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LESSON
15.3
Name
Constructing
Exponential Functions
Class
Date
15.3 Constructing Exponential
Functions
Essential Question: What are discrete exponential functions and how do you represent them?
Common Core Math Standards
Resource
Locker
The student is expected to:
Explore
F-LE.2
Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or
two input-output pairs (include reading these from a table). Also F-IF.2,
F-IF.7e
Understanding Discrete Exponential Functions
Recall that a discrete function has a graph consisting of isolated points.

Mathematical Practices
The table represents the cost of tickets to an annual event as a function of the number t of
tickets purchased. Complete the table by adding 10 to each successive cost. Plot each ordered
pair from the table.
MP.6 Precision
Tickets t
Cost ($)
(t, f(t))
1
10
(1, 10)
2
20
(2, 20)
3
30
4
40
5
50
Language Objective
Explain to a partner what the graph of a discrete exponential function
looks like.
Possible answer: Discrete exponential functions are
exponential functions whose domains are limited,
such as the set of integers, making the graph appear
as isolated points instead of a continuous curve.
4, 40
5, 50
40
30
20
10
x
0
1
2
3
4
5
Tickets
Module 15
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DO NOT Key=NL-A;CA-A
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Lesson 3
693
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F-IF.7e
A1_MNLESE368187_U6M15L3.indd 693
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ing Discrete
Explore
Understand
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Resource
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Also F-IF.2,
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Exponentia
HARDCOVER PAGES 545554
Functions
.
isolated points
of
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has a graph
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a function
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a discrete
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an annua
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by adding
represents
lete the table
The table
ased. Comp
tickets purch
(t, f(t))
the table.
pair from
Cost ($)
(1, 10)
Tickets t
10
1
(2, 20)
20
2
3, 30
30
3
4, 40
Turn to these pages to
find this lesson in the
hardcover student
edition.
Recall that

(
40
4
5
y
g Compan
(
)
(
)
5, 50
50
)
y
60
n Mifflin
Harcour t
Cost ($)
Publishin
50
40
30
© Houghto
20
x
10
0
1
3
2
4
5
Tickets
Lesson 3
693
Module 15
Lesson 15.3
L3.indd
7_U6M15
SE36818
A1_MNLE
693
)
)
)
50
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo
and why the population of invasive species might
grow exponentially. Then preview the Lesson
Performance Task.
3, 30
y
60
Cost ($)
Essential Question: What are discrete
exponential functions and how do you
represent them?
© Houghton Mifflin Harcourt Publishing Company
ENGAGE
(
(
(
693
3/24/14
11:20 AM
3/24/14 11:15 AM
B
The number of people attending an event doubles each year. The table represents the total
attendance at each annual event as a function of the event number n. Complete the table by
multiplying each successive attendance by 2. Plot each ordered pair from the table.
Event Number n
Attendance
(n, g(n))
1
20
(1, 20)
2
40
(2, 40)
3
80
4
160
5
320
360
(
(
(
3, 80
4, 160
5, 320
EXPLORE
Understanding Discrete Exponential
Functions
)
)
)
INTEGRATE TECHNOLOGY
Have students complete the Explore activity in either
the book or online lesson.
QUESTIONING STRATEGIES
y
How are the tables for the functions
different? They are different because
successive output values of the first function have a
common difference, while successive output values
of the second function have a common ratio.
320
Attendance
280
240
200
160
120
80
© Houghton Mifflin Harcourt Publishing Company
40
x
0
1
2
3
4
5
Event Number
C
Complete the table.
Function
Linear?
Discrete?
f(t)
Yes
Yes
g(n)
No
Yes
Reflect
1.
Communicate Mathematical Ideas What are the limitations on the domains of these functions? Why?
The domains include only whole numbers. You cannot buy a fraction of a ticket to an
event, nor stage a fraction of an event.
Module 15
694
Lesson 3
PROFESSIONAL DEVELOPMENT
A1_MNLESE368187_U6M15L3.indd 694
Learning Progressions
3/24/14 11:15 AM
In this lesson, students learn how to identify and represent exponential functions.
Some key understandings for students are as follows:
• An exponential function can be represented by an equation of the form
f (x)= abx, where a, b, and x are real numbers, a ≠ 0, b > 0, and b ≠ 1.
