 12.24
Draw a line from (0,0) to (4,10). Draw the point at the if you have to.) What are the coordinates of that point?
Exercise 12.25
Draw a line from (3,2) to (5,4). What are the coordinates of the midpoint?
Exercise 12.26
Look back at your diagram of a line going from (x1 , y1 ) to (x2 , y2 ). What are the coordinates of the midpoint of that line?
2 This
209
Exercise 12.27
Find the distance from the point (3,7) to the line
x = 2. x = 2.
Exercise 12.28
Find the distance from the generic point (x,y ) to the line
Exercise 12.29
Find the distance from the point (3,7) to the line
x = 2. x = 2.
5, and which are exactly 10 units
Exercise 12.30
Find the distance from the generic point (x,y ) to the line
Exercise 12.31
Find the coordinates of all the points that have away from the origin.
y -coordinate
Exercise 12.32
Draw all the points you can nd which are exactly 3 units away from the point (4,5).
Draw as many points as you can which are exactly 5 units away from (0,0) and ll in the shape. What shape is it?
Exercise 12.34
Now, let's see if we can nd the the sentence:
equation for that shape. How do we do that? Well, for any point
(x,y ) to be on the shape, it must be exactly ve units away from the origin. So we have to take
The point (x,y ) is exactly ve units away from the origin
every point on our shape, and no other points. (Stop for a second and discuss this point, make sure it makes sense.) OK, but how do we do that?
and translate it into math. Then we will have an equation that describes
a. To the right is a drawing of our point (x,y ), 5 units away from the origin. On the drawing,
I have made a little triangle as usual. How long is the vertical line on the right side of the triangle? Label it in the picture.
b. How long is the horizontal line at the bottom of the triangle? Label it in the picture.
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c. Now, all three sides are labeled. Just write down the Pythagorean Theorem for this triangle,
and you have the equation for our shape!
d. Now, let's see if it worked. A few points that are obviously part of our shapethat is,
they are obviously 5 units away from the originare the points (5,0) and (4,3). Plug them both into your equation from the last part and see if they work.
e. A few points that are clearly not part of our shape are (1,4) and (2,7). Plug them both
into your equation for the shape to make sure they don't work!
Exercise 12.35
OK, that was all the points that were 5 units away from the origin. Now we're going to nd an equation for the shape that represents all points that are exactly 3 units away from the point (4, 1). Go through all the same steps we went through abovedraw the point (4,1) and an arbitrary point (x,y ), draw a little triangle between them, label the distance from (x,y ) to (4,1) as being 3, and write out the Pythagorean Theorem. Don't forget to test a few points!
Exercise 12.36
By now you probably get the idea. Sowithout going through all that workwrite down the equation for all the points that are exactly 7 units away from the point (5,3).
Exercise 12.37
And nally, the generalization as always: write down the equation for all the points that are exactly
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Write down the equation for a circle (*aka All the Points Equidistant from a Given Point) with center (3,6) and radius 9. (Although the wording is dierent, this is exactly like the problems you did on the in-class assignment.)
Exercise 12.39
Now, let's take it the other way.
(x 4) + (y + 8) = 49
a. b. c. d.
What is the center of the circle? What is the radius? Draw the circle. Find two points on the circle (by looking at your drawing) and plug them into the equation to make sure they work. (Show your work!)
2x2 + 2y 2 + 8x + 24y + 60 = 0 is also the equation for a circle. But in order to graph it, we need to put it 2 2 2 into our canonical form (x h) + (y k) = r . In order to do that, we have to complete the square. . .
twice! Here's how it looks.
x2 and
Collect and
and
bring
the
other side.
x2 + 4x + 4 + y 2 + 12y + 36 = 30 + 4+ 36 (x + 2) + (y + 6) = 10
2 2
Complete
the
square
in
both
the radius is
10.
Table 12.1
Exercise 12.40
(x h) + (y k) =
b. What are the center and radius of the circle? c. Draw the circle. d. Find two points on the circle (by looking at your drawing) and plug them into original
equation to make sure they work. (Show your work!)
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= 3).
same distance from (0,3) that they are from the line (y
= 3).
One of the points is very obvious. You can get two more of them, exactly, with a bit of thought. After that you have to start playing around. Feel free to use some sort of measuring device (such as your ngernail, or a pencil eraser). When you think you have the whole shape, call me and let me look.
y = 3x2 30x 70
a. b. c. d.
5 This 6 This
Put into the standard form of a parabola. Vertex: Opens (up/down/right/left): Graph it
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Exercise 12.42
x = y2 + y
a. b. c. d.
Put into the standard form of a parabola. Vertex: Opens(up/down/right/left): Graph it
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Exercise 12.43
Find the equation for a parabola that goes through the points (0,2) and (0,8).
We have talked about the geometric denition of a parabola: all the points in a plane that are the same
focus) that they are from a given line (the directrix). And we have talked
2
y = a (x h) + k 2 parabola: x = a (y k) + h
denition of a parabola, and the equation
What we haven't done is connect these two thingsthe denition, and turn it into an equation.
for a parabola. We're going to do it the exact same way we did it for a circlestart with the geometric
In the drawing above, I show a parabola whose focus is the origin (0,0) and directrix is the line On the parabola is a point (x,y ) which represents
y = 4.
Exercise 12.44
d1
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Exercise 12.45
d2
= 4).
What is
d2?
Exercise 12.46
What denes the parabola as suchwhat makes (x,y ) part of the parabolais that these two distances are the
d1 = d2
Exercise 12.47
Simplify your answer to #3; that is, rewrite the equation in the standard form.
a. b. c. d.
across from one to the other (the horizontal line in the drawing)? down from one to the other (the vertical line in the drawing)? midpoint of the diagonal line?
Exercise 12.49
What is the distance from the point (1024,3) to the line
y = 1? x
and
Exercise 12.50
Find all the points that are exactly 4 units away from the origin, where the are the
coordinates
same.
Exercise 12.51
Find the equation for a parabola whose vertex is
(3, 1),
(0, 4).
Exercise 12.52
2x2 + 2y 2 6x + 4y + 2 = 0
a: Put this equation in the standard form for a circle. b: What is the center? c: What is the radius?
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Exercise 12.53
1 x = 4 y2 + y + 2
a: b: c: d:
Put this equation in the standard form for a parabola. What direction does it open in? What is the vertex? Graph it on the graph paper.
Exercise 12.54
Find the equation for a circle whose diameter stretches from (2,2) to (5,6).
Exercise 12.55
We're going to nd the equation of a parabola whose focus is (3,2) and whose directrix is the line
x = 3.
denition of a parabola.
In the drawing above, I show the focus and the directrix, and an arbitrary point (x,y ) on the parabola.
a: d1 is the distance from the point (x,y ) to the focus (3,2). What is d1? b: d2 is the distance from the point (x,y ) to the directrix (x = 3). What is d2? c: What denes the parabola as suchwhat makes (x,y ) part of the parabolais that
Write the equation for the parabola.
d1 = d2.
d: Simplify your answer to part (c); that is, rewrite the equation in the standard form.
On the drawing below are the points (3,0) and (3,0). We're going to draw yet another shapenot a circle or a parabola or a line, which are the three shapes we know about. In order to be on our shape, the point (x,y ) must have the following property:
plus
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We're going to start this one the same way we did our other shapes: intuitively. Your object is to nd all the points that have that particular property. Four of them are. . .well, maybe none of them are exactly obvious, but there are four that you can get exactly, with a little thought. After that, you have to sort of dope it out as we did before. When you think you know the shape, don't call it out! Call me over and I will tell you if it's right.
10
In class, we discussed how to draw an ellipse using a piece of cardboard, two thumbtacks, a string, and a pen or marker.
Do this. Bring your drawing in as part of your homework. (Yes, this is a real part of your homework!) Exercise 12.57
(x2)2 9
y2
(1) 4
=1
a. Is it horizontal or vertical? b. What is the center? c. What is a? d. What is b? e. What is c? f. Graph it. Exercise 12.58
4x2 2 9 + 25y = 1 This sort of looks like an ellipse in standard form, doesn't it?
But it isn't. Because we have no room in our standard form for that 4 and that 25for numbers
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a. Rewrite the left-hand term, 4x , by dividing the top and bottom of the fraction by 4. Leave 9
the bottom as a fraction; don't make it a decimal.
25y 2 ,
c. Now, you're in standard form. what is the center? d. How long is the major axis? e. How long is the minor axis? f. What are the coordinated of the two foci? g. Graph it. Exercise 12.59
1 18x2 + 2 y 2 + 108x + 5y + 170 = 0
a. Put in standard form. b. Is it horizontal or vertical? c. What is the center? d. How long is the major axis? e. How long is the minor axis? f. What are the coordinates of the two foci? g. Graph it. Exercise 12.60
The major axis of an ellipse runs from (5,-6) to (5,12). One focus is at (5,-2). Find the equation for the ellipse.
Exercise 12.61
The foci of an ellipse are at (-2,3) and (2,3) and the ellipse contains the origin. Find the equation for the ellipse.
