MCR 3U1 Unit 3

Lesson 1: Properties of a Quadratic Function

Quadratic functions can be expressed in different algebraic forms:

Standard Form: ax 2 + bx + c
Factored Form: a( x − r )( x − s)

Vertex Form: f ( x) = a( x − h) 2 + k

▪ If a is positive then the parabola opens upwards. If a is negative then the parabola opens
downwards.
▪ The vertex is located at the point (h, k). If a is negative then the vertex is a maximum. If a is
positive then the vertex is a minimum.
▪ The axis of symmetry is the x value of the vertex (or x = h).
▪ The zeroes (x-intercepts) of the parabola can be found by writing the quadratic function in
factored form.
▪ To calculate the y-intercept, substitute x = 0 into the quadratic equation.

Example 1: Determine the type of function x y 1st Differences 2nd Differences

represented by the table of values, and then state 0 -32
the algebraic equation for the function. 1 -14
2 0
3 10
4 16
5 18
6 16
7 10
8 0
9 -14

Example 2: Graph the function f ( x) = 3( x − 3) 2 − 3 . State the vertex, the axis of symmetry,
the y-intercept, the x-intercepts, the direction of the opening, the domain and the range.
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Example 3: The height of a rocket above the ground is modeled by the quadratic function
h(t ) = −4t 2 + 32t , where h(t) is the height in metres t seconds after the rocket was launched.
a) Graph the quadratic function.

b) How long will the rocket be in the air? How do you know?

c) How high will the rocket be after 3 seconds?

d) What is the maximum height that the rocket will reach?

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Lesson 2: Determining Maximum and Minimum Values of a Quadratic Function

***Completing the Square***

The vertex form of quadratic function can be used to solve word problems involving minimum and
maximum values.

Example 1: A golfer attempts to hit a golf ball over a gorge from a platform above the ground.
The function that models the height of the ball is h(t) = -5t2 + 40t + 100, where h(t) is the
height in meters at time t seconds after contact. There are power lines 185 m above the
ground. Will the golf ball hit the power lines?

Group the terms containing x.

Factor the coefficient of x 2 from the first two terms.

Complete the square inside the brackets.

Write the perfect square trinomial as the square of a binomial.

Expand to remove the square brackets.

Simplify.

Example 2 – Complete the square.

C(x) = 1.8x2 – 14.4x + 156.5

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The solutions to the next 3 questions will be done in class.

1. A ball thrown vertically with a velocity of 18 m/s is h metres above the ground after t
seconds, where h = −5t 2 + 18t . What is the maximum height of the ball, and when does it reach
that height?
2. A rectangular lot is bordered on one side by a stream and on the other three sides by 600 m
of fencing. Find the dimensions of the lot if its area is a maximum.
3. A theatre seats 2000 people and charges $10 for a ticket. At this price, all the tickets can be
sold. A survey indicates that if the ticket price is increased, the number sold will decrease by
100 for every dollar of increase. What ticket price would result in the greatest revenue?
MCR 3U1 Unit 3
Lesson 3: Determining Maximum and Minimum Values of a Quadratic Function

***Method # 2 – Factoring ***

Same questions as yesterday, different method to find the maximum height

Example 1: A golfer attempts to hit a golf ball over a gorge from a platform above the ground.
The function that models the height of the ball is h(t) = -5t2 + 40t + 100, where h(t) is the
height in meters at time t seconds after contact. There are power lines 185 m above the
ground. Will the golf ball hit the power lines?

Lesson 4: Operations with Radicals

The following properties are used to simplify radical expressions:

1. ab = a  b , a  0 , b  0

a a
2. = , a  0, b  0
b b

A radicand is the expression under the radical (square root) sign.

An entire radical : a radical with coefficient of 1; for example, √12
A mixed radical: a radical with coefficient other than 1: for example 2 √3

A radical is in simplest form when:

▪ The radicand has no perfect square factors other than 1.
▪ The radicand does not contain a fraction.
▪ No radical appears in the denominator of a fraction.

Example 1: Simplify the following radicals using the perfect-square method.

MCR 3U1 Unit 3

33 3
a) 75 b) 24 c) d)
3 16

Note: you can also work “backwards” and express a mixed radical as an entire radical.
Change the mixed radical: 2 3

Example 2: To multiply radical expressions, multiply the whole numbers and multiply the
radicals. Simplify the radical product where possible.
a) 2  10 b) 4 3  −2 7 c) 2 5  3 10

You can also express radicals in simplest radical form by adding or subtracting like radicals. For
example:
4 3+2 3− 3
= 6 3 −1 3
=5 3

Example 3: Simplify the following expressions.

a) 8 5 − 3 7 + 7 7 − 4 5 b) 20 + 45 c) 5 27 + 4 48
MCR 3U1 Unit 3

To multiply any radical expression, use the distributive property and add or subtract like radicals.
Example 4: Simplify each radical expression.
a) 3 ( 6 − 1) b) (2 3 − 1)(3 3 + 2)

Remember that you should never leave a radical in the denominator because it is bad form! You
can easily simplify a radical expression with a radical in the denominator. Just multiply the
numerator and the denominator by this monomial radical.

