LESSON
13.2
Name
Modeling Exponential
Growth and Decay
Class
Date
13.2 Modeling Exponential Growth
and Decay
Essential Question: How can you use exponential functions to model the increase or
decrease of a quantity over time?
A1.9.D …graph exponential functions that model growth and decay and identify key
features… Also A1.9.A, A1.9.C, A1.9.E
Texas Math Standards
The student is expected to:
Explore 1
A1.9.D
Describing End Behavior of a Growth Function
Graph exponential functions that model growth and decay and identify
key features, including y-intercept and asymptote, in mathematical and
real-world problems. Also A1.9.A, A1.9.C, A1.9.E
When you graph a function ƒ(x) in a coordinate plane, the x-axis represents the independent variable and
the y-axis represents the dependent variable. Therefore, the graph of ƒ(x) is the same as the graph of the
equation y = ƒ(x). You will use this form when you use a calculator to graph functions.
Mathematical Processes

Use a graphing calculator to graph the exponential growth
x
function ƒ(x) = 200(1.10) , using Y 1 for ƒ(x). Use a viewing
window from -20 to 20 for x, with a scale of 2, and
from -100 to 1000 for y, with a scale of 50. Sketch the curve on the axes provided.

To describe the end behavior of the function, you describe the function values as
x increases or decreases without bound. Using the TRACE feature, move the cursor to the
right along the curve. Describe the end behavior as x increases without bound.
A1.1.A
Apply mathematics to problems arising in everyday life, society, and
the workplace.
Language Objective
1.F, 2.C, 3.C, 3.D
As x increases without bound, the graph grows larger at an exponentially increasing rate.
Compare and contrast exponential growth and exponential
decay functions.
Essential Question: How can you use
exponential functions to model the
increase or decrease of a quantity
over time?
An exponential function represents growth when
b > 1 and decay when 0 < b < 1. You can use
t
y = a(1 + r) where a > 0 to represent exponential
t
growth and y = a(1 - r) where a > 0 to represent
exponential decay.
PREVIEW: LESSON
PERFORMANCE TASK
Using the TRACE feature, move the cursor to the left along the curve. Describe the end
behavior as x decreases without bound.
As x decreases without bound, the graph approaches, but never hits, 0.
Reflect
1.
Describe the domain and range of the function using inequalities.
Domain: {x| - ∞ < x < ∞} Range: {y| y > 0}
2.
Identify the y-intercept of the graph of the function.
The y-intercept of the graph of the function is (0, 200).
3.
An asymptote of a graph is a line the graph approaches more and more closely. Identify an asymptote of
this graph.
The line y = 0 is an asymptote of this graph.
4.
Discussion Why is the value of the function always greater than 0?
Since a positive number is being multiplied by another positive number, it is impossible
for the growth function to go below the x-axis.
Module 13
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A1_MTXESE353879_U5M13L2.indd 617
HARDCOVER PAGES 457466
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Lesson 2
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Reflect
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View the Engage section online. Discuss what an
archeologist might learn from studying the bones,
tools, weapons, and other objects left by prehistoric
people. Then preview the Lesson Performance Task.

© Houghton Mifflin Harcourt Publishing Company
ENGAGE
© Houghto
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1.
2.
617
Module 13
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9_U5M13
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Lesson 13.2
Resource
Locker
617
13/11/14
7:24 PM
13/11/14 7:24 PM
Explore 2
Describing End Behavior of a Decay Function
EXPLORE 1
Use the form from the first Explore exercise to graph another function on your calculator.
A
Use a graphing calculator to graph the exponential decay function
x
ƒ(x) = 500(0.8) , using Y 1 for ƒ(x). Use a viewing window from -10 to 10 for
x, with a scale of 1, and from -500 to 5000 for y, with a scale of 500. Sketch
the curve on the axes provided.
Describing End Behavior of a Growth
Function
B
Using the TRACE feature, move the cursor to the right along the curve.
Describe the end behavior as x increases without bound.
INTEGRATE TECHNOLOGY
Students will use graphing calculators to
complete the Explore activities that investigate
the end behavior of exponential functions.
As x increases without bound, the graph approaches, but never hits, 0.
C
Using the TRACE feature, move the cursor to the left along the curve. Describe the end
behavior as x decreases without bound.
As x decreases without bound, the graph grows larger at an exponentially increasing rate.
CONNECT VOCABULARY
Connect the word bound in the phrase “as x increases
without bound” to the more familiar word boundary.
To increase without bound means to increase without
reaching an end point or boundary. Another way to
state the same idea is “as x approaches infinity.”
Reflect
5.
Discussion Describe the domain and range of the function using inequalities.
Domain: {x| - ∞ < x < ∞}
Range: {y| y > 0}
6.
7.
Identify the y-intercept of the graph of the function.
The y-intercept of the graph of the function is (0, 500).
QUESTIONING STRATEGIES
line approaches, but never hits, y = 0.
Describe the x-intercepts of the exponential
growth function. There are no x-intercepts
for this function because there are no values of x for
which f(x) = 0.
© Houghton Mifflin Harcourt Publishing Company
Identify an asymptote of this graph. Why is this line an asymptote?
