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.

Vertical Shifts: A vertical shift takes place when a function is shifted up or down. The following examples look at the difference between the graph of f(x) and the graph of f(x) + C.

Example 3: Graph the function tex2html_wrap_inline69 and graph the function tex2html_wrap_inline71 on the same rectangular coordinate system. Answer the following questions about each graph:

1.In what quadrants is the graph of the function tex2html_wrap_inline69 located? In what quadrants is the graph of the function tex2html_wrap_inline71 located?

2.What is the x-intercept and the y-intercept of the graph of the function tex2html_wrap_inline69 ? What is the x-intercept and the y-intercept of the graph of the function tex2html_wrap_inline71 ?

3.Find the point (2, f(2)) on the graph of tex2html_wrap_inline69 and find (2, g(2)) on the graph of tex2html_wrap_inline71 . What do these two points have in common? What is the difference between the two points.

4.Describe the relationship between the two graphs.

5.How would you move the graph of tex2html_wrap_inline69 so that it is superimposed on the graph of tex2html_wrap_inline71 ? After you move the graph, where would the point (0, 1) on tex2html_wrap_inline69 be located?

6.Describe the difference between the two equations.

1.You can see that the both graphs are located in quadrants I and II. Therefore, both function values will always be positive.

2.You can see that from the graph that neither of the graphs crosses the x-axis; therefore, neither of the graphs has an x-intercept.

Note that the graph of f(x) crosses the y-axis at 1 because tex2html_wrap_inline91 ; the graph of g(x) crosses the y-axis at 4 because .

3.The point tex2html_wrap_inline95 , rounded to (2, 7.4) for graphing purposes, is located on the graph of tex2html_wrap_inline69 .

The point , rounded to (2, 10.4) for graphing purposes, is located on the graph of tex2html_wrap_inline71 . For each x-coordinate, the y-coordinates differ by 3.

4.Both graphs have the same shape. The graph of tex2html_wrap_inline71 is nothing more than the graph of tex2html_wrap_inline69 shifted up three units.

5.Shift (move) the graph of tex2html_wrap_inline69 up 3 units. Every point on the graph of tex2html_wrap_inline69 would be moved up 3 units. Therefore, the point (0, 1) would wind up at tex2html_wrap_inline111 or tex2html_wrap_inline113 after the move.

6.The only difference between the two equations is the + 3. The equations could be rewritten as follows: tex2html_wrap_inline115 since tex2html_wrap_inline69 . This means that for every value of x, the function g(x) will always be 3 units larger than the function f(x).

Example 4: Graph the function tex2html_wrap_inline69 and the function tex2html_wrap_inline121 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_inline69 located? In what quadrants is the graph of the function tex2html_wrap_inline121 located?

2.What is the x-intercept and the y-intercept of the graph of the function tex2html_wrap_inline69 ? What is the x-intercept and the y-intercept of the graph of the function tex2html_wrap_inline121 ?

3.Find the point (2, f(2)) on the graph of tex2html_wrap_inline69 and find (2, g(2)) on the graph of tex2html_wrap_inline121 . What do these two points have in common?

4.Describe the relationship between the two graphs.

5.Describe how you would shift (move) the graph of tex2html_wrap_inline69 so that it is superimposed on the graph of tex2html_wrap_inline121 . Where would the point (0, 1) on the graph of tex2html_wrap_inline69 be located after the move?

6.What is the difference between the two equations?

1.The graph of tex2html_wrap_inline69 is located in quadrants I and II. The graph of tex2html_wrap_inline121 is located in quadrants I, III, and IV. This means that f(x) will always be positive, and g(x) can be positive, negative, and zero.

2.The graph of tex2html_wrap_inline69 does not cross the x-axis. There is no x value such that tex2html_wrap_inline69 will ever equal 0.

The graph of tex2html_wrap_inline121 crosses the x-axis at 1.60943791243, rounded to 1.6 for graphing purposes:

Let and solve for x. Add 5 to both sides of the equation tex2html_wrap_inline153 and we have tex2html_wrap_inline155 . Take the natural logarithms of both sides of tex2html_wrap_inline155 : or

3.The point tex2html_wrap_inline95 , rounded to (2, 7.4) for graphing purposes, is located on the graph of tex2html_wrap_inline69 .

4.The point , rounded to (2, 2.4) for graphing purposes, is located on the graph of tex2html_wrap_inline121 . For each x-coordinate, the y-coordinates differ by 5.

5.Both graphs have the same shape. The graph of tex2html_wrap_inline121 is nothing more than the graph of tex2html_wrap_inline69 shifted down 5 units.

6.Shift (move) the graph of tex2html_wrap_inline69 down five units so that it is superimposed on the graph of tex2html_wrap_inline121 . When we move the graph of tex2html_wrap_inline69

down 5 units so that it is superimposed on the graph of . tex2html_wrap_inline121 , every point on the graph of tex2html_wrap_inline69 is shifted down 5 units. Therefore, the point (0, 1) will be shifted down to tex2html_wrap_inline185 or tex2html_wrap_inline187 .

7.The difference between the two equations is the - 5 because tex2html_wrap_inline69 . The only substitution we made was to replace with f(x) in the equation tex2html_wrap_inline121 . If we knew that the point (c, d) is located on the graph of f(x), we can conclude that the point (c, d - 5) is located on the graph of g(x). For every value of x, the function g(x) will always be 5 units less than the function f(x).

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

[Exponential Rules] [Logarithms]

[Algebra] [Trigonometry ] [Complex Variables]

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Author: Nancy Marcus

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