LESSON
13.2
Name
Exponential Decay
Functions
Class
Date
13.2 Exponential Decay Functions
Essential Question: How is the graph of g(x) = ab x–h + k where 0 < b < 1 related to the
graph of f(x) = b x?
Resource
Locker
Common Core Math Standards
The student is expected to:
Explore 1
F.BF.3
and f(x) =
Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx),
and f(x + k) for specific values of k (both positive and negative); find the
value of k given the graphs. Experiment with cases and illustrate an
explanation of the effects on the graph using technology. Also F-IF.7e,
F-LE.2, A-REI.11


MP.4 Modeling
Language Objective
x
To begin, fill
in the table in order to find points along the function
x
ƒ(x) = __12 . You may need to review the rules of the properties of
exponents, including negative exponents.
()
What does the end behavior of this function appear
to be as x increases?
f(x) approaches 0.
Work with a partner to compare and contrast exponential decay and
exponential growth functions.

ENGAGE
x
x
8
-2
4
-1
2
1
y
6
f(x) =
-3
0
Plot the points on the graph and
draw a smooth curve through
them.
2
4
3
(__12 )
x
1
__1
2
__1
4
__1
8
2
© Houghton Mifflin Harcourt Publishing Company
Possible answer: The graph of g(x) = ab x - h + k
involves transformations of the graph of ƒ(x) = b x.
In particular, the graph of g(x) is a vertical stretch or
compression of the graph of ƒ(x) by a factor of ∣a∣, a
reflection of the graph across the x-axis if a < 0, and
a translation of the graph h units horizontally and k
units vertically.
1
(___
10 )
(__12 )
Exponential functions with bases between 0 and 1 can be transformed in a manner similar to exponential functions
1
with bases greater than 1. Begin by plotting the parent functions of two of the more commonly used bases: __12 and __
.
10
Mathematical Practices
Essential Question: How is the graph
of g(x) = ab x - h + k where 0 < b < 1
related to the graph of ƒ(x) = b x?
Graphing and Analyzing f(x) =
x
-4
0
-2
2
x
4
Complete the table for ƒ(x) =
1 x.
_
10

( )

Plot the points on the graph and
draw a smooth curve through
them.
y
1000
-2
100
-1
10
1
6
2
4
1
(___
10 )
-3
0
8
f(x) =
3
x
1
___1
10
1
___
100
1
____
1000
2
x
be
ges must
EDIT--Chan
DO NOT Key=NL-A;CA-A
Correction
2
4
Lesson 2
651
gh “File info”
made throu
Date
Class
nenti
13.2 Expo
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is the graph x
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ion: How
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Essential
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F.BF.3 For
A.REI.11
F.IF.7e, F.LE.2,
A2_MNLESE385900_U6M13L2.indd 651
nctions
al Decay Fu
Name
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HARDCOVER PAGES 473484
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Locker
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-3
the rules
fill in the
Explore
1
x
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to review
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ents.
ƒ(x) = 2
negative expon
including
exponents,
appear
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ior of this
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What does
increases?
to be as x
()


Harcour t
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
the table
Complete
1 x.
_
10
( )
__1
2
__1
4
__1
2
8
3
4
-2
f(x) =
x
x
2
0
4
1000
-3
100
-2
for ƒ(x) =
10
-1
1
8
2
6
3
4
2
-4
-2
1
0
y
and
on the graph
Plot the points curve through
h
draw a smoot
them.
(___101 )
x
2

1
0
y
6
-4
Turn to these pages to
find this lesson in the
hardcover student
edition.
2
-1
1
aches 0.
f(x) appro
and
on the graph
Plot the points curve through
h
draw a smoot
them.
x
4
-2

Publishin
View the Engage section online. Discuss the
photo and why it might be important to know the
amount of a radioactive isotope remaining in
the environment. Then preview the Lesson
Performance Task.
