chapter 6: quadratic functions - mr....

Chapter 6: Quadratic Functions Section 6.1

Chapter 6: Quadratic Functions

Section 6.1 Exploring Quadratic Relations

Terminology:

Quadratic Relations:

A relation that can be written in the standard form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, where 𝑎 ≠ 0;

A function is considered a quadratic function if its highest power when expanded is of

degree 2 (ie a squared term)

Ex. 𝑦 = 4𝑥2 + 2𝑥 + 1

Parabola:

The shape of the graph of any quadratic relation. (Parabolas are ∪-shape or ∩-shape)

Identifying a Quadratic Function from an Equation

To determine if a function is quadratic, you must identify if the function has a highest power of 2

when expanded. Any function that does not have a degree of 2, is not quadratic.

Ex. Which functions are quadratic?

(a) 𝑦 = 2𝑥 − 3 (b) 𝑦 = 5 − 2𝑥2 + 3𝑥 (c) 𝑦 = (𝑥 + 2)(𝑥 − 3)

(d) 𝑦 = 𝑥3 − 2𝑥2 + 3𝑥 − 5 (e) 𝑦 = 3𝑥(𝑥 + 7)

Features of a Quadratic Functions in Standard Form

Given the quadratic function in standard form:

𝑦 = 2𝑥2 + 5𝑥 + 3

Will produce the graph shown to the right:

What do you notice about the shape of the graph?


Exploring the Effects of the 𝑎 – value:

If we change the a-value of the equation, how will this impact the graph in each situation?

Original Function New Function How has it Changed?

𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = −2𝑥2 + 5𝑥 + 3

𝑦 = 2𝑥2 + 5𝑥 + 3

𝑦 =1

2𝑥2 + 5𝑥 + 3

𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = 3𝑥2 + 5𝑥 + 3

𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = −

3

4𝑥2 + 5𝑥 + 3


Exploring the Effects of the 𝑐 – value:

If we change the c-value of the equation, how will this impact the graph in each situation?


𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = 2𝑥2 + 5𝑥 + 6

𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = 2𝑥2 + 5𝑥 − 4

𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = 2𝑥2 + 5𝑥


Exploring the Effects of the 𝑏 – value:

If we change the b-value of the equation, how will this impact the graph in each situation?


𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = 2𝑥2 + 9𝑥 + 3

𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = 2𝑥2 − 5𝑥 + 3

𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = 2𝑥2 + 2𝑥 + 3

𝑦 = 2𝑥2 + 5𝑥 + 3 𝑦 = 2𝑥2 + 3


Summary of Quadratic Function in Standard Form

The degree of all quadratic functions is 2.

The standard form of a quadratic function is 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, where 𝑎 ≠ 0.

The graph of any quadratic function is a parabola with a single vertical line of symmetry.

A quadratic function is written in standard form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, has the following

characteristics:

The highest or lowest point on the graph of the quadratic function lies on its

vertical line of symmetry.

If a is positive, the parabola opens up. If a is negative, the parabola opens down.

𝑎 > 0 𝑎 < 0

Changing the value of b changes the location of the parabola’s line of symmetry.

The constant term, c, is the value of the parabola’s y-intercept.


Section 6.2: Properties of Graphs of Quadratic Functions

Terminology:

Vertex:

The point at which the quadratic function reaches its maximum or minimum value.

Axis of Symmetry:

A line that separates a two dimensional figure into two identical parts.

Ex. A parabola has a vertical axis of symmetry passing through the vertex.

Maximum Value:

The greatest value of the dependent variable in a relation.

Minimum Value:

The least value of the dependent variable in a relation.

Domain:

All possible x-values that a graph passes through.

Range:

All possible y-values that a graph passes through.

Using Symmetry to Estimate the Coordinates of the Vertex

Ex. Nicole plays on her school’s volleyball team. At a recent match, her

friend, Marko, took some time-lapse photographs while she warmed up.

He set his camera to take pictures every 0.25 seconds. He started his

camera at the moment the ball left her arms during a bump and stopped

the camera at the moment that the ball hit the floor. Marko wanted to

capture a photo of the ball at its greatest height. However, after looking

at the photographs, he could not be sure that he had done so. He decided

to place the information from his photographs in a table of values.

From his photographs, Marko observed that Nicole struck the ball at a height of 2 ft above the

ground. He also observed that it took about 1.25 seconds for the ball to reach the same height on

the way down. When did the volleyball reach its greatest height?