An asymptote of this graph is the line y = 0. It is an asymptote of the graph because the
Are there any asymptotes for this function
other than the line y = 0? Explain. No; the
graph gets closer and closer to y = 0 as x decreases
without bound, and it is ever-increasing as x
increases without bound.
EXPLORE 2
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618
Lesson 2
Describing End Behavior of a Decay
Function
PROFESSIONAL DEVELOPMENT
A1_MTXESE353879_U5M13L2.indd 618
Math Background
The end behavior of a function f(x) is a description of what happens to a function
f(x) as x increases or decreases without bound. The value of an exponential
growth function f(x) = ab x, where b > 1 and a > 0, increases without bound as
x increases without bound, and approaches 0 as x decreases without bound. The
value of an exponential decay function f(x) = ab x, where 0 < b < 1 and a > 0,
approaches 0 as x increases without bound, and increases without bound as x
decreases without bound. Such end behavior can also be described using the
notation f(x) → ∞ as x → -∞ and f(x) → 0 as x → ∞, where the symbol ∞
represents infinity.
13/11/14 7:24 PM
QUESTIONING STRATEGIES
Before you graph it, how can you tell that the
x
function f(x) = 500(0.8) will decrease as x
increases? A positive number is being multiplied x
times by a number between 0 and 1, so it will get
smaller as x increases.
Modeling Exponential Growth and Decay
618
Modeling Exponential Growth
Explain 1
QUESTIONING STRATEGIES
Recall that a function of the form y = ab x represents exponential growth when a > 0 and b > 1. If b is replaced by
t
1 + r and x is replaced by t, then the function is the exponential growth model y = a(1 + r) , where a is the initial
amount, the base (1 + r) is the growth factor, r is the growth rate, and t is the time interval. The value of the model
increases with time.
What does it mean for a function to approach
0 as x increases without bound? Is 0 a value of
the function? The value of the function gets closer
and closer to 0 as x increases, but it never reaches 0.
Example 1

EXPLAIN 1
Write an exponential growth function for each situation. Graph each
function and state its domain, range and an asymptote. What does the
y-intercept represent in the context of the problem?
A painting is sold for $1800, and its value increases by 11%
each year after it is sold. Find the value of the painting in 30 years.
Write the exponential growth function for this situation.
y = a(1 + r)
t
Modeling Exponential Growth
= 1800(1 + 0.11)
t
= 1800(1.11)
t
QUESTIONING STRATEGIES
t
= 1800(1.11)
30
≈ 41,206.13
After 30 years, the painting will be worth approximately $41,206.13.
Create a table of values to graph the function.
t
y
0
1800
(t, y)
y
(0, 1800)
8
4148.20
(8, 4148.20)
16
9559.60
(16, 9559.60)
24
22,030
(24, 22,030)
32
50,770
(32, 50,770)
Determine the domain, range and an asymptote of the function.
Value (dollars)
Why doesn’t the graph of an exponential
growth model intersect the y-axis at the
origin? When x = 0, y is the initial value of the
increasing quantity.
y = 1800(1.11)
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Tetra
Images/Corbis
How do you use a percent increase amount
when writing an exponential growth
equation? Give an example. Convert the percent
increase to a decimal and add it to 1 to find the
number that is raised to a power. Possible answer:
For a 4% increase, the decimal 1.04 will be raised
to a power.
Find the value in 30 years.
44,000
(32, 50,770)
33,000
22,000
(24, 22,030)
11,000 (0, 1800)
(16, 9560)
t
(8, 4148)
0
16 24 32
8
Time (years)
The domain is the set of real numbers t such that t ≥ 0.
The range is the set of real numbers y such that y ≥ 1800.
An asymptote for the function is y = 0.
The y-intercept is the value of y when t = 0, which is the value of the painting when it was sold.
Module 13
619
Lesson 2
COLLABORATIVE LEARNING
A1_MTXESE353879_U5M13L2.indd 619
Small Group Activity
Divide students into groups of three or four. Have students research several banks,
either locally or online, for interest rates on savings accounts, money market
accounts, and CDs. Have them work as a group to develop exponential growth
models for each type of account at each bank on an initial deposit of $500. Ask
students to make a chart comparing the rates and potential earnings after 5 years
for the different types of accounts at each bank.
619
Lesson 13.2
13/11/14 7:24 PM
B
A baseball trading card is sold for $2, and its value increases by 8% each year after it is sold.
Find the value of the baseball trading card in 10 years.
Write the exponential growth function for this situation.
y = a(1 + r)
t
(1 + 0.08 )
2 ( 1.08 )
= 2
=
t
t
Find the value in 10 years.
y = a(1 + r)
t
( 1.08 )
2 ( 1.08 )
= 2
=
t
10
≈ 4.32
4.32
After 10 years, the baseball trading card will be worth approximately $
.
Create a table of values to graph the function.
(t, y)
y
6
(0, 2)
0
2
3
2.52
6
3.17
9
4.00
12
5.04
Value (dollars)
t
(3, 2.52)
(6, 3.17)
(9, 4.00)
(12, 5.04)
4.5
3
1.5
0
y
(12, 5.04)
(9, 4.00)
(6, 3.17)
(3, 2.52)
(0, 2)
t
3
9
6
Time (years)
12
Determine the domain, range, and an asymptote of the function.
