Exponential Functions

Exponential Functions
At the beginning of the quarter we saw how certain relationships between
variables could be modeled with a variety of functions. When you studied linear
functions, you saw how growth always proceeded at a constant rate, regardless of the size
of the variables. This rate was called slope. Sometimes however, the amount of growth
changes as the size of the population changes. This occurs when there is a feedback loop.
A feedback loop occurs when you have two (or more) variables and each variable affects
the other. An example of a positive feedback loop is what occurs between the size of a
population and the number of births in that population. More people leads to more births
which lead to more people, etc. Another example is money in an account. As the
account gets larger, the amount of interest earned increases, which leads to more money
in the account, which leads to more interest, etc.
An exponential function is used to model situations in which there is a positive
feedback loop. An exponential function is shown as y = abx or f(x) = abx, where a is the
initial size of the population, b is the growth factor which indicates the proportion by
which the population will increase every period. The variable x, which is an exponent,
represents the number of periods. The variable y or f(x) is the total amount that the
population becomes after x periods.
As an example, consider a small business that sees a hundred percent growth
every year. This corresponds to a doubling of their first year’s earnings. If the small
business earned $1,000 in profit the first year, then the exponential equation that would
model this is f(x) = 1,000(2)x-1, where x is the number of years the business has existed.
Number of years in business Substitution into function
f(x) = 1,000(2)1-1
f(x) = 1,000(2)0
f(x) = 1,000(2)2-1
f(x) = 1,000(2)1
f(x) = 1,000(2)3-1
f(x) = 1,000(2)2
f(x) = 1,000(2)4-1
f(x) = 1,000(2)3
Total Amount of profit after
that many years in business
f(x) = 1000
f(x) = 2000
f(x) = 4000
f(x) = 8000
Notice how the total amount of profit made doubles each year.
The variable b in the function determines if there is exponential growth or
exponential decay. When b is greater than 1, the function will grow. When b is between
0 and 1, the function will decay. These are shown in the graphs below. Note that b
cannot be equal to 1, nor can it be negative.
f(x) = 2x
f(x) = (1/2)x
Exponential equations are used for the growth of investments in which money
remains in the account leading to compounded interest. The formula is Pt  P0 1  
 k
Pt is the value after t years
P0 is the starting value
r is the annual interest rate (APR)
t is the number of years for the entire investment
k is the number of compounding periods in a year
Example: $5,000 is invested at 3.1% interest, compounded monthly. How much money
is in the account after 8 years?
 0.031 
Pt  5000 1 
12 
Pt  6405.25