GRAPHS OF EXPONENTIAL FUNCTIONS

GRAPHS OF EXPONENTIAL FUNCTIONS

By Nancy Marcus

In this section we will illustrate, interpret, and discuss the graphs of exponential functions. We will also illustrate how you can use graphs to HELP you solve exponential problems and check your answers.

Reflection across the y-axis: The graph of f(x) versus the graph of f(-x)

Example 2:

Graph the function tex2html_wrap_inline43 and the function tex2html_wrap_inline45 on the same rectangular coordinate system and answer the following questions about each graph:

1.In what quadrants is the graph of the function tex2html_wrap_inline43 located? In what quadrants is the graph of the function tex2html_wrap_inline45 located?

2.What is the x-intercept and the y-intercept on the graph of the function tex2html_wrap_inline43 ? What is the x-intercept and the y-intercept on the graph of the function tex2html_wrap_inline45 ?

3.Find the point (2, f(2)) on the graph of tex2html_wrap_inline43 and find (-2, g(-2)) on the graph of tex2html_wrap_inline45 .

4.What do these two points have in common?

5.Describe the relationship between the two graphs.

6.Describe the difference in the two equations.

7.How would you physically move the graph of tex2html_wrap_inline43 so that it is superimposed on the graph of tex2html_wrap_inline45 ?

Where would the point tex2html_wrap_inline63 , rounded to (1, 2.7) for graphing purposes, be located after such a move?

8.What is the difference between the two equations?

1.You can see that the both graphs are located in quadrants I and II. Therefore, the value of both functions is positive.

2.You can see that neither of the graphs cross the x-axis; therefore, neither of the graphs has an x-intercept. You can see that both of the graphs cross the y-axis at 1; therefore, both graphs have a y-intercept of 1.

3.The point tex2html_wrap_inline65 , rounded to (2, 7.4) for graphing purposes, is located on the graph of tex2html_wrap_inline43 .

The point tex2html_wrap_inline69 , rounded to (-2, 7.4) for graphing purposes, is located on the graph of tex2html_wrap_inline45 .

4.Note that both points have the same y-coordinate and their x-coordinates differ by a minus sign.

5.The graphs are mirror images of each other over the y-axis. This is another way of saying that the graphs are symmetric to each other with respect to the y-axis. The shapes are the same. The graph of tex2html_wrap_inline45 is a reflection over the y-axis of the graph of tex2html_wrap_inline43 . This is also the definition of an even function.

6.Since tex2html_wrap_inline43 , substitute f(x) for tex2html_wrap_inline81 in the equation tex2html_wrap_inline45 . If we rewrite tex2html_wrap_inline45 as tex2html_wrap_inline87 before the substitution, we have tex2html_wrap_inline89 . For every x, the function values are reciprocals of one another. For x values that differ by a minus sign, the function values are the same.

7.Mentally fold the graph of tex2html_wrap_inline43 over the y-axis so that it is superimposed on the graph of tex2html_wrap_inline45 . In the move, every point is moved to the left twice it's distance from the y-axis. In other words, if a point (a, b) is located on tex2html_wrap_inline43 in quadrant I, the point would be a units from the y-axis. When you fold the graph of tex2html_wrap_inline43 over the y-axis, the point (a, b) would be located in quadrant II at (- a, b). The distance between + a and - a is 2a.

After the move, the point (2, 7.4) on the graph of tex2html_wrap_inline43 would be located at (- 2, 7.4) on the graph of tex2html_wrap_inline45

If you would like to review another example, click on Example.

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Author: Nancy Marcus

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