The range is the set of real numbers y such that y ≥ 2 .
An asymptote for the function is
y=0
.
The y-intercept is the value of y when t = 0, which is the
value of the card when it was sold
.
Reflect
8.
Find a recursive rule that models the exponential growth of y = 1800( 1.11 )t.
a 1 = 1800, a n = 1.11a n-1
9.
Find a recursive rule that models the exponential growth of y = 2(1.08) .
a 1 = 2, a n = 1.08a n-1
Module 13
© Houghton Mifflin Harcourt Publishing Company
The domain is the set of real numbers t such that t ≥ 0 .
t
620
Lesson 2
DIFFERENTIATE INSTRUCTION
A1_MTXESE353879_U5M13L2.indd 620
Critical Thinking
13/11/14 7:24 PM
To help students understand the difference between exponential growth and linear
growth, point out that in exponential growth or decay, the amount added or
subtracted in each time period is proportional to the amount already present. For
exponential growth, this means that as the amount becomes greater, the amount
of increase in each time period also becomes greater. Contrast this to linear
growth, in which the amount of increase remains constant.
Modeling Exponential Growth and Decay
620
Your Turn
10. Write and graph an exponential growth function, and state the domain and
range. Tell what the y-intercept represents. Sara sold a coin for $3, and its
value increases by 2% each year after it is sold. Find the value of the coin
in 8 years.
t
8
y = a(1 + r)
y = 3(1.02)
Modeling Exponential Decay
= 3(1.02)
≈ 3.51
t
QUESTIONING STRATEGIES
4
Value (dollars)
EXPLAIN 2
3
y
(6, 3.38)
(2, 3.12)
(4, 3.25) (8, 3.51)
(0, 3)
2
1
After 8 years, the coin will be worth approximately $3.51.
What is the domain for a function that
represents the exponential growth or decay of
a population? Explain. The domain is all
non-negative real numbers, because the model
applies to all future times (positive values of t) but
not past times (negative values of t).
The domain is the set of real numbers t such that t ≥ 0.
t
0
The range is the set of real numbers y such that y ≥ 3.
2
4
6
Time (years)
8
The y-intercept is the value of y when t = 0, which is the value of the coin when it was sold.
Modeling Exponential Decay
Explain 2
Recall that a function of the form y = ab x represents exponential decay when a > 0 and 0 < b < 1. If b is replaced
t
by 1 - r and x is replaced by t, then the function is the exponential decay model y = a(1 - r) , where a is the initial
amount, the base (1 - r) is the decay factor, r is the decay rate, and t is the time interval.
What is the range for a function describing the
exponential growth or decay of a
population? For a growth function, the range
consists of all numbers greater than or equal to the
starting value of the population. For a decay
function, the range includes all numbers between
the starting value and 0.
Example 2

Write an exponential decay function for each situation. Graph each
function and state its domain and range. What does the y-intercept
represent in the context of the problem?
The population of a town is decreasing at a rate of 3% per year. In 2005, there were 1600
people. Find the population in 2013.
Write the exponential decay function for this situation.
y = a(1 - r)
t
© Houghton Mifflin Harcourt Publishing Company
= 1600(1 - 0.03)
t
= 1600(0.97)
t
Find the value in 8 years.
y = 1600(0.97)
t
= 1600(0.97)
8
≈ 1254
After 8 years, the town’s population will be about 1254 people.
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621
Lesson 2
LANGUAGE SUPPORT
A1_MTXESE353879_U5M13L2.indd 621
Connect Vocabulary
Remind students that exponential growth refers to an increasing function and
exponential decay refers to a decreasing function. While students are probably
familiar with the word growth, they may be less familiar with decay. Explain that,
in everyday use, it can mean rot, or loss of strength and health. Have students list
key words that can indicate growth or decay in descriptions of real-world
situations. Examples of key words for growth include increases, goes up, rises, gains.
Examples of key words for decay include decreases, goes down, falls, loses value,
declines, depreciates.
621
Lesson 13.2
13/11/14 7:24 PM
Create a table of values to graph the function.
1600
y
0
1600
8
1254
16
983
24
770
32
604
Population
(t, y)
t
(0, 1600)
(8, 1254)
y
(0, 1600)
(8, 1254)
1200
(16, 983)
(16, 983)
800
(24, 770)
(32, 604)
400
t
0
(24, 770)
8
16 24 32
Time (years since 2005)
(32, 604)
Determine the domain and range of the function.
The domain is the set of real numbers t such that t ≥ 0. The range is the set of real numbers y such
that 0 ≤ y ≤ 1600.
The y-intercept is the value of y when t = 0, the number of people before it started to lose population.
B
The value of a car is depreciating at a rate of 5% per year. In 2010, the car was worth
$32,000. Find the value of the car in 2013.
Write the exponential decay function for this situation.
y = a(1 - r)
t
(
)
32,000 ( 0.95 )
t
= 32,000 1 - 0.05
t
=
Find the value in 3 years.
y = a(1 - r)
t
= 32,000
( 0.95 ) =
( 0.95 ) ≈
3
t
32,000
27,436
t
y
0
32,000
1
30,400
2
28,880
3
27,436
(t, y)
(0, 32,000)
(1, 30,400)
(2, 28,880)
(3, 27,436)
Value (dollars)
Create a table of values to graph the function.
