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Advanced Algebra: Exponential Functions
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:
 
xmin = -5, xmax = 5
ymin = -3, ymax = 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:
ExamplesExamples:
What point do all exponent graphs have in common?
What do all of the first 3 graphs, the graphs where b > 1, have in common?
What do all of the second 3 graphs, the graphs where 0 < b < 1, have in common?
What is the domain of an exponential function?
What is the range of an exponential function?

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
 
 
ExamplesExamples:
If you put $500 in a savings account at 6.4% interest compounded monthly, how much money would you have if you left it in for 10 years?
If you put the same $500 in a savings account where 6.4% interest was compounded CONTINUOUSLY, how much would you have after 10 years?

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
 
 
ExamplesExample:
The population of Niceville in 1985 was 12,500. Each year the population grows at, approximately, the rate of 3.7 % per year. What is the expected population in the year 2010 if the present growth rate continues?
 
Hint: Let f(x) = population, x = number of years since 1985, and b = 1.037 (each year 3.7% is added to the whole population).



K Dodd

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