Graphing Quadratic Functions

Chapter 6.1

Quadratic Functions

Music managers handle publicity and other business issues for the artists they manage. One group’s manager has found that based on past concerts, the predicted income for a performance is

where x, is the price per ticket in dollars.

7500400050)( 2 xxxP

The graph of our functions is shown here

7500400050)( 2 xxxP

20

40

60

80

20 40 60 80

Price per ticket

Inco

me

(th

ousa

nds

of

dolla

rs)

Notice that at first the income increases, as the price per ticket increases, but as the price continues to increase, the income declines.

How is this graph useful?

Quadratic Functions

A quadratic function is described by an equation of the following form:

cbxaxxf 2)(

The shape of the graph of any quadratic function is called a parabola

One way to graph parabolas

If you have no idea how to graph a function, your first game plan should be…?

PLOT POINTS!!

PLOT POINTS!!

PLOT POINTS!!

PLOT POINTS!!

PLOT POINTS!!

Example 1

Plot points to graph the function:

13)( 2 xxxf

x

f(x)

-3 -2 -1 0 1

-1 -3 -3 -1 3

Another way to graph quadratics

You need 3 things:– Axis of symmetry– Y-intercept– Vertex

y-intercept

(0, c)

vertex

axis of symmetry x = -b/2a

cbxaxxf 2)(

Example 2

Find axis, vertex and y-int to graph:

14)( 2 xxxf

y-intercept:

axis:

vertex:

(0, 1)

)1(2

)4(

2

4

2

x = -2

(-2, 5)

a

b

2

(-2, ?)

1)2(4)2()2( 2 f184)2( f

5)2( f

Example 3

Find axis, vertex and y-int to graph:

2)( xxf y-intercept:

axis:

vertex:

(0, 0)

)1(2

0 0

x = 0

(0, 0)

a

b

2

(0, ?)

20)0( f0)0( f

FIND MORE POINTS!!

Example 4

Find axis, vertex and y-int to graph:

12)( 2 xxf

y-intercept:

axis:

vertex:

(0, -1)

)1(2

0 0

x = 0

(0, -1)

a

b

2

(0, ?)

1)0(2)0( 2 f

10)0( f

1)0( f

Example 5

Find axis, vertex and y-int to graph:

34)( 2 xxxf

y-intercept:

axis:

vertex:

(0, 3)

)1(2

4

2

x = -2

(-2, 7)

a

b

2

(-2, ?)

3)2(4)2()2( 2 f384)2( f

7)2( f

Maximum and minimum values

The y-coordinate of the vertex gives the maximum or minimum value for a quadratic function.

Minimum

Maximum

a>0

a<0

Example 1

32)( 2 xxxf• Does this function have a maximum or a minimum value? (max)

• What is the max value? 4

a

b

2

)1(2

2

2

2

13)1(2)1()1( 2 f

321)1( f4)1( f

Vertex: (1, 4)

Example 2

A souvenir shop sells about 200 coffee mugs each month for $6.00 each. The shop owner estimates that for each $0.50 increase in the price, he will sell about 10 fewer mugs per month.

a.) How much should the owner charge for each mug in order to maximize the monthly income from their sales?

b.) What is the maximum income?

Example 2

In words: Income equals the # of mugs sold multiplied

by the price per mug.

In variables: Let x = the number of $0.50 price increasesThen price is… x5.06 And # sold is… x10200

Example 2

Equation: Income = Mugs x Price

)(xI )10200( x )5.06( x25601001200)( xxxxI

1200405)( 2 xxxI