Exercise 12.62
We traditionally say that the Earth is 93 million miles away from the sun. However, if it were
always 93 million miles away, that would be a circle (right?). In reality, the Earth travels in an ellipse, with the sun at one focus. According to one Web site I found,
There is a 6% dierence in distance between the time when we're closest to the sun (perihelion) and the time when we're farthest from the sun (aphelion). Perihelion occurs on January 3 and at that point, the earth is 91.4 million miles away from the sun. At aphelion, July 4, the earth is 94.5 million miles from the sun. (http://geography.about.com/library/weekly/aa121498.htm)
Write an equation to describe the orbit of the Earth around the sun. Assume that it is centered on the origin and that the major axis is horizontal. (*Why not? There are no axes in space, so you can put them wherever it is most convenient.) Also, work in units of numbers you are given are simply 91.4 and 94.5.
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Figure 12.1
11
Here is the geometric denition of an ellipse. There are two points called the foci: in this case, (-3,0) and (3,0). A point is on the ellipse if the I'll use 10. Note that the foci second.
sum of its distances to both foci is a certain constant: in this case, dene the ellipse, but are not part of it.
d1
d2
to the
Exercise 12.63
Calculate the distance
d1 d2
Exercise 12.64
Calculate the distance (by drawing a right triangle, as always).
Exercise 12.65
Now, to create the equation for the ellipse, write an equation asserting that the equals 10. Now simplify it. We did problems like this earlier in the year (radical equations, the harder variety that have two radicals). The way you do it is by isolating the square root, and then squaring both sides. In this case, there are two square roots, so you will need to go through that process twice.
sum of
d1
and
d2
Exercise 12.66
Rewrite your equation in #3, isolating one of the square roots.
Exercise 12.67
Square both sides.
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Exercise 12.69
Square both sides again.
Exercise 12.70
Multiply out, cancel, combine, and get it to look like the standard form for an ellipse.
Exercise 12.71
Now, according to the machinery of ellipses, what should that equation look like? Horizontal or vertical? Where should the center be? What are started with?
a, b,
and
c?
12
On the drawing below are the points (5,0) and (5,0). We're going to draw yet another shapeour nal Take the distance from (x,y ) to (5,0), and the distance from (x,y ) to (5,0). Those two distances must
dier by 6. (In other words, this distance minus that distance must equal
6.) x-axis
that work.
We're going to start this one the same way we did our other shapes: intuitively. Your object is to nd all the points that have that particular property. Start by nding the two points on the After that, you have to sort of dope it out as we did before. When you think you know the shape, don't call it out! Call me over and I will tell you if it's right.
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13
Complete the following chart, showing the similarities and dierences between ellipses and hyperbolas. .899999999999999
Ellipse
How to identify an equation with this shape Has an
Hyperbola
2 2
Equation in standard form: horizontal How can you tell if it is horizontal? Draw the shape here. Label
(xh)2 (yk)2 a2 b2
=1
a, b, and
(h,k )
on the drawing.
a b c
a, b,
or
c?
a, b,
and
c.
Table 12.2
Exercise 12.73
y2
1 4
( )
(x2)2 9
=1
a. Is it horizontal or vertical? b. What is the center? c. What is a? d. What is b? e. What is c? f. Graph it. Make sure the box and asymptotes can be clearly seen in your graph. Exercise 12.74
2x2 + 8x 4y 2 + 4y = 6
a. Put in standard form. b. Is it horizontal or vertical? c. What is the center? d. What is a? e. What is b? f. What is c? g. Graph it. Make sure the box and asymptotes can be clearly seen in your graph.
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Exercise 12.76
A hyperbola has vertices at (1,2) and (1,22), and goes through the origin.
a. Find the equation for the hyperbola. b. Find the coordinates of the two foci.
14
Vertical
Equations
(yk)2 a2 (xh)2 b2
(xh)2 a2 (xh)2 a2
(yk)2 b2 (yk)2 b2
=1 =1
(xh)2 b2 (yk)2 a2
=1 =1
a2 = b2 + c2 , (a > b) c2 = a2 + b2
Table 12.3
Exercise 12.77
Identify each equation as a
a. Vertical line b. Horizontal line c. Vertical parabola d. Horizontal parabola e. Circle f. Vertical ellipse g. Horizontal ellipse h. Vertical hyperbola i. Horizontal hyperbola Exercise 12.79
The United States Capitol building contains an elliptical room. It is 96 feet in length and 46 feet in width.
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a. Write an equation to describe the shape of the room. Assume that it is centered on the
origin and that the major axis is horizontal.
b. John Quincy Adams discovered that if he stood at a certain spot in this elliptical chamber,
he could overhear conversations being whispered a the opposing party leader's desk. This is because both the desk, and the secret listening spot, were was Adams standing from the desk?
c. How far was Adams standing from the edge of the room closest to him? Exercise 12.80
A comet zooms in from outer space, whips around the sun, and zooms back out. Its path is one branch of a hyperbola, with the sun at one of the foci. Just at the vertex, the comet is 10 million miles from the center of the hyperbola, and 15 million miles from the sun. Assume the hyperbola is horizontal, and the
a. Find the equation of the hyperbola. b. When the comet is very far away from the sun, its path is more or less a line. As you
might guess, that is represented by the asymptotes of the hyperbola. (One asymptote as it come in, another as it goes out.) Write the equation for the line that describes the path of the comet after it has
left the sun and gotten far out of our solar system.
Exercise 12.81
Extra Credit:
Consider a hyperbola with foci at (5,0) and (5,0). In order to be on the hyperbola, a point must have the following property: its distance to one focus, standard form (2 points).
equation for this hyperbola by using the geometric denition of a hyperbola (3 points). Then simplify it to
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Chapter 13
Date: January...
1 2 3 5 10 d
Number of Songs
100 103 106
Table 13.1
d)
Exercise 13.2
You begin an experiment with 10 amoebas in a petrie dish. Each minute, each amoeba splits, so the total number of amoebas doubles. Fill in the following table.
Minutes elapsed
0 1 2 5 m
1 This
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Look at a calendar for this month. Look at the column that represents all the Thursdays in this month.
a. What are the dates? b. What kind of sequence do these numbers represent? c. If it is arithmetic, what is d, the common dierence? If geometric, what is r , the common
ratio?
d. If that sequence continued, what would be the 100th term? Exercise 13.4
How many terms are in the arithmetic sequence 25, 28, 31, 34,...,61?
Exercise 13.5
Suppose that indicate if it is
a, b, c, d. . .
a. b. c. d.
Exercise 13.6
Find
10, 30,
2x + 8
tn+1 = tn + d
leads
tn = t1 + d (n 1).
tn+1 = rtn .
Exercise 13.8
Suppose a gallon of gas cost $1.00 in January, and goes up by 3% every month throughout the year.
a. Find the cost of gas, rounded to the nearest cent, each month of the year. (Use your
calculator for this one!)
b. Is this sequence arithmetic, geometric, or neither? c. If it keeps going at this rate, how many months will it take to reach $10.00/gallon? d. How about $1000.00/gallon?
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Exercise 13.9
In an arithmetic sequence, each term is the previous term plus a constant. In a geometric sequence, each term is the previous term times a constant. Is it possible to have a sequence which is both arithmetic
and geometric?
S=
7 n=1 10
a. Write out all the terms of this series. b. Does this represent an arithmetic series, a geometric series, or neither? c. Find the sum. Exercise 13.11
True or false?
a. b.
Exercise 13.12
Write the following series in series notation.
arithmetic series
3 + 5 + 7 + 9 + 11 + 13 + 15 + 17,
by using
a trick that works on all arithmetic series. Use that same trick to nd the sum of the following
10 + 13 + 16 + 19 + 22...100
(Note that you will rst have to gure out how many terms there arethat is, an arithmetic sequence!)
which term in
this series 100 is. You can do that by using our previously discovered formula for the
nth
term of
Exercise 13.14
Now I want you to take that same trick, and apply it to the have an arithmetic series of dierence is
an .
The common
d,
t1 + d,
t1 + 2d,
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geometric series
by using
r,
starting with
t1
tn ,
which is
t1 rn1 .
r,
t1 r,
a. Find the general formula for the sum of this series b. Use that general formula to add up 1 + 2 + 4 + 8 + 16 + 32. Did it come out right? 1 1 1 1 c. Use that general formula to add up 2 + 4 + 1 + 16 + 32 . Did it come out right? 8 Exercise 13.17
Suppose a ball is dropped from a height of 1 ft. It bounces back up. But each time it bounces, it reaches only
9
10
a. The ball falls. Then it bounces up and falls down again (second bounce). Then it bounces up and falls down again (third bounce). How high does it go after each of these bounces? b. How high does it go after the 100th bounce? c. How far does it travel before the fourth bounce? (*You don't need any fancy math to
do this part, just write out all the individual trips and add them up.)
2 + 4 + 6 + 8...2n = n (n + 1). n.
a. First, show that this formula works when n = 1. b. Now, show that this formula works for (n + 1), assuming that it works for any given Exercise 13.19
Use mathematical induction to prove that
n x=1
x2 =
n(n+1)(2n+1) . 6
Exercise 13.20
In a room with that there are
n people (n 2), every person shakes hands once with every other person. Prove n2 n handshakes. 2
n x=1
Exercise 13.21
Find and prove a formula for
x3 .
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note: You will have to play with it for a while to nd the formula. Just write out the rst four
or ve terms, and see if you notice a pattern. Of course, that won't the induction is for!
