Exploring Exponential Growth and Decay Models

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Exploring Exponential Growth and Decay Models

• Sections 8.5 and 8.6

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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

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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.

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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.

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What does an exponential function look like?
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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 .

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Exponential Growth

• An exponential function models growth whenever
  its base gt 1. (Why?)
• If the base b is larger than 1, then b is
  referred to as the growth factor.

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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)
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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
w 2.18(1.05)t
a 2.18
t 10
r 0.05
w 2.18(1 0.05)t
w 2.18(1.05)10
w 2.18(1.05)t
w 3.551 million workers
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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.
A new Amount
P Principal (initial amount)
r interest rate (as a decimal)
t time (in years)
n number of compounding periods/year
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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.

t 5

• b. Monthly.

r 0.06
a. n 4
b. n 12
2023.28
2020.28

• ?

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What does Exponential Growth look like?

• Consider y 2x

Table of Values
x 2x y
-3 2-3
-2 2-2 ¼
-1 2-1 ½
0 20 1
1 21 2
2 22 4
3 23 8
Graph
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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.

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Investigation Tournament Play
After round x Number of teams in tournament (y)
0 64
1
2
3
4
5
6

• 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.

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Exponential Decay

• An exponential function models decay whenever its
  0 lt base lt 1. (Why?)
• If the base b is between 0 and 1, then b is
  referred to as the decay factor.

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What does Exponential Decay look like?

• Consider y (½)x

Graph
Table of Values
x (½)x y
-2 ½-2 4
-1 ½-1 2
0 ½0 1
1 ½1 ½
2 ½2 ¼
3 ½3 1/8
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End Behavior

• Notice the end behavior of the first
  graph-exponential growth. Go back and look at
  your graph.

as you move to the right, the graph goes up
without bound.
as you move to the left, the graph levels
off-getting close to but not touching the x-axis
(y 0).
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End Behavior

• Notice the end behavior of the second
  graph-exponential decay. Go back and look at
  your graph.

as you move to the right, the graph levels
off-getting close to but not touching the x-axis
(y 0).
as you move to the left, the graph goes up
without bound.
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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.

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Asymptotes

• Notice that the left side of the graph gets
  really close to y 0 as .
• 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.

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Asymptotes

• Notice the right side of the graph gets really
  close to y 0 as
• .
• 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.

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Lets 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 cant the base be negative?
• Why cant the base be zero?
• Why cant the base be one?

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ExamplesDetermine if the function represents
exponential growth or decay.

• 1.
• 2.
• 3.

Exponential Growth
Exponential Decay
Exponential Decay
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Example 4 Writing an Exponential Function

• Write an exponential function for a graph that
  includes (0, 4) and (2, 1). (Well write out
  each step.)

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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. Well show the solution step by step.)

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Example 5 Writing an Exponential Function

• Write an exponential function for a graph that
  includes (2, 2) and (3, 4).

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Whats coming up tomorrow?

• Applications of growth and decay functions using
  percent increase and decrease
• Translations of y abx
• The number e
• Continuously Compounded Interest

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Homework Problems

• Worksheet