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PARACHUTE REVIEW

Math Topic: exponential functions
Key Terms: growth, decay, and parameters 

The Math behind the Observations:

In the parachute applet the function, , is an example of a class of functions known as exponential functions. From experience in pre-calculus and calculus you may recall that changes in the parameters a, b, and c can affect whether the exponential function is increasing (growth) or decreasing (decay); whether the function is stretched vertically or horizontally, etc. We now explore some of these effects using various static pictures of the applet.

I. a=1,c=0; explore changes in b ( growth versus decay ):

Let b>0:� In the screen capture below b = 2 and thus our equation is ; the function is increasing, indicating exponential growth.

Let b=0: In this next screen capture, b=0 and thus, our function is now , for a constant velocity.

Let b>0: In the screen capture below b = -2 and our resulting function, , is decreasing indicating exponential decay.

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II. a=1,b=2 and changes in c: ( vertical shifts)

In the following 3 screen captures we see that changes in the values of c result in vertical shifts of the basic exponential function. This applies to growth as well as decay functions. However, we explore just the growth cases ( i.e. where b>0).

��������� a=1,b=2 and c>0:

a=1,b=2,c=0:

a=1,b=2, c<0:

III. Finally, we explore the effects of changes in a: (vertical stretching)

In the following three screen captures we set c=0, b=2, and let a take on different values.

b=2,a=3:

b=2,c=0, and a=1:

b=2,a=0:

b=2,a=-2:

b=2, a =-3:

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USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil