Intermediate Algebra (11th edition)

NOW TRY

534 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

2

• 1
  
  
  • 2
  
  
  
  


• 1
  2

x

0

• 2


• 3

2

x

y

f f (x) x^22

F F (x)

F F (x) (x (^) ++ 33 )^22 – – 22
x –– 33

• 5


• 4


• 3


• 2


• 1

FIGURE 8

Vertex:

Axis:

Domain:

Range: 3 - 2, q 2

1 - q, q 2

x=- 3

1 - 3, - 22

F 1 x 2 = 1 x+ 322 - 2

x

00

1

2 - 2

• 
  

  (^12)

  
  


• 
  

  1 - (^12)

  
  


• 
  

  2 - 2

  
  

ƒ 1 x 2

–2^02

–2

–4

x

y

f(x) = –^1 x^2

2

FIGURE 9

Vertex:

Axis:

Domain:

Range: 1 - q, 0 4

1 - q, q 2

x= 0

1 0, 0 2

ƒ 1 x 2 =-^12 x^2

NOW TRY

General Characteristics of

1. The graph of the quadratic function defined by

with

is a parabola with vertex and the vertical line as axis.

2. The graph opens up if ais positive and down if ais negative.

3. The graph is wider than that of if

The graph is narrower than that of ƒ 1 x 2 = x^2 if |a| 7 1.

ƒ 1 x 2 = x^206 |a| 6 1.

1 h, k 2 x=h

F 1 x 2 a 1 xh 22 k, a0,

F 1 x 2 a 1 xh 22 k 1 a 02

NOW TRY
EXERCISE 3
Graph.
Give the vertex, axis, domain,
and range.

ƒ 1 x 2 = 1 x+ 122 - 2

NOW TRY ANSWERS
3.

vertex: ; axis: ;
domain: ; range:
4.

vertex: ; axis: ;

domain: ; range: 1 - q, q 2 1 - q, 0 4

1 0, 0 2 x= 0

1 - q, q 2 3 - 2, q 2

1 - 1, - 22 x=- 1

x

y

0

–1

–2

2

f(x) = (x + 1)^2 – 2

x

y

0

–2 2

f(x) = –3x^2

–3

NOW TRY
EXERCISE 4
Graph. Give
the vertex, axis, domain,
and range.

ƒ 1 x 2 =- 3 x^2

OBJECTIVE 3 Use the coefficient of to predict the shape and direction

in which a parabola opens. Not all parabolas open up, and not all parabolas have

the same shape as the graph of

Graphing a Parabola That Opens Down

Graph

This parabola is shown in FIGURE 9. The coefficient affects the shape of the

graph—the makes the parabola wider since the values of increase more slowly

than those of and the negative sign makes the parabola open down. The graph

is not shifted in any direction. Unlike the parabolas graphed in Examples 1–3,the

vertex here has the greatestfunction value of any point on the graph.

x^2 B,

1

2 x

(^1) A 2
2

-

1

2

ƒ 1 x 2 =-

1

2 x

(^2).

EXAMPLE 4

ƒ 1 x 2 = x^2.

x^2

Vertex and Axis of a Parabola

The graph of is a parabola.

• The graph has the same shape as the graph of


• The vertex of the parabola is


• The axis is the vertical line x=h.

1 h, k 2.

ƒ 1 x 2 =x^2.

F 1 x 2  1 xh 22 k