Time (s)

Height (ft)

0.00 2 0.25 6 0.50 8 0.75 8 1.00 6 1.25 2


Ex. Given the coordinates of a parabola, as shown in the table,

determine the coordinates of the vertex.

Determining the Coordinates of the Vertex from Standard Form

There exists a relationship between the value of the x-coordinate of the vertex and the

values of coefficients in standard form.

To determine the value of the x component of the vertex (p) we use the formula:

𝑥 = −𝑏

2𝑎

To determine the y component of the vertex (q) we simply plug in the result of the above

formula into the given equation in standard form.

Ex: For each of the following state the vertex, directionality, y-intercept, and range:

(a) 𝑓(𝑥) = −𝑥2 + 2𝑥 + 8 (d) 𝑓(𝑥) = 2𝑥2 − 12𝑥 + 25

(b) 𝑓(𝑥) = 𝑥2 − 2𝑥 (e) 𝑓(𝑥) = −3𝑥2 + 9𝑥 + 12

𝒙 𝒚 0 0 3 6 4 7.5 8 7.5 9 6

12 0


(c) (𝑥) = 4𝑥2 − 2𝑥 + 5 (f) 𝑓(𝑥) = 𝑥2 + 10𝑥 − 4

Determining the Maximum Value of a Quadratic

Ex. Some children are playing at the local splash pad. The water jets spray water from ground

level. The path of water from one of these jets forms an arch that can be defined by the function:

𝑓(𝑥) = −0.12𝑥2 + 3𝑥

Where 𝑥 represents the horizontal distance from the opening in the ground in feet and 𝑓(𝑥) is

the height of the sprayed water, also measured in feet. What is the maximum height of the arch

of water, and how far from the opening in the ground can the water reach? State the range for

the jet of water.

Ex. A skier’s jump was recorded in frame-by-frame analysis and placed in one picture. The

skier’s coach used the picture to determine the quadratic function that relates the skier’s height

above the ground, 𝑦, measured in metres, to the time, 𝑥, in seconds that the skier was in the air:

𝑦 = −4.9𝑥2 + 15𝑥 + 1

Determine the skier’s maximum height, to the nearest tenth of a metre, and state the range of

the function for this context.


Graphing a Quadratic Function in Standard Form

Ex: For each of the functions determine the following functions then Sketch:

1. The Vertex

2. The Domain and Range

3. The Direction of Opening

4. The Equation of the Axis of Symmetry

(a) 𝑦 = 2𝑥2 + 4𝑥 − 1

(a) 𝑦 = 2𝑥2 + 4𝑥 − 1


(b) 𝑦 = −1

4𝑥2 + 2𝑥 − 3

(c) 𝑦 =1

2𝑥2 − 2𝑥 − 2

x- 15 - 10 - 5 5 10 15

y

- 15

- 10

- 5

5

10

15


(d) 𝑦 = −3𝑥2 − 6𝑥

x- 15 - 10 - 5 5 10 15

y

- 15

- 10

- 5

5

10

15


Section 6.3: Factored Form of a Quadratic Function

Terminology:

Factored Form of a Quadratic Function:

A quadratic function is in factored form when it is written in the form:

𝑦 = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠)

Where 𝑎, 𝑟, and 𝑠 are real numbers and 𝑎 ≠ 0.

Zero:

In a function, a value of the variable that makes the value of the function equal to zero.

Zeros are also referred to as solutions, roots, and x-intercepts. The zeros of an equation

can be determined by setting an equation equal to zero (or by substituting 𝑦 = 0 into an

equation).

Y-Intercept:

The value at which a function passes through the y-axis. The x-coordinate of y-intercept

is always zero and hence can be found from an equation by plugging 𝑥 = 0 into an

equation.

Determining the Zeros of an Equation in Factored Form

Ex. Determine the zeros of each equation

(a) 𝑦 = 3(𝑥 + 2)(𝑥 − 5) (b) 𝑦 = −1

2(𝑥 + 3)(𝑥 + 9)

(c) 𝑓(𝑥) =1

7(𝑥)(𝑥 − 4) (d) 𝑓(𝑥) = 5(𝑥 − 1)(𝑥 − 1)


(e) 𝑦 = (𝑥 + 10)(𝑥 − 10) (f) 𝑦 = 2(3𝑥 + 4)(𝑥 − 5)

Determining the Zeros of Quadratic Functions in Standard Form

Determine the zeros of a function by using partial factoring

(a) 𝑦 = 𝑥2 + 8𝑥 + 15 (b) 𝑦 = −3𝑥2 + 6𝑥 − 3

(c) 𝑦 = 𝑥2 − 6𝑥 − 40 (d) 𝑦 = 2𝑥2 + 14𝑥 + 12

Graphing a Function in Factored Form

Graph each of the functions.

