# Calculator Investigation of Exponential Function Graphs

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```					                            Exponential Function Investigation
Go to the site http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=4 where you
will find an applet which allows you to explore the graphs of exponential functions.

Explore the graphs of exponential functions of the form y = ax by setting the 'M' and 'k' sliders
to 1; (M=1) and (k=1).

Set the ‘a’ slider to 1 (a=1).
1. The resulting graph is that of y = 1. What type of graph is this?

2. Why is the graph of y = 1 not that of an exponential function?

3. For a function of the form y = ax to be exponential, what are the restrictions on ‘a’?
In other words, what value(s) can ‘a’ not have?

Grab the ‘a’ slider and slide it left and right.

1. What is the y-intercept of the graph? Why? (Hint: Recall that x=0 when the graph crosses the y-axis.)

2. Does the graph ever cross the x-axis? In other words, does the graph have an x-intercept?
Why or why not?
(Hint: Keep in mind that when a graph crosses the x-axis, y = 0 giving the equation 0 = ax.)

3. What two quadrants does the graph never enter?
What does this suggest about the range (values of y)?

x       y = ax                  Use your calculator to complete the given table of values for the
exponential function y = 2x.
3       23 =
In the table at the left, what do you notice about the value of y?
0.5      20.5 =
0       20 =
-0.5      2-0.5 =
-3       2-3 =
-5       2-5 =

Adjust the ‘a’ slider but restrict its setting to values greater than 1.

1. As the value of x gets larger and larger, what happens to the value of y?

2. As the value x gets smaller and smaller (closer to negative infinity), what value does y approach?

This would suggest that a horizontal asymptote occurs at y = ____.
(Hint: Recall your study of limits of infinity in lesson 6 of Polynomial and Rational Function Unit)
Adjust the ‘a’ slider but restrict its setting to positive values less than 1.

1. As the value of x gets smaller and smaller, what happens to the value of y?

2. As the value x gets larger and larger (closer to positive infinity), what value does y approach.

This would suggest that a horizontal asymptote occurs at y = ____.
(Hint: Recall your study of limits of infinity in lesson 6 of Polynomial and Rational Function Unit)

Conclusion
The graph of an exponential function of the form y = ax, where a>0 and a      ____, is a smooth curve with a
y-intercept of _____ and a horizontal asymptote of y = _____. The functions domain is _______________
and its range is ____________________________________________________.

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