• For an exponential function ƒ(x), if x is in the domain of the function, then the
ratio of ƒ(x+1) to ƒ(x) is a constant ratio.
In the next module, students will learn more about exponential functions,
including exponential growth and decay functions.
Constructing Exponential Functions
694
Representing Discrete Exponential Functions
Explain 1
EXPLAIN 1
An exponential function is a function whose successive output values are related by a constant ratio. An exponential
function can be represented by an equation of the form ƒ(x) = ab x, where a, b, and x are real numbers, a ≠ 0, b > 0,
and b ≠ 1. The constant ratio is the base b.
Representing Discrete Exponential
Functions
When evaluating exponential functions, you will need to use the properties of exponents, including zero and negative
exponents.
Recall that, for any nonzero number c:
c 0 = 1, c ≠ 0
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Point out that the constant b in a function of
1 , c ≠ 0.
c -n = _
cn
Example 1
the form f (x) = abx, where a and b are real numbers,
a ≠ 0, b > 0, and b ≠ 1, corresponds to the common
ratio, r, in a geometric sequence. Explain to students
that if the domain of the function is the set of positive
integers, the points of the graph of a discrete
exponential function represent a geometric sequence.
The x-values are the term numbers of the sequence
and the y-values are the corresponding terms.

2 = __
2 = 2.
reciprocal of the denominator: 1∙ __
1 1
2
© Houghton Mifflin Harcourt Publishing Company
__1
( )
1 x with a domain of ⎧⎨-1, 0, 1, 2, 3, 4⎫⎬
ƒ(x) = 3 ⋅ _
⎭
⎩
2
-1
1 = 3⋅ _
1 = 3⋅2= 6
ƒ(-1) = 3 ⋅ _
2
1
_
2
QUESTIONING STRATEGIES
How do you evaluate a complex fraction, such
1 ? Multiply the numerator by the
as __
Complete the table for each function using the given domain. Then graph
the function using the ordered pairs from the table.
( )
()
(x, f(x))
x
f(x)
-1
6
(-1, 6)
0
3
(0, 3)
1
1
1_
2
2
3
_
4
3
3
_
8
4
3
_
16
(1, 1_12 )
(2, _34 )
(3, _38 )
3
(4, _
16 )
y
6
5
4
3
2
1
x
-1
0
1
Module 15
2
3
695
4
5
Lesson 3
COLLABORATIVE LEARNING
A1_MNLESE368187_U6M15L3.indd 695
Peer-to-Peer Activity
Have students work in pairs. Instruct one student in each pair to construct
a table of values and graph for the function defined by h(x) = 3x with the domain
⎧
⎫
⎨-2, -1, 0, 2⎬. Instruct the other student to do the same for the function defined
⎩
⎭
by g(x) = 3x with the same domain. Have the students compare and contrast the
tables of values and the graphs.
695
Lesson 15.3
8/21/14 4:18 PM
B
( )
4 x; domain = ⎧⎨-2, -1, 0, 1, 2, 3⎫⎬
ƒ(x) = 3 _
⎩
⎭
3
()
4
ƒ(-2) = 3 _
3
-2
( )
3
x
2
9
3
27
11
1
= 3⋅ _
= 3 ⋅ _2 = 3 ⋅ _ = _ = 1_
2
16
4
16
16
4
_
2
f(x)
-2
11
1__
16
-1
1
2_
4
0
3
1
4
2
1
5_
3
3
1
7_
9
(
(
(x, f(x))
)
_ )
( )
( )
( _)
__
-2, 1 16
11
-1,
214
0,
3
1,
4
2,
513
(3, 7_91 )
y
8
6
4
© Houghton Mifflin Harcourt Publishing Company
2
x
-4
-2
0
2
4
Reflect
2.
What If What would happen to the function ƒ(x) = ab x if a were 0? What if b were 1?
If a = 0 then ƒ(x) = 0 for all x, and if b = 1, then ƒ(x) = a for all x. Neither of these
constant functions are exponential functions.
3.
Discussion Why is a geometric sequence a discrete exponential function?
A sequence is a function. When the input values increase by 1, the successive output
values are related by a constant ratio, the common ratio. The general recursive rule for a
geometric sequence is the form of an exponential function.