(a) 𝑦 = 2(𝑥 + 1)(𝑥 − 2)


(b) 𝑦 = −1

4(𝑥 − 8)(𝑥 − 3)

(c) 𝑦 =1

2(𝑥 + 6)(𝑥 − 2)

x- 15 - 10 - 5 5 10 15

y

- 15

- 10

- 5

5

10

15


(d) 𝑦 = −3(𝑥 + 1)(𝑥 − 2)

x- 15 - 10 - 5 5 10 15

y

- 15

- 10

- 5

5

10

15

(e) 𝑦 =1

5(𝑥 + 5)(𝑥 − 7)


x- 15 - 10 - 5 5 10 15

y

- 15

- 10

- 5

5

10

15

Determining the Equation of Quadratic Function in Factored Form from a Graph

To determine the equation of a quadratic function in factored form

𝑦 = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠)

we need to determine the x-intercepts and y-intercept. The x-intercepts fill in the 𝑟 and 𝑠

values and then use the y-intercept to solve for 𝑎.


(a)

(b)


(c)

(d)


Max/Min Word Problems using Factored Form Quadratics

Ex. The members of a Ukrainian church hold a fundraiser every Friday night in the summer.

They usually charge $6 for a plate of perogies. They known from previous Fridays, that 120

plates of perogies can be sold at $6 but, for every $1 the price is increased, 10 fewer plates will be

sold. What should the members charge if they want to raise as much money as they can?

Ex. A career and technology class at a high school operates a small T-shirt business out of the

school. Over the last few years, the shop has had monthly sales of 300 T-shirts at a price of $15

per T-shirt. The students have learned that for every $2 increase in price, they will sell 20 fewer

T-shirts each month. What should they charge for their T-shirt to maximize their monthly

revenue? What will their monthly revenue be?

Ex. Paulette owns a store that sells used video games in Red Deer. She charges $10 for each used

game. At this price she sells 70 games per week. She has learned that for every $1 increase in

price, she sells 5 fewer games per week. At what price should Paulette sell her games to

maximize her sales? What is her maximum revenue?


Ex. You wish to build a rectangular garden next to your house. You intend to use your house as

one of the sides of the garden. If you buy 30 ft of fencing to go around the edges of your garden,

determine the length and width that will result in your garden having a maximum area.

Ex. Nathan is planning on building a sand box next to his house to play in. He plans to use his

house as one side of the sandbox. He has purchased 20 ft of board to create the sides of his

sandbox. Determine the length and width of the sandbox to produce a maximum area.

Ex. Mr. Hutchings is planning on creating a kennel for his classroom to store some of his

troublesome Math 2201 students when they need a time out. The design for the kennel is shown

below. If he has 34 ft of boards to create the kennel, what are the measurements of each kennel

room that will produce the maximum area?

Kennel


Section 6.4: Vertex Form of a Quadratic Function

Terminology:

Vertex Form of a Quadratic:

A quadratic function is in factored form when it is written in the form:

𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘

Where 𝑎, 𝑟, and 𝑠 are real numbers and 𝑎 ≠ 0. In vertex form, the values of h and k are

the coordinates of the vertex of the function (h, k).

Examples of vertex form are shown below, state the vertex of each:

(a) 𝑦 = (𝑥 + 1)2 − 2 Vertex:__________

(b) 𝑦 = (𝑥 − 6)2 + 7 Vertex:__________

(c) 𝑦 = 2(𝑥 + 4)2 − 4 Vertex:__________

(d) 𝑦 = −1

2(𝑥 + 10)2 Vertex:__________

(e) 𝑦 = 4(𝑥 − 3)2 + 6 Vertex:__________

(f) 𝑦 = −3𝑥2 + 5 Vertex:__________

(g) 𝑦 =1

9(𝑥 − 1.5)2 − 4 Vertex:__________

(h) 𝑦 = 𝑥2 Vertex:__________

Sketching a Graph of a Quadratic from Vertex Form

To sketch a graph of a quadratic from vertex form, we need two pieces of information:

1. The coordinates of the vertex

2. Any other set of coordinates relating to the graph of the function, usually

something simple like the y-intercept is recommended!