Module 13
0
-2
y
g Compan
PREVIEW: LESSON
PERFORMANCE TASK
-4
___1
10
1
___
100
1
____
1000
x
0
2
4
Lesson 2
651
Module 13
L2.indd
0_U6M13
SE38590
A2_MNLE
651
Lesson 13.2
651
3/31/14
10:00 PM
3/31/14 10:01 PM
F
Fill in the following table of properties:
f(x) =
(__12 )
x
f(x) =
⎧
⎫
⎨x ⎢ -∞ < x < ∞⎬
⎩
⎭
Domain
⎧
⎨y ⎢
⎩
Range
y > 0 ⎫⎬
⎭
End behavior as x → ∞
f(x) →
End behavior as x → -∞
f(x) → ∞
y-intercept
Graphing and Analyzing
ƒ(x) =
x
1
___
and ƒ(x) = 10
( )
⎧
⎨y ⎢
⎩
⎫
y > 0 ⎬⎭
0
G
Both of these functions [decrease/increase] throughout the domain.

1
Of the two functions, ƒ(x) = _
decreases faster.
1
QUESTIONING STRATEGIES
What is the decay factor in an exponential
decay function? the base b of an exponential
decay function y = ab x where a > 0
( )
10
What is the parent function for exponential
decay functions? ƒ(x) = b x where 0 < b < 1
is the parent function for the family of exponential
decay functions with base b.
Reflect
(_)
Graphing Combined Transformations of f(x) = b x
Where 0 < b < 1
When graphing transformations of ƒ (x) = b x where 0 < b < 1, it is helpful to consider the effect of the
transformation on two reference points, (0, 1) and -1, __b1 , as well as the effect on the asymptote, y = 0. The table
(
)
shows these reference points and the asymptote y = 0 for ƒ (x) = b and the corresponding points and asymptote for
the transformed function, g(x) = ab x-h + k.
Module 13
x
652
© Houghton Mifflin Harcourt Publishing Company
Make a Conjecture Look at the table of properties for the functions. What do you
notice? Make a
x
conjecture about these properties for exponential functions of the form ƒ(x) = (__n1 ) , where n is a constant.
The domain, range, end behavior, and y-intercept are the same for both functions. These
x
1
same properties apply to all exponential functions of the form f(x) = n .
Explain 1
x
Make sure that students are comfortable using their
calculators to graph exponential functions. They may
need to practice putting in the appropriate domains
and ranges.
0
x
1.
(__12 )
INTEGRATE TECHNOLOGY
f(x) → ∞
1
0
EXPLORE
x
⎧
⎫
⎨x⎢ -∞ < x < ∞ ⎬
⎩
⎭
f(x) →
0
1
(___
10 )
EXPLAIN 1
Graphing Combined Transformations
of ƒ(x) = b x Where 0 < b < 1
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 As a starting point for graphing combined
transformations, students should be aware that the
graph of ƒ(x) = ab x always passes through the points
(0, a) and (1, ab).
Lesson 2
PROFESSIONAL DEVELOPMENT
A2_MNLESE385900_U6M13L2 652
Math Background
10/16/14 11:23 AM
Students will graph most of the exponential functions in this lesson by hand. They
will see that the graphs of exponential decay functions approach the positive
x-axis as x increases without bound, so the x-axis is an asymptote for the graph of
any function of the form ƒ(x) = b x where b > 0 and b ≠ 1. Students will also
transform the graphs of exponential functions and discover how the
transformations affect the asymptote, y-intercept, and rate of increase or decrease,
and write transformed functions for graphs based upon the asymptote and two
points on the graph, the reference points.
Exponential Decay Functions
652
QUESTIONING STRATEGIES
What is the horizontal asymptote of the graph
ƒ(x) = a · b x - h + k where b = __12 ? The
horizontal asymptote of the graph ƒ(x) is y = k.
f(x) = b x
g(x) = ab x-h + k
(0, 1)
(h, a + k)
(-1, _b1 )
(h - 1, _ba + k)
y=0
y=k
First reference point
Second reference point
For the graph of ƒ(x) = a · b x - h + k where
b = __12 and (h, k) is at the origin, what are the
reference points? (0, a) and (-1, 2a)
Asymptote
Example 1

The graph of a parent exponential function is shown. Use the reference
points and the asymptote shown for the parent graph to graph the
given transformed function. Then describe the domain and range of the
transformed function using set notation.
