Transcript exponential function

Exploring Exponential Growth
and Decay Models
Sections 8.5 and 8.6
What am I going to learn?
Concept of an
exponential function
Models for exponential
growth
Models for exponential
decay
Meaning of an asymptote
Finding the equation of
an exponential function
Recall
Independent variable is another name for
domain or input, which is typically but not
always represented using the variable, x.
Dependent variable is another name for
range or output, which is typically but not
always represented using the variable, y.
What is an exponential function?
Obviously, it must have something to do
with an exponent!
An exponential function is a function
whose independent variable is an
exponent.
What does an exponential function
look like?
Dependent
Variable
Just some
number
that’s not 0
Why not 0?
Base
Exponent
and
Independent
Variable
The Basis of Bases
The base of an exponential function carries
much of the meaning of the function.
The base determines exponential growth
or decay.
The base is a positive number; however, it
cannot be 1. We will return later to the
reason behind this part of the definition .
Exponential Growth
An exponential function models growth
whenever its base > 1. (Why?)
If the base b is larger than 1, then b is
referred to as the growth factor.
Exponential Growth Models
When you deposit money into a bank savings account,
the bank pays you interest for using your money.
The interest the bank pays you is added into your
account, and you earn interest on the interest.
This is called compound interest. Compound interest is
an Exponential Growth Function.
Exponential Growth Function
y = a(1 + r)t
a = Initial Amount
r = Growth Rate
1 + r = Growth Factor
t = Time (usually in years)

Example: In 1980 about 2,180,000 U.S. workers
worked at home. During the next ten years, the number
of workers working at home increased 5% per year.
a. Write a model giving the number w (in millions) of
workers working at home t years after 1980.
b. Find the number of workers working at home in 1990.
a. y = a(1 + r)t
b. y = a(1 + r)t
a = 2.18
r = 0.05
w = 2.18(1 + 0.05)t
w = 2.18(1.05)t
t = 10
w = 2.18(1.05)t
w = 3.551 million workers
w = 2.18(1.05)10

The Exponential Growth Function works fine if all we need to find
the new amount only once during the growth period.
But, interest earned in bank accounts is typically computed
monthly. The interest earned this month is added to your account
and will earn interest next month, and so on.
This is called compound interest. We need a new formula to
compute compound interest.
r

A  P 1  
n

nt
A = new Amount
P = Principal (initial amount)
r = interest rate (as a decimal)
t = time (in years)
n = number of compounding periods/ye

Example: You deposit $1,500 in an account that pays
6% annual interest. Find the balance after 5 years if the
interest is compounded:
P = 1500
a. Quarterly.
nt
r

t=5
A

P
1



b. Monthly.
n
r = 0.06

a. n = 4
b. n = 12
0.06 

A  1500 1 

4


 45
A  1500 1  0.015
A  1500 1.015
= $2020.28
20
20
0.06 

A  1500 1 

12


125 
A  1500 1  0.005
A  1500 1.005
= $2023.28
60
60

What does Exponential Growth look like?
Consider y = 2x
Table of Values:
x
2x
y
2
22
4
3
23
8
Cool
Fact:
1
-3
2-3
8
All
-2
2-2
¼
exponential
-1 growth
2-1
½
0
20 look
1
functions
1
1 like 2this!
2
Graph:
Investigation: Tournament Play
The NCAA holds an annual basketball
tournament every March.
The top 64 teams in Division I are invited
to play each spring.
When a team loses, it is out of the
tournament.
Work with a partner close by to you and
answer the following questions.
Investigation: Tournament Play
After round x
Fill in the following
chart and then graph
the results on a piece
of graph paper.
Then be prepared to
interpret what is
happening in the
graph.
0
1
2
3
4
5
6
Number of
teams in
tournament
(y)
64
Exponential Decay
An exponential function models decay
whenever its 0 < base < 1. (Why?)
If the base b is between 0 and 1, then b is
referred to as the decay factor.
What does Exponential Decay look
like?
Consider y = (½)x
Graph:
Cool xFact:
x (½)
y
All
-2exponential
½-2
4
-1
-1 ½
2
decay
0 functions
½0
1
1 look
½1 like
½
2
2
½this!
¼
Table of Values:
3
½3
1/8
End Behavior
Notice the end behavior of the first graph-exponential
growth. Go back and look at your graph.
 , which means
As x  , f  x   _______
as you move to the right, the graph goes up without bound.
________________________________________
As x  , f  x   _______ , which means
0
as you move to the left, the graph levels off-getting close to but
not touching the x-axis (y = 0).
_______________________________________
End Behavior
Notice the end behavior of the second graphexponential decay. Go back and look at your
graph.
As x  , f  x   _______ , which means
0
as you move to the right, the graph levels off-getting close to but
not touching the x-axis (y = 0).
________________________________________

As x  , f  x   _______ , which means
as you move to the left, the graph goes up without bound.
________________________________________
Asymptotes
One side of each of the graphs appears to
flatten out into a horizontal line.
An asymptote is a line that a graph
approaches but never touches or
intersects.
Asymptotes
Notice that the left
side of the graph gets
really close to y = 0
as x .
We call the line y = 0
an asymptote of the
graph. Think about
why the curve will
never take on a value
of zero and will never
be negative.
Asymptotes
Notice the right side of
the graph gets really
close to y = 0 as
x  .
We call the line y = 0
an asymptote of the
graph. Think about why
the graph will never take
on a value of zero and
will never be negative.
Let’s take a second look at the base
of an exponential function.
(It can be helpful to think about the base as the object that is
being multiplied by itself repeatedly.)
Why can’t the base be negative?
Why can’t the base be zero?
Why can’t the base be one?
Examples
Determine if the function represents exponential
growth or decay.
1.
2.
3.
y  5(3)
y4

1
5
y  2(4)
x
x
x
Exponential Growth
Exponential Decay
Exponential Decay
Example 4 Writing an Exponential Function
Write an exponential function for a graph
that includes (0, 4) and (2, 1). (We’ll write
out each step.)
Example 5 Writing an Exponential Function
Write an exponential function for a graph that includes (2,
2) and (3, 4). (Do each step on your own. We’ll show the
solution step by step.)
(1) y  ab
x
(2) 2  ab
2
(3) a  2
b
3
(4) 4  ab
2 3
(5) 4  2 b
b
2
Use the general form.
Substitute using (2, 2).
Solve for a.
Substitute using (3, 4).
Substitute in for a.
Example 5 Writing an Exponential Function
Write an exponential function for a graph that
includes (2, 2) and (3, 4).
(6) 4  2b  b  2
2 1
(7) a  2 
2
2
1
x
(8) y   (2)
2
Simplify.
Backsubstitute to get a.
Plug in a and b into the general
formula to get equation.
What’s coming up tomorrow?
Applications of growth and decay functions
using percent increase and decrease
Translations of y = abx
The number e
Continuously Compounded Interest
Homework Problems
Worksheet