The teacher sees the wishing star high in the night sky, and makes the mistake of wishing for whiteboard markers. The next day (let's call it day 1), the marker fairy arrives and gives the The day after that (day 2), the marker fairy gives the teacher a marker that workshooray! as many new markers as the day before.
teacher three markers. The day after that, nine new markers. . .and so on. . .each day, three times
a. If
is the number of new markers the fairy brings that day, is the list
of all
b. Give a recursive denition for the sequence: that is, a formula for mn+1 based on mn . c. Give an explicit denition for the sequence: that is, a formula that I can use to quickly
nd
mn
n,
terms.
d. On day 30 (the end of the month) how many markers does the fairy bring? e. After that day 30 shipment, how many total markers has the fairy brought? Exercise 13.23
You start a dot-com business. Like all dot-com businesses, it starts great, and then starts going downhill fast. Specically, you make $10,000 the rst day. Every day thereafter, you make $200 less than the previous dayso the second day you make $9,800, and the third day you make $9,600, and so on. You might think this pattern stops when you hit zero, but the pattern just keeps right on goingthe day after you make $0, you on.
lose $200, and the day after that you lose $400, and so
a. If
b. How much money do you make on the 33rd day? c. On the day when you lose $1,000 in one day, you nally close up shop. What day is that? d. Your accountant needs to gure out the total amount of money you made during the life
of the business. Express this question in summation notation.
2 3
2 9
2
27
2
81
+ ... ). d (if
a. b. c. d. e.
6 This
Is this an arithmetic series, a geometric series, or neither? Write this series in summation notation (with a What is
t1 ,
and what is
(if it is geometric) or
it is arithmetic)?
What is the sum of the rst 4 terms of this series? What is the sum of the rst
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n:
1 12
1 23
1 34
+ ... +
1 n(n+1)
n n+1
Extra credit:
An arithmetic series starts with
t1
and goes up by
to nd the general formula for the sum of this series, as a function of
t1 , n,
and
d.
Chapter 14
Probability
14.1 How Many Groups?
Exercise 14.1
A group of high school students is being divided into groups based on two characteristics: class (Freshman, Sophomore, Junior, or Senior) and hair color (blond, dark, or red). For instance, one group is the red-haired Sophomores. How many groups are there, total?
Exercise 14.2
According to some sources, there are approximately 5,000 species of frog. Within each species, there are three types: adult female, adult male, and tadpole. If we divide frogs into groups according to both species and typeso one group is adult females of the species Western Palearctic Water Froghow many groups are there?
Exercise 14.3
Suppose I roll a normal, 6-sided die, and ip a normal, 2-sided coin, at the same time. So one possible result is 4 on the die, heads on the coin.
a. How many possible results are there? b. If I repeat this experiment 1,000 times, roughly how many times would you expect to see
the result 4 on the die, heads on the coin?
c. If I repeat this experiment 1,000 times, roughly how many times would you expect to see the result any even number on the die, heads on the coin? d. Now, let's come back to problem #1. I could have asked the question If you choose 1,000
students at random, how many of them will be red-haired Sophomores? would The answer
The following tree diagram represents all the possible outcomes if you ip a coin three times.
1 This 2 This
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232
Figure 14.1
a. One possible outcome is First ip heads, second ip tails, third ip heads. Locate and
circle this outcome on the diagram. Then, in the space below, answer the question: what is the probability of that particular outcome?
b. c. d. e.
What is the probability that all three ips will be the same? What is the probability that What is the probability that
exactly one of the coins will end up heads? at least one of the coins will end up heads?
Suppose there were a thousand people in a room. (A really big room.) Each one of those people pulled out a coin and ipped it three times. Roughly how many people would you be able to say, All three of my ips came out the same?
Exercise 14.5
There are seven dierent types of star. In order of decreasing temperature, they are: O, B, A, F, G, K, and M. (Some astronomers remember this based on the mnemonic: Oh, be a ne girl: kiss me.) Within each stellar type, stars are placed into ten subclasses, numbered from 0 to 9. Our own sun is a type G, subclass 2.
a. How many dierent type-and-subclass categories are there? (In other words, if you drew
the tree diagramwhich I am not recommendinghow many leaves would there be?)
b. Of these type-and-subclass categories, how many of them have a letter (type) that is a vowel, and a number (subclass) that is a multiple of 3? c. If you surveyed a thousand randomly chosen stars, how many of them would you expect
to be G2 like our own sun?
Exercise 14.6
According to the U.S. Census Bureau, the U.S. population crossed the 300 Million mark in the year 2006. In that year, three out of four people in the U.S. were considered white; one out (*Hispanic or Latino was not considered a separate of four belonged to minority ethnic groups.
ethnic group in this study.) Children (under age 18) made up approximately one quarter of the population. Males and females were equally distributed.
a. In a room full of a hundred people randomly chosen from the U.S. 2006 population, how
many of them would you expect to be white?
b. Of those, how many would you expect to be children? c. Of those, how many would you expect to be boys? d. So, what is the probability that a randomly chosen person in the U.S. in 2006 was a white
boy?
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e. What assumptionnot necessarily true, and not stated in the problemdo you have to
make in order to believe your answer to part (d) is accurate?
a. What is the probability that you will roll a 3? b. Explain, in your own words, what your answer to part (a) means. Do not use the words
probability or chance. A typical 9-year-old should be able to understand your explanation.
c. What is the probability that you will roll an even number? d. What is the probability that you will roll a number less than 3? e. What is the probability that you will roll a number less than 4? f. What is the probability that you will roll a number less than 7? Exercise 14.8
Suppose you roll
a. b. c. d.
Draw a tree diagram showing all the possible outcomes for both rolls. What is the probability that you will roll 3 on the rst die, 2 on the second? What is the probability that you will roll 3 on one die, 2 on the other? What is the probability that the sum of both dice will be a 5?
Eli and Beth each chooses a letter of the alphabet, completely at random.
a. What is the probability that Eli and Beth both choose the letter A? b. What is the probability that Eli and Beth both choose vowels? c. What is the probability that Eli chooses a letter that appears somewhere in the word Eli,
and Beth chooses a letter that appears somewhere in the word Beth?
Exercise 14.10
In your hand, you hold two regular 6-sided dice. One is red, and one is blue. You throw them both.
a. If both dice roll 1 that is sometimes referred to as snake eyes. What is the probability
of snake eyes?
b. What is the probability that both dice will roll even numbers? c. What is the probability that both dice will roll the same as each other?
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a. b. c. d.
What is the probability that the top card in the rst deck is the ace of spades? What is the probability that the top card in the second deck is the ace of spades? What is the probability that the top card in the rst What is the probability that the top card in
two decks are both the ace of spades? all ve decks are the ace of spades?
Exercise 14.12
According to http://www.m-ms.com/, a bag of Milk Chocolate (plain) M&Ms brown, 14% yellow, 13% red, 24% blue, 20% orange, and 16% green M&Ms. M&Ms
contains 13%
contains 12% brown, 15% yellow, 12% red, 23% blue, 23% orange, and 15% green.
that it will be brown?
A bag of Peanut
a. If you choose one M&M at random from a Milk Chocolate bag, what is the probability b. If you choose one M&M at random from a Milk Chocolate bag, what is the probability
that it will be brown or yellow?
c. If you choose one M&M at random from a Peanut bag, what is the probability that it will
be brown or yellow?
d. If you choose one M&M at random from a Milk Chocolate bag, and one M&M at random
from a Peanut bag, what is the probability that they will both be brown or yellow?
Exercise 14.13
What is the probability that a married couple were both born...
Each morning, before they go o to work in the mines, the seven dwarves line up and Snow White kisses each dwarf on the top of his head. In order to avoid any hint of favoritism, she kisses them in random order each morning.
note: No two parts of this question have exactly the same answer.
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a. b. c. d.
What is the probability that the dwarf named Bashful gets kissed rst on Monday? What is the probability that Bashful gets kissed rst both Monday What is the probability that Bashful does (Monday Friday)?
and Tuesday? not get kissed rst, either Monday or Tuesday? What is the probability that Bashful gets kissed rst at least once during the week
e. What is the probability that, on Monday, Bashful gets kissed rst and Grumpy second? f. What is the probability, on Monday, that the seven dwarves will be kissed in perfect
alphabetical order?
g. What is the probability that, on Monday, Bashful and Grumpy get kissed before any other
dwarves?
Exercise 14.16
The drawing shows a circle with a radius of 3" inside a circle with a radius of 4". If a dart hits somewhere at random inside the larger circle, what is the probability that it will fall somewhere in the smaller circle?
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Exercise 14.17
A bag has 26 tiles in it, each with a dierent letter of the alphabet.
a. You pick one tile out of the bag, look at it, and write it down. Then you put it back in the
bag, which is thoroughly mixed up. Then you pick another tile out of the back, look at it, and write it down. What is the probability that your rst letter was A and your second letter was T?
b. Same bag, dierent plan. This time you pick the rst tile, but do not put it back in
the bag. Then you pick a second tile and place it next to the rst? Now what is the probability that your rst letter was A and your second letter was T?
c. In the second case, what is the probability that your two letters, together, could make the
word AT?
