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 |
9(( |
|
−1 |
3(( |
|
0 |
1 |
|
1 |
0.33 |
|
2 |
0.11 |
|
3 |
0.037 |
) −3 = 3 3 )
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.
-
-
- h ( x ) = e x (Use the e or e x key on your calculator.)
Solutions
-
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
-
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
-
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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