Parabola
Parabola
Parabola
Brief Overview:
Students will use a trajectory as a means of learning about a quadratic function.
Students will model a parabolic path and find the equation of a parabola from
given points by solving a system of equations and by using quadratic regression.
Students will study the effects of the constants a, b, and c through an interactive
graphing calculator application and learn to find the coordinates of the vertex.
NCTM Content Standard/National Science Education Standard:
Algebra
Students will understand patterns, relations, and functions.
Students will represent and analyze mathematical situations and structures using
algebraic symbols.
Students will use mathematical models to represent and understand quantitative
relationships.
Students will analyze changes in various contexts.
Geometry
Students will analyze characteristics and properties of two-and three-dimensional
geometric shapes and develop mathematical arguments about geometric
relationships.
Students will specify locations and describe spatial relationships using coordinate
geometry and other representational systems.
Students will apply transformations and use symmetry to analyze mathematical
situations.
Data Analysis and Probability
Students will formulate questions that can be addressed with data and collect,
organize, and display relevant data to answer them.
Students will develop and evaluate inferences and predictions that are based on
data.
Communication
Students will communicate their mathematical thinking coherently and clearly to
peers, teachers, and others.
Students will analyze and evaluate the mathematical thinking and strategies of
others.
Student will use the language of mathematics to express mathematical ideas
precisely.
Connections
Students will recognize and use connections among mathematical ideas.
Representation
Students will create and use representations to organize, record, and communicate
mathematical ideas.
Students will use representations to model and interpret physical, social, and
mathematical phenomena.
Grade/Level:
Algebra I and II, grades 9-11
Duration/Length:
Three or four class periods, each approximately 50 minutes in length.
Student Outcomes:
The students will be able to:
Collect trajectory data and use technology to predict a quadratic curve of best fit.
Identify key characteristics of the behavior of the graph of a quadratic equation.
Use the equation of a parabolic path to investigate the horizontal and vertical
position over an interval.
Solve a system of linear equations (using three points) to determine the exact
equation of a parabolic graph.
Development/Procedures:
Lesson 1
3x + 2y = 18
Lesson 2
Lesson 3
2x y = 5
Preassessment Students will now know how to create a table of
values for a quadratic equation. (What causes the graph to
go up or down?) The teacher will distribute the Behavior
of Parabolas Worksheet (WS #3). To complete this
worksheet, the student uses the TI list and graphing
utilities with data generated from quadratic equations.
Launch The teacher will link the behavior of parabolas to a
trajectory caused by catapulting an object.
Teacher Facilitation - The teacher will ask questions to tie Lesson
1s material to the current lesson. (Why does a projected
object take the path of a parabola? Which upward or
downward does this path might have?) In addition, student
pairs will discuss how key concepts of vertex and xintercepts might have practical meaning in a trajectory
situation.
Student Application - For tactile and visual learning, groups of
four students will investigate the trajectories of Styrofoam
balls (of two sizes) and record the paths of the catapulted
balls. The teacher will distribute the Catapult Lab
Investigation Worksheet (WS #4) and model the procedure
described in the worsheet.
Embedded Assessment - This is included in the analysis section of
the Catapult Lab Investigation Worksheet (WS #4).
Reteaching/Extension Students will complete the Post Lab
Activity Worksheet (WS #5) and the Finding the Exact
Quadratic Worksheet (WS #6).
Elton Holmes
Easton High School
Talbot County Public Schools
Carol Leibee
Home and Hospital
Montgomery County Public Schools
Name ____________________
Date _____________________
Introduction to Parabolas
#1
Identify the marked coordinates of the following parabola in ordered pair notation.
1).
10
8
7
6
5
4
point
A
B
C
D
E
F
G
2
1
0
-4
-3
-2
-1
-1
-2
2).
5
point
A
B
C
D
0
-4
-3
-2
-1
y
2
-1
-2
-3
-4
yA
-5
-6
3). If a parabola opens down (like a lampshade), does it have a maximum or minimum?
4). If a parabola opens up (like a tulip), does it have a maximum or minimum?
1).
10
8
7
6
5
4
3
2
1
point
A
B
C
D
E
F
G
x
-3
-2
-1
0
1
2
3
y
9
4
1
0
1
4
9
x
-3
-1
0
2
y
-5
3
4
0
0
-4
-3
-2
-1
-1
-2
2).
By
point
A
B
C
D
0
-4
-3
-2
-1
y
2
-1
-2
-3
-4
yA
-5
-6
3). If a parabola opens down (like a lampshade), does it have a maximum or minimum?
4). If a parabola opens up (like a tulip), does it have a maximum or minimum?
