Intermediate Algebra (11th edition)

The graph of any quadratic function is a parabola with a vertical axis.

NOTE We use the variable yand function notation interchangeably. Although

we use the letter ƒ most often to name quadratic functions, other letters can be used.

We use the capital letter Fto distinguish between different parabolas graphed on the

same coordinate axes.

Parabolas have a special reflecting property that makes

them useful in the design of telescopes, radar equipment, so-

lar furnaces, and automobile headlights. (See the figure.)

OBJECTIVE 2 Graph parabolas with horizontal and vertical shifts.Parabolas

need not have their vertices at the origin, as does the graph of

Graphing a Parabola ( Vertical Shift)

Graph

The graph of has the same shape as that of but is

shifted, or translated, 2 units down, with vertex Every function value is 2

less than the corresponding function value of Plotting points on both

sides of the vertex gives the graph in FIGURE 6.

ƒ 1 x 2 = x^2.

1 0, - 22.

F 1 x 2 =x^2 - 2 ƒ 1 x 2 = x^2

F 1 x 2 =x^2 - 2.

EXAMPLE 1

ƒ 1 x 2 =x^2.

ƒ 1 x 2

532 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

Quadratic Function

A function that can be written in the form

for real numbers a, b, and c, with aZ 0,is a quadratic function.

ƒ 1 x 2 ax^2 bxc

Headlight

• 2


• 1
  0
  1
  2

2

• 1


• 2


• 1
  2

4

1

0

1

4

x

– 220

2

x

y

f f (x) x^22

f f (x) x^22

F F (x) x^22 – – 22

F F (x) xx^22 – – 22

FIGURE 6

This parabola is symmetric about its axis so the plotted points are “mirror

images” of each other. Since xcan be any real number, the domain is still. The

value of y 1 or F 1 x 22 is always greater than or equal to -2,so the range is 3 - 2, q 2.

1 - q, q 2

x= 0,

Vertex:

Axis:

Domain:

Range:

The graph of

is shown for comparison.

ƒ 1 x 2 =x^2

3 - 2, q 2

1 - q, q 2

x= 0

1 0, - 22

F 1 x 2 =x^2 - 2

NOW TRY

NOW TRY
EXERCISE 1
Graph. Give
the vertex, axis, domain,
and range.

ƒ 1 x 2 =x^2 - 3

NOW TRY ANSWER

1. 
   

vertex: axis: ;

domain: ; range: 1 - q, q 2 3 - 3, q 2

1 0, - 32 ; x= 0

x

y

–2 0 2

1

6

–3

f(x) = x^2 – 3