Exercise 14.18
A deck of cards has 52 cards, 13 of each suit. Assume there are no Jokers.
parts of this question have exactly the same answer.) a. If you draw a card at random, what is the probability of getting the Ace of Spades? b. If you draw two cards at random, what are the odds that the rst will be the Ace of Spades
and the second will be the King of Spades?
c. If you draw two cards at random, in how many dierent ways can you draw those two
cards?
d. Based on your answers to (b) and (c), if you draw two cards at random, what is the
probability that you will get those two cards?
e. If you draw three cards at random, what are the odds that the rst will be the Ace of
Spades, the second will be the King of Spades, and the third the Queen of Spades?
237
f. If you draw three cards at random, in how many dierent ways can you draw those
three cards?
g. Based on your answers to (e) and (f ), if you draw three cards at random, what is the
probability that you will get those three cards?
Exercise 14.19
Jack and Jill were born in the same year.
a. What is the probability that they were born on the same day? b. What is the probability that Jack's birthday comes rst? c. Assuming that Jack and Jill do not have the same birthday, what is the probability that their mother has the same birthday as one of them? d. So...if three random people walk into a room, what is the probability that no two of them
will have the same birthday?
e. If three random people walk into a room, what is the probability that at least two of
them will have the same birthday?
They
are going to show two movies back-to-back for their Superhero End-of-School Blowout.
a. List all possible pairings. To keep it short, use letters to represent the movies: for instance, BX means Batman followed by X-Men. (XB is a dierent pairing, and should
be listed separately.) BB, of course, means they will show Batman twice! Try to list them in a systematic way, to make sure you don't miss any.
b. How many pairings did you list? c. Now, list all possible pairings that do not show the same movie twice. Once again,
try to list them in a systematic way, to make sure you don't miss any.
a. First, the Lead Singer is chosen, and comes up onto the stage. How many possible choices
are there?
How many
c. Next, from the remaining audience members, a Rhythm Guitarist is chosen. How many
possible choices are there?
How many
6 This
238
g. Multiply your answers in parts (a)(f ) to nd the number of possible Rock Bands that
could be created?
h. How could this calculation be expressed more compactly, with factorials? Exercise 14.22
The school math club is going to elect a President, a Vice President, a Secretary, and a Treasurer. How many possible ocer lists can be drawn up if...
a. There are only four people in the math club? b. There are twenty people in the math club? Exercise 14.23
A license plate consists of exactly eight characters. uppercase letter, or a numerical digit. Assume that each character must be an
a. How many possible license plates are there? b. Now assume that you are also allowed to have blank spaces, which count as part of your
eight letters. How many possible license plates are there?
Exercise 14.24
According to Robert de Boron's version of the story, King Arthur's Round Table was large enough to accommodate 50 chairs. One chair was always left empty, for the knight who would fulll the Grail Quest. So, if 49 knights approach the table one day, how many dierent ways can they seat themselves?
Exercise 14.25
Invent, and solve, your own permutations problem. It should be a scenario that is quite dierent from all the scenarios listed above, but it should logically lead to the same method of solving.
The comedy troupe Monty Python had six members: John Cleese, Eric Idle, Graham Chapman, Michael Palin, Terry Gilliam, and Terry Jones. Suppose that on the way to lming an episode of their Flying Circus television show, they were forced to split up into two cars: four of them could t in the Volkswagen, and two rode together on a motorcycle. How many dierent ways could they split up?
a. List all possible pairs that might go on the motorcycle. (Note that Cleese-Idle and
Idle-Cleese are the same pair: it should be listed once, not twice.)
b. List all possible groups of four that might ride in the car. (Same note.) c. If you listed properly, you should have gotten the same number of items in parts (a) and
(b). (After all, if Cleese and Idle ride on the motorcycle, we know who is in the car!)
6 2
.
It is also 6 choose 4, or
6 4
.
Given that
7 This
239
d. Write an algebraic generalization to express the rule discussed in part (c). Exercise 14.27
A Boston Market
Side Item Sampler allows you to choose any three of their fteen side items.
a. To answer this combinations question, begin with a related permutations question. Suppose
you had three plates labeled Plate 1, Plate 2, and Plate 3, and you were going to put a dierent side item in each plate.
i. How many items could you put in Plate 1? ii. For each such choice, how many items could you put in Plate 2? iii. For each such choice, how many items could you put in Plate 3? iv. So, how many Plate 1 Plate 2 Plate 3 permutations could you create? b. The reason you haven't answered the original question yet is that, in a real Side Item
Sampler, the plates are not numbered. Corn. So...in the space below, list For instance, Sweet CornMashed Potatoes Creamed Spinach is the same meal as Creamed SpinachMashed PotatoesSweet
c. How many possible arrangements did you list? (In other words, how many times did we
originally count every possible meal?)
d. Divide your answer to a(iv) by your answer to (b) to nd out how many Side Item Samplers
can be created.
Exercise 14.28
How many three-note chords can be made by...
a. Using only the eight natural notes (the white keys on a piano)? b. Using all twelve notes (black and white keys)? Exercise 14.29
The United States Senate has 100 members (2 from each state). Suppose the Senate is divided evenly: 50 Republicans, and 50 Democrats.
a. b. c. d. e.
How many possible 3-man committees can the Republicans make? Express your answer to part (a) in terms of factorials. How many possible 47-man committees can the Republicans make? How many 10-man committees can the Democrats make? How many committees can be formed that include 5 Democrats and 5 Republicans?
Exercise 14.30
Invent, and solve, your own combinations problem. It should be a scenario that is quite dierent from all the scenarios listed above, but it should logically lead to the same method of solving.
(Hotsy, Totsy, Potsy, or Mac) and what color stroller you will be racing (red, green, blue, or yellow).
8 This
240
c. What is the chance that the computer will choose a baby whose name rhymes with Dotsy? d. What is the chance that the computer will choose a baby whose name rhymes with Dotsy and a red stroller? e. All four babies are racing in red strollers today. One possible outcome is that Hotsy
will come in rst, followed by Totsy, then Potsy, then Mac. How many total possible outcomes are there?
Exercise 14.32
The weatherman predicts a 20% chance of rain on Tuesday. will have a leaky roof on Tuesday?
your roof will leak. (If it doesn't rain, of course, your roof is safe.) What is the chance that you
Exercise 14.33
A game of Yahtzee! begins by rolling ve 6-sided dice.
a. b. c. d. e.
What is the chance that all ve dice will roll 6? What is the chance that all ve dice will roll the same as each other? What is the chance that all ve dice will roll 5 or 6? What is the chance that What is the chance that
no dice will roll 6? at least one die will roll 6?
Exercise 14.34
How many three-letter combinations can be made from the 26 letters in the alphabet? We can ask this question three dierent ways, with three dierent answers.
a. First, assume any three-letter combination is valid: NNN, for instance. (This gives you
the actual number of possible three-letter words.)
b. Second, assume that you cannot use the same letter twice. (Here you can imagine that
you have a bag of les, one for each letter, and you are drawing three of them out in order to make a word.)
c. Third, assume that you still cannot use the same letter twice, but order doesn't matter:
CAT and ACT are the same. (Here you can imagine that as you pull the tiles, you are creating unscramble-the-word puzzles instead of words.)
Exercise 14.35
In a Sudoku puzzle, a 3-by-3 grid must be populated with each of the digits 1 through 9. Every digit must be used once, which means that no digit can be repeated. How many possible 3-by-3 grids can be made?
241
Extra credit:
How many dierent three-digit numbers can you make by rearranging the following digits?
Table 14.1
242
INDEX
Ex.
Ex.