()
1
g(x) = 3 _
2
x-2
-2
1
b=_
2
a=3
Identify parameters:
h=2
k = -2
Find reference points:
(h, a + k) = (2, 3 - 2) = (2, 1)
(h - 1, _ab + k) = (2 - 1, ___3 - 2) = (1, 4)
1
2
y
Find the asymptote: y = -2
4
Plot the points and draw the asymptote. Then connect the points with a
smooth curve that approaches the asymptote without crossing it.
⎧
⎫
Domain: ⎨x ⎢ -∞ < x < ∞ ⎬
⎩
⎭
-2
© Houghton Mifflin Harcourt Publishing Company
( )
1
g(x) = - _
10
x+2
x
0
2
4
-2
y = -2
⎧
⎫
Range: ⎨ y ⎢ y > -2 ⎬
⎩
⎭

2
y=0
+8
Identify parameters:
a = -1
_
1
b = 10
h = -2
k= 8
Find reference points:
(h,
a+k
(h - 1, _ab + k) =
(
) = (-2, -1 + 8) = (-2, 7)
)(
-1
-2 -1, _ + 8 = -3 , -2
1
__
10
Module 13
653
)
Lesson 2
COLLABORATIVE LEARNING
A2_MNLESE385900_U6M13L2 653
Peer-to-Peer Activity
Have pairs of students work together to create a graphic organizer to compare and
contrast exponential growth functions and exponential decay functions.
653
Lesson 13.2
08/06/15 1:29 PM
Find the asymptote:
y
y=8
y= 8
CONNECT VOCABULARY
8
Connect the terms decay and growth to life. When
something alive grows, it tends to become taller and
larger. When something alive decays, it tends to get
smaller; it takes up less space.
Plot the points and draw the asymptote. Then connect the points with
a smooth curve that approaches the asymptote without crossing it.
⎫
⎧
Domain: ⎨x ⎢ - ∞ < x < ∞ ⎬
⎭
⎩
2
⎫
⎧
Range: ⎨y ⎢ y < 8 ⎬
⎭
⎩
-4
-2
0
2
y=0
4
x
Reflect
2.
Which parameters make the domain and range of g(x) differ from those of the parent function? Write the
transformed domain and range for g(x) in set notation.
None of the parameters alter the domain, which is all real numbers for both the parent
and transformed functions. The parameter a alters the range if it is less than 0, and the
⎧
⎫
⎭
parameter k alters the finite end of the range. Domain: ⎨⎩x|-∞ < x < ∞⎬;
⎧
⎧
⎫
⎫
Range (a > 0): ⎩⎨ y| y > k⎬⎭; Range (a < 0): ⎨⎩y| y < k⎬⎭
Your Turn
Graph the given transformed function. Then describe the domain and range of the
transformed function using set notation.
3.
()
1
g (x) = 3 _
3
x+2
-4
Identify parameters:
1
a = 3; b = ; h = -2; k = -4
3
Find reference points:
y
_
(h - 1, _a + k) =
b
(
)
3
-2 - 1, _ -4
_1
3
Find the asymptote: y = -4
⎧
= (-3, 5)
2
x
-4
-2
0
2
4
-2
-4
y = -4
⎫
⎭
Domain: ⎨⎩x ⎢-∞ < x < ∞⎬
⎧
⎫
Range: ⎨⎩y ⎢ y > -4⎬⎭
Module 13
654
© Houghton Mifflin Harcourt Publishing Company
(h, a + k) = (-2, 3 - 4) = (-2, - 1)
Lesson 2
DIFFERENTIATE INSTRUCTION
A2_MNLESE385900_U6M13L2 654
Communicating Math
10/06/15 2:22 AM
Have students make up their own functions in the form y = ab x - h + k for
different a, h, and k values, and then discuss the transformations with each other.
Exponential Decay Functions
654
Explain 2
EXPLAIN 2
Writing Equations for Combined Transformations
of f(x) = b x where 0 < b < 1
Given a graph of an exponential function, g(x) = abx - h + k, the reference points and the asymptote can be used to
identify the transformation parameters in order to write the function rule.
