Advanced Mathematical Concepts
Chapter 11
Lesson 11-2 Exponential Functions
Power Function – a function of the form y = xp where p is a real number
Exponential Function – a function of the form y = bx where b is a positive real number
Characteristics of the graphs of y = bx
b>1
x ɛ (-∞,∞)
y ɛ (0,∞)
(0,1)
Continuous, one-to-one, always
increasing
Negative x-axis
None
Domain
Range
y-intercept
Behavior
Horizontal Asymptote
Vertical Asymptote
0<b<1
x ɛ (-∞,∞)
y ɛ (0,∞)
(0,1)
Continuous, one-to-one, always
decreasing
Positive x-axis
none
Note: when b=1, y= bx is a horizontal line y=1
Example 1
a. Graph the exponential functions y = 3x, y = 3x + 5, and y = 3x - 4 on the same set of axes.
Compare and contrast the graphs.
All of the graphs are continuous, increasing, and one-toone. They have the same domain, and no vertical
asymptote.
The y-intercept and the horizontal asymptotes for each
graph are different from the parent graph y = 3x. While
y = 3x and y = 3x + 5 have no x-intercept, y = 3x - 4 has
an x-intercept.
  , y = 4  14  , and y = -3  14 
b. Graph the exponential functions y = 1
4
Compare and contrast the graphs.

The graphs of y = 1
4
x

x
x
x
on the same set of axes.
x
and y = 4 1 are decreasing
4
x
and the graph of y = -3 1 is increasing. All of the
4
graphs are continuous and one-to-one. They have the
same domain and horizontal asymptote. They have no
vertical asymptote.

The y-intercepts for each graph are different from the
x
parent graph y = 1 .
4

Example 2
PHYSICS A ball is dropped from a height of 20 meters on to pavement. On each bounce, the ball
bounces to a height that is 40% less than its height on the previous bounce. The height of the ball
can be modeled by the equation y = 20(0.6)t, where y is the height of the ball in meters, and t is the
number of times the ball bounces.
a. Find the height of the ball after its fourth bounce.
Advanced Mathematical Concepts
Chapter 11
b. Graph the height function.
a. y = 20(0.6)t
y = 20(0.6)4
y = 2.592
b.
t=4
The height of the ball after the fourth
bounce is about 2.6 meters.
Exponential Growth and Decay Formula – N = N0(1+r)t where N is the final amount, N0 is the
initial amount, r is the rate of growth or decay per time period, and t is the number of periods.
Example 3
POPULATION Between 1990 and 2000, the population of Florida had an annual growth rate of
about 2.14%. If the state’s population was 12,937,926 in 1990, approximately what was Florida’s
population in 2000?
N = N0(1 +r)t
N = 12,937,926(1 + 0.0214)10
N = 15,989,069.78
N0 = 12,937,926, r = 0.0214, t = 10
Use a calculator.
In 2000, the population of Florida was approximately 15,989,070.
 
nt
Compound Interest Formula – A = P 1  r where P is the principal or initial investment, A is the
n
final amount of the investment, r is the annual interest rate, n is the number of times interest is paid
(compounded each year), and t is the number of years.
Example 4
FINANCE Determine the amount of money in a savings account providing an annual rate of 3%
compounded daily if Sandra made a one-time deposit of $8500 in to the account and left it there for
5 years.
 
A = 8500 1  0.03 
365
A = P 1 r
n
nt
3655
A = 9875.530189
After 5 years, the $8500 investment will have a value of $9875.53.
Advanced Mathematical Concepts
Example 5
Graph y > 3x - 2.
First, graph y = 3x - 2. Since the points on this curve are not
in the solution of the inequality, the graph of y = 3x - 2 is
shown as a dashed curve.
Then, use (0, 0) as a test to determine which area to shade.
y = 3x - 2

0 > 30 - 2
0 >1-2
0 > -1
Since (0, 0) satisfies the inequality, the region that contains
(0, 0) should be shaded.
Assignment: Pages 708-711
5,8,11,14,17,19-21,24,27,28,31,32
Chapter 11