Module 15
696
Lesson 3
DIFFERENTIATE INSTRUCTION
A1_MNLESE368187_U6M15L3.indd 696
Cognitive Strategies
8/21/14 4:18 PM
Encourage students to isolate the necessary information from any problem
scenario they are given. It may help to circle or underline values or to write the
important information on a separate sheet of paper.
Constructing Exponential Functions
696
Your Turn
EXPLAIN 2
Make a table for the function using the given domain. Then graph the function using
the ordered pairs from the table.
Constructing Exponential Functions
from Verbal Descriptions
4.
()
x
3 ; domain = ⎧⎨-3, -2, -1, 0, 1, 2⎫⎬
(x) = 4 _
⎩
⎭
2
y
x
8
MODELING
-3
6
Have students fold a piece of paper in half repeatedly.
Have them make a table representing the number of
layers of paper after 0 folds, 1 fold, 2 folds, and so on.
-2
4
-1
2
x
-4
-2
0
2
4
QUESTIONING STRATEGIES
How do you find the value of b in order to
write an exponential function in the form
ƒ(x) = abx ? Choose a number x in the domain of
the function. Then divide f(x + 1) by f(x) to get b.
Explain 2
f(x)
32
5
= 1_
(_
27 )
27
16
7
= 1_
(_
9
9)
8
2
_
_
(3) = 23
(x, f(x))
5
(-3, 1_
27 )
(-2, 1_79 )
(-1, 2_23 )
0
4
(0, 4)
1
6
(1, 6)
2
9
(2, 9)
Constructing Exponential Functions
from Verbal Descriptions
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Digital
Vision/Getty Images
You can write an equation for an exponential function f(x) = ab x by finding or calculating the values of a and b.
The value of a is the value of the function when x = 0. The value of b is the common ratio of successive function
ƒ(x + 1)
values, b = _______
. For discrete functions with integer or whole number domains, these will be successive values
ƒ(x)
of the function.
Example 2

Write an equation for the function.
When a piece of paper is folded in half, the total thickness
doubles. Suppose an unfolded piece of paper is 0.1 millimeter
thick. The total thickness t(n) of the paper is an exponential
function of the number of folds n.
The value of a is the original thickness of the paper before any
folds are made, or 0.1 millimeter.
Because the thickness doubles with each fold, the value of b
(the constant ratio) is 2.
The equation for the function is t(n) = 0.1(2) .
n
Module 15
697
Lesson 3
LANGUAGE SUPPORT
A1_MNLESE368187_U6M15L3.indd 697
Connect Vocabulary
In this lesson there are several key words that begin with the prefix in-. Learning
the meaning of prefixes helps all students of math and science, since many key
terms in both disciplines originated in Latin or have Latin prefixes. In this lesson,
the words increase and input both start with the prefix in-, which means in, into,
on, near, or towards. The prefix in- can also mean not, as in the word inverse.
Another key prefix in this lesson is non-, which means not. The term nonzero, for
example, means not zero.
697
Lesson 15.3
8/21/14 4:18 PM

A savings account with an initial balance of $1000 earns 1% interest per month. That means
that the account balance grows by a factor of 1.01 each month if no deposits or withdrawals
are made. The account balance in dollars B(t) is an exponential function of the time t in
months after the initial deposit.
EXPLAIN 3
Constructing Exponential Functions
from Input-Output Pairs
Let B represent the balance in dollars as a function of time t in months.
1000 .
The value of a is the original balance,
The value of b is the factor by which the balance changes every month,
)
(
The equation for the function is B(t) = 1000 1.01 .
1.01 .
t
AVOID COMMON ERRORS
When calculating the value of b, students may divide
in the wrong order. Remind them that they should
always divide an output value by the preceding
output value. For example, if they are using (4, 27)
and (5, 9), they should divide the function value of
the second pair by the function value of the first pair:
Reflect
5.
Why is the exponential function in the paper-folding example discrete?
The paper can only be folded a whole number of times.
Your Turn
6.
A piece of paper that is 0.2 millimeters thick is folded. Write an equation for the thickness t of the paper in
millimeters as a function of the number n of folds.
Initial thickness (n = 0)
a = 0.2
Change per fold: double
b=2
t(n) = 0.2 ⋅ 2 n
Equation:
Explain 3
9 = __
1.
b = ___
27 3
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 After students have found an equation for an
Constructing Exponential Functions
from Input-Output Pairs
You can use given two successive values of a discrete exponential function to write an equation for the function.