Name ____________________
Date _____________________
Graphing Parabolas
#2
Use the given quadratic equation to create a table of values, graph the function, and circle
the vertex (maximum/minimum) and the zeros ( x -intercepts) on the graph.
1). y = 2x 2
-2
x
y
-1
10
9
8
7
6
5
4
3
2
1
0
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-1
x
y
2). y = x 2 + 6x
10
9
8
7
6
5
4
3
2
1
0
-10
-8
-6
-4
-2
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
10
3. y = x 2 + 4x + 12
-2
-1
x
y
10
9
8
7
6
5
4
3
2
1
0
-10
-8
-6
-4
-2
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
10
x
y
-2
8
-1
2
0
0
1
2
2
8
10
9
8
7
6
5
4
3
2
1
0
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
minimum
and zero
x
y
2). y = x 2 + 6x
-1
-7
0
0
1
5
4
8
6
0
10
9
maximum
8
7
6
5
4
3
2
1
0
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
zeros
10
3. y = x 2 + 4x + 6
-2
x
-6
y
-1
1
0
6
1
9
2
10
3
9
4
6
5
1
6
-6
12
maximum
10
0
-3
-2
-1
-2
-4
-6
zeros
-8
Name ____________________
Date _____________________
Behavior of Parabolas
#3
A). Investigating y = 1x 2 + x + 1
Select and record three integers between -6 and -1, three integers between 1 and 6, and
zero as x-values for the table below.
x
Compare/contrast the two quadratic equations investigated. What might have caused the
differences in the shape of the data for each example?
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
y
21
7
3
1
3
7
43
B). Investigating y = 1x 2 + x + 1
Select and record three integers between -6 and -1, three integers between 1 and 6, and
zero as x-values for the table below. (answers may vary)
x
--6
-4
-2
0
1
3
6
y
-41
-19
-5
1
1
-5
-29
Name ____________________
Date _____________________
Setup
NOTE: Each group should set up the lab station in the same manner so class averages can
be calculated. All spoons should be taped the same distance from the edge of the block.
The vertical scale is height in inches.
Figure 1
y
3 inches
6 inches
9 inches
spoon
block
stand
Procedure
1. Orient the paper vertically.
2. Make the origin the lower left corner.
3. Draw three vertical lines on the grid at 3, 6, and 9 inches, each in a different color.
4. Hang and tape graph paper from the desk edge.
5. Tape spoon so that the tip of the handle is 4 inches from the edge of the wood
block.
6. Put catapult (with spoon attached) in front of graph as shown.
7. Place large ball in spoon.
8. Release the ball from spoon and observe the projectile path. Each of the other
three students must mark the spot where the foam ball crosses a given colored
line.
9. Record the heights in data table.
10. Repeat steps 8 and 9 twice more for a total of three trials.
11. Repeat steps 7 to 10 using small Styrofoam ball.
at x= 0
0
0
0
0
at x = 9
at x = 9
at x= 0
0
0
0
0
Analysis:
1). Enter the four data points for the forward distance under L1 and the corresponding
four data points for the group average data for vertical distance under L2 using the small
ball data.
2). Enter the corresponding four data points for the class average data for vertical
distance under L3.
3). Graph L1 and L2 as a scatter plot and create a parabolic curve of best fit using
QuadReg on the graphing calculator. Confirm your curve with the data points. Record
your equation in the appropriate cell below (to the nearest tenth).
4). Repeat this procedure with L1 and L3.
5). Repeat this entire process for the large ball and record the curve equation.
Ball Size
Small
Large
Averaged Data
Group
Group
20
18
16
14
12
10
0
0
10
Distance (inches)
12
14
16
18
20
Setup
NOTE: Each group should set up the lab station in the same manner. Any deviation
should be discouraged if class averages will be calculated. All spoons should be taped
the same distance from the edge of the block.
In Figure 1 the dots represent the students observations as the ball crosses the vertical
lines. There are three students who each observe the location of the ball as it passes one
of the vertical lines. The dotted line represents the actual path of the ball (only three
points of which are recorded).
The vertical scale is height in inches.
Figure 1
y
y
y
3 inches
6 inches
9 inches
spoon
block
stand
Procedure
1. Orient the paper vertically.
2. Make the origin the lower left corner.
3. Draw three vertical lines on the grid at 3, 6, and 9 inches, each in a different color.
4. Hang and tape graph paper from the desk edge.
5. Tape spoon so that the tip of the handle is 4 inches from the edge of the wood
block.
6. Put catapult (with spoon attached) in front of graph as shown.
7. Place large ball in spoon.
8. Release the ball from spoon and observe the projectile path. Each of the other
three students must mark the spot where the foam ball crosses a given colored
line.
