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Graphs of Exponential Functions

Author: Sophia
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what's covered
In this lesson, you will graph exponential functions. Specifically, this lesson will cover:

Table of Contents

1. Graphing an Exponential Function When b > 1

From the models we have seen so far in this course, we know that an exponential function f open parentheses x close parentheses equals a times b to the power of x when b greater than 1 increases as x increases, but in all of our models, the starting value, a comma was positive. In this section, we’ll take a look at the behavior of graphs of exponential functions that also consider a less than 0.

Recall that we called the value of a the “starting value,” which was in the context of modeling situations. When referring to graphs of exponential functions, notice that f open parentheses 0 close parentheses equals a times b to the power of 0 equals a. This means that the y-intercept of an exponential function f open parentheses x close parentheses equals a times b to the power of x is open parentheses 0 comma space a close parentheses.

Consider three exponential functions: f open parentheses x close parentheses equals 2 to the power of x comma g open parentheses x close parentheses equals 3 times 2 to the power of x comma and h open parentheses x close parentheses equals short dash 5 times 2 to the power of x. The tables of values for each function are shown here.

bold italic x bold italic f open parentheses bold x close parentheses bold equals bold 2 to the power of bold x bold italic g open parentheses bold x close parentheses bold equals bold 3 bold times bold 2 to the power of bold x bold italic h open parentheses bold x close parentheses bold equals bold short dash bold 5 bold times bold 2 to the power of bold x
-3 0.125 0.375 -0.625
-2 0.25 0.75 -1.25
-1 0.5 1.5 -2.5
0 1 3 -5
1 2 6 -10
2 4 12 -20
3 8 24 -40

The graphs and descriptions of each function are shown below. Note the vertical scales in each.

bold italic f open parentheses bold x close parentheses bold equals bold 2 to the power of bold x bold italic g open parentheses bold x close parentheses bold equals bold 3 bold times bold 2 to the power of bold x bold italic h open parentheses bold x close parentheses bold equals bold short dash bold 5 bold times bold 2 to the power of bold x
A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from 0 to 9. The graph has a curve that rises from the upper left quadrant, along the negative x-axis from above, passing through the marked points at (−3, 0.125), (−2, 0.25), (−1, 0.5), (0, 1), (1, 2), (2, 4), (3, 8), and extends upward into the upper right quadrant. A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from 0 to 45, in increments of 5. The graph has a curve that rises from the upper left quadrant along the negative x-axis from above, passes through the marked points at (−3, 0.375), (−2, 0.75), (−1, 1.5), (0, 3), (1, 6), (2, 12), (3, 24) and extends upward into the upper right quadrant. A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from −80 to 10, in increments of 10. The graph has a curve that descends from the lower left quadrant, along the negative x-axis from below, passing through the marked points at (−3, −0.625), (−2, −1.25), (−1, −2.5), (0, −5), (1, −10), (2, −20), (3, −40) and extends downward into the fourth quadrant.
As x rightwards arrow short dash infinity comma f open parentheses x close parentheses rightwards arrow 0 from above and as x rightwards arrow infinity comma f open parentheses x close parentheses rightwards arrow infinity.

Since f open parentheses x close parentheses equals 2 to the power of x equals 1 times 2 to the power of x comma a equals 1 and the y-intercept is open parentheses 0 comma space 1 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of f open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of f open parentheses x close parentheses is the set of all positive numbers, or using interval notation, open parentheses 0 comma space infinity close parentheses. 		As x rightwards arrow short dash infinity comma g open parentheses x close parentheses rightwards arrow 0 from above and as x rightwards arrow infinity comma g open parentheses x close parentheses rightwards arrow infinity.