3 6
Example 5: State the following expression in simplified form: .
4 10
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Lesson 5: Solving Quadratic Equations

A quadratic equation can be written in the form ax 2 + bx + c = 0 , where a  0 .

The terms of a quadratic equation are named as follows:

ax2 + bx + c = 0

Quadratic Linear Constant

term term term

Solving Quadratics
There are many methods for solving quadratic equations. By solving the equation, we are
determining the roots (zeroes/x-intercepts/solutions) of the quadratic function.

1. Reducing an equation with no linear terms to the form x 2 = c .

Example 1: x 2 − 5 = 0

2. Graphing – the roots of the equation are where the graph crosses the x-axis.
Example 2:

y


− − − −     

−

−

−

−

3. Factoring, using any of the given methods.

Example 3: x 2 + 7 x + 10 = 0
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4. Completing the square.
Example 4: Complete the square to determine the maximum value: − 5 x 2 + 30 x + 10 = 0 .

5. Using the quadratic formula.

Example 5: Solve − 5 x 2 + 30 x − 20 using the quadratic formula.

Note: You should only use the quadratic formula when you cannot factor the equation. Also,
you may want to leave your final answer as an exact value, depending on the situation.

Example 6: Represent and solve a problem using a quadratic equation. You are given that a
rectangle has an area of 330 m2. One side is 7 m longer than the other. What are the dimensions
of the rectangle?
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Lesson 6: The Zeros of a Quadratic Function

If you were asked to state the number of zeros (or x-intercepts) for an equation, what would
you do?

For example, determine the number of zeros for the following:

a) 4 x 2 + 12 x − 16 = 0 b) x 2 − 4 x + 7 = 0

Is there an easier way to find zeros?

There is. We can use the nature of the roots of a quadratic. There are 3 possibilities for the
roots of a quadratic in the form ax 2 + bx + c = 0 .

1. Two different real roots

2. Two equal real roots
3. No real roots

*** Draw Pictures from pg. 184***

− b  b 2 − 4ac
Part of the quadratic formula, x = , can help determine the nature of the roots
2a
without solving for them. The expression that we can take from the quadratic formula is the
part underneath the square root sign: b 2 − 4ac . This part is called the discriminant.

By the nature of the roots we mean:

▪ Whether or not the equation has real roots.
▪ If there are real roots, whether they are different or equal.

We can make the following conclusions about the nature of the roots of ax 2 + bx + c = 0 .
▪ If b 2 − 4ac  0 , there are two different real roots.
▪ If b 2 − 4ac = 0 , there are two equal real roots.
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▪ If b 2 − 4ac  0 , there are no real roots.

Example 1: Determine the nature of the roots for the following expressions.
a) 4 x 2 + 2 x − 5 b) 3x 2 − x + 7

Example 2: For what values of k will the function f ( x) = kx 2 − 4 x + k have no zeros?

Example 3: Student council decides it wants to raise money for a local charity. They decide to
sell little sticky note graph pads to everyone in the school. The equation f ( x) = −0.5x 2 + 4 x − 11
will determine their profit on the graph pads. Will they raise money for the charity?
MCR 3U1 Unit 3

Lesson 7: Linear-Quadratic Systems

Recall: A SYSTEM of equations is two or more equations.

To SOLVE a system means to find:

I. all intersection points
II. all values that “satisfy” both equations

To solve GRAPHICALLY means to graph the two equations and find the point(s) of intersection.

A line intersects a curve in one of three ways:

1. Not at all (no solution)
2. Once (a “tangent,” or one solution)
3. Twice (two solutions)

System 1
This system has two solutions. The parabola
and the line intersect at the points (-4.3, -1.3)
and (-1.7, 2.2)

System 2
This system has one solution. The parabola
and the line intersect at the point (-3, -3).
The line is a tangent to the parabola.

System 3
This system has no solution. The parabola and
the line do not intersect at any points.
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To solve a linear-quadratic system algebraically:

1. Set the two equations equal to each other.
2. Put the resulting equation into standard form.
3. Solve for the variable in the quadratic equation. (You may have none, one, or two solutions.)

Example 1: Determine the points of intersection of the functions f ( x) = −2 x 2 + 10x + 3 and

g ( x) = 4 x + 7 .

In addition to solving linear-quadratic systems, we can also predict the number of points of
intersection by using the discriminant.

Example 2: State the number, if any, of points of intersection for the following system:
f ( x) = 4 x 2 + x − 3 and g ( x) = 5 x − 4 .