© Houghton Mifflin Harcourt Publishing Company
After 3 years, the car’s value will be $ 27,436 .
y (1, 30,400)
32,000
(3, 27,436)
(0, 32,000)
(2, 28,880)
24,000
16,000
8,000
t
0
1
2
3
4
Time (years since 2010)
Determine the domain and range of the function.
The domain is the set of real numbers t such that t ≥ 0 . The range is the set of real numbers y such
that 0 ≤ y ≤ 32,000 .
The y-intercept, 32,000, is the value of y when t = 0, the
Module 13
A1_MTXESE353879_U5M13L2.indd 622
622
original value of the car.
Lesson 2
2/14/15 6:58 PM
Modeling Exponential Growth and Decay
622
Reflect
EXPLAIN 3
11. Find a recursive rule that models the exponential decay of y = 1600(0.97) .
a 1 = 1,600, a n = 0.97a n-1
t
Comparing Exponential Growth
and Decay
12. Find a recursive rule that models the exponential decay of y = 32,000(0.95) .
a 1 = 32,000, a n = 0.95a n-1
t
AVOID COMMON ERRORS
13. The value of a boat is depreciating at a rate of 9% per year. In 2006,
the boat was worth $17,800. Find the worth of the boat in 2013.
Write an exponential decay function for this situation. Graph the
function and state its domain and range. What does the y-intercept
represent in the context of the problem?
Some students may forget to convert the percent
growth rate to decimal form. Remind them that
growth rate must be written as a decimal because the
percent sign means “parts out of 100.”
y = a(1 - r) = 17,800(0.91)
t
t
y = 17,800(0.91) ≈ 9198.35
7
Value (dollars)
Your Turn
y
20,000 (0, 17,800)
(2, 14,740)
15,000
(8, 8370.5)
5,000
t
After 7 years, the boat will be worth approximately $9,198.35.
The domain is the set of real numbers t such that t ≥ 0. The
range is the set of real numbers y such that 0 ≤ y ≤ 17,800.
QUESTIONING STRATEGIES
(6, 10,108)
(4, 12,206)
10,000
0
2
4
6
8
Time (years since 2006)
The y-intercept is 17,800, the value of y when t = 0, which is the original value
of the boat.
When you graph two functions on the same
coordinate grid, what does the intersection
point of the graphs represent? the time when the
two functions are equal in value
Explain 3
Comparing Exponential Growth and Decay
Example 3
© Houghton Mifflin Harcourt Publishing Company

Use the graphs provided to write the equations of the
functions. Then describe and compare the behaviors of
both functions.
The graph shows the value of two different shares of stock over the
period of 4 years since they were purchased. The values have been
changing exponentially.
The graph for Stock A shows that the value of the stock is decreasing as
time increases.
The initial value, when t = 0, is 16. The value when t = 1 is 12. Since
12 ÷ 16 = 0.75, the function that represents the value of Stock A after
t
t years is A(t) = 16(0.75) . A(t) is an exponential decay function.
Value (dollars per share)
Graphs can be used to describe and compare exponential growth and exponential decay models over time.
y
(0, 16)
16
12
Stock A
(1, 12)
8
4
Stock B
(1, 3)
(0, 2)
t
0
1
2
3
4
Time (years since purchase)
The graph for Stock B shows that the value of the stock is increasing as time increases.
The initial value, when t = 0, is 2. The value when t = 1 is 3. Since 3 ÷ 2 = 1.5, the
t
function that represents the value of Stock B after t years is B(t) = 2(1.5) . B(t) is an
exponential growth function.
The value of Stock A is going down over time. The value of Stock B is going up over time.
The initial value of Stock A is greater than the initial value of Stock B. However, after about
3 years, the value of Stock B becomes greater than the value of Stock A.
Module 13
A1_MTXESE353879_U5M13L2.indd 623
623
Lesson 13.2
623
Lesson 2
13/11/14 8:54 PM
The graph shows the value of two different shares of stocks over the period of 4 years since
they were purchased. The values have been changing exponentially.
The graph for Stock A shows that the value of the stock is
decreasing
as time increases.
The initial value, when t = 0, is 100 . The value when t = 1
is 50 . Since 50 ÷ 100 = 0.5 , the function that
represents the value of Stock A after t years is A(t) =
100 ( 0.5 ) .
t
Value (dollars per share)
B
A(t) is an exponential decay function.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
y
100 (0, 100)
Stock A
75
When modeling real-world examples of exponential
growth and decay, ensure that students understand
the meaning of the real-world values on the graph.
Point out that the independent variable represents
time, while the meaning of the dependent variable
can vary with the situation. Remind students to label
the axes of their graphs with the correct quantities
and units. Students should also recognize that the
y-intercept is the value of the independent variable at
time t = 0, which may be called its initial value or
starting value.
(1, 50)
50
Stock B
25
(1, 3)
(0, 1.5)
0
2
1
3
4
Time (years since purchase)
The graph for Stock B shows that the value of the stock is increasing as time increases.