9. Record the heights in data table.
10. Repeat steps 8 and 9 twice more for a total of three trials.
11. Repeat steps 7 to 10 using small Styrofoam ball.
at x= 0
0
0
0
0
at x= 0
0
0
0
0
Analysis:
1). Enter the four data points for the forward distance under L1 and the corresponding
four data points for the group average data for vertical distance under L2 using the small
ball data.
2). Graph L1 and L2 as a scatter plot and create a parabolic curve of best fit using
QuadReg on the graphing calculator. Confirm your curve with the data points. Record
your equation in the appropriate cell below (to the nearest tenth).
3). Repeat this entire process for the large ball and record the curve equation.
Ball Size
Small
Large
Averaged Data
Group
Group
18
16
14
12
10
Small Ball
Large Ball
0
0
10
12
14
16
Name ____________________
Date _____________________
2). Using your group fitted curve of best fit equations, predict the forward distance of the
balls when the projectiles hits the ground.
3). Elton and Carol used the same catapult to launch Styrofoam balls. Carols ball was
twice as large as Eltons. In this graph, label which path would most likely represent
Carols ball. Explain your answer.
10
0
0
2). Using your group fitted curve of best fit equations, predict the forward distance of the
ball when the projectile hits the ground.
y = f ( x ) = .41x 2 + 5.2 x + .19 = 0
x 12.72
x 14.68
3). Elton and Carol used the same catapult to launch Styrofoam balls. Carols ball was
twice as large as Eltons. In this graph, label which path would most likely represent
Carols ball. Explain your answer.
10
0
0
Name ____________________
Date _____________________
16a + 4b = 6
100a + 10b = 3
3). Use your equation to find the height of the ball when the horizontal distance is 12
feet.
Let y = f (x) = .2x 2 + 2.3x . f (12) = .2(12) 2 + 2.3(12) = 1.2 feet. Therefore
when the horizontal distance is 12 ft, the vertical distance is 1.2 feet below ground zero.
Name ____________________
Date _____________________
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
Changing A in y=Ax2+Bx+C
Key: A=-1 _____ A=-2 A=-3 -----9. Highlight A=-1 > type 2 >ENTER. This will change the value of A so A=-2. Use a
dotted line to sketch the new view on the screen above.
10. Change the value of A so A=-3. Use a dashed line to sketch the new view on the
screen above.
11. Describe what happens when you press the cursor right.
12. Describe what happens when you press the cursor left.
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
Changing B in y=Ax2+Bx+C
Key: B=1 _____ B=2 ...... B=3 -----14. Highlight B=1 > type 2 > ENTER. This will change the value of B so B=2. Use a
dotted line to sketch the new view on the screen above.
15. Change the value of B so B=3. Use a dashed line to sketch the new view on the
screen above.
16. Describe what happens when you press the cursor right.
17. Describe what happens when you press the cursor left.
-4
-3
-2
-1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
Changing C in y=Ax2+Bx+C
Key: C=1 _____ C=2 ...... C=3 -----19. Highlight C=1 > type 2 > ENTER. This will change the value of C so C=2. Use a
dotted line to sketch the new view on the screen above.
20. Change the value of C so C=3. Use a dashed line to sketch the new view on the
screen above.
21. Describe what happens when you press the cursor right.
22. Describe what happens when you press the cursor left.
0
-10
-8
-6
-4
-2
-2
-4
-6
-8
-10
Changing A in y=Ax2+Bx+C
10
9. Highlight A=-1 > type 2 >ENTER. This will change the value of A so A=-2. Use a
dotted line to sketch the new view on the screen above.
10. Change the value of A so A=-3. Use a dashed line to sketch the new view on the
screen above.
11. Describe what happens when you press the cursor right.
As A increases from -1 toward 0; the downward opening parabola widens, and the
vertex moves diagonally upward and rightward. At A = 0, there is a straight line.
As A increases from 0; the upward opening parabola narrows, and the vertex moves
diagonally upward and rightward.
12. Describe what happens when you press the cursor left.
As A decreases from a positive value toward 0; the upward opening parabola
widens, and the vertex moves diagonally downward and leftward. At A = 0, there is
a straight line. As A decreases below 0; the downward opening parabola narrows,
and the vertex moves diagonally downward and leftward.
13. Change the value of A back to A=-1.
This should be the view on your screen.
10
0
-10
-8
-6
-4
-2
10
-2
-4
-6
-8
-10
Changing B in y=Ax2+Bx+C
14. Highlight B=1 > type 2 > ENTER. This will change the value of B so B=2. Use a
dotted line to sketch the new view on the screen above.
