# 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.

Stretch and Shrink: The following examples discuss the difference between the graph of f(x) and the graph of Cf(x):

Example 13: Graph the function and the function on the same rectangular coordinate system. and answer the following questions about each graph:

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

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

3.Find the point (2, f(2)) on the graph of and find (2, g(2)) on the graph of . What do these two points have in common?

4.Describe the relationship between the two graphs.

5.How would you moved the graph of so that it would be superimposed on the graph of ? When you moved the graph, where would the point (0, 1) on be after the move?

1.You can see that the both graphs are located in quadrants I and II.

2.You can see that neither of the graphs crosses the x-axis; therefore, neither of the graphs has an x-intercept. Notice that the graph of f(x) crosses the y-axis at 1 because . The graph of g(x) crosses the y-axis at 3 because .

3.The point is located on the graph of . The point is located on the graph of .

4.Both graphs have the same shape. It appears that the the graph of is a result of stretching the graph of . For example, for every value of x the value of g(x) is 3 times larger than the value of f(x).

The point (0, 1) on the graph of would first be moved up 3 units to (0,3). If (a, b) were a point on the graph of f(x), then the point (a, 3b) would be a point on the graph of g(x).

Example 14:

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

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

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

3.Find the point (2, f(2)) on the graph of and find (2, g(2)) on the graph of . What do these two points have in common?

4.Describe the relationship between the two graphs.

5.Describe how you would move the graph of moved so that it would be superimposed on the graph of . . Where would the point (0, 1) on the graph of wind up on after the move?

1.Both graphs are located in quadrants I and II.

Neither graph crosses the x-axis; therefore, neither graph has an x-intercept. This means that neither equation has a real solution.

2.The graph of crosses the y-axis at 1, and the graph of crosses the y-axis at because .

3.The point is located on the graph of . The point is located on the graph of .

Both graphs have the same shape. The graph of appears to shrink. It is like you just let the graph kind of slip down a little.

4.Notice that the graph of g(x) is below the graph of f(x). This is because for every value of x, the value of g(x) is less than the value of f(x) by one-third.

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

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