apples, 1
absolute value, 2.4(66), 2.5(68), 2.6(69) absolute values, 2.7(70), 2.8(71), 2.9(72), 2.10(74) algebra, (1), (3), 1.1(23), 1.2(23), 1.3(24), 1.4(25), 1.5(26), 1.6(27), 1.7(29), 1.8(30), 1.9(32), 1.10(33), 1.11(37), 1.12(41), 1.13(45), 1.14(50), 1.15(52), 1.16(54), 1.17(55), 1.18(57), 1.19(58), 1.20(58), 2.1(63), 2.2(64), 2.3(65), 2.4(66), 2.5(68), 2.6(69), 2.8(71), 2.9(72), 2.10(74), 3.1(75), 3.2(76), 3.3(77), 3.4(79), 3.5(80), 3.6(81), 4.1(83), 4.2(84), 4.3(86), 4.4(87), 4.5(88), 4.7(90), 4.8(91), 4.9(92), 4.10(93), 4.11(94), 4.12(96), 4.13(97), 4.14(99), 4.15(100), 4.16(101), 4.17(102), 4.19(105), 5.1(107), 5.2(108), 5.3(109), 5.4(110), 5.5(111), 5.6(113), 5.7(114), 5.8(116), 5.9(118), 6.1(121), 6.2(122), 6.3(124), 6.4(125), 6.5(127), 6.6(127), 6.7(129), 6.8(130), 7.1(133), 7.2(134), 7.3(134), 7.4(135), 7.5(136), 7.6(137), 8.1(139), 8.2(139), 8.3(140), 8.4(141), 8.5(141), 8.6(143), 8.7(145), 8.8(146), 8.9(148), 8.10(149), 9.1(153), 9.2(155), 9.3(157), 9.4(158), 9.5(159), 9.6(160), 9.7(161), 9.8(161), 10.1(165), 10.2(166), 10.3(167), 10.4(169), 10.5(169), 10.6(170), 10.7(172), 10.8(174), 10.9(175), 10.10(176), 10.11(178), 10.12(180), 10.13(181), 10.14(182), 10.15(183), 10.16(185), 10.17(186), 11.1(191), 11.2(192), 11.3(194), 11.4(196), 11.5(197), 11.6(199), 11.7(201), 12.1(207), 12.2(208), 12.3(209), 12.4(211), 12.5(212), 12.6(212), 12.7(214), 12.8(215), 12.9(216), 12.10(217), 12.11(219), 12.12(220), 12.13(221), 12.14(222), 13.1(225), 13.2(226), 13.3(227), 13.4(227), 13.5(228),
13.6(229), 14.1(231), 14.2(231), 14.3(233), 14.4(233), 14.5(234), 14.6(237), 14.7(238), 14.8(239) Algebra 2, (3) answer sheet, 1.3(24) arithmetic sequences, 13.1(225), 13.2(226) arithmetic series, 13.4(227)
B C
binomials, 4.1(83), 4.2(84) calculator, 10.13(181), 11.6(199) circles, 12.3(209), 12.4(211), 12.8(215) combinations, 14.7(238) completing the square, 4.7(90), 4.8(91) complex, 9.2(155), 9.3(157), 9.4(158), 9.5(159), 9.6(160), 9.7(161), 9.8(161) complex numbers, 9.2(155), 9.3(157), 9.4(158), 9.5(159), 9.6(160), 9.7(161), 9.8(161) composite functions, 1.15(52), 1.16(54) conic sections, 12.1(207), 12.2(208), 12.3(209), 12.4(211), 12.5(212), 12.6(212), 12.7(214), 12.8(215), 12.9(216), 12.10(217), 12.11(219), 12.12(220), 12.13(221), 12.14(222) conics, 12.1(207), 12.2(208), 12.3(209), 12.4(211), 12.5(212), 12.6(212), 12.7(214), 12.8(215), 12.9(216), 12.10(217), 12.11(219), 12.12(220), 12.13(221), 12.14(222) coordinate plane, 12.1(207), 12.2(208)
data modeling, 11.7(201) denition, 12.7(214), 12.11(219) dependent, 1.12(41) dependent variable, 1.5(26) determinants, 10.14(182) direct variation, 11.1(191), 11.2(192), 11.3(194), 11.4(196), 11.5(197), 11.6(199) distance, 3.1(75), 12.1(207), 12.2(208), 12.5(212), 12.8(215), 12.9(216),
INDEX
12.12(220) division, 7.5(136) imaginary numbers, 9.1(153), 9.2(155),
243
9.3(157), 9.4(158), 9.5(159), 9.6(160), 9.7(161), 9.8(161) independent, 1.12(41) independent variable, 1.5(26) induction, 13.5(228) inequalities, 2.1(63), 2.2(64), 2.3(65), 2.6(69), 2.7(70), 2.8(71), 2.9(72), 2.10(74), 4.17(102), 4.18(104) introduction, 1.1(23) inverse, 1.17(55), 1.18(57), 10.7(172), 10.8(174), 10.9(175) inverse functions, 1.17(55), 1.18(57), 1.19(58) inverse matrix, 10.7(172), 10.8(174) inverse variation, 11.2(192), 11.3(194), 11.4(196), 11.5(197), 11.6(199)
ellipses, 12.9(216), 12.10(217), 12.11(219), 12.14(222) equation, 12.7(214), 12.11(219) equations, 2.4(66), 2.5(68), 4.2(84), 4.3(86), 4.4(87), 4.5(88), 4.7(90), 4.8(91), 4.10(93), 4.12(96), 4.13(97), 4.14(99), 4.16(101), 4.18(104), 8.8(146), 8.9(148) exponential, 5.7(114), 5.8(116) exponential curves, 5.7(114), 5.8(116) exponents, 5.1(107), 5.2(108), 5.3(109), 5.4(110), 5.5(111), 5.6(113), 5.7(114), 5.8(116), 5.9(118), 8.2(139)
factoring, 4.3(86), 4.4(87) felder, (3) fractional exponents, 5.5(111), 5.6(113) fractions, 5.5(111), 5.6(113) function game, 1.1(23), 1.2(23), 1.3(24), 1.4(25) functions, (1), 1.1(23), 1.2(23), 1.3(24), 1.4(25), 1.5(26), 1.6(27), 1.7(29), 1.8(30), 1.9(32), 1.10(33), 1.11(37), 1.12(41), 1.13(45), 1.14(50), 1.15(52), 1.16(54), 1.17(55), 1.18(57), 1.19(58), 1.20(58), 4.12(96), 4.13(97), 4.14(99), 4.15(100), 4.16(101)
leader's sheet, 1.2(23) linear equations, 10.15(183), 10.16(185) lines, 1.13(45) log, 6.1(121), 6.2(122), 6.3(124), 6.4(125), 6.5(127), 6.6(127), 6.7(129), 6.8(130) logarithms, 6.1(121), 6.2(122), 6.3(124), 6.4(125), 6.5(127), 6.6(127), 6.7(129), 6.8(130)
generalizations, 1.6(27), 1.7(29) geometric sequences, 13.1(225), 13.2(226) geometric series, 13.4(227) graphing, 1.8(30), 1.9(32), 1.10(33), 1.11(37), 1.13(45), 1.14(50), 2.8(71), 2.9(72), 3.2(76), 4.12(96), 4.13(97), 4.14(99), 4.15(100), 4.16(101) groupings, 14.1(231) groups, 14.1(231)
H I
homework, 7.1(133), 7.3(134) horizontal parabolas, 12.6(212) hyperbolas, 12.12(220), 12.13(221), 12.14(222) i, 9.1(153), 9.2(155), 9.3(157), 9.4(158), 9.5(159), 9.6(160), 9.7(161), 9.8(161) identity, 10.7(172), 10.8(174) identity matrix, 10.7(172), 10.8(174) imaginary, 9.1(153), 9.2(155), 9.3(157), 9.4(158), 9.5(159), 9.6(160), 9.7(161), 9.8(161)
N P
notation, 13.3(227) parabolas, 12.6(212), 12.7(214), 12.8(215) permutation, 1.9(32), 1.10(33), 1.11(37), 14.6(237) permutations, 14.7(238) polynomials, 7.5(136)
244
INDEX
probability, 14.1(231), 14.2(231), 14.3(233), 14.4(233), 14.5(234), 14.6(237), 14.7(238), 14.8(239) problems, 1.4(25) proof by induction, 13.5(228) proofs, 13.5(228) properties, 5.1(107), 5.2(108), 6.3(124), 6.4(125) 7.3(134), 7.4(135), 7.6(137) ratios, 7.1(133), 7.2(134), 7.3(134), 7.4(135), 7.5(136), 7.6(137) roots, 8.1(139), 8.2(139), 8.3(140), 8.4(141), 8.5(141), 8.6(143), 8.7(145), 8.8(146), 8.9(148), 8.10(149)
sequences, 13.1(225), 13.2(226), 13.3(227), 13.4(227), 13.5(228), 13.6(229) series, 13.1(225), 13.2(226), 13.3(227), 13.4(227), 13.5(228), 13.6(229) simplication, 8.4(141) simplify, 8.4(141) simultaneous equations, 3.1(75), 3.2(76), 3.3(77), 3.4(79), 3.5(80), 3.6(81) slope, 1.14(50)
quadratic equations, 4.3(86), 4.4(87), 4.7(90), 4.8(91), 4.12(96), 4.13(97), 4.14(99), 9.7(161) quadratic functions, 4.14(99), 4.15(100) quadratics, 4.1(83), 4.2(84), 4.3(86), 4.4(87), 4.5(88), 4.7(90), 4.8(91), 4.9(92), 4.10(93), 4.11(94), 4.12(96), 4.13(97), 4.14(99), 4.15(100), 4.16(101), 4.17(102), 4.18(104), 4.19(105), 9.7(161)
TAPPS, 1.19(58) time, 3.1(75) transformations, 10.10(176), 10.11(178) translation, 1.9(32), 1.10(33), 1.11(37) tree diagrams, 14.2(231)
radicals, 8.1(139), 8.2(139), 8.3(140), 8.4(141), 8.5(141), 8.6(143), 8.7(145), 8.8(146), 8.9(148), 8.10(149) radii, 1.20(58) radius, 1.20(58) rate, 3.1(75) ratio, 11.1(191), 11.2(192), 11.3(194), 11.4(196), 11.5(197), 11.6(199), 11.7(201) rational equations, 7.3(134), 7.4(135), 7.6(137) rational expressions, 7.1(133), 7.2(134), 7.4(135), 7.6(137) rational numbers, 7.1(133), 7.2(134),
variable, 1.4(25) variables, 1.12(41) variation, 11.1(191), 11.2(192), 11.3(194), 11.4(196), 11.5(197), 11.6(199), 11.7(201) vertical parabolas, 12.6(212)
ATTRIBUTIONS
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Attributions
Collection:
Edited by: Kenny M. Felder URL: http://cnx.org/content/col10686/1.5/ License: http://creativecommons.org/licenses/by/2.0/ Module: "The Philosophical Introduction No One Reads" By: Kenny M. Felder URL: http://cnx.org/content/m19111/1.2/ Page: 1 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "How to use Advanced Algebra II" Used here as: "How to Use Advanced Algebra II" By: Kenny M. Felder URL: http://cnx.org/content/m19435/1.6/ Pages: 3-17 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework The Function Game: Introduction" Used here as: "The Function Game: Introduction" By: Kenny M. Felder URL: http://cnx.org/content/m19125/1.1/ Page: 23 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework The Function Game: Leader's Sheet" Used here as: "The Function Game: Leader's Sheet" By: Kenny M. Felder URL: http://cnx.org/content/m19126/1.1/ Pages: 23-24 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework The Function Game: Answer Sheet" Used here as: "The Function Game: Answer Sheet" By: Kenny M. Felder URL: http://cnx.org/content/m19124/1.1/ Pages: 24-25 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework Homework: The Function Game" Used here as: "Homework: The Function Game" By: Kenny M. Felder URL: http://cnx.org/content/m19121/1.2/ Pages: 25-26 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Function Homework Homework: Horizontal and Vertical Permutations II" Used here as: "Homework: Horizontal and Vertical Permutations II" By: Kenny M. Felder URL: http://cnx.org/content/m31952/1.1/ Pages: 37-41 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/3.0/ Module: "Function Homework Sample Test: Function I" Used here as: "Sample Test: Function I" By: Kenny M. Felder URL: http://cnx.org/content/m19122/1.1/ Pages: 41-45 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework Lines" Used here as: "Lines" By: Kenny M. Felder URL: http://cnx.org/content/m19113/1.1/ Pages: 45-50 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework Homework: Graphing Lines" Used here as: "Homework: Graphing Lines" By: Kenny M. Felder URL: http://cnx.org/content/m19118/1.2/ Pages: 50-52 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework Composite Functions" Used here as: "Composite Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19109/1.1/ Pages: 52-54 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework Problems: Composite Functions" Used here as: "Homework: Composite Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19107/1.1/ Pages: 54-55 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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248 Module: "Function Homework Inverse Functions" Used here as: "Inverse Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19112/1.1/ Pages: 55-57 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework Homework: Inverse Functions" Used here as: "Homework: Inverse Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19120/1.3/ Pages: 57-58 