15. Change the value of B so B=3. Use a dashed line to sketch the new view on the
screen above.
16. Describe what happens when you press the cursor right.
As B increases from 1, the vertex of the parabola moves diagonally upward and
rightward.
17. Describe what happens when you press the cursor left.
As positive B decreases toward 0, the vertex of the parabola moves diagonally
downward and leftward. At B=0 the parabola is at its lowest. As B decreases below
0 the parabola moves diagonally upward and leftward.
0
-10
-8
-6
-4
-2
10
-2
-4
-6
-8
-10
Changing C in y=Ax2+Bx+C
Key: C=1 _____ C=2 ...... C=3 -----19. Highlight C=1 > type 2 > ENTER. This will change the value of C so C=2. Use a
dotted line to sketch the new view on the screen above.
20. Change the value of C so C=3. Use a dashed line to sketch the new view on the
screen above.
21. Describe what happens when you press the cursor right.
As C increases the parabola moves up.
22. Describe what happens when you press the cursor left.
As C decreases the parabola moves down.
Name ____________________
Date _____________________
Target Practice
#8
Directions: Use the TI Transform program to answer questions below.
Set-Up
1). Turn off Stat plots and Y=.
2). Set the following parameters on your calculator.
Y=Ax2+Bx+C
Window [-1, 10, 1; -1, 10, 1] (this is just the starting window; it
will change as you work)
Window>Settings>: >||; A=-1; B=0; C=0; Step=.01
MODE>FLOAT>2>ENTER (this will round answers to nearest hundredth)
MODE>CONNECTED>ENTER
2nd ZOOM (FORMAT) > AxesOn >ENTER
2nd ZOOM (FORMAT) > GridOn >ENTER
3). The vertex of your parabola should now be at (0, 0).
4). If you use TRACE, in order to return to the TRANSFORM screen push GRAPH. This
will show the A, B, and C values.
Directions
Use TI Transform to identify a parabola (there may be more than one possible) to fit each
condition in the left column below. In some cases coefficients of the quadratic equation
are given. In other cases, conditions are described in terms of a catapult.
On the appropriate graph, sketch the parabola (aka catapult path) you select.
Write the parabolas equation.
Conditions
If A=-1 and B=6, find C so that
the parabola ascends through (0,
0) and descends through (6, 0).
If C=0, B=6, find A so that the
catapult has a maximum point
(3, 9).
If A=-.5 and C=0, find B so that
the catapult has a maximum
point of (4, 8).
A catapult begins at (0, 0), and
there is a wall 4.5 units high
exactly 8 units from the launch
site. Find a parabola that goes
over the wall and lands as close
as possible to the far side of the
Graph
Equation
wall.
Graph
C=0
Equation
y=-x2+6x
A= -1
y=-x2+6x
B=4
y=-.5x2+4x
Sample A
A=-1
B=8.6
C=0
max of (8, 4.8)
Sample B
y=-x2+8.6x
(A)
A=-1
B=8.57
C=0
max of (8, 4.56) y=-x2+8.57x
(B)
Name ____________________
Date _____________________
Summative Assessment
#9
1). The jump rope held by Alice and Bonita has its lowest point closer to Alice. (How
could this be? )
A. Make a sketch labeling the position of the girls, the shape of the jump rope and
indicate the lowest point.
B. Where would the lowest point be if the students were of equal heights?
2).
y = 2x
y = x 2
y = 1+ x
y = 1+ x 2
4). Graph the following quadratic equations of the form y = ax 2 + bx + c using the
TI-Transform/TI-Interact Application.
A). Set the graphing calculator using the following window [-7, 7, 1; -10, 10,1]. Use the
default values of the coefficients:
Describe the change when only the single modification is made to the parent function.
A=4
A=-2
A=1/3
A=-1/5
B=3
B=-4
C=2
C=-4
_____________________________________________
_____________________________________________
_____________________________________________
_____________________________________________
_____________________________________________
_____________________________________________
_____________________________________________
_____________________________________________
BONUS: Find any A, B, and C such that the vertex of the graph of y = ax 2 + bx + c
falling in quadrant and the graph is inverted.
Lowest point
B. Where would the lowest point be if the students were of equal heights?
The lowest point would be exactly in the middle of the two students.
2).
y = 2x
y = x 2
y = 1+ x
y = 1+ x 2
Describe the change when only the single modification is made to the parent function.
A=4
A=-2
A=1/3
A=-1/5
B=3
B=-4
C=2
C=-4
BONUS: Find any A, B, and C such that the vertex of the graph of y = ax 2 + bx + c
falling in quadrant and the graph is inverted.
sample answer
A = 1
B=6
C = 3