The initial value, when t = 0, is 1.5 . The value when t = 1 is 3 . Since 3 ÷ 1.5 = 2 , the
( 2 ) . B(t) is an exponential
t
function that represents the value of Stock B after t years is B(t) = 1.5
growth function.
The value of Stock A is going down over time. The value of Stock B is going up over time.
The initial value of Stock A is greater than the initial value of Stock B. However, after about
3 years,
the value of Stock B becomes greater than the value of Stock A.
Reflect
14. Discussion In the function B(t) = 1.5(2) , is it likely that the value of B can be accurately predicted
in 50 years?
The exponential function grows much more quickly than any stock can reasonably grow, it
t
is unlikely that the value of B can be accurately predicted in 50 years by using the function.
t(0) = 150
t(0) = 5
t(1) = 45
45 ÷ 150 = 0.3
t(1) = 15
A(t) = 150(0.3)
t
15 ÷ 5 = 3
B(t) = 5(3)
t
Value
(dollars per share)
15. The graph shows the value of two different shares of stocks over the
period of 4 years since they were purchased. The values have been
changing exponentially. Use the graphs provided to write the equations
of the functions. Then describe and compare the behaviors of both
functions.
Stock B:
Stock A:
y
150 (0, 150)
Stock A
100
50
(0, 5)
0
(1, 45)
(1, 15)
Stock B
t
1
3
2
Time
(years since purchase)
© Houghton Mifflin Harcourt Publishing Company
Your Turn
The value of Stock A is going down over time. The value of Stock B is going up over time.
The initial value of Stock A is greater than the initial value of Stock B. However, after about
1.5 years, the value of Stock B becomes greater than the value of Stock A.
Module 13
A1_MTXESE353879_U5M13L2.indd 624
624
Lesson 2
2/14/15 6:58 PM
Modeling Exponential Growth and Decay
624
Elaborate
ELABORATE
16. If b > 1 in a function of the form y = ab x, is the function an example of exponential growth or an example
of exponential decay?
If b >1, the function is an example of exponential growth.
QUESTIONING STRATEGIES
17. What is an asymptote of the function y = 35( 1.1 )x ?
x
An asymptote of the function y = 35(1.1) is y = 0.
Describe three real-world situations that can
be described by exponential growth or
exponential decay functions. Possible answers:
interest earned on an investment, population
growth or decline, radioactive decay
18. Essential Question Check-In What equation should be used when modeling an exponential function
that models a decrease in a quantity over time?
When modeling an exponential function that models a decrease in a quantity over time,
the equation y = a(1 - r) should be used.
t
SUMMARIZE
How do you write an exponential growth or
decay function? The formula for growth is
t
y = a(1 + r) , and the formula for decay is
t
y = a(1 - r) , where y represents the final amount,
a represents the original amount, r represents the
rate of growth expressed as a decimal, and t
represents time.
Evaluate: Homework and Practice
Graph the function on a graphing calculator, and state its domain, range,
end behavior, and an asymptote.
1.
ƒ(x) = 300(1.16)
x
Domain: {x| - ∞ < x < ∞} Range: {y| y > 0}
End behavior: As x → -∞, y → 0 and as
x → ∞, y → ∞ Asymptote: y = 0
© Houghton Mifflin Harcourt Publishing Company
3.
ƒ(x) = 65(1.64)
x
x
4.
ƒ(x) = 57(0.77)
x
Domain: {x| - ∞ < x < ∞} Range: {y| y > 0}
End behavior: As x → -∞, y → ∞ and as
x → ∞, y → 0 Asymptote: y = 0
Write an exponential function to model each situation. Then find the value of the
function after the given amount of time.
5.
Annual sales for a company are $155,000 and
increases at a rate of 8% per year for 9 years.
6.
The value of a textbook is $69 and decreases at
a rate of 15% per year for 11 years.
y = 69(0.85)
y = 155,000(1.08)
t
t
= 69(0.85)
y = 155,000(1.08)
11
9
A1_MTXESE353879_U5M13L2 625
Lesson 13.2
ƒ(x) = 800(0.85)
Domain: {x| - ∞ < x < ∞} Range: {y| y > 0}
End behavior: As x → -∞, y → ∞ and as
x → ∞, y → 0 Asymptote: y = 0
Domain: {x| - ∞ < x < ∞} Range:{y| y > 0}
End behavior: As x → -∞, y → 0 and as
x → ∞, y → ∞ Asymptote: y = 0
= 155,000(1.999)
= 69(0.167)
= $309,845.72
= $11.52
Module 13
625
2.
• Online Homework
• Hints and Help
• Extra Practice
625
Lesson 2
23/02/14 12:34 PM
7.
A new savings account is opened with $300 and
gains 3.1% yearly for 5 years.
8.
y = 300(1.031)
y = 7800(0.92)
= 300(1.16)
= 7800(0.61)
t
EVALUATE
t
= 300(1.031)
= 7800(0.92)
5
6
= $349.47
9.
The value of a car is $7800 and decreases at a
rate of 8% yearly for 6 years.
= $4729.57
The starting salary at a construction company is
fixed at $55,000 and increases at a rate of 1.8%
yearly for 4 years.
