Section 3.1
Exponential Functions
Example
The exponential equation f  x   13.49 .967   1 predicts the number of O-rings
x
that are expected to fail at the temperature x o F on the space shuttles. The
O-rings were used to seal the connections between different sections of the shuttle
engines. Use a calculator to find the number expected to fail at the temperature of
40 degrees.
Graphing Exponential
Functions
x
x
1
Graph the following two equations: f(x)=   , f(x)=  4 
4
Draw the asymptotes.
Example

















Transformations of Exponential
Functions
Use the graph of f(x)=4 x to obtain the graph of g(x)=4 x  3.
What is the domain and range of each function?
Example

















Example
Use the graph of f(x)=4 x to obtain the graph of g(x)=4 x2
Find the domain and range for the g(x) function.

















Example
Use the graph of f(x)=4 x to obtain the graph of g(x)=2  4 x
Find the domain and range for the g(x) function.

















The Natural Base e
n
 1
The values of 1+  for increasingly
 n
large values of n. As n   the
approximate value of e to nine decimal
places is e  2.718281827.
The irrational number e, approximately
2.72, is called the natural base. The
function f(x)=e x is called the natural
exponential function.
Example
The population of a small polynesian island can be modeled by
the equation f(x)=8e.8 x . Round off your answers to the
nearest integer.
a. How many people originally went to this small island? (x=0)
b. How many people will be living on this island 10 years later
due to general population growth with no new immigration.
Compound Interest
Compound interest is interest computed on your original investment
as well as on any accumulated interest. The sum of money is called
the principal, P. It is invested at an annual percentage rate r, and
compounded once a year.
Sometimes interest is compounded semiannually, monthly, quarterly, etc. This introduces
the variable n for the number of compounding periods per year. The formula is then adjusted
to take this into account the number of compunding periods in a year. If there are n compuonding
r
periods per year, in each time period the interest rate is i = and there are nt time periods in t years.
n
The results in the following formula for the balance, A, after t years is
 r
A=P 1  
 n
nt
Continuous Compounding of Interest
Example
Find the accumulated value of an investment of $15,000
for 2 years at an interest rate of 3.2% if the money is
a. compounded semiannually
b. compounded quarterly
c. compounded monthly
d. compounded continuously
For the graph of f(x)=4x3  1, What is the domain and range?
(a) D : ( 1, ), R : ( 1, )
(b) D : ( , ), R : ( 1, )
(c) D : ( , ), R : ( , )
(d) D :[ 1, ), R :[ 1, )
Find the accumulated value of a CD of $20,000 for 3 years
at an interest rate of 3.1% if the money is compounded continuously?
(a) 50,690
(b) 218,760,384
(c) 21,949
(d) 21,860