Since g open parentheses x close parentheses equals 3 times 2 to the power of x comma a equals 3 and the y-intercept is open parentheses 0 comma space 3 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of g open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of g open parentheses x close parentheses is the set of all positive numbers, or using interval notation, open parentheses 0 comma space infinity close parentheses. 		As x rightwards arrow short dash infinity comma h open parentheses x close parentheses rightwards arrow 0 from below and as x rightwards arrow infinity comma h open parentheses x close parentheses rightwards arrow short dash infinity.

Since h open parentheses x close parentheses equals short dash 5 times 2 to the power of x comma a equals short dash 5 and the y-intercept is open parentheses 0 comma space short dash 5 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of h open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of h open parentheses x close parentheses is the set of all negative numbers, or using interval notation, open parentheses short dash infinity comma space 0 close parentheses.

In summary, the graph of f open parentheses x close parentheses equals a times b to the power of x when b greater than 1 can be generalized into two shapes: one where a less than 0 and one where a greater than 0.

General Shape and Behavior When bold italic a bold less than bold 0 General Shape and Behavior When bold italic a bold greater than bold 0
A graph with an x-axis and a y-axis intersecting at the origin. The graph has a curve that descends along the negative x-axis from below, opens downward in the lower left quadrant, passes through the marked point (0, a) on the negative y-axis near the origin and extends downward into the fourth quadrant. A graph with an x-axis and a y-axis intersecting at the origin. The graph has a curve that rises along the negative x-axis from above, opens upward in the upper left quadrant, passes through the marked point (0, a) on the positive y-axis near the origin and extends upward into the first quadrant.
The graph decreases over its entire domain.
Domain: open parentheses short dash infinity comma space infinity close parentheses
Range: open parentheses short dash infinity comma space 0 close parentheses
Horizontal asymptote: y equals 0 		The graph increases over its entire domain.
Domain: open parentheses short dash infinity comma space infinity close parentheses
Range: open parentheses 0 comma space infinity close parentheses
Horizontal asymptote: y equals 0

try it
Consider the function f open parentheses x close parentheses equals 3 open parentheses 1.25 close parentheses to the power of x.
Is the graph of f   (x  ) increasing or decreasing?
It is increasing since a equals 3 and 1.25 greater than 1.
What are the domain and range of f   (x  )?
The domain is open parentheses short dash infinity comma space infinity close parentheses and the range is open parentheses 0 comma space infinity close parentheses.
What is the y-intercept of the graph of f   (x  )?
The y-intercept is open parentheses 0 comma space 3 close parentheses.
What is the horizontal asymptote of f   (x  )?
Its equation is y equals 0.

Let’s do a similar exploration, but this time when 0 less than b less than 1.

2. Graphing an Exponential Function When 0 < b < 1

Consider three exponential functions: f open parentheses x close parentheses equals open parentheses 1 half close parentheses to the power of x comma g open parentheses x close parentheses equals 4 open parentheses 1 half close parentheses to the power of x comma and h open parentheses x close parentheses equals short dash 3 open parentheses 1 half close parentheses to the power of x.

The tables of values for each function are shown here:

bold italic x bold italic f open parentheses bold x close parentheses bold equals open parentheses bold 1 over bold 2 close parentheses to the power of bold x bold italic g open parentheses bold x close parentheses bold equals bold 4 open parentheses bold 1 over bold 2 close parentheses to the power of bold x bold italic h open parentheses bold x close parentheses bold equals bold short dash bold 3 open parentheses bold 1 over bold 2 close parentheses to the power of bold x
-3 8 32 -24
-2 4 16 -12
-1 2 8 -6
0 1 4 -3
1 0.5 2 -1.5
2 0.25 1 -0.75
3 0.125 0.5 -0.375

The graphs and descriptions of each function are shown below. Note the vertical scales in each.

