Intermediate Algebra (11th edition)

SECTION 11.1 Additional Graphs of Functions 637

The graphs of these elementary functions can be shifted, or translated, just as we

did with the graph of ƒ 1 x 2 = x^2 in Section 9.5.

Applying a Horizontal Shift

Graph Give the domain and range.

The graph of is obtained by shifting the graph of two units

to the right. In a similar manner, the graph of is found by shifting the

graph of y=|x|two units to the right, as shown in FIGURE 4.

ƒ 1 x 2 =|x- 2 |

y= 1 x- 222 y= x^2

ƒ 1 x 2 = |x- 2 |.

EXAMPLE 1

0

f(x) = ⏐x – 2⏐

y

x

2

–2 (2, 0)

FIGURE 4

xy

02

11

20

31

42

Compare this table

of values to that

with FIGURE 1.

Domain:

Range: 3 0, q 2

1 - q, q 2

NOW TRY

Applying a Vertical Shift

Graph Give the domain and range.

The graph is found by shifting the graph of y= three units up. See FIGURE 5.

1

x

ƒ 1 x 2 =

1

x+3.

EXAMPLE 2

y

x

4

3

1

0

f(x) =^1 x+ 3

FIGURE 5

xy

0

1

2

• 2 2.5


• 1
  -^12
  -^13

xy

6

5

14

2 3.5

1

2

1

3

Compare this table

of values to that

with FIGURE 2.

Domain:

Range:

Vertical asymptote:

Horizontal asymptote: y= 3

x= 0

1 - q, 3 2 ́ 1 3, q 2

1 - q, 0 2 ́ 1 0, q 2

NOW TRY

Applying Both Horizontal and Vertical Shifts

Graph Give the domain and range.

The graph of is obtained by shifting the graph of one

unit to the left and four units down. Following this pattern, we shift the graph of

one unit to the left and four units down to get the graph of See

FIGURE 6 on the next page.

ƒ 1 x 2 = 2 x+ 1 - 4.

y= 2 x

y= 1 x+ 122 - 4 y= x^2

ƒ 1 x 2 = 2 x+ 1 - 4.

EXAMPLE 3

NOW TRY

EXERCISE 1

Graph Give the
domain and range.

ƒ 1 x 2 =x+^13.

NOW TRY ANSWERS

1. 
   

domain:

range: 1 - q, 0 2 ́ 1 0, q 2

1 - q, - 32 ́ 1 - 3, q 2 ;

x

y

0

2

–3

f(x) =x +^13

NOW TRY

EXERCISE 2

Graph Give
the domain and range.

ƒ 1 x 2 = 2 x+2.

2. 
   

domain: ; range: 3 0, q 2 3 2, q 2

2

0

y

x

1

f(x) = x + 2