ASSIGNMENT GUIDE
y = 55,000(1.018)
t
= 55,000(1.018)
4
= 55,000(1.074)
= $59,068.21
10. The value of a piece of fine jewelry is $280 and
decreases at a rate of 3% yearly for 7 years.
11. The population of a town is 24,000 and is
increasing at a rate of 6% per year for 3 years.
y = 280(0.97)
y = 24,000(1.06)
t
t
= 280(0.97)
= 24,000(1.06)
7
= 24,000(1.191)
= $226.24
= 28,584
12. The value of a new stadium is $3.4 million and decreases at a rate of 2.39% yearly for 10 years.
y = 3.4(0.9761)
t
= 3.4(0.785)
= $2.67 million
y = a(1 - r) = 192,000(0.93)
t
t
y = 192,000(0.93) ≈ 99,918,93
200,000
Value (dollars)
Write an exponential function for each situation. Graph each function
and state its domain and range. Determine what the y-intercept
represents in the context of the problem. On answer graphs,
coordinates may be rounded.
13. The value of a boat is depreciating at a rate of 7% per year. In 2004, the
boat was worth $192,000. Find the value of the boat in 2013.
150,000
100,000
9
50,000
y
(0, 192,000)
(4, 143,626)
(8, 107,440)
(12, 80,370)
t
After 9 years, the boat will be worth approximately $99,918.93.
0
4
8
12
Time (years since 2004)
Domain: {x|0 < x < ∞} Range: {y| y > 0}
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©CoolKengzz/Shutterstock
= 3.4(0.9761)
10
Module 13
Depth of Knowledge (D.O.K.)
Mathematical Processes
1 Recall
1.C Select tools
5–12
1 Recall
1.A Everyday life
13–20
2 Skills/Concepts
1.D Multiple representations
21–24
2 Skills/Concepts
1.C Select tools
1 Recall
1.C Select tools
3 Strategic Thinking
1.G Explain and justify arguments
26–28
Exercises 1, 3
Explore 2
Describing End Behavior of a Decay
Function
Exercises 2, 4
Example 1
Modeling Exponential Growth
Exercises 5, 7, 9,
11, 14, 16, 18, 20,
25
Example 2
Modeling Exponential Decay
Exercises 6, 8, 10,
12–13, 15, 17, 19,
26–28
Example 3
Comparing Exponential Growth
and Decay
Exercises 21–24
Students may want to check their results on a
graphing calculator. They can enter the
function in the Y = menu and then press GRAPH.
Then they can press the TABLE key to get a list of
x- and y-values, and can scroll up or down to see
more values. This method is especially convenient
when seeking the value of the function for more than
one input value, because it is an alternative to
repeatedly typing the expression into the calculator.
Lesson 2
626
1–4
25
Explore 1
Describing End Behavior of a
Growth Function
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Technology
The y-intercept is 192,000, the original value of the boat in 2004.
Exercise
Practice
3
= 280(0.808)
A1_MTXESE353879_U5M13L2.indd 626
Concepts and Skills
13/11/14 7:23 PM
Modeling Exponential Growth and Decay
626
14. The value of a collectible baseball card is increasing at a
rate of 0.5% per year. In 2000, the card was worth $1350.
Find the value of the card in 2013.
VISUAL CUES
Focus on visual cues as you discuss how to recognize
exponential growth and decay from a graph. Students
should recognize that a rising line represents growth
and a falling line represents decay. The steeper the
line, the greater the rate of growth or decay.
y = a(1 + r) = 1350(1.005)
t
t
y = 1350(1.005) ≈ 1440.43
13
After 13 years, the card will be worth
approximately $1440.43.
Domain: {x|0 < x < ∞} Range: {y| y > 0}
The y-intercept is 1350.00, the original value of
the card in 2000.
Value (dollars)
AVOID COMMON ERRORS
Some students may forget to add 1 to the rate of
growth in the exponential growth model. Remind
students that 1 + r is the growth factor in the model.
15. The value of an airplane is depreciating at a rate of 7% per year. In
2004, the airplane was worth $51.5 million. Find the value of the
airplane in 2013.
y = a(1 - r) = 51.5(0.93)
After 9 years, the airplane will be worth approximately
$26.8 million.
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©RoseOfSharon/Alamy
Domain: {x|0 < x < ∞} Range: {y| y > 0}
Value (dollars)
y = 51.5(0.93) ≈ 26.8
9
500
t
0
5
10 15
Time (years since 2000)
y (0, 51.5)
40 (2, 44.5)
30
(6, 33.3)
t
16. The value of a movie poster is increasing at a rate of 3.5% per year.
In 1990, the poster was worth $20.25. Find the value of the poster
in 2013.
t
t
y = 20.25(1.035)
23
≈ 44.67
After 23 years, the poster will be worth approximately
$44.67.
Domain: {x|0 < x < ∞} Range: {y| y > 0}
The y-intercept is 20.25, the original value of the poster
in 1990.
A1_MTXESE353879_U5M13L2.indd 627
627
(10, 24.9)
(8, 28.8)
20
0
y = a(1 + r) = 20.25(1.035)
(4, 38.5)
10
The y-intercept is 51.5, the original value of the airplane
in 2004.