bold italic f open parentheses bold x close parentheses bold equals open parentheses bold 1 over bold 2 close parentheses to the power of bold x bold italic g open parentheses bold x close parentheses bold equals bold 4 open parentheses bold 1 over bold 2 close parentheses to the power of bold x bold italic h open parentheses bold x close parentheses bold equals bold short dash bold 3 open parentheses bold 1 over bold 2 close parentheses to the power of bold x
A graph with an x-axis ranging from −7 to 7 and a y-axis ranging from 0 to 10. The graph has a curve that descends from the upper left quadrant, passes through the marked points at (−3, 8), (−2, 4), (−1, 2), (0, 1), (1, 0.5), (2, 0.25), (3, 0.125) and extends into the first quadrant along the positive x-axis from above. A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from 0 to 40, in increments of 5. The graph has a curve that descends from the upper left quadrant, passes through the marked points at (−3, 32), (−2, 16), (−1, 8), (0, 4), (1, 2), (2, 1), (3, 0.5) and extends into the upper right quadrant along the positive x-axis from above. A graph with an x-axis ranging from −5 to 5 and a y-axis ranging from −35 to 5, in increments of 5. The graph has a curve that rises from the lower left quadrant, passes through the marked points at (−3, −24), (−2, −12), (−1, −6), (0, −3), (1, −1.5), (2, −0.75), (3, −0.375) and extends upward into the fourth quadrant along the positive x-axis from below.
As x rightwards arrow infinity comma f open parentheses x close parentheses rightwards arrow 0 from above and as x rightwards arrow short dash infinity comma f open parentheses x close parentheses rightwards arrow infinity.

Since f open parentheses x close parentheses equals open parentheses 1 half close parentheses to the power of x equals 1 times open parentheses 1 half close parentheses to the power of x comma a equals 1 and the y-intercept is open parentheses 0 comma space 1 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of f open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of f open parentheses x close parentheses is the set of all positive numbers, or using interval notation, open parentheses 0 comma space infinity close parentheses. 		As x rightwards arrow infinity comma g open parentheses x close parentheses rightwards arrow 0 from above and as x rightwards arrow short dash infinity comma g open parentheses x close parentheses rightwards arrow infinity.

Since g open parentheses x close parentheses equals 4 open parentheses 1 half close parentheses to the power of x comma a equals 4 and the y-intercept is open parentheses 0 comma space 4 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of g open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of g open parentheses x close parentheses is the set of all positive numbers, or using interval notation, open parentheses 0 comma space infinity close parentheses. 		As x rightwards arrow infinity comma h open parentheses x close parentheses rightwards arrow 0 from below and as x rightwards arrow short dash infinity comma h open parentheses x close parentheses rightwards arrow short dash infinity.

Since h open parentheses x close parentheses equals short dash 3 open parentheses 1 half close parentheses to the power of x comma a equals short dash 3 and the y-intercept is open parentheses 0 comma space short dash 3 close parentheses.

The graph has horizontal asymptote y equals 0.

The domain of h open parentheses x close parentheses is the set of all real numbers, or using interval notation, open parentheses short dash infinity comma space infinity close parentheses.

The range of h open parentheses x close parentheses is the set of all negative numbers, or using interval notation, open parentheses short dash infinity comma space 0 close parentheses.

In summary, the graph of f open parentheses x close parentheses equals a times b to the power of x when 0 less than b less than 1 can be generalized into two shapes: one where a less than 0 and one where a greater than 0.

General Shape and Behavior When bold italic a bold less than bold 0 General Shape and Behavior When bold italic a bold greater than bold 0
A graph with an x-axis and a y-axis intersecting at the origin. The graph has a curve that rises upward from the lower left quadrant, opens downward, passes through the marked point (0, a) on the negative y-axis near the origin and extends into the lower right quadrant along the positive x-axis from below. A graph with an x-axis and a y-axis intersecting at the origin. The graph has a curve that descends from the upper left quadrant, passes through the marked point (0, a) on the positive y-axis near the origin and extends into the first quadrant along the positive x-axis from above.
The graph increases over its entire domain.
Domain: open parentheses short dash infinity comma space infinity close parentheses
Range: open parentheses short dash infinity comma space 0 close parentheses
Horizontal asymptote: y equals 0 		The graph decreases over its entire domain.
Domain: open parentheses short dash infinity comma space infinity close parentheses
Range: open parentheses 0 comma space infinity close parentheses
Horizontal asymptote: y equals 0

watch
This video covers the example of graphing f open parentheses x close parentheses equals 1 half open parentheses 3 over 4 close parentheses to the power of short dash x end exponent and describing the graph’s characteristics.