We then place those coordinates on the graph, create a third point using the axis of

symmetry as we did in the previous sections, and then sketch the graph of the function

passing through those three points.

Ex. Sketch the graph of each function in vertex form.

(a) 𝑦 = 2(𝑥 + 3)2 − 10


(b) 𝑦 = −1

2(𝑥 − 4)2 + 5

(c) 𝑦 = 5(𝑥 − 1)2 − 5


(d) 𝑦 = −6𝑥2 − 4

(e) 𝑦 =1

10(𝑥 + 5)2


Determining the Equation of a Quadratic in Vertex Form

When determining the equation of a quadratic in vertex form, we need the location of the vertex

and one other point. The additional point is usually one that is easily read or one that is clearly

identified. We plug the vertex (h, k) into the equation 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘, then use the additional

point (𝑥, 𝑦) into the equation and solve for a.

Ex. Determine the equation in vertex form for each graph

(a)

(b)


(c)

(d)


Using Vertex and Directionality to Determine the Number of Zeros

To determine the number of zeros that a quadratic function has, we use the vertex and the

directionality of a given function. Drawing a sketch often helps to determine the number of

zeros. Once you have a sketch, you just count the number of x-intercepts that your sketch has.

Note that there are only three possibilities for a quadratic function; Two Zeros, One Zero, or No

Zeros.

Ex. Determine the number of zeros for each function:

(a) 𝑦 = (𝑥 + 5)2 − 6

(b) 𝑦 = −(𝑥 − 6)2 + 7

(c) 𝑦 = 2(𝑥 + 4)2

(d) 𝑦 = −1

2(𝑥 + 10)2 − 6

(e) 𝑦 = 4(𝑥 − 3)2 + 6

(f) 𝑦 = −3(𝑥 + 1)2

(g) 𝑦 =1

9(𝑥 − 1.5)2 − 4

(h) 𝑦 = 𝑥2


Word Problems and Vertex Form

To solve word problems involving vertex form we often have to create the vertex form

equation 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 and then use the equation to answer the question. As usual

we will need two coordinates to solve the equation, one of which needs to be the vertex

of the equation.

Ex. A soccer ball is kicked from the ground. After 2 s, the ball reaches its maximum

height of 20 m. It lands on the ground at 4 seconds.

(a) Determine the equation of the quadratic function in vertex form that models the height

of the ball after it is kicked.

(b) What was the height of the ball after 1 second?

(c) When was the ball at the same height on the way down?

Ex, When playing a game of baseball, a player hits after it is pitched. It reached a maximum

height of 24.2 m after 6.5 seconds. The ball was in there air for 13.8 seconds.

(a) Determine the equation of the quadratic function in vertex form that models the height

of the ball after it is hit.

(b) Determine the height of the ball after 9 seconds.

(c) When was the ball at the same height on the way up?


Ex. The underside of a concrete underpass forms a parabolic arch. The arch is 30.0 m wide at

the base and 10.8 m high in the centre. What would be the headroom at the edge of a sidewalk

that starts 1.8 m from the base of the underpass? Would this amount of headroom be safe for

walking?

Ex. The Lions Gate Bridge in Vancouver is a suspension bridge. The main span, between the two

towers, is 472 m long. Large cables are attached to the top of both towers, 50 m above the road.

Each large cable forms a parabola. The road is suspended from the large cables by a series of

vertical cables. The shortest vertical cable measures about 2 m from the road. Use this

information to determine the quadratic function in vertex form to model this situation. Use your

equation to determine the height of the vertical cable that is 300 m away from the first tower.


(20,0)(0,0)

Path of Soccer Ball

y

x

Ex A soccer ball is kicked into the air and first lands on the ground 20 m away. If a maximum height of 8 m is reached, determine the vertex form of the quadratic function representing the path of the soccer ball while it is in the air.

Ex. The entrance to a garden party is in the shape of an arch. If the arch is to be 2 metres wide

and 2 metres tall write the quadratic equation that describes its shape. Write the equation that

models this situation in vertex form.

chapter 6: quadratic functions - mr....

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