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 Shifts: This is where the graph is shifted to the right or to the left on the rectangular coordinate axis. The following examples discuss the difference between the graph of f(x) and f(x + C).

Example 5: Graph the function tex2html_wrap_inline58 and the graph function tex2html_wrap_inline60 on the same rectangular coordinate system. Answer the following questions about each graph:

1.In what quadrants is the graph of the function located? In what quadrants is the graph of the function tex2html_wrap_inline60 located?

2.What is the x-intercept and the y-intercept on the graph of the function tex2html_wrap_inline58 ? What is the x-intercept and the y-intercept on the graph of the function tex2html_wrap_inline60 ?

3.Find the point (2, f(2)) on the graph of tex2html_wrap_inline58 and find (2, g(2)) on the graph of tex2html_wrap_inline60 . What do these two points have in common?

4.Describe the relationship between the two graphs.

5.How would you shift (move) the graph of tex2html_wrap_inline58 so that it is superimposed on the graph of tex2html_wrap_inline60 ? After you move the graph, where would the point (0, 1) on . 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. 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. This is because .

The graph of g(x) crosses the y-axis at 0.00673794699909 (just above the x-axis). This is because .

3.The point tex2html_wrap_inline84 , rounded to (2, 7.4) for graphing purposes, is located on the graph of tex2html_wrap_inline58 .

The point tex2html_wrap_inline88 , rounded to (2, 0) for graphing purposes, is located on the graph of tex2html_wrap_inline60 , just above the x-axis.

4.Both graphs have the same shape. The graph of tex2html_wrap_inline60 is nothing more than the graph of tex2html_wrap_inline58 shifted to the right 6 units.

5.Shift (move) the graph of tex2html_wrap_inline58 to the right 6 units. Every point on the graph of tex2html_wrap_inline58 is thus moved to the right 6 units. The point (0, 1) is shifted to the right 6 units to or tex2html_wrap_inline102 .

6.The only difference in the two equations is that the exponent in the g(x) equation is 6 units less than the exponent in the f(x) equation.

Example 6: Graph the function and the function 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_inline58 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 tex2html_wrap_inline58 ? What is the x-intercept and the y-intercept on the graph of the function tex2html_wrap_inline106 ?

3.Find the point (2, f(2)) on the graph of tex2html_wrap_inline58 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 tex2html_wrap_inline58 so that it is superimposed on the graph of tex2html_wrap_inline106 . Where would the point (0, 1) on the graph of tex2html_wrap_inline58 wind up on after the move?

6.Describe the difference between the two equations.

1.The graph of tex2html_wrap_inline58 is located in quadrants I and II. The graph of tex2html_wrap_inline106 is also located in quadrants I and II. This means that both functions values will always be positive.

2.Neither graphs cross the x-axis; therefore, neither graph has an x-intercept.

The graph of tex2html_wrap_inline58 crosses the y-axis at 1, and the graph of tex2html_wrap_inline106

crosses the y-axis at 54.5981500331, rounded to (0, 54.6) for graphing purposes, because tex2html_wrap_inline132 .

3.The point tex2html_wrap_inline84 , rounded to (2, 7.4) for graphing purposes, is located on the graph of . The point tex2html_wrap_inline138 , rounded to (2, 403.4) for graphing purposes, is located on the graph of tex2html_wrap_inline106 .

4.Both graphs have the same shape. The graph of tex2html_wrap_inline106 is nothing more than the graph of tex2html_wrap_inline58 shifted to the left 4 units.

5.Shift (move) the graph of to the left 4 units. It will then be superimposed on the graph of . When you move the graph of tex2html_wrap_inline58 to the left 4 units, every point on the graph of tex2html_wrap_inline58 is shifted to the left 4 units. Therefore, the point (0, 1) will be shifted left to or .

6.The only difference in the two equations is in their exponents. The exponent in the g(x) equation is 4 units greater than the exponent in the f(x) equation.

Hint: Many students have a hard time determining whether the shift is to the left or to the right. One easy way to determine the answer is to set the exponent equal to zero and solve. If the answer is negative, the shift is to the left. If the answer is positive, the shift is to the right.

For example, in the equation tex2html_wrap_inline158 , the shift would be to the right because x-10=0 when x = 10. In the equation , the shift would be to the left because x+10=0 when x = -10.

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

[Exponential Rules] [Logarithms]

[Algebra] [Trigonometry ] [Complex Variables]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Nancy Marcus

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour