Topic 3 Vertex Form of Quadratic Function Unit 6 Topic 3

Topic 3Vertex Form of Quadratic Function

Unit 6 Topic 3

Explore1.The equations and graphs of four quadratic functions are shown below. Under each graph write the coordinates of the vertex.

Try this on your own first!!!!

Explore1.The equations and graphs of four quadratic functions are shown below. Under each graph write the coordinates of the vertex.

(1, 3) ( 1,2)

(2,1) ( 3, 4)

Explore2. Based on the equation, how could you predict whether the graph opens upward or downward?

opens up opens down

3. Based on the equation, how could you predict the coordinates of the vertex?

0 0 a a2( )y x h ka

2

( , k)

( )

( )

,

h ky a x

h

x y

Notice that h is the opposite sign and k is the same sign as in the equation.

InformationA quadratic function is a relation that can be written in the standard form

For a quadratic function in vertex form, , where a ≠ 0, the graph

• has the shape of a parabola• has its vertex at (h, k)• has an axis of symmetry with the equation x = h

In , where a ≠ 0, the value of a determines whether the parabola opens upward or downward.

• If a > 0, the parabola opens upward.• If a < 0, the parabola opens downward.

2y ax bx c

InformationThe location of the vertex and the direction the parabola opens determine the number of x-intercepts. The graph of a quadratic function can have two x-intercepts, one x-intercept or no x-intercepts, as illustrated in the table below.

Example 1Identifying the Characteristics of a Quadratic Function in Vertex Form

Sketch a graph of each quadratic function and state the following characteristics.

Use technology to check your answers.


Example 1a: SolutionsThe vertex form tells us the vertex is (h, k). In this case, then, the vertex is at (4, 5).

min

up

(4,5)

5

4

5

none

y

x

x

y

Example 1b: Solutions

min

up

( 2, 6)

6

2

6

2

y

x

x

y

Example 1c: Solutions

max

down

( 6,0)

0

6

0

1

y

x

x

y

Example 2Finding Maximum Revenue

The ERCP is planning a Chili Cook-off as a fundraising event. a) The students were trying to decide how much to charge for one bowl of chili. One student claimed, “The more we charge the more money we make”. Do you agree or disagree? Explain.


Example 2a: Solutions

The ERCP is planning a Chili Cook-off as a fundraising event. a) The students were trying to decide how much to charge for one bowl of chili. One student claimed, “The more we charge the more money we make”. Do you agree or disagree? Explain.

VaryPossible Answer: Disagree because, as the bowls become more and more expensive, less people will buy it.


Revenue is the product of price sales. The revenue, R, in dollars, can be modelled by the quadratic function , where p is the price for a bowl of chili. What price produces the greatest revenue? What is the maximum revenue?

2150( 3) 600R p

Price ($)x

yR

eve

nu

e

($)

[0,10,1]

[0,700,50]

X

Y

2nd trace4: maximum

3.00 600x y

xPrice ($)

yRevenue ($)

A price of $3.00 will produces a maximum revenue of $600.

As we proceed through this question, refer to the steps in your workbook.

Example 3Determining the Vertex Form Equation From a Graph

The graph below has a vertex at (1, 6) and passes through the point (0, 4). Determine the vertex form equation of each graph.a)



Step 1: Use the coordinates of the vertex to find h and k in .2(x h)y a k

2(

(1, 6)

( , k)

6)1y x

h

a


Example 3a: SolutionsStep 2: Substitute the coordinates of a point on the graph into vertex form, , and solve for a.

2(x h)y a k

2

2

( 1) 6

(0, 4)

( ,

( 1)

)

4 0

4 1 6

2

6 6

6

y a x

a

x y

a

a

2)2( 1 6y x

Example 3Determining the Vertex Form Equation From a Graph

The graph below has a vertex at (4, 3) and passes through the point (0, 5). Determine the vertex form equation of each graph.b)




2(

( 4, 3)

( , k)

3)4y x

h

a


Example 3b: SolutionsStep 2:

2

2

(0, 5)

( , )

5 0

5 16 3

8 16

8 16

( 4) 3

( 4) 3

1

2

3 3

16 16

x y

a

a

x

a

y

a

a

a

24 31

2( )y x

Example 4Determining the Height of a Football Field Goal Kick

A field goal kicker in a CFL game attempts to kick a field goal. At a distance of 20 m from the kicker, the football reaches a maximum height of 40 m above ground. The football hits the ground at a distance of 40 m from the kicker.a) Let y represent the height of the football, in metres, and x represent the horizontal distance from the kicker, in metres. Determine the vertex form equation that describes the relationship between height and distance from the kicker.





Horizontal distance (m)x

yH

eig

ht

(m)

2

(20, 40)

( , k)

2 40( )0y

h

a x


Step 2:

2

2

(40, 0)

( , )

0 40

0 4

40

( 20) 40

( 20) 4

40

4

00 40

40 400

40 400

1

10

00 40

0

0

x

y x

y

a

a

a

a

a

a

2( 20) 40

01

1y x

Example 4b) A field goal is scored if the football goes over a 3 m high cross bar and between the goal posts. If the kicker aimed the football between the goal posts, did he kick the football high enough to clear the cross bar, at a distance of 39 m?


Example 4b: SolutionA field goal is scored if the football goes over a 3 m high cross bar and between the goal posts. If the kicker aimed the football between the goal posts, did he kick the football high enough to clear the cross bar, at a distance of 39 m?

2nd trace1: valuex = 39

y = 3.9 m

Yes, he did kick the football high enough to clear the cross bar of 3 m. The height of the ball was 3.9 m.

[0,50,5]

[0,40,5]

X

Y

Need to Know:

• The vertex form of a quadratic equation is, where a ≠ 0.▫ If , the parabola is opening upward.▫ If , the parabola is opening downward.▫ The vertex of the parabola is (h, k). ▫ The equation for the axis of symmetry is .

• The vertex of the graph and direction the graph opens determine the number of x-intercepts. ▫ A graph has two x-intercepts if the graph crosses the x-

axis twice.▫ A graph has one x-intercept if the graph touches the x-

axis.▫ A graph has zero x-intercepts if the graph does not cross

the x-axis.

Need to Know:

• The number of x-intercepts a graph has depends on where the vertex is located and the direction the parabola opens.▫ A graph has 2 x-intercepts if the graph crosses the x-

axis twice.▫ A graph has 1 x-intercept if the graph touches the x-

axis once.▫ A graph has 0 x-intercepts if the graph does not cross

the x-axis.You’re ready! Try the homework from this section.

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Topic 3 Vertex Form of Quadratic Function Unit 6 Topic 3