Exponential and Logarithmic
Functions
Section 11-2
Exponential Functions
Power Function verses Exponential
Function
 You have worked with power functions throughout your
math career. Those are functions where the base is a variable
and the exponent is a real number.
 Now we are going to look at exponential functions where the
base is a positive real number and the exponent is a variable.
x
 Exponential functions are written y = b This is the parent
graph of an exponential function. The same techniques used
to transform the graphs of other functions can be applied to
graphs of exponential functions.
Example
x
 Consider the graph of y = 2 where
x is an integer. This is a function since
there is a unique y-value for each xvalue. The graph seems to be
increasing.
 Now suppose the domain is
expanded to include all rational
numbers.
 We can also include irrational
num bers, thus “filling in the graph
for all real numbers.
Characteristics of graphs of y = b
x
Example # 1
x
x
 Graph the exponential functions y = 2 , y = 2 + 3, and
x
y = 2 -2 on the same set of axes. Compare and contrast the graphs.
Solution: Set up table of values
for each one and plot points.
Compare:
All of the graphs are continuous,
increasing, and one-to-one. They
have the same domain and no vertical
asymptote. The y intercepts and the
horizontal asymptotes for each graph are
different from the parent graph y = 2
x
x
Y = 2 -2 is the only one of the three that has an x intercept.
Example # 2
1 x
1
 Graph the exponential functions y = ( ) , y = 5(
)
3
3
1 x
x
and y = -1 ( ) on the same set of axes. Compare and
3
contrast the graphs.
Again set up a table of values for each graph and plot the points.
Compare: All of the graphs are
decreasing, continuous, and
one-to-one. They have the same
domain and horizontal asymptote.
They have no vertical asymptote or
x intercept. The y intercepts for each graph are different from the parent graph.
Exponential Growth and Decay
 Many real life situations involve exponential growth (increase
population, bacteria, money market account, AIDS, etc) or
exponential decay (cooling, carbon dating, etc)
 When you know the rate at which the growth or decay is
occurring, you can use the equation N = N (1+r)t where N
is the final amount, N is the initial amount, r is the rate of
growth or decay per time period, and t is the number of time
periods.
o
o
Example # 3
 The average growth rate of the population of Union City is 7.5% per year and is
represented by the formula
y = A (1.075)x , where x is the number of years and y is the most recent
population of the city.T he city’s population A is now 70, 300 people.W hat is
the expected population in 10 years?
Solution: y = 70300 (1.075)10 ≈ 144,891
Compound Interest
 The general equation for exponential growth is modified for
finding the balance in an account that earns compound
interest.
r
A = P ( 1 + n
n t
)
where P is the principal or initial
investment. A is the final amount of the investment, r is the
annual interest rate, n is the number of times interest is paid,
or compounded each year, and t is the number of years.
 Annual n = 1
Semi-annual n= 2 Monthly n = 12
Quarterly n =4 Weekly n =52
Daily n = 365
Example # 4
 How much should Dario invest now in a money market
account if he wishes to have $9000 in the account at the end
of 10 years? The account provides an APR of 6%
compounded quarterly. (APR= Annual percentage return)
 Solution: Compounded quarterly means every three months
and there are 4 times interest is paid.
4 ( 10 )
4 ( 10 )
9000 = P ( 1 + .06/4)
= P (1.015)
= P (1.81402)
S0 P = 9000 / 1.81402 = $4961.36
Example # 5
x
 G raph y ≥ 3 + 1
 First graph the equation using a table of values.
 The curve is solid because it includes the points on the curve
 Next test a point inside the curve to see if it holds true for
the inequality. If it does, shade that region. If not, shade the
opposite region.
HW # 26
Section 11-2
Pp. 708-711
# 11-21 odds, 25, 29,32