Module 13
Lesson 13.2
t
t
Students may not be familiar with the word
depreciate. Explain that to depreciate is to decrease in
value. One meaning of the word appreciate is the
opposite of depreciate: to increase in value.
(15, 1455)
(0, 1350) (10, 1419)
1,000
50
50
Value (dollars)
CONNECT VOCABULARY
627
(5, 1384)
y
1,500
2
4
6
8
10
Time (years since 2004)
y
40
(15, 33.93)
30
(5, 24.05)
20
(0, 20.25)
(20, 40.29)
(10, 28.57)
10
t
0
5
10 15 20 25
Time (years since 1990)
Lesson 2
13/11/14 8:57 PM
17. The value of a couch is decreasing at a rate of 6.2% per year. In 2007,
the couch was worth $1232. Find the value of the couch in 2014.
y = a(1 - r) = 1232(0.938)
t
t
y = 1232(0.938) ≈ 787.10
7
After 7 years, the couch will be worth approximately $787.10.
Value (dollars)
1600
1200
(2, 1083.97)
800
t
t
y = 34,567(1.022) ≈ 50,041
17
After 17 years, the town will have about 50,041 people.
Population
y = a(1 + r) = 34,567(1.022)
Domain: {x|0 < x < ∞} Range: {y| y > 0}
0
2
4
6
8
Time (years since 2007)
y = 131,000(0.946) ≈ 75,194
10
After 10 years, the house will be worth about $75,194.
30,000 (0, 34,567)
t
y = 113(1.0494) ≈ 201.54
12
After 12 years, the bank account will hold about $201.54.
Domain: {x|0 < x < ∞} Range: {y| y > 0}
The y-intercept is 113, the original amount in the account
in 2005.
A1_MTXESE353879_U5M13L2.indd 628
628
y
(0, 131,000)
(2, 117,234)
(4, 104,915)
60,000
30,000
t
0
200
Value (dollars)
y = a(1 + r) = 113(1.0494)
3
6
9
12 15
Time (years since 2001)
120,000
(1, 123,926)
90,000
(3, 110,903)
20. An account is gaining value at a rate of 4.94% per year. The
account held $113 in 2005. What will the bank account hold
in 2017?
Module 13
Some students may fail to write percents as decimals
before applying the exponential growth model.
Encourage them to write out the decimal equivalent
of a growth or decay rate before adding it to 1 or
subtracting it from 1 in the exponential equation.
20,000
150,000
The y-intercept is 131,000, the original value of the
house in 2009.
t
AVOID COMMON ERRORS
(12, 44,882)
(6, 39,388)
1
2
3
4
5
Time (years since 2009)
y
(8, 166.19)
160
(0, 113)
120
80
(4, 137.04)
(6, 150.91)
(2, 124.44)
© Houghton Mifflin Harcourt Publishing Company
Domain: {x|0 < x < ∞} Range: {y| y > 0}
40,000
t
Value (dollars)
t
(15, 47,910)
(9, 42,046)
(3, 36,899)
y
0
19. A house is losing value at a rate of 5.4% per year. In 2009,
the house was worth $131,000. Find the value of the house
in 2019.
t
(6, 839.13)
10,000
The y-intercept is 34,567, the original population of the
town in 2001.
y = a(1 - r) = 131,000(0.946)
The population of a town is decreasing at the
rate of 1% a year. You are asked to find the
population after four years. Solving the exponential
growth function, you get an answer of 1248.77. What
answer will you record? Explain. Round the
answer to 1249 because there cannot be a
fraction of a person.
(4, 953.72)
(8, 738.30)
The y-intercept is 1232, the original value of the couch
in 2007.
50,000
QUESTIONING STRATEGIES
400
Domain: {x|0 < x < ∞} Range: {y| y > 0}
18. The population of a town is increasing at a rate of 2.2% per
year. In 2001, the town had a population of 34,567. Find the
population of the town in 2018.
y
(0, 1232.00)
40
t
0
2
4
6
8
10
Time (years since 2005)
Lesson 2
13/11/14 8:59 PM
Modeling Exponential Growth and Decay
628
Use a calculator to graph the functions. Describe and compare each pair of functions.
PEERTOPEER DISCUSSION
21. A(t) = 13(0.6) and B(t) = 4(3.2)
t
Ask students to discuss with a partner how to write
an exponential growth or decay model from a verbal
description of a real-world situation. They should
consider which key words tell whether the situation
involves growth or decay, what the starting value a is,
and how to use the growth or decay rate r in
the equation.
t
The value of A(t) is decreasing. The value of B(t) is increasing. The initial
value of A(t) is greater than the initial value of B(t). However, for t greater
than about 0.7, the value of B(t) becomes greater than the value of A(t).
22. A(t) = 9(0.4) and B(t) = 0.6(1.4)
t
t
The value of A(t) is decreasing. The value of B(t) is increasing. The initial
value of A(t) is greater than the initial value of B(t). However, for t greater
than about 2.2, the value of B(t) becomes greater than the value of A(t).
23. A(t) = 547(0.32) and B(t) = 324(3)
t
t
The value of A(t) is decreasing. The value of B(t) is increasing. The initial
value of A(t) is greater than the initial value of B(t). However, for t greater
than about 0.2, the value of B(t) becomes greater than the value of A(t).