Example 3
(3, 12) and (4, 24)
24 = 2.
Find b by dividing the function value of the second pair by the function value of the first: b = _
12
Evaluate the function for x = 3 and solve for a.
Write the general form.
ƒ(x) = ab x
Substitute the value for b.
ƒ(x) = a ⋅ 2 x
Substitute a pair of input-output values.
12 = a ⋅ 2 3
Simplify.
12 = a ⋅ 8
Solve for a.
Use a and b to write an equation for
the function.
Module 15
A1_MNLESE368187_U6M15L3.indd 698
© Houghton Mifflin Harcourt Publishing Company

exponential function, have them check both pairs in
the equation to verify that the equation is correct.
Write an equation for the function that includes the points.
3
a= _
2
3 ⋅ 2x
ƒ(x) = _
2
698
Lesson 3
8/21/14 4:18 PM
Constructing Exponential Functions
698
B
QUESTIONING STRATEGIES
9 ÷3=
Find b by dividing the function value of the second pair by the first: b = _
4
Write the general form.
ƒ(x) = ab x
If an exponential function includes the points
(4, 64) and (5, 32), what is the common ratio?
Explain how you found it. The common ratio is 0.5.
64 = __
1 = 0.5 .
I found it by dividing 32 by 64: ___
32 2
Substitute the value for b.
Substitute a pair of input-output values.
ELABORATE
CRITICAL THINKING
.
4
x
(_34)
1
( )
3
3 = a⋅ _
4
3= a⋅
Solve for a.
a=
ƒ(x) =
_3
4
4
()
3
4 _
4
x
Your Turn
SUMMARIZE THE LESSON
(-2, _52 ) and (-1, 2)
2 =5
b=2÷_
5
a
2 = a ⋅ 5 -1; 2 = _; a = 10
5
f(x) = 10 ⋅ 5 x
Elaborate
© Houghton Mifflin Harcourt Publishing Company
8.
Explain why the following statement is true: For 0 < b < 1 and a > 0, the function ƒ(x) = ab decreases
as x increases.
When you multiply a number a by a fraction b between 0 and 1, the product is less than
x
the original number a. When you multiply by this fraction b repeatedly, the results
decrease further.
9.
Explain why the following statement is true: For b > 1 and a > 0, the function ƒ(x) = ab increases as
x increases.
When you multiply a number a by a number b greater than 1, the product is greater
x
than the original number a. When you multiply by this number b repeatedly, the results
increase further.
10. Essential Question Check-In What property do all pairs of adjacent points of a discrete exponential
function share?
The ratio of function values is the same for any two pairs.
Module 15
A1_MNLESE368187_U6M15L3.indd 699
Lesson 15.3
_3
Write an equation for the function that includes the points.
7.
699
ƒ(x) = a ⋅
Simplify.
Use a and b to write an equation for
the function.
Have students give an example of an increasing
exponential function and of a decreasing exponential
function.
What are discrete exponential functions and
how can you represent them? A discrete
exponential function is a function of the form
f (x) = abx , with a ≠ 0, b > 1, and b ≠ 1, such that
the graph of the function is a set of isolated points.
For example, the domain of the function could be
the set of whole numbers. Such a function can be
represented by a set of input and output values in a
table or by a graph of ordered pairs.
( )
9
(1, 3) and 2, _
4
699
Lesson 3
8/21/14 4:18 PM
EVALUATE
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
Complete the table for each function using the given domain.
Then graph the function using the ordered pairs from the table.
1.
1 ⋅ 4 x; domain = ⎧⎨-2, -1, 0, 1, 2⎫⎬.
ƒ(x) = _
⎩
⎭
2
x
y
8
-2
6
-4
2.
-2
2
0
_1
1
2
2
8
0
2
4
()
32
ASSIGNMENT GUIDE
(-1, _18)
(0, _12)
_1
4
8
2
(1, 2)
(2, 8)
x
1 ; domain = ⎧⎨0, 1, 2, 3, 4, 5⎫⎬.
