We define an exponential function as a “function that has a variable in the exponent."

 

To get a first-hand understanding of how these functions behave, let's sketch the graphs of several exponential functions on our calculator and examine their x- and y-intercepts.

 

Use the following window:

 

x_min = -5, x_max = 5
y_min = -3, y_max = 10

 

Remember how you locate x- and y-intercepts on a graph in the calculator:

 

x-intercept: zeros, roots
y-intercept: x = 0 → y = ?

 

Recall that to sketch the first graph of , you would input the function in the “y =” screen as: y = 2^x. If you need further practice using these commands, review the lesson on Using the Calculate Menu-Part I.

 

To test your understanding, answer these questions:

Now, let’s sketch a new function, . The number “e” is the base of a natural logarithm and can be found as a second function to the natural log key, , on your calculator. Therefore, “e” is referred to as the natural base.

The actual definition of  “e” is the value of as (or as gets infinitely large).

 

“e” is approximately equal to 2.7182818 and is a good model for exponential growth.

 

We will now investigate some applications of exponential functions.

 Formulas for compound interest

• Interest is compounded “n” times per year.

 

A is the amount of money in the account
P is the principal
r is the interest rate
n is the number of times per year that the interest is compounded
t is the time in years

 

• Continuously compounded interest

A is the amount of money in the account
P is the principal
r is the interest rate
t is the number of years that the interest is compounded

Summary: Exponential Growth and Decay

 

a is the initial amount
b is the base, or growth/decay factor
                        → growth: b > 1
                        → decay: 0 < b < 1