Writing Equations for Combined
Transformations of ƒ(x) = b x where
0<b<1
Example 2
Write the function represented by this graph and state the domain and
range using set notation.
y=4

INTEGRATE TECHNOLOGY
4
y
2
(-3, 1)
Students can check the equations they write by
graphing the functions on their graphing calculators.
Have them use the TRACE or TABLE feature to
identify coordinates of points in the resulting graph.
-4
-2
0
x
2
4
-2
(-4, -2)
-4
Find k from the asymptote: k = 4.
The first reference point is at (-3,1).
Equate point value with parameters-based expression.
(-3, 1) = (h, a + k)
Use the x-coordinate to solve for h.
h = -3
Use the y-coordinate to solve for a.
a=1-k
= -3
The second reference point is at (-4, -2).
Equate point value with parameters-based expression.
-3 + 4 = -2
_
b
-3 = -6
_
b
-3
b=_
-6
1
=_
2
© Houghton Mifflin Harcourt Publishing Company
Equate y-coordinate with parameters.
Solve for b.
( )
1
g (x) = -3 _
2
A2_MNLESE385900_U6M13L2.indd 655
Lesson 13.2
x+3
b
)
+4
⎧
⎫
Domain: ⎨x⎟ -∞ < x < ∞⎬
⎩
⎭
⎧
⎫
Range:⎨y⎟ y < 4⎬
⎩
⎭
Module 13
655
(
a +k
(-4, -2) = h - 1, _
655
Lesson 2
3/31/14 4:59 PM
B
4
y
(1, 4)
QUESTIONING STRATEGIES
3
For a given value of k and a first reference
point of (6, 8), how do you find the values of h
and a? The value of h is 6 and the value of a is 8 - k.
2
1
x
-1
0
-1
y = -1
1
3
For a given value of k and a and a second
reference point of (4, 10), how do you find the
values of b? Set the value of __ba + k to 10, substitute
the values of a and k, and solve for b.
4
(2, - 1)
2
Find k from the asymptote: k = -1 .
The first reference point is at
(
_1
a = -2
h= 2
=
The second reference point is at
_1
2
_
-1 = 4
b
)
1
__
2 , - 2 , so
-k
(2
) (
1 = h,
, -_
2
a+k
)
_a + k
)
_1
2
( 1 , 4 ), so ( 1 , 4) = (h -1,
b
1
_
2 = 5
_
b
__1
2
b=_
5
= __
10
1
_1
2
(_101 )
x- 2
-
© Houghton Mifflin Harcourt Publishing Company
g(x) =
1
⎧
⎫
Domain: ⎨x⎟ -∞ < x < ∞⎬
⎩
⎭
⎧
⎫
Range: ⎨y⎟ y > -1⎬
⎩
⎭
Module 13
A2_MNLESE385900_U6M13L2 656
656
Lesson 2
8/20/14 1:56 PM
Exponential Decay Functions
656
Reflect
4.
Compare the y-intercept and the asymptote of the function shown in this table to the function plotted in
Example 2A.
x
-5
-4
-3
-2
-1
0
1
2
g(x)
-10
-4
-4
1
_
2
1
1_
4
5
1_
8
13
1_
16
29
1_
32
( _58 to 1_58 ), and the
The y-intercept appears to have moved down by 2 units from 3
asymptote appears to have moved down by 2 units as well (from 4 to 2).
5.
Compare the y-intercept and the asymptote of the function shown in this table to the function plotted in
Example 2B.
x
-3
-2
-1
0
1
2
g(x)
49
4
-0.5
-0.95
-0.995
–0.9995
The y-intercept is not apparent in the graph but must be larger than 10, while the table
shows that the y- intercept is at -0.95. The asymptote of both functions appears to be -1.
Your Turn
Write the function represented by this graph and state the domain and range using set
notation.
6.
