
Example of an Exponential Function
A biologist grows bacteria in a culture. If initially there were three
grams of bacteria, after one day there are six grams of bacteria, and
after two days, there are twelve grams, how many grams will there be at the
end of the week?
Solution:
We draw a t chart
t 
P(t) 
0 
3 = 3(2^{0}) 
1 
6 = 3(2^{1}) 
2 
12 = 3(2^{2}) 
We see that the general formula is
P(t) = 3(2^{t})
Hence after one week, we calculate
P(7) = 3(2^{7}) = 384 grams of bacteria.
We call P(t) and exponential function with base 2.

Graphing Exponentials
Below is the graph of y = 2^{x}.
It turns out that for any b > 1 the graph of y = b^{t} looks
similar.
Notice that

the left horizontal asymptote at 0

The yintercept is 1

The graph is always increasing.
Shifting
techniques can also be used to graph variations of this curve.
Example
Graph y = 2^{x}
Solution:
We
see that the graph is reflected about the yaxis:

Three Properties of Exponents

b^{x }b^{y} = b^{x+y
}

b^{x }/ b^{y} = b^{xy
}

(b^{x})^{y} = b^{xy
}
^{
}
Definition
b^{x } = 1 / b^{x}

Example
Simplify
3^{4}(3)^{1}/[(3^{2})^{3}]
Solution
3^{4}(3)^{1}/[(3^{2})^{3}]
= 3^{4}(3)^{1}/ 3^{6
}= 3^{4} /(3^{1}3^{6})
= 3^{4 }/ 3^{7
} =
1/3^{3} = 1/27

Applications
Money and Compound Interest
We have the formula for compound interest
where A corresponds to the amount in the account after
t years in a bank
that gives an annual interest rate r compounded n times per year.
Example
Suppose we have $2,000 to put into a savings account at a 4% interest
rate compounded monthly. How much will be in the account after 2 years?
We have
P = 2,000, r = .04, n = 12 and
t = 2
We want A.
A = 2000(1 + .04/12)^{12(2)} = $2,166.29.
Continuous Interest
For continuously compounded interest, we have the formula:
Inflation Example
With an 8% rate of inflation in the health industry, how much will health
insurance cost in 45 years if currently I pay $200 per month?
Solution
We have
r = .08, P = 200, and
t = 45
So that
A = 200e^{(.08)(45)} = $7319 per month!