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# Graphs of Exponential Functions Help

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## Introduction to Graphs of Exponential Functions

A basic exponential function is of the form f ( x ) = a x , where a is any positive number except 1. The graph of f ( x ) = a x comes in two shapes depending whether 0 < a < 1 ( a is positive but smaller than 1) or a > 1. Figure 9.2 is the graph of and Figure 9.3 is the graph of f ( x ) = 2 x .

Fig. 9.2

Fig. 9.3

Sketch the graph of f ( x ) = a x by plotting points for x = −3, x = − 2, x = −1, x = 0, x = 1, x = 2, and x = 3. If a is too large or too small, points for x = −3 and x = 3 might be too awkward to graph because their y -values are too large or too close to 0. Before we begin sketching graphs, we will review the following exponent properties.

#### Examples

Sketch the graphs.

• f ( x ) = 2.5 x
• We will begin with x = −3, −2, −1,0, 1,2, and 3 in a table of values.

Table 9.1

Fig. 9.4

Table 9.2

 x f ( x ) −3 27(( ) −3 = 3 3 ) −3 9(( ) −2 = 3 2 ) −1 3(( ) −1 = 3 1 ) 0 1 1 0.33 2 0.11 3 0.037

Fig. 9.5

Transformations of the graphs of exponential functions behave in the same way as transformations of other functions.

#### Examples

• The graph of f(x) = −2 x is the graph of y = 2 x reflected about the x -axis (flipped upside down).
• The graph of g(x) = 2− x is the graph of y = 2 x reflected about the y -axis (flipped sideways).
• The graph of h(x) = 2 x +1 is the graph of y = 2 x shifted to the left 1 unit.
• The graph of f(x) = −3 + 2 x is the graph of y = 2 x shifted down 3 units.

#### Practice

Sketch the graphs.

1.
2.
3. h ( x ) = e x (Use the e or e x key on your calculator.)

#### Solutions

1.

Table 9.3

 x f ( x ) −3 0.30 (( ) −3 = ( ) 3 = ) −2 0.44 (( ) −2 = ( ) 2 = ) −1 0.67 (( ) −1 = ( ) 0 1 1 1.5 2 2.25 3 3.375

Fig. 9.6

2.

Table 9.4

 x f ( x ) −3 3.375 (( ) −3 = ( ) 3 )) −2 2.25(( ) −2 = ( ) 2 ) −1 1.5(( ) −1 = ) 0 1 1 0.67 2 0.44 3 0.30

Fig. 9.7

3.

Table 9.5

 x f ( x ) −3 0.05 −2 0.14 −1 0.37 0 1 1 2.72 2 7.39 3 20.09

Fig. 9.8

Find practice problems and solutions for these concepts at: Exponents and Logarithms Practice Test.

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