Asymptote: y = -4, so k = -4
y
First reference point: (1, -2)
4
(1, -2) = (h, a + k)
2
© Houghton Mifflin Harcourt Publishing Company
x
-2
0
2
-2
y = -4
4
6
8
h= 1
a = -2 - (-4) = 2
Second reference point: (0, 4)
(0, 4) = h - 1, a + k
b
2
-4=4
b
2
=8
b
b= 1
4
x-1
1
g(x) = 2
-4
4
(
_
_
)
_
_
(_)
⎧
⎫
⎭
Domain: ⎨⎩x⎢ -∞ < x < ∞⎬
⎧
⎫
Range: ⎨⎩y ⎢y > -4⎬⎭
Module 13
A2_MNLESE385900_U6M13L2 657
657
Lesson 13.2
657
Lesson 2
8/20/14 1:58 PM
Explain 3
Modeling with Exponential Decay Functions
EXPLAIN 3
An exponential decay function has the form ƒ (t) = a(1 - r) where a > 0 and r is a constant percent decrease
(expressed as a decimal) for each unit increase in time t. That is, since ƒ(t + 1) = (1 – r) · ƒ(t) = ƒ(t) – r · ƒ(t) , the
value of the function decreases by r · ƒ (t) on the interval [t, t + 1]. The base 1 – r of an exponential decay function is
called the decay factor, and the constant percent decrease r, in decimal form, is called the decay rate.
t
Example 3

Modeling With Exponential Decay
Functions
Given the description of the decay terms, write the exponential decay
t
function in the form f (t) = a (1 - r) and graph it with a graphing
calculator.
QUESTIONING STRATEGIES
The value of a truck purchased new for $28,000 decreases by 9.5% each year. Write an
exponential function for this situation and graph it using a calculator. Use the graph to
predict after how many years the value of the truck will be $5000.
“Purchased new for $28,000...”
a = 28,000
“...decreases by 9.5% each year.”
r = 0.095
Substitute parameter values.
V T (t) = 28,000 ( 1 - 0.095 )t
Simplify.
V T (t) = 28,000 ( 0.905 )t
How is the decay factor related to the percent
of decrease in value? The sum of the
decay factor and the percent of decrease is 1. For
example, for a percent of decrease of 23%, the
decay factor is 0.77.
Graph the function with a graphing calculator. Use
WINDOW to adjust the graph settings so that you can see
the function and the function values that are important.
t
The intersection is at the point (17.26, 5000), which means after 17.26 years, the truck will have a value
of $5000.
Module 13
658
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Transtock
Inc./Superstock
Find when the value reaches $5000 by finding the intersection between V T( t ) = 28, 000 (0.905)
and V T( t ) = 5000 on the calculator.
Lesson 2
LANGUAGE SUPPORT
A2_MNLESE385900_U6M13L2.indd 658
Graphic Organizers
3/31/14 5:03 PM
Have each pair of students complete a compare and contrast Venn diagram to
show the similarities and differences between exponential decay and exponential
growth functions. Encourage students to discuss and show the similarities and
differences between their graphs, their equations, and so on.
Exponential Decay Functions
658
B
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Explain the concept of depreciation and how
it is used in the business world. Discuss the difference
between an item losing the same dollar amount of
value each year, known as straight-line depreciation,
and the item losing the same percent of its value each
year, a type of depreciation which is based on an
exponential decay model.
The value of a sports car purchased new for $45,000 decreases by 15% each year. Write
an exponential function for the depreciation of the sports car, and plot it along with the
previous example. After how many years will the two vehicles have the same value if they are
purchased at the same time?
“Purchased new for $45,000...”
a = 45,000
“...decreases by 15% each year.”
r = 0.15
(1 = 45,000 ( 0.85 )
Substitute parameter values.
V c ( t ) = 45,000
Simplify.
Vc (t)
)
t
0.15
t
Add this plot to the graph for the truck value from Example A and find the intersection of the two functions to
determine when the values are the same.
The intersection point is
After
7.567
(
)
7.567 , 13,155 .
years, the values of both vehicles will be
$ 13,155 .
Reflect
7.
What reference points could you use if you plotted the value function for the sports car on graph paper?
Confirm that the graph passes through them using the calculate feature on a graphing calculator.
The transformation parameters are a = 45,000, h = 0, and k = 0. The parent function is
a
b = 0.85. The reference points are (h, a + k) = (0, 45,000) and h - 1, + k = (-1, 52,941).
b
Using the calculate feature confirms the graph passes through (0, 45,000) and (–1, 52,941).
(
© Houghton Mifflin Harcourt Publishing Company
8.
___
Second year: rate of change =
45,000 ∙ (0.85) - 45,000 ∙ 0.85
___
= -$5737.50 per year
2
2 -1
The absolute value of the rate of change decreased during the second interval. This means
that the car depreciates less each year than the year before.
A2_MNLESE385900_U6M13L2 659
Lesson 13.2
)
Using the sports car from example B, calculate the average rate of change over the course of the first year
and the second year of ownership. What happens to the absolute value of the rate of change from the first
interval to the second? What does this mean in this situation?
f (t 1) - f (t 0)
Average rate of change during the interval from t 0 to t 1 = ________
t1 - t0
45,000 ∙ 0.85 - 45,000
= -$6750 per year
First year: rate of change =
1-0
Module 13
659
_
659
Lesson 2
08/06/15 1:30 PM
Your Turn
9.
ELABORATE
On federal income tax returns, self-employed people can depreciate the value of business equipment.
Suppose a computer valued at $2765 depreciates at a rate of 30% per year. Use a graphing calculator to
determine the number of years it will take for the computer’s value to be $350.
QUESTIONING STRATEGIES
v (t) = 2765 (1 - 0.3) = 2765 (0.7)
t
t
How do you rewrite exponential decay
functions to answer questions about the
functions? Properties of exponents can be used to
rewrite exponential functions to show specific
growth or decay factors.
Intersect with v (t) = 350
Using a graphing calculator, the intersection point is at (5.79, 350).
It will take about 5.79 years for the value of the computer to drop to $350.
Elaborate
( )
()
x
x
1
10. Which transformations of ƒ(x) = __12 or ƒ(x) = __
change the function’s end behavior?
10
Vertical translations change the asymptote and thus the lower end behavior as x increases.
SUMMARIZE THE LESSON
Reflections across the x-axis change the end behavior as x decreases, from approaching
What does the graph of an exponential decay
function look like? An exponential decay
function is a function of the form y = ab x with a > 0
and 0 < b < 1. Exponential decay models describe
situations in which a quantity decreases by a fixed
percent each time period. The graph of an
exponential decay function is a curve that falls from
left to right and gets less and less steep as x
increases. The x-axis, or a line parallel to it, is a
horizontal asymptote of the graph.
positive infinity to approaching negative infinity.
11. Which transformations change the location of the graph’s y-intercept?
Vertical translations, horizontal translations, vertical stretches/compressions, and
reflections across the x-axis all change the y-intercept.
12. Discussion How are reference points and asymptotes helpful when graphing transformations of
()
x
( )
x
1
or when writing equations for transformed graphs?
ƒ(x) = __12 or ƒ(x) = __
10
Reference points and asymptotes are easy to transform and have a simple relationship to
the function parameters (a, h, and k) associated with the transformation. The point (0, 1)
(
)
(
)
a
1
becomes (h, a + k), the point -1, _
becomes h - 1, _
+ k , and the asymptote y = 0
b
becomes y = k.
b
a is the starting value, or the value at t = 0.
r is the decay rate, or what fraction of the value is lost per unit of time.
(1 - r) is the decay factor, or what fraction of the previous value remains after the passage
of a unit of time.
14. Essential Question Check-In How is the graph of ƒ(x) = b x used to help graph the function
g(x) = ab x - h + k?
The graph of g (x) = ab x-h + k can be taken from the basic shape of the parent function,
© Houghton Mifflin Harcourt Publishing Company
13. Give the general form of an exponential decay function based on a known decay rate and describe its
parameters.
t
f(t) = a (1 - r)
f(x) = b x, with transformations applied based on the parameters a, h, and k.
Module 13
A2_MNLESE385900_U6M13L2.indd 660
660
Lesson 2
3/22/14 1:40 PM
Exponential Decay Functions
660
EVALUATE
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
Describe the transformation(s) from each parent function and give the domain and
range of each function.