try it
Consider the function f open parentheses x close parentheses equals short dash 5 open parentheses 5 over 6 close parentheses to the power of x.
Is the graph of f   (x  ) increasing or decreasing?
It is increasing since a equals short dash 5 comma which is negative, and 5 over 6 less than 1.
What are the domain and range of f   (x  )?
The domain is open parentheses short dash infinity comma space infinity close parentheses and the range is open parentheses short dash infinity comma space 0 close parentheses.
What is the y-intercept of the graph of f   (x  )?
The y-intercept is open parentheses 0 comma space short dash 5 close parentheses.
What is the horizontal asymptote of f   (x  )?
Its equation is y equals 0.

3. Graphing Exponential Functions With Base e

Recall that a function of the form f open parentheses x close parentheses equals a times e to the power of k x end exponent can be rewritten as f open parentheses x close parentheses equals a times open parentheses e to the power of k close parentheses to the power of x.

Therefore, this is identical to an exponential function of the form f open parentheses x close parentheses equals a times b to the power of x comma where b equals e to the power of k.

Therefore, we can determine the shape of the graph of f open parentheses x close parentheses equals a times e to the power of k x end exponent in a similar manner.

EXAMPLE

Consider the function f open parentheses x close parentheses equals 4 e to the power of short dash 2 x end exponent.

Note that f open parentheses x close parentheses can be written f open parentheses x close parentheses equals 4 open parentheses e to the power of short dash 2 end exponent close parentheses to the power of x. Since e to the power of short dash 2 end exponent almost equal to 0.1353 comma which is less than 1, this function has the form f open parentheses x close parentheses equals a times b to the power of x, where a greater than 0 and 0 less than b less than 1.

Its graph is shown in the figure below. Note that the y-intercept is open parentheses 0 comma space 4 close parentheses comma indicating that a equals 4.

A graph with an x-axis ranging from −4 to 7 and a y-axis ranging from 0 to 8. The graph has a curve that descends from the upper left quadrant close to the point (−0.5, 8), passes through the marked point at (0, 4), and extends along the positive x-axis from above after the point (2, 0.1).

try it
Consider the function f open parentheses x close parentheses equals short dash 2 e to the power of 0.61 x end exponent. This function is equivalent to f open parentheses x close parentheses equals a times b to the power of x.
What are the values of a and b? Round your answers to two decimal places as needed.
Note that f open parentheses 0 close parentheses equals short dash 2 comma which means a equals short dash 2.

Now, rewrite the function as f open parentheses x close parentheses equals short dash 2 open parentheses e to the power of 0.61 end exponent close parentheses to the power of x.

This means that the base, b, is equal to e to the power of 0.61 end exponent comma which is approximately equal to 1.84.

To bring it all together, a equals short dash 2 and b almost equal to 1.84. 
Describe the end behavior of f   (x  ).
To the left, the graph approaches 0 from below and to the right, the graph decreases indefinitely.

summary
In this lesson, you learned that the shape of the graph of an exponential function f open parentheses x close parentheses equals a times b to the power of x is affected by the sign of a and the value of b. The graph of an exponential function when b > 1 and the graph of an exponential function when 0 < b < 1 can each be generalized into two shapes: one where a less than 0 and one where a greater than 0 (a total of four possible general shapes). You also learned that you can graph exponential functions with base e (of the form f open parentheses x close parentheses equals a e to the power of k x end exponent) using this convention, where b equals e to the power of k.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

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