JOURNAL
t
t
24. A(t) = 2(0.6) and B(t) = 0.2(1.4)
Have students write a journal entry in which they
describe how to graph an exponential growth or
decay function, and how to find the equation for a
real-world situation involving exponential growth
or decay.
The value of A(t) is decreasing. The value of B(t) is increasing. The initial
value of A(t) is greater than the initial value of B(t). However, for t greater
than about 2.7, the value of B(t) becomes greater than the value of A(t).
25. Identify the y-intercept of each of the exponential functions.
a. 3123(432,543)
(0, 3123)
d. 76(89, 047, 832)
b. 0
(0, 0)
e. 1
x
c. 45(54)
x
x
(0, 76)
(0, 1)
(0, 45)
© Houghton Mifflin Harcourt Publishing Company
H.O.T. Focus on Higher Order Thinking
26. Explain the Error A student was asked to find the value of a $2500 item after 4 years. The
item was depreciating at a rate of 20% per year. What is wrong with the student’s work?
4
2500(0.2)
$4
t
t
The exponential decay function is y = 2500(1 - 0.2) , or y = 2500(0.8) . The
student forgot to subtract the rate of depreciation from 1 before solving.
27. Make a Conjecture The value of a certain car can be modeled by the function
t
y = 18000(0.76) , where t is time in years. Will the value of the function ever be 0?
The value of the function will never be 0 because the right side of the
function is a product of positive numbers. Although the value can
become extremely close to 0, it can never equal 0.
28. Communicate Mathematical Ideas Explain how a graph of an exponential function may
resemble the graph of a linear function.
A graph of an exponential function may appear to be a linear function if only a small
part of the graph is shown and the values in that part are changing slowly.
Module 13
A1_MTXESE353879_U5M13L2.indd 629
629
Lesson 13.2
629
Lesson 2
11/17/14 4:41 PM
Lesson Performance Task
CONNECT VOCABULARY
Some students may not be familiar with the terms
archeologist, artifacts, or radiometric dating. Explain
that an archeologist studies prehistoric people and
cultures by analyzing items that remain for us to
find, such as their bones, weapons, artifacts, and
tools. Artifacts are objects that the people made, such
as pottery. Radiometric dating, also called radioactive
dating, is a method of determining the age of an
object based on the rate of decay of an element in
the object.
Archeologists have several methods of determining the age of
recovered artifacts. One method is radioactive dating.
All matter is made of atoms. Atoms, in turn, are made of protons,
neutrons, and electrons. An “element” is defined as an atom with a
given number of protons. Carbon, for example, has exactly 6 protons.
Carbon atoms can, however, have different numbers of neutrons.
These are known as “isotopes” of carbon. Carbon-12 has 6 neutrons,
carbon-13 has 7 neutrons, and carbon-14 has 8 neutrons. All carbonbased life forms contain these different isotopes of carbon.
Carbon-12 and carbon-13 account for over 99% of all the carbon in
living things. Carbon-14, however, accounts for approximately 1 part
per trillion or 0.0000000001% of the total carbon in living things. More importantly, carbon-14
is unstable and has a half-life of approximately 5700 years. This means that, within the span of
5700 years, one-half of any amount of carbon will “decay” into another atom. In other words, if
you had 10 g of carbon-14 today, only 5 g would remain after 5700 years.
But, as long as an organism is living, it keeps taking in and releasing carbon-14, so the level of
it in the organism, as small as it is, remains constant. Once an organism dies, however, it no
longer ingests carbon-14, so the level of carbon-14 in it drops due to radioactive decay. Because
we know how much carbon-14 an organism had when it was alive, as well as how long it takes
for that amount to become half of what it was, you can determine the age of the organism by
comparing these two values.
QUESTIONING STRATEGIES
What is the growth factor (the base raised to a
power) in the exponential model? Explain. It
is 1 - 0.5 = 0.5, because the amount of carbon-14
decreases by half during every time period.
n
Carbon-14 Concentration In
Parts Per Quadrillion
C(n) = 1000(0.5)
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©worker/
Shutterstock
Use the information presented to create a function that will model the amount of carbon-14 in
a sample as a function of its age. Create the model C(n) where C is the amount of carbon-14 in
parts per quadrillion (1 part per trillion is 1000 parts per quadrillion) and n is the age of the
sample in half-lives. Graph the model.
1000
800
600
400
200
0
2
4
6
8
Age in Half-Lives
Module 13
630
Lesson 2
EXTENSION ACTIVITY
A1_MTXESE353879_U5M13L2.indd 630
Have students research how very large numbers such as million, billion, trillion,
and quadrillion compare to each other. Then have students rewrite the model
C(n) in different units and tell how the graph would change.
Students may find that since 1 trillion, or 10 12, is equal to 1000 billion, or 10 3 ∙ 10 9,
1 part per trillion is the same as 0.001 parts per billion. Therefore, the model
could be C(n) = 0.001(0.5)n. The graph would differ only in that the scale of the
vertical axis would be in parts per billion rather than parts per quadrillion.
13/11/14 7:23 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Modeling Exponential Growth and Decay
630