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Transformations of Graphs of Exponential Functions

Author: Sophia
PDF Version

what's covered
In this lesson, you will use transformations to graph exponential functions and examine their properties. Specifically, this lesson will cover:

Table of Contents

1. Horizontal Translations of Exponential Functions

Given the graph of y equals f open parentheses x close parentheses comma recall the following for a positive value of k  :

  • The graph of y equals f open parentheses x minus k close parentheses shifts the graph of f open parentheses x close parentheses to the right k units.
  • The graph of y equals f open parentheses x plus k close parentheses shifts the graph of f open parentheses x close parentheses to the left k units.

EXAMPLE

Consider the functions f open parentheses x close parentheses equals 3 to the power of x and g open parentheses x close parentheses equals 3 to the power of x minus 2 end exponent. Since g open parentheses x close parentheses equals f open parentheses x minus 2 close parentheses comma it follows that the graph of g open parentheses x close parentheses is obtained by shifting the graph of f open parentheses x close parentheses to the right by 2 units. Their graphs are shown below.



The domain of f is open parentheses short dash infinity comma space infinity close parentheses and the range of f is open parentheses 0 comma space infinity close parentheses. It turns out that the domain of g is open parentheses short dash infinity comma space infinity close parentheses and the range of g is open parentheses 0 comma space infinity close parentheses. In general, shifting the graph of a function horizontally does not alter the domain or range of f.

try it
Consider the functions f open parentheses x close parentheses equals open parentheses 3 over 4 close parentheses to the power of x and g open parentheses x close parentheses equals open parentheses 3 over 4 close parentheses to the power of x plus 3 end exponent.
Explain how the graph of g is related to the graph of f.
The graph of g is obtained by shifting the graph of f to the left 3 units.

2. Vertical Translations of Exponential Functions

Given the graph of y equals f open parentheses x close parentheses comma recall the following for a positive value of k  :

  • The graph of y equals f open parentheses x close parentheses plus k shifts the graph of f open parentheses x close parentheses up k units.
  • The graph of y equals f open parentheses x close parentheses minus k shifts the graph of f open parentheses x close parentheses down k units.

EXAMPLE

Consider the functions f open parentheses x close parentheses equals open parentheses 2 over 3 close parentheses to the power of x and g open parentheses x close parentheses equals open parentheses 2 over 3 close parentheses to the power of x plus 4. Since g open parentheses x close parentheses equals f open parentheses x close parentheses plus 4 comma it follows that the graph of g open parentheses x close parentheses is obtained by shifting the graph of f open parentheses x close parentheses up 4 units. Their graphs are shown below.



The domains of both f and g are open parentheses short dash infinity comma space infinity close parentheses. However, take a closer look at the ranges of f and g  :

  • The range of f is open parentheses 0 comma space infinity close parentheses.
  • The range of g is open parentheses 4 comma space infinity close parentheses.
When the graph shifts vertically, the range is shifted as well.

Notice also that the horizontal asymptote of f is y equals 0 comma while the horizontal asymptote of g is y equals 4.

big idea
When the graph of the exponential function f open parentheses x close parentheses equals b to the power of x is shifted up k units, its range becomes open parentheses k comma space infinity close parentheses and the horizontal asymptote becomes y equals k.

When the graph of the exponential function f open parentheses x close parentheses equals b to the power of x is shifted down k units, its range becomes open parentheses short dash k comma space infinity close parentheses and the horizontal asymptote becomes y equals short dash k.

try it
Consider the functions f open parentheses x close parentheses equals open parentheses 1.2 close parentheses to the power of x and g open parentheses x close parentheses equals open parentheses 1.2 close parentheses to the power of x minus 3.
Explain how the graph of g is related to the graph of f.
The graph of g is obtained by shifting the graph of f down 3 units.
What is the range of g  ?
The range is open parentheses short dash 3 comma space infinity close parentheses.
What is the equation of the horizontal asymptote of g  ?
The equation is y equals short dash 3.

3. Vertical Compressions and Stretches of Exponential Functions

Given the graph of y equals f open parentheses x close parentheses and a positive number a colon

  • The graph of y equals a times f open parentheses x close parentheses is vertical compression of the graph of y equals f open parentheses x close parentheses if 0 less than a less than 1.
  • The graph of y equals a times f open parentheses x close parentheses is a vertical stretch of the graph of y equals f open parentheses x close parentheses if a greater than 1.

EXAMPLE

Consider the functions f open parentheses x close parentheses equals open parentheses 2 close parentheses to the power of x comma g open parentheses x close parentheses equals 3 open parentheses 2 close parentheses to the power of x comma and h open parentheses x close parentheses equals 1 fourth open parentheses 2 close parentheses to the power of x. Notice that functions g and h are constant multiples of function f. The graphs of g and h are each shown with the graph of f below.

bold italic g open parentheses bold x close parentheses bold equals bold 3 open parentheses bold 2 close parentheses to the power of bold x bold italic h open parentheses bold x close parentheses bold equals bold 1 over bold 4 open parentheses bold 2 close parentheses to the power of bold x

The functions f, g, and h all have domain open parentheses short dash infinity comma space infinity close parentheses and range open parentheses 0 comma space infinity close parentheses.

