We have seen in past courses that exponential
functions are used to represent growth
and decay.


Where a = initial
amount
b =
growth/decay factor
x = time
y = ending
amount 


Growth
When a > 0
and
b
> 1,
the function models
growth.
(b is called the growth factor)
(a
represents the initial amount) 


Decay
When a
> 0 and
0 < b < 1,
the function
models decay.
(b is called
the decay factor)
(a represents the initial
amount) 

Let's look at examples of these exponential functions at work.
1. 
Population:
The population of the popular town of
Smithville in 2003 was estimated to be
35,000 people with an annual rate of
increase (growth)
of about 2.4%.
a.) What is the
growth
factor for Smithville?
After one year the
population would be 35,000 + 0.024(35000).
By factoring we see that this is 35,000(1 +
0.024) or 35,000(1.024).
The growth factor is 1.024.
(Remember
that the growth factor is greater than 1.)
b.) Write an equation to model future
growth.
where
y is the population; x is
the number of years since 2003


c.) Use your
equation to estimate the population in 2007
to the nearest hundred people.




Where y_{0} =
initial amount
k = constant
of proportionality
t = time
y = ending
amount 


Growth
models
growth
since
e > 1. 
Exponential
functions with the base e are often used to
describe continuous growth or decay. 

Decay
models decay
since
which is between 0 and 1 

4. 
More Money:
Most banks
compound interest more than once a year.
When interest is compounded n times
per year for t years at an interest
rate of r, the principal, P,
grows to the amount A given by the
formula:

If you allow n
to get larger and larger, you will
discover that as n increases,
approaches
e. 
If compounding
takes place without interruption
(called compounded continuously),
this formula becomes:



6. 
Bacteria Growth:
A certain strain of bacteria
that is growing on your kitchen counter
doubles every 5 minutes. Assuming that
you start with only one bacterium, how many
bacteria could be present at the end of 96
minutes?