ƒ(x) = 9 _
⎩
⎭
3
f(x)
8
0
9
6
1
3
4
2
1
2
3
_1
4
_1
5
1
__
x
f(x)
-1
9
x
0
2
4
6
8
()
(x, f(x))
(0, 9)
(1, 3)
(2, 1)
(3, _13)
(4, _19)
1
(5, __
27 )
3
9
27
x
2 ; domain = ⎧⎨-1, 0, 1, 2, 3, 4⎫⎬.
ƒ(x) = 6 _
⎩
⎭
3
y
8
6
0
6
4
1
4
2
2
2
2_
3
3
7
1_
9
4
5
1__
27
x
0
2
4
6
8
Module 15
Exercise
A1_MNLESE368187_U6M15L3.indd 700
(x, f(x))
(-1, 9)
(0, 6)
(1, 4)
(2, 2_23)
(3, 1_79)
5
(4, 1__
27 )
Depth of Knowledge (D.O.K.)
Mathematical Practices
1 Recall of Information
MP.6 Precision
5–8
2 Skills/Concepts
MP.4 Modeling
9–21
2 Skills/Concepts
MP.2 Reasoning
1 Recall of Information
MP.2 Reasoning
3 Strategic Thinking
MP.3 Logic
23–24
Practice
Explore
Understanding Discrete
Exponential Functions
Exercise 19
Example 1
Representing Discrete
Exponential Functions
Exercises 1–4,
22–23
Example 2
Constructing Exponential Functions
from Verbal Descriptions
Exercises 5–8,
20–21
Example 3
Constructing Exponential Functions
from Input-Output Pairs
Exercises 9–18,
24
Lesson 3
700
1–4
22
Concepts and Skills
© Houghton Mifflin Harcourt Publishing Company
x
y
3.
1
__
32
-1
x
(x, f(x) )
(-2, __1 )
f(x)
3/24/14 11:15 AM
Constructing Exponential Functions
700
4.
QUESTIONING STRATEGIES
()
x
4 ; domain = ⎧⎨-3, -2, -1, 0, 1⎫⎬
ƒ(x) = 6 _
⎩
⎭
3
Why is the restriction b ≠ 1 in the definition
of an exponential function necessary? What
would happen if b = 1? 1 raised to any power is 1,
so the function would always have the same value.
It would not increase or decrease.
8
y
6
4
2
x
-8
-6
-4
-2
0
(x, f(x))
17
)
(-3, 2__
x
f(x)
-3
17
2__
32
-2
3
3_
-1
1
4_
2
0
6
(0, 6)
1
8
(1, 8)
8
32
(-2, 3_38)
(-1, 4_12)
Write an equation for each function.
5.
Business A recent trend in advertising is viral marketing. The goal is to convince
viewers to share an amusing advertisement by e-mail or social networking. Imagine
that the video is sent to 100 people on day 1. Each person agrees to send the video to
5 people the next day, and to request that each of those people send it to 5 people the
day after they receive it. The number of viewers v(n) is an exponential function of the
number n of days since the video was first shown.
a = 100
b=5
v(n) = 100 ⋅ 5 n
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©David J.
Green - technology/Alamy
6.
A pharmaceutical company is testing a new antibiotic. The number of bacteria
present in a sample when the antibiotic is applied is 100,000. Each hour, the number
of bacteria present decreases by half. The number of bacteria remaining r(n) is an
exponential function of the number n of hours since the antibiotic was applied.
a = 100,000
1
b=_
2
7.
a=5
b = 0.99, (power remaining after 1% lost)
p(n) = 5(0.99)
A1_MNLESE368187_U6M15L3 701
Lesson 15.3
n
Optics A laser beam with an output of 5 milliwatts is
directed into a series of mirrors. The laser beam loses 1% of
its power every time it reflects off of a mirror. The power p(n)
is a function of the number n of reflections.
Module 15
701
()
1
r(n) = 100,000 _
2
n
701
Lesson 3
6/8/15 12:35 PM
8.
The NCAA basketball tournament begins with 64 teams, and after each round, half
the teams are eliminated. The number of remaining teams t(n) is an exponential
function of the number n of rounds already played.
a = 64
1
b=_
2
AVOID COMMON ERRORS
Students may forget to find the value of both a and b
when writing a function rule for an exponential
function of the form ƒ(x) = ab x. Remind students
that they need to determine the common ratio and
the initial value of an exponential function in order
to write the function rule.
()
1
t(n) = 64 _
2
n
Write an equation for the function that includes the points.
9.