Also, each function has horizontal asymptote y equals 0.

4. Reflections of Exponential Functions

Given the graph of y equals f open parentheses x close parentheses colon

  • The graph of y equals short dash f open parentheses x close parentheses is the reflection of the graph of y equals f open parentheses x close parentheses across the x-axis.
  • The graph of y equals f open parentheses short dash x close parentheses is the reflection of the graph of y equals f open parentheses x close parentheses across the y-axis.

EXAMPLE

Consider the functions f open parentheses x close parentheses equals 3 open parentheses 2 close parentheses to the power of x and g open parentheses x close parentheses equals short dash 3 open parentheses 2 close parentheses to the power of x. Since g open parentheses x close parentheses equals short dash f open parentheses x close parentheses comma it follows that the graph of g open parentheses x close parentheses is obtained by reflecting the graph of f open parentheses x close parentheses around the x-axis. Their graphs are shown below.



The domains of both f and g are open parentheses short dash infinity comma space infinity close parentheses. However, take a closer look at the ranges of f and g:

  • The range of f is open parentheses 0 comma space infinity close parentheses.
  • The range of g is open parentheses short dash infinity comma space 0 close parentheses.
Note: since these functions both have the form f open parentheses x close parentheses equals a times b to the power of x comma the aspects of these functions were discussed earlier.

Here is an example which illustrates a reflection across the y-axis.

EXAMPLE

Consider the functions f open parentheses x close parentheses equals 3 open parentheses 2 close parentheses to the power of x and h open parentheses x close parentheses equals 3 open parentheses 2 close parentheses to the power of short dash x end exponent. Since h open parentheses x close parentheses equals f open parentheses short dash x close parentheses comma it follows that the graph of h open parentheses x close parentheses is obtained by reflecting the graph of f open parentheses x close parentheses around the y-axis. Their graphs are shown below.



The domains of both f and h are open parentheses short dash infinity comma space infinity close parentheses comma and the ranges of both f and h are open parentheses 0 comma space infinity close parentheses.

Note: the equation for h open parentheses x close parentheses can be rewritten using properties of exponents:

h open parentheses x close parentheses equals 3 open parentheses 2 close parentheses to the power of short dash x end exponent equals 3 open parentheses 2 to the power of short dash 1 end exponent close parentheses to the power of x equals 3 open parentheses 1 half close parentheses to the power of x

Thus, the equation of the exponential function obtained from reflecting across the y-axis has the same y-intercept, but its base is the reciprocal of the original base.

5. Combining Transformations

Similar to what we did earlier in the course, a series of transformations can be applied to one function to create another function.

EXAMPLE

Consider the function g open parentheses x close parentheses equals 5 minus 2 e to the power of x minus 1 end exponent in comparison with f open parentheses x close parentheses equals e to the power of x.

Notice the following transformations starting with the graph of f open parentheses x close parentheses equals e to the power of x colon

  • x was replaced with x minus 1 comma which shifts the graph of f to the right 1 unit.
  • The “-2” multiplied to the exponential will cause the graph to reflect around the x-axis, as well as stretch vertically by a factor of 2.
  • The constant term “5” causes the graph to shift upward by 5 units. This also means that the horizontal asymptote is y equals 5.
The graphs of f and g are shown below:



Note that the range of g is open parentheses short dash infinity comma space 5 close parentheses.

watch
In this video, we describe the sequence of transformations that are applied to f open parentheses x close parentheses equals open parentheses 2 over 3 close parentheses to the power of x to obtain the graph g open parentheses x close parentheses equals short dash 7 plus 6 open parentheses 2 over 3 close parentheses to the power of short dash x minus 5 end exponent.

try it
Consider the functions f open parentheses x close parentheses equals open parentheses 1.4 close parentheses to the power of x and g open parentheses x close parentheses equals 2 plus 0.75 open parentheses 1.4 close parentheses to the power of x plus 3 end exponent.
Describe the sequence of transformations that are applied to f to obtain the graph of g.
Since there is a coefficient of 0.75, the graph of f open parentheses x close parentheses gets vertically compressed by a factor of 0.75.

Since the exponent is x plus 3 instead of x, this means that the graph of f open parentheses x close parentheses will also shift 3 units to the left.

Since there is a constant 2 added to the function, this means that the graph of f open parentheses x close parentheses will shift upward 2 units.

These transformations can be applied to f to obtain the graph of g.

summary
In this lesson, you learned that exponential functions can undergo several transformations: horizontal and vertical translations, vertical compressions and stretches, and reflections around the x- and y-axes. Some transformations, such as vertical translations and reflections across the x-axis, produce a function with a different range than the original function. You also learned that you can combine transformations by applying a series of transformations to one function to create another function.

SOURCE: THIS WORK IS ADAPTED FROM PRECALCULUS BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/BOOKS/PRECALCULUS/PAGES/1-INTRODUCTION-TO-FUNCTIONS

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