1.
()
x
1 +3
g(x) = _
2
ASSIGNMENT GUIDE
Concepts and Skills
Practice
Example 1
Graphing Combined
Transformations of ƒ(x) = b x where
0 < b < 1.
Exercises 5–8
Example 2
Writing Equations for Combined
Transformations of ƒ(x) = b x where
0<b<1
Exercises 12–13
Example 3
Modeling With Exponential Decay
Functions
Exercises 14–16
3.
2.
Horizontal translation left by 4.
⎧
⎫
Domain: ⎨⎩x| -∞ < x < ∞⎬
⎭
⎧
⎫
Range: ⎨⎩y | y > 3⎬⎭
Domain: ⎨⎩x| -∞ < x < ∞⎬
( )
1
g(x) = - _
10
x-1
⎧
⎧
⎫
⎭
⎫
Range: ⎨⎩y | y > 0⎬⎭
+2
4.
()
x+3
1
g(x) = 3 _
2
-6
Reflection across the x-axis, translation
Vertical stretch by a factor of 3, translation
right by 1 and up by 2.
left by 3 and down by 6.
⎧
⎫
Domain: ⎨⎩x| -∞ < x < ∞⎬
⎭
⎧
⎫
Range: ⎨⎩y | y < 2⎬⎭
Domain: ⎨⎩x| -∞ < x < ∞⎬
⎧
⎧
⎫
⎭
⎫
Range: ⎨⎩y | y > -6⎬⎭
Graph the given transformed function. Then describe the domain and range of the
transformed function using set notation.
5.
()
1
g(x) = -2 _
2
x-1
+2
Identify parameters: a = -2; b =
© Houghton Mifflin Harcourt Publishing Company
x
-2
0
2
4
-2
(h - 1, __ba + k) = 1 - 1, ___
__ + 2 = (0, -2)
1
2
-2
Find the asymptote: y = 2
-4
Domain: ⎨⎩x| -∞ < x < ∞⎬
⎧
Exercise
⎧
⎫
Range: ⎨⎩y | y < 2⎬⎭
Module 13
A2_MNLESE385900_U6M13L2 661
2
(h, a + k) = (1, -2 + 2) = (1, 0)
2
-4
_1 ; h = 1; k = 2
Find reference points:
4
Lesson 13.2
x+4
Vertical translation up by 3.
y
661
( )
1
g(x) = _
10
Lesson 2
661
Depth of Knowledge (D.O.K.)
⎫
⎭
Mathematical Practices
1–4
2 Skills/Concepts
MP.2 Reasoning
5–8
2 Skills/Concepts
MP.4 Modeling
9–10
2 Skills/Concepts
MP.2 Reasoning
11–13
2 Skills/Concepts
MP.4 Modeling
14
3 Strategic Thinking
MP.6 Precision
15–17
3 Strategic Thinking
MP.2 Reasoning
10/06/15 2:24 AM
6.
()
1
g(x) = _
4
x+2
+3
Identify parameters: a = 1; b =
y
4
y=3
2
x
7.
-2
0
()
1
x-_
2
1
1 _
g(x) = _
2 3
2
Identify parameters: a =
1
2
Find the asymptote: y = 2
3
⎧
⎧
⎫
Range: ⎨⎩y | y > 2⎬⎭
x+2
+7
8
y=7
6
1
2
2
x
0
(h, a + k) = (-2, -3 + 7) = (-2, 4)
-3
+ 7 = (-3, 1)
(h - 1, _ba + k) = (-2 -1, _
__
)
4
-2
1 ; h = -2 ; k = 7
Identify parameters: a = -3; b = _
2
Find reference points:
2
Find the asymptote: y = 7
⎧
⎫
⎭
Domain: ⎨⎩x| -∞ < x < ∞⎬
⎧
⎫
Range: ⎨⎩y | y < 7⎬⎭
662
© Houghton Mifflin Harcourt Publishing Company
y
A2_MNLESE385900_U6M13L2 662
⎫
⎭
Domain: ⎨⎩x| -∞ < x < ∞⎬
()
