Intermediate Algebra (11th edition)

Graphing an Exponential Function

Graph Then compare it to the graph of

Choose some values of x, and find the corresponding values of Plotting

these points and drawing a smooth curve through them gives the darker graph

shown in FIGURE 5. This graph is typical of the graphs of exponential functions of

the form where The larger the value of a,the faster the graph

rises.To see this, compare the graph of with the graph of in

FIGURE 5.

F 1 x 2 = 5 x ƒ 1 x 2 = 2 x

F 1 x 2 = ax, a 7 1.

ƒ 1 x 2.

ƒ 1 x 2 = 2 x. F 1 x 2 = 5 x.

EXAMPLE 1 1 a 712

SECTION 10.2 Exponential Functions 581

0

1

2

3

• 3

1

2

4

8

x

0

2

4

6

8

x

y

1

8

• 2
  
  
  • 22
  
  
  
  

1

4

– (^112)

f (x) 22 x

f (x) 22 x

F (x) 55 x

FIGURE 5

The vertical line test assures us that the graphs in FIGURE 5represent functions.

FIGURE 5also shows an important characteristic of exponential functions with

As x gets larger, y increases at a faster and faster rate. NOW TRY

a 7 1:

CAUTION The graph of an exponential function approachesthe x-axis, but

does nottouch it.

Graphing an Exponential Function

Graph

Again, find some points on the graph. The graph, shown in FIGURE 6, is very

similar to that of (FIGURE 5) except that here as x gets larger, y decreases.

This graph is typical of the graph of a function of the form where

06 a 6 1.

ƒ 1 x 2 = ax,

ƒ 1 x 2 = 2 x

g 1 x 2 = a

1

2

b

x

.

EXAMPLE 2 106 a 612

1

x

0

2

4

6

8

x

y

• 22

1

2

(^214)
(^318)
x

g (x)^12

x

g (x)^12

8

4

2

1

• 3


• 2


• 1
  0

FIGURE 6 NOW TRY

Exponential function

with base

Domain:

Range:

The function is one-to-

one, and its graph rises

from left to right.

1 0, q 2

1 - q, q 2

a> 1

Exponential function

with base

Domain:

Range:

The function is one-to-

one, and its graph falls

from left to right.

1 0, q 2

1 - q, q 2

0 <a< 1

NOW TRY
EXERCISE 1
Graph y= 4 x.

NOW TRY ANSWERS

1. 
   

x

y

4

–1 0 1

y = 4x

NOW TRY

EXERCISE 2

Graph g 1 x 2 = a

1

10

b

x

.

2. 
   

x

y

1

10

–1 0 1

10

g(x) =(^1 (x