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework TAPPS Exercise: How Do I Solve That For y?" Used here as: "TAPPS Exercise: How Do I Solve That For y?" By: Kenny M. Felder URL: http://cnx.org/content/m19123/1.1/ Page: 58 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Function Homework Sample Test: Functions II" Used here as: "Sample Test: Functions II" By: Kenny M. Felder URL: http://cnx.org/content/m19117/1.1/ Pages: 58-62 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Inequalities and Absolute Value Homework Inequalities" Used here as: "Inequalities" By: Kenny M. Felder URL: http://cnx.org/content/m19158/1.1/ Pages: 63-64 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Inequalities and Absolute Value Homework Homework: Inequalities" Used here as: "Homework: Inequalities" By: Kenny M. Felder URL: http://cnx.org/content/m19154/1.2/ Pages: 64-65 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Inequalities and Absolute Value Homework Inequality Word Problems" Used here as: "Inequality Word Problems" By: Kenny M. Felder URL: http://cnx.org/content/m19163/1.1/ Pages: 65-66 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Inequalities and Absolute Value Homework Absolute Value Equations" Used here as: "Absolute Value Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19148/1.1/ Pages: 66-68 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Inequalities and Absolute Value Homework Homework: Absolute Value Equations" Used here as: "Homework: Absolute Value Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19151/1.2/ Pages: 68-69 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Inequalities and Absolute Value Homework Absolute Value Inequalities" Used here as: "Absolute Value Inequalities" By: Kenny M. Felder URL: http://cnx.org/content/m19149/1.1/ Pages: 69-70 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Inequalities and Absolute Value Homework Homework: Absolute Value Inequalities" Used here as: "Homework: Absolute Value Inequalities" By: Kenny M. Felder URL: http://cnx.org/content/m19155/1.2/ Pages: 70-71 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Inequalities and Absolute Value Homework Graphing Inequalities and Absolute Values" Used here as: "Graphing Inequalities and Absolute Values" By: Kenny M. Felder URL: http://cnx.org/content/m19150/1.1/ Pages: 71-72 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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250 Module: Values" Used here as: "Homework: Graphing Inequalities and Absolute Values" By: Kenny M. Felder URL: http://cnx.org/content/m19153/1.2/ Pages: 72-73 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ "Inequalities and Absolute Value Homework Homework:
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Graphing Inequalities and Absolute
Module: "Inequalities and Absolute Value Homework Sample Test: Inequalities and Absolute Values" Used here as: "Sample Test: Inequalities and Absolute Values" By: Kenny M. Felder URL: http://cnx.org/content/m19166/1.1/ Page: 74 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Simultaneous Equations Homework Distance, Rate, and Time" Used here as: "Distance, Rate, and Time" By: Kenny M. Felder URL: http://cnx.org/content/m19288/1.1/ Pages: 75-76 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Simultaneous Equations Homework Homework: Simultaneous Equations by Graphing" Used here as: "Homework: Simultaneous Equations by Graphing" By: Kenny M. Felder URL: http://cnx.org/content/m19291/1.2/ Pages: 76-77 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Simultaneous Equations Homework Simultaneous Equations" Used here as: "Simultaneous Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19293/1.1/ Pages: 77-78 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Simultaneous Equations Homework Homework: Simultaneous Equations" Used here as: "Homework: Simultaneous Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19289/1.2/ Pages: 79-80 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Simultaneous Equations Homework The Generic Simultaneous Equations" Used here as: "The Generic Simultaneous Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19294/1.1/ Pages: 80-81 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Simultaneous Equations Homework Sample Test: 2 Equations and 2 Unknowns" Used here as: "Sample Test: 2 Equations and 2 Unknowns" By: Kenny M. Felder URL: http://cnx.org/content/m19292/1.1/ Pages: 81-82 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Multiplying Binomials" Used here as: "Multiplying Binomials" By: Kenny M. Felder URL: http://cnx.org/content/m19247/1.1/ Pages: 83-84 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Homework: Multiplying Binomials" Used here as: "Homework: Multiplying Binomials" By: Kenny M. Felder URL: http://cnx.org/content/m19253/1.2/ Pages: 84-86 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Factoring" Used here as: "Factoring" By: Kenny M. Felder URL: http://cnx.org/content/m19243/1.1/ Pages: 86-87 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Homework: Factoring Expressions" Used here as: "Homework: Factoring Expressions" By: Kenny M. Felder URL: http://cnx.org/content/m19248/1.3/ Pages: 87-88 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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252 Module: "Quadratic Homework Introduction to Quadratic Equations" Used here as: "Introduction to Quadratic Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19246/1.1/ Pages: 88-89 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Homework: Introduction to Quadratic Equations" Used here as: "Homework: Introduction to Quadratic Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19251/1.2/ Pages: 89-90 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Completing the Square" Used here as: "Completing the Square" By: Kenny M. Felder URL: http://cnx.org/content/m19242/1.1/ Pages: 90-91 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Homework: Completing the Square" Used here as: "Homework: Completing the Square" By: Kenny M. Felder URL: http://cnx.org/content/m19249/1.2/ Pages: 91-92 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework The Generic Quadratic Equation" Used here as: "The Generic Quadratic Equation" By: Kenny M. Felder URL: http://cnx.org/content/m19262/1.1/ Pages: 92-93 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Homework: Solving Quadratic Equations" Used here as: "Homework: Solving Quadratic Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19256/1.2/ Pages: 93-94 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Quadratic Homework Sample Test: Quadratic Equations I" Used here as: "Sample Test: Quadratic Equations I" By: Kenny M. Felder URL: http://cnx.org/content/m19259/1.1/ Pages: 94-95 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Graphing Quadratic Functions" Used here as: "Graphing Quadratic Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19245/1.1/ Pages: 96-97 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Graphing Quadratic Functions II" Used here as: "Graphing Quadratic Functions II" By: Kenny M. Felder URL: http://cnx.org/content/m19244/1.1/ Pages: 97-99 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Homework: Graphing Quadratic Functions II" Used here as: "Homework: Graphing Quadratic Functions II" By: Kenny M. Felder URL: http://cnx.org/content/m19250/1.2/ Pages: 99-100 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Solving Problems by Graphing Quadratic Functions" Used here as: "Solving Problems by Graphing Quadratic Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19260/1.1/ Pages: 100-101 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Homework: Solving Problems by Graphing Quadratic Functions" Used here as: "Homework: Solving Problems by Graphing Quadratic Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19255/1.2/ Pages: 101-102 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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254 Module: "Quadratic Homework Quadratic Inequalities" Used here as: "Quadratic Inequalities" By: Kenny M. Felder URL: http://cnx.org/content/m19257/1.2/ Pages: 102-104 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/3.0/ Module: "Quadratic Homework Homework: Quadratic Inequalities" Used here as: "Homework: Quadratic Inequalities" By: Kenny M. Felder URL: http://cnx.org/content/m19254/1.2/ Pages: 104-105 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Quadratic Homework Sample Test: Quadratics II" Used here as: "Sample Test: Quadratics II" By: Kenny M. Felder URL: http://cnx.org/content/m19258/1.1/ Pages: 105-106 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Exponents Homework Rules of Exponents" Used here as: "Rules of Exponents" By: Kenny M. Felder URL: http://cnx.org/content/m19104/1.1/ Pages: 107-108 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Exponents Homework Homework: Rules of Exponents" Used here as: "Homework: Rules of Exponents" By: Kenny M. Felder URL: http://cnx.org/content/m19101/1.2/ Pages: 108-109 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Exponents Homework Extending the Idea of Exponents" Used here as: "Extending the Idea of Exponents" By: Kenny M. Felder URL: http://cnx.org/content/m19096/1.1/ Pages: 109-110 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Exponents Homework Homework: Extending the Idea of Exponents" Used here as: "Homework: Extending the Idea of Exponents" By: Kenny M. Felder URL: http://cnx.org/content/m19098/1.2/ Pages: 110-111 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Exponents Homework Fractional Exponents" Used here as: "Fractional Exponents" By: Kenny M. Felder URL: http://cnx.org/content/m19097/1.1/ Pages: 111-113 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Exponents Homework Homework: Fractional Exponents" Used here as: "Homework: Fractional Exponents" By: Kenny M. Felder URL: http://cnx.org/content/m19100/1.2/ Pages: 113-114 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Exponents