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 x-axis: The graph of f(x) versus the graph of -f(x).

Example 1: Graph the function tex2html_wrap_inline43 and graph the function tex2html_wrap_inline45 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_inline43 located?

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

3.What is the x-intercept and the y-intercept of the graph of the function tex2html_wrap_inline43 ?

4.What is the x-intercept and the y-intercept of the graph of the function tex2html_wrap_inline45 ?

5.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 .

6.What do these two points have in common?

7.Describe the relationship between the two graphs.

8.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 (0, 1) on tex2html_wrap_inline43 be located after such a move?

9.What do the two equations have in common?

1.You can see that the graph of the function tex2html_wrap_inline43 is located in quadrants I and II above the x-axis.

2.You can see that the graph of the function tex2html_wrap_inline45 is located in quadrants III and IV below the x-axis.

3.Note that the graph of tex2html_wrap_inline43 does not cross the x-axis anywhere, and crosses the y-axis at 1.

4.Note that the graph of tex2html_wrap_inline45 does not cross the x-axis anywhere, and crosses the y-axis at - 1.

5.The point tex2html_wrap_inline73 , rounded to tex2html_wrap_inline75 for graphing purposes, is located on the graph of tex2html_wrap_inline43 . The point tex2html_wrap_inline79 , rounded to tex2html_wrap_inline81 for graphing purposes, is located on the graph of tex2html_wrap_inline45 .

Note that both points have the same x-coordinate and the y-coordinate's differ by a minus sign.

6.Mentally fold the coordinate system at the x-axis. Note that when you fold the coordinate system at the x-axis, the graph above the x-axis, tex2html_wrap_inline43 , is superimposed on the graph below the x-axis, tex2html_wrap_inline45 . This means both graphs are symmetric to each other with respect to the x-axis.

What exactly does that mean? Well for one thing, it means if there is a point (a, b) on the graph of tex2html_wrap_inline43 , we know that the point (a, - b) is located on the graph of tex2html_wrap_inline45 . The vertical distance between the two points is 2b.

7.The shapes are the same. The graph of tex2html_wrap_inline45 is a reflection over the x-axis of the graph of tex2html_wrap_inline43 .

Mentally fold the graph of tex2html_wrap_inline43 over the x-axis so that it is superimposed on the graph of tex2html_wrap_inline45 . Every point on the graph of tex2html_wrap_inline43 would be shifted down twice it's distance from the x-axis. For example, the point (a, 8) is located 8 units up from the x-axis. If we shifted the point (a, 8) down 16 units, it would wind up at (a,- 8), 8 units below the x-axis.

If the point (b, 11) is located on the graph of tex2html_wrap_inline43 , it would be shifted down 22 units to (b, -11) when the graph is reflected over the x-axis.

8.The point (0, 1) on the graph of tex2html_wrap_inline43 would wind up at (0, - 1) after the graph was reflected over the x-axis.

9.Since tex2html_wrap_inline43 , substitute f(x) for tex2html_wrap_inline111 in the equation tex2html_wrap_inline45 to have g(x)=-f(x). This means that for every value of x, the function values will differ by a minus sign.

If you would like to review any of the following, click on Example:

Reflection over the y-axis: The graph of f(x) versus the graph of f(-x)Example.

Vertical shifts: The graph of f(x) versus the graph of f(x) + C Example.

Horizontal shifts: The graph of f(x) versus the graph of f(x + C)Example.

Combination horizontal shift and reflection across the y-axis: The graph of f(x) versus the graph of f(- x + C) or f(C - x) Example.

Combination horizontal and vertical shifts: The graph of f(x) versus the graph of f(x + A) + B Example.

Combination horizontal and vertical shifts and reflections: The graph of f(x) versus the graphs of -f(x) + C. f(-x) + C, f(x + C) + D Example.

Stretch and Shrink: The graph of f(x) versus the graph of C(x) Example.

Stretch and Shrink: The graph of f(x) versus the graph of f(Cx) Example.

Combination of stretch, shrink, reflection, horizontal, and vertical shifts: Example.

Solving an equation from a graph: Example.

[Exponential Rules] [Logarithms]

[Algebra] [Trigonometry ] [Complex Variables]

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Author: Nancy Marcus

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