(2, 100) and (3, 1000)
10. (-2, 4) and (-1, 8)
8
b=
4
=2
_
1000
b = ____
100
= 10
8 = a ⋅ 2 -1
a
8=
2
a = 16
100 = a ⋅ 10 2
_
100 = 100a
a=1
f(x) = 16 ⋅ 2 x
f(x) = 10 x
11.
(1, _54 ) and (2, _32 )
12.
(_23)
(_45)
(-3, _161 ) and (-2, _83 )
(_38)
1
(__
16 )
b = ___
b =___
2 _
⋅5
=_
3 4
3 __
⋅ 16
=_
8 1
_4 = a ⋅ (_5)1
5
6
_4 = a ⋅ _5
5
6
4 _
a=_
⋅6
5 5
24
= __
25
x
24 _
f(x) = __
⋅ (5)
1
__
= a ⋅ 6 -3
16
1
1
__
= a ⋅ ___
16
216
1
a = __
⋅ 216
5
=_
6
16
= 13.5
© Houghton Mifflin Harcourt Publishing Company
25
=6
f(x) = 13.5 ⋅ 6 x
6
Use two points to write an equation for the function.
13.
x
f(x)
1
2
2
2
_
7
3
2
_
49
4
2
_
343
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(_27)
b = ___
2
1
=_
7
()
1
2=a⋅ _
7
1
1
2=a⋅_
7
a = 14
1
f(x) = 14 ⋅ _
7
()
702
x
Lesson 3
3/24/14 11:15 AM
Constructing Exponential Functions
702
Use points (-2, 53) and (-1, 530)
14.
x
f(x)
-4
0.53
-3
5.3
-2
53
-1
530
530
b = ___
53
= 10
530 = a ⋅ (10)
-1
1
530 = a ⋅ __
10
a = 5300
f(x) = 5300 ⋅ (10)
x
15.
8
2
6
b=_
3
(1, 6)
6
4
Use points (0, 3) and (1, 6)
y
=2
a=3
(0, 3)
f(x) = 3 ⋅ 2 x
x
-6
-4
0
-2
2
16.
8
y
Use points (1, 8) and (2, 4)
(1, 8)
4
b=_
8
1
=_
2
6
4
© Houghton Mifflin Harcourt Publishing Company
2
x
0
2
4
()
1
8=a _
2
(2, 4)
6
8
a
8=_
2
a = 16
1
()
1
f(x) = 16 ⋅ _
2
x
17. The height h(n) of a bouncing ball is an exponential function of the number n of
bounces. One ball is dropped and on the first bounce reaches a height of 6 feet. On
the second bounce it reaches a height of 4 feet.
4
b=_
6
2
=_
3
()
2
6=a _
3
1
2
6=a⋅_
3
3
a=6⋅_
2
=9
()
2
h(n) = 9 ⋅ _
3
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Lesson 15.3
n
703
Lesson 3
8/21/14 4:18 PM
18. A child starts a playground swing from standing and doesn’t use
her legs to keep swinging. On the first swing she swings forward
by 18 degrees, and on the second swing she only comes 13.5
degrees forward. The measure in degrees of the angle m(n) is an
exponential function of the number n of swings.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Point out to students that, unlike linear
13.5
b = ___
18
3
=_
4
functions, the letter b in an exponential function of
the form ƒ(x) = ab x does not represent the
y-intercept.
()
3
18 = a _
4
1
3
18 = a ⋅ _
4
4
a = 18 ⋅ _
3
= 24
3
m(n) = 24 _
4
()
n
19. Make a Prediction A town’s population has been declining in recent years. The
table shows the population since 1980. Is this data consistent with an exponential
function? Explain. If so, predict the population for 2010 assuming the trend holds.
Year
Population
1980
5000
1990
4000
2000
3200
2010
2560
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Steve Hix/
Somos Images/Corbis
To check for exponential behavior, find the ratios between successive
values of the function.
4000
4 ____
4
____
=_
, 3200 = _
5000
5 4000
5
4
The ratio between successive function values is _
, so this is consistent with
5
an exponential function.
4
3200 ⋅ _
= 2560
5
20. A piece of paper has a thickness of 0.15 millimeters. Write an equation to describe
the thickness t(n) of the paper when it is repeatedly folded in thirds, where n is the
number of foldings.
b=3
a = 0.15
t(n) = 0.15 ⋅ 3 n
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Lesson 3
6/12/15 9:53 AM
Constructing Exponential Functions
704
21. Probability The probability of getting heads on a single coin flip
is __12 . The probability of getting nothing but heads on a series of coin
flips decreases by __12 for each additional coin flip. Write an exponential
function for the probability p(n) of getting all heads in a series of n coin
flips.
JOURNAL
Have students describe a specific discrete exponential
function in at least two different ways. They can
describe it with words, a function rule, a table, and/or
a graph.
1
b=_
2
_1 = a(_1)
2
1
2
_1 = a ⋅ _1
2
2
a=1
n
p(n) = 1
2
(_)
22. Multipart Classification Determine whether each of the functions is
exponential or not.
a. ƒ(x) = x 2
b. ƒ(x) = 3 ⋅ 2 x
1
c. ƒ(x) = 3 ⋅ _ x
2
d. ƒ(x) = 1.001 x
e. ƒ(x) = 2 ⋅ x 3
1
f. ƒ(x) = _ ⋅ 5 x
10
Exponential
Not exponential
Exponential
Not exponential
Exponential
Not exponential
Exponential
Not exponential
Exponential
Not exponential
Exponential
Not exponential
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©mj007/
Shutterstock
H.O.T. Focus on Higher Order Thinking
23. Explain the Error Biff observes that in every math test he has taken this year, he
has scored 2 points higher than the previous test. His score on the first test was 56.
He models his test scores with the exponential function s(n) = 28 · 2 n where s(n) is
the score on his nth test. Is this a reasonable model? Explain.
No; Biff’s test scores do not increase by a constant ratio. Instead they go
up by 2 points each test. His scores are not consistent with an exponential
function.
24. Find the Error Kaylee needed to write the equation of an exponential function from
points on the graph of the function. To determine the value of b, Kaylee chose the
ordered pairs (1, 6) and (3, 54) and divided 54 by 6. She determined that the value of
b was 9. What error did Kaylee make?
She should use ordered pairs with x-values that differ by 1. The value
Kaylee found is not the constant ratio of successive values.
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Lesson 15.3
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Lesson 3
6/10/15 12:49 AM
Lesson Performance Task
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 Have students explain why they would choose
In ecology, an invasive species is a plant or animal species newly introduced to an ecosystem,
often by human activity. Because invasive species often lack predators in their new habitat,
their populations typically experience exponential growth. A small initial population grows to
a large population that drastically alters an ecosystem. Feral rabbits that populate Australia and
zebra mussels in the Great Lakes are two examples of problematic invasive species that grew
exponentially from a small initial population.
3 or its decimal equivalent, 1.5,
to use the fraction __
2
when solving Part C of the Lesson Performance Task.
An ecologist monitoring a local stream has been collecting samples of an unfamiliar fish species
over the past four years and has summarized the data in the table.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 After students have completed the graph of
Here are the results so far:
Year
Average Population Per Mile
2009
32
2010
48
2011
72
2012
108
the populations versus time, discuss how the value of
b, when a is positive, causes the graph to increase
over time. Then discuss what happens to the graph
t
when students let b = __23 in f (t) = 32 __23 .
2013
2014
()
a. Look at the data in the table and confirm that the growth pattern is exponential.
48
_ = 1.5
32
72
_ = 1.5
48
108
_ = 1.5
72
Since there is a common ratio, r = 1.5, the growth pattern is exponential.
b. Write the equation that represents the average population per square meter as
a function of years since 2009.
()
t
3
The function is f(t) = 32 _
.
2
© Houghton Mifflin Harcourt Publishing Company
c. Predict the average populations expected for 2013 and 2014.
2013: 162 individuals per mile
d. Graph the population versus time since 2009 and include
the predicted values.
Average population per mile
2014: 243 individuals per mile
250 P
200
150
100
50
t
0
1
2
3
4
5
Time (years)
Module 15
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Lesson 3
EXTENSION ACTIVITY
A1_MNLESE368187_U6M15L3.indd 706
Have students research the feral rabbit population in Australia or the zebra mussel
population in the Great Lakes. Have students find out how the population was
introduced into the new habitat and describe efforts to control the population.
3/24/14 11:15 AM
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.
Constructing Exponential Functions
706