Homework Real Life Exponential Curves" Used here as: "Real Life Exponential Curves" By: Kenny M. Felder URL: http://cnx.org/content/m19103/1.1/ Pages: 114-116 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Exponents Homework Homework: Real life exponential curves" Used here as: "Homework: Real life exponential curves" By: Kenny M. Felder URL: http://cnx.org/content/m19102/1.2/ Pages: 116-118 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Exponents Homework Sample Test: Exponents" Used here as: "Sample Test: Exponents" By: Kenny M. Felder URL: http://cnx.org/content/m19105/1.1/ Pages: 118-120 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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256 Module: "Logs Homework Introduction to Logarithms" Used here as: "Introduction to Logarithms" By: Kenny M. Felder URL: http://cnx.org/content/m19175/1.1/ Pages: 121-122 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Logs Homework Homework: Logs" Used here as: "Homework: Logs" By: Kenny M. Felder URL: http://cnx.org/content/m19176/1.2/ Pages: 122-124 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Logs Homework Properties of Logarithms" Used here as: "Properties of Logarithms" By: Kenny M. Felder URL: http://cnx.org/content/m19269/1.1/ Pages: 124-125 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Logs Homework Homework: Properties of Logarithms" Used here as: "Homework: Properties of Logarithms" By: Kenny M. Felder URL: http://cnx.org/content/m19177/1.2/ Pages: 125-127 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Logs Homework Using the Laws of Logarithms" Used here as: "Using the Laws of Logarithms" By: Kenny M. Felder URL: http://cnx.org/content/m19184/1.1/ Page: 127 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Logs Homework So What Are Logarithms Good For, Anyway?" Used here as: "So What Are Logarithms Good For, Anyway?" By: Kenny M. Felder URL: http://cnx.org/content/m19181/1.2/ Pages: 127-129 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Logs Homework Homework: What Are Logarithms Good For, Anyway?" Used here as: "Homework: What Are Logarithms Good For, Anyway?" By: Kenny M. Felder URL: http://cnx.org/content/m19268/1.2/ Pages: 129-130 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Logs Homework Sample Test" Used here as: "Sample Test" By: Kenny M. Felder URL: http://cnx.org/content/m19180/1.1/ Pages: 130-131 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Rational Expressions Homework Rational Expressions" Used here as: "Rational Expressions" By: Kenny M. Felder URL: http://cnx.org/content/m19278/1.1/ Page: 133 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Rational Expressions Homework Homework: Rational Expressions" Used here as: "Homework: Rational Expressions" By: Kenny M. Felder URL: http://cnx.org/content/m19275/1.1/ Page: 134 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Rational Expressions Homework Rational Equations" Used here as: "Rational Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19279/1.1/ Pages: 134-135 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Rational Expressions Homework Homework: Rational Expressions and Equations" Used here as: "Homework: Rational Expressions and Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19277/1.1/ Pages: 135-136 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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258 Module: "Rational Expressions Homework Dividing Polynomials" Used here as: "Dividing Polynomials" By: Kenny M. Felder URL: http://cnx.org/content/m19276/1.1/ Page: 136 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Rational Expressions Homework Sample Test: Rational Expressions" Used here as: "Sample Test: Rational Expressions" By: Kenny M. Felder URL: http://cnx.org/content/m19274/1.1/ Page: 137 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Radicals Homework Radicals (aka* Roots)" Used here as: "Radicals (aka* Roots)" By: Kenny M. Felder URL: http://cnx.org/content/m19420/1.1/ Page: 139 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Radicals Homework Radicals and Exponents" Used here as: "Radicals and Exponents" By: Kenny M. Felder URL: http://cnx.org/content/m19419/1.1/ Pages: 139-140 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Radicals Homework Some Very Important Generalizations" Used here as: "Some Very Important Generalizations" By: Kenny M. Felder URL: http://cnx.org/content/m19422/1.1/ Page: 140 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Radicals Homework Simplifying Radicals" Used here as: "Simplifying Radicals" By: Kenny M. Felder URL: http://cnx.org/content/m19421/1.1/ Page: 141 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Radicals Homework Homework: Radicals" Used here as: "Homework: Radicals" By: Kenny M. Felder URL: http://cnx.org/content/m19270/1.1/ Pages: 141-143 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Radicals Homework A Bunch of Other Stu About Radicals" Used here as: "A Bunch of Other Stu About Radicals" By: Kenny M. Felder URL: http://cnx.org/content/m19263/1.1/ Pages: 143-145 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Radicals Homework Homework: A Bunch of Other Stu About Radicals" Used here as: "Homework: A Bunch of Other Stu About Radicals" By: Kenny M. Felder URL: http://cnx.org/content/m19264/1.1/ Pages: 145-146 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Radicals Homework Radical Equations" Used here as: "Radical Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19272/1.1/ Pages: 146-148 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Radicals Homework Homework: Radical Equations" Used here as: "Homework: Radical Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19271/1.1/ Pages: 148-149 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Radicals Homework Sample Test: Radicals" Used here as: "Sample Test: Radicals" By: Kenny M. Felder URL: http://cnx.org/content/m19273/1.1/ Pages: 149-151 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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260 Module: "Imaginary Numbers Homework Imaginary Numbers" Used here as: "Imaginary Numbers" By: Kenny M. Felder URL: http://cnx.org/content/m19129/1.1/ Pages: 153-155 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Imaginary Numbers Homework Homework: Imaginary Numbers" Used here as: "Homework: Imaginary Numbers" By: Kenny M. Felder URL: http://cnx.org/content/m19130/1.1/ Pages: 155-156 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Imaginary Numbers Homework Complex Numbers" Used here as: "Complex Numbers" By: Kenny M. Felder URL: http://cnx.org/content/m19128/1.1/ Pages: 157-158 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Imaginary Numbers Homework Homework: Complex Numbers" Used here as: "Homework: Complex Numbers" By: Kenny M. Felder URL: http://cnx.org/content/m19132/1.1/ Pages: 158-159 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Imaginary Numbers Homework Me, Myself, and the Square Root of i" Used here as: "Me, Myself, and the Square Root of i" By: Kenny M. Felder URL: http://cnx.org/content/m19134/1.1/ Pages: 159-160 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Imaginary Numbers Homework The Many Merry Cube Roots of -1" Used here as: "The Many Merry Cube Roots of -1" By: Kenny M. Felder URL: http://cnx.org/content/m19131/1.1/ Pages: 160-161 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Imaginary Numbers Homework Homework: Quadratic Equations and Complex Numbers" Used here as: "Homework: Quadratic Equations and Complex Numbers" By: Kenny M. Felder URL: http://cnx.org/content/m19127/1.1/ Page: 161 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Imaginary Numbers Homework Sample Test: Complex Numbers" Used here as: "Sample Test: Complex Numbers" By: Kenny M. Felder URL: http://cnx.org/content/m19133/1.1/ Pages: 161-163 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Introduction to Matrices" Used here as: "Introduction to Matrices" By: Kenny M. Felder URL: http://cnx.org/content/m19206/1.1/ Pages: 165-166 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Homework: Introduction to Matrices" Used here as: "Homework: Introduction to Matrices" By: Kenny M. Felder URL: http://cnx.org/content/m19205/1.1/ Pages: 166-167 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Multiplying Matrices I" Used here as: "Multiplying Matrices I" By: Kenny M. Felder URL: http://cnx.org/content/m19207/1.1/ Pages: 167-169 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Homework: Multiplying Matrices I" Used here as: "Homework: Multiplying Matrices I" By: Kenny M. Felder URL: http://cnx.org/content/m19196/1.1/ Page: 169 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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262 Module: "Matrices Homework Multiplying Matrices II" Used here as: "Multiplying Matrices II" By: Kenny M. Felder URL: http://cnx.org/content/m19208/1.1/ Pages: 169-170 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Homework: Multiplying Matrices II" Used here as: "Homework: Multiplying Matrices II" By: Kenny M. Felder URL: http://cnx.org/content/m19201/1.1/ Pages: 170-171 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework The Identity and Inverse Matrices" Used here as: "The Identity and Inverse Matrices" By: Kenny M. Felder URL: http://cnx.org/content/m19213/1.1/ Pages: 172-174 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Homework: The Identity and Inverse Matrices" Used here as: "Homework: The Identity and Inverse Matrices" By: Kenny M. Felder URL: http://cnx.org/content/m19194/1.1/ Page: 174 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework The Inverse of the Generic 2x2 Matrix" Used here as: "The Inverse of the Generic 2x2 Matrix" By: Kenny M. Felder URL: http://cnx.org/content/m19214/1.1/ Pages: 175-176 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Using Matrices for Transformation" Used here as: "Using Matrices for Transformation" By: Kenny M. Felder URL: http://cnx.org/content/m19221/1.1/ Pages: 176-178 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Matrices Homework Homework: Using Matrices for Transformation" Used here as: "Homework: Using Matrices for Transformation" By: Kenny M. Felder URL: http://cnx.org/content/m19190/1.1/ Pages: 178-179 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Sample Test : Matrices I" Used here as: "Sample Test : Matrices I" By: Kenny M. Felder URL: http://cnx.org/content/m19210/1.1/ Pages: 180-181 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Homework: Calculators" Used here as: "Homework: Calculators" By: Kenny M. Felder URL: http://cnx.org/content/m19188/1.1/ Pages: 181-182 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Homework: Determinants" Used here as: "Homework: Determinants" By: Kenny M. Felder URL: http://cnx.org/content/m19193/1.1/ Pages: 182-183 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Solving Linear Equations" Used here as: "Solving Linear Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19212/1.1/ Pages: 183-185 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Matrices Homework Homework: Solving Linear Equations" Used here as: "Homework: Solving Linear Equations" By: Kenny M. Felder URL: http://cnx.org/content/m19204/1.1/ Pages: 185-186 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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264 Module: "Matrices Homework Sample Test: Matrices II" Used here as: "Sample Test: Matrices II" By: Kenny M. Felder URL: http://cnx.org/content/m19209/1.1/ Pages: 186-187 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Modeling Data with Functions Homework Direct Variation" Used here as: "Direct Variation" By: Kenny M. Felder URL: http://cnx.org/content/m19228/1.1/ Pages: 191-192 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Modeling Data with Functions Homework Homework: Inverse Variation" Used here as: "Homework: Inverse Variation" By: Kenny M. Felder URL: http://cnx.org/content/m19227/1.1/ Pages: 192-194 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Modeling Data with Functions Homework Homework: Direct and Inverse Variation" Used here as: "Homework: Direct and Inverse Variation" By: Kenny M. Felder URL: http://cnx.org/content/m19225/1.1/ Pages: 194-196 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Modeling Data with Functions Homework From Data Points to Functions" Used here as: "From Data Points to Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19224/1.1/ Pages: 196-197 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Modeling Data with Functions Homework Homework: From Data Points to Functions" Used here as: "Homework: From Data Points to Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19232/1.1/ Pages: 197-199 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Modeling Data with Functions Homework Homework: Calculator Regression" Used here as: "Homework: Calculator Regression" By: Kenny M. Felder, Kenny M. Felder URL: http://cnx.org/content/m19231/1.1/ Pages: 199-200 Copyright: Kenny M. Felder, Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Modeling Data with Functions Homework Sample Test: Modeling Data with Functions" Used here as: "Sample Test: Modeling Data with Functions" By: Kenny M. Felder URL: http://cnx.org/content/m19222/1.1/ Pages: 201-205 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Distance" Used here as: "Distance" By: Kenny M. Felder URL: http://cnx.org/content/m19081/1.1/ Pages: 207-208 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Homework: Distance" Used here as: "Homework: Distance" By: Kenny M. Felder URL: http://cnx.org/content/m19086/1.1/ Pages: 208-209 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework All the Points Equidistant from a Given Point" Used here as: "All the Points Equidistant from a Given Point" By: Kenny M. Felder URL: http://cnx.org/content/m19078/1.1/ Pages: 209-210 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Homework: Circles" Used here as: "Homework: Circles" By: Kenny M. Felder URL: http://cnx.org/content/m19084/1.1/ Page: 211 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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266 Module: "Conics Homework All the Points Equidistant from a Point and a Line" Used here as: "All the Points Equidistant from a Point and a Line" By: Kenny M. Felder URL: http://cnx.org/content/m19079/1.1/ Page: 212 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Homework: Vertical and Horizontal Parabolas" Used here as: "Homework: Vertical and Horizontal Parabolas" By: Kenny M. Felder URL: http://cnx.org/content/m19091/1.1/ Pages: 212-214 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Parabolas: From Denition to Equation" Used here as: "Parabolas: From Denition to Equation" By: Kenny M. Felder URL: http://cnx.org/content/m19092/1.1/ Pages: 214-215 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Sample Test: Distance, Circles, and Parabolas" Used here as: "Sample Test: Distance, Circles, and Parabolas" By: Kenny M. Felder URL: http://cnx.org/content/m19094/1.2/ Pages: 215-216 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Conics Homework Distance to this point plus distance to that point is constant" Used here as: "Distance to this point plus distance to that point is constant" By: Kenny M. Felder URL: http://cnx.org/content/m19083/1.1/ Pages: 216-217 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Homework: Ellipses" Used here as: "Homework: Ellipses" By: Kenny M. Felder URL: http://cnx.org/content/m19088/1.1/ Pages: 217-219 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Conics Homework The Ellipse: From Denition to Equation" Used here as: "The Ellipse: From Denition to Equation" By: Kenny M. Felder URL: http://cnx.org/content/m19095/1.1/ Pages: 219-220 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Distance to this point minus distance to that point is constant" Used here as: "Distance to this point minus distance to that point is constant" By: Kenny M. Felder URL: http://cnx.org/content/m19082/1.2/ Page: 220 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Homework: Hyperbolas" Used here as: "Homework: Hyperbolas" By: Kenny M. Felder URL: http://cnx.org/content/m19089/1.1/ Pages: 221-222 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Conics Homework Sample Test: Conics 2 (Ellipses and Hyperbolas)" Used here as: "Sample Test: Conics 2 (Ellipses and Hyperbolas)" By: Kenny M. Felder URL: http://cnx.org/content/m19093/1.1/ Pages: 222-223 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Sequences and Series Homework Arithmetic and Geometric Sequences" Used here as: "Arithmetic and Geometric Sequences" By: Kenny M. Felder URL: http://cnx.org/content/m19285/1.1/ Pages: 225-226 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Sequences and Series Homework Homework: Arithmetic and Geometric Sequences" Used here as: "Homework: Arithmetic and Geometric Sequences" By: Kenny M. Felder URL: http://cnx.org/content/m19284/1.1/ Pages: 226-227 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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268 Module: "Sequences and Series Homework Homework: Series and Series Notation" Used here as: "Homework: Series and Series Notation" By: Kenny M. Felder URL: http://cnx.org/content/m19280/1.1/ Page: 227 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Sequences and Series Homework Homework: Arithmetic and Geometric Series" Used here as: "Homework: Arithmetic and Geometric Series" By: Kenny M. Felder URL: http://cnx.org/content/m19282/1.1/ Pages: 227-228 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Sequences and Series Homework Homework: Proof by Induction" Used here as: "Homework: Proof by Induction" By: Kenny M. Felder URL: http://cnx.org/content/m19281/1.1/ Pages: 228-229 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Sequences and Series Homework Sample Test: Sequences and Series" Used here as: "Sample Test: Sequences and Series" By: Kenny M. Felder URL: http://cnx.org/content/m19283/1.1/ Pages: 229-230 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Probability Homework How Many Groups?" Used here as: "How Many Groups?" By: Kenny M. Felder URL: http://cnx.org/content/m19236/1.1/ Page: 231 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Probability Homework Homework: Tree Diagrams" Used here as: "Homework: Tree Diagrams" By: Kenny M. Felder URL: http://cnx.org/content/m19234/1.1/ Pages: 231-233 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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Module: "Probability Homework Introduction to Probability" Used here as: "Introduction to Probability" By: Kenny M. Felder URL: http://cnx.org/content/m19237/1.1/ Page: 233 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Probability Homework Homework: The Multiplication Rule" Used here as: "Homework: The Multiplication Rule" By: Kenny M. Felder URL: http://cnx.org/content/m19233/1.1/ Pages: 233-234 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Probability Homework Homework: Trickier Probability Problems" Used here as: "Homework: Trickier Probability Problems" By: Kenny M. Felder URL: http://cnx.org/content/m19235/1.1/ Pages: 234-237 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Probability Homework Homework: Permutations" Used here as: "Homework: Permutations" By: Kenny M. Felder URL: http://cnx.org/content/m19241/1.1/ Pages: 237-238 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Probability Homework Homework: Permutations and Combinations" Used here as: "Homework: Permutations and Combinations" By: Kenny M. Felder URL: http://cnx.org/content/m19240/1.1/ Pages: 238-239 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/ Module: "Probability Homework Sample Test: Probability" Used here as: "Sample Test: Probability" By: Kenny M. Felder URL: http://cnx.org/content/m19238/1.1/ Pages: 239-241 Copyright: Kenny M. Felder License: http://creativecommons.org/licenses/by/2.0/
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