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.

Horizontal and vertical shifts: The next examples discuss the difference between the graph of f(x) and the graph of f(x + A) + B.

Example 9: Graph the function tex2html_wrap_inline59 and the function tex2html_wrap_inline61 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_inline59 located? In what quadrants is the graph of the function tex2html_wrap_inline61 located?

2.What is the x-intercept and the y-intercept on the graph of the function tex2html_wrap_inline59 ? What is the x-intercept and the y-intercept on the graph of the function tex2html_wrap_inline61 ?

3.Find the point (2, f(2)) on the graph of tex2html_wrap_inline59 and find (2, g(2)) on the graph of tex2html_wrap_inline61 . What do these two points have in common?

4.Describe the relationship between the two graphs.

5.How would you physically shift (move) the graph of tex2html_wrap_inline59 so that it would be superimposed on the graph of tex2html_wrap_inline61 ? After you move the graph, where would the point (0, 1) be located?

6.Describe what you can tell about the relationship between the graphs from just their equations.

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

2.You can see that neither of the graphs cross 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 tex2html_wrap_inline79 . The graph of g(x) crosses the y-axis at 5.00247875218 because tex2html_wrap_inline81 .

3.The point tex2html_wrap_inline83 , rounded to (2, 7.4) for graphing purposes, is located on the graph of tex2html_wrap_inline59 . The point , rounded to (2, 5) for graphing purposes, is located on the graph of tex2html_wrap_inline61 .

4.Both graphs have the same shape. It appears from the graph, that the graph of tex2html_wrap_inline61 is a result of shifting the graph of tex2html_wrap_inline59 to the right and upward.

5.After we move the graph of tex2html_wrap_inline59 to the right 6 units and up 5 units, it is superimposed on the graph of tex2html_wrap_inline61 . The point (0, 1) on the graph of tex2html_wrap_inline59 would first be shifted to the right 6 units and up 5 units to (0 + 6, 1 + 5) or (6, 6).

6.Since the exponents differ in each equation by a constant, there will be a horizontal shift. By setting x - 6 to 0, you can tell that the shift is to the right 6 units. You can also tell that the equations differ by a constant. This means there will also be vertical shift of 5 units up

Example 10: Graph the function tex2html_wrap_inline59 and the function tex2html_wrap_inline103 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_inline59 located? In what quadrants is the graph of the function tex2html_wrap_inline103 located?

2.What is the x-intercept and the y-intercept on the graph of the function tex2html_wrap_inline59 ? What is the x-intercept and the y-intercept on the graph of the function tex2html_wrap_inline103 ?

3.Find the point (2, f(2)) on the graph of tex2html_wrap_inline59 and find (2, g(2)) on the graph of tex2html_wrap_inline103 . What do these two points have in common?

4.Describe the relationship between the two graphs.

5-.Describe how you would physically move (shift) the graph of tex2html_wrap_inline59 so that it would be superimposed on the graph of tex2html_wrap_inline103 . Where would the point (0, 1) on the graph of tex2html_wrap_inline59 wind up on after the move?

1.The graph of tex2html_wrap_inline59 is located in quadrants I and II. The graph of tex2html_wrap_inline103 is also located in quadrants I, III, and IV.

2.The graph of tex2html_wrap_inline59 does not cross the x-axis because there is no value of x that would cause tex2html_wrap_inline129 to equal zero.

The graph of tex2html_wrap_inline103 crosses the x-axis at -9.09861228867 because that is the solution when we set tex2html_wrap_inline133 :

tex2html_wrap_inline135

3.The graph of tex2html_wrap_inline59 crosses the y-axis at 1, and the graph of crosses the y-axis at -2.99966453737 because tex2html_wrap_inline139 .

4.The point tex2html_wrap_inline83 , rounded to (2, 7.4) for graphing purposes, is located on the graph of tex2html_wrap_inline59 .

The point tex2html_wrap_inline145 , rounded to (2, -3) for graphing purposes. is located on the graph of tex2html_wrap_inline103 .

5.Both graphs have the same shape. The graph of appears to the right and above the graph of tex2html_wrap_inline59 .

Whenever the exponents differ by a constant, there is a horizontal shift in the graphs. Whenever the constant terms differ in the equations, there is a vertical shift in the graphs.

From the equation, you can see that the horizontal shift is to the right 8 units, and the vertical shift is down 3 units. It does no make any difference what you do first.

Therefore, shift the graph of tex2html_wrap_inline59 to the right 8 units and down 3 units. The point (0, 1) would be moved to the right 8 units and down 3 units to (0 + 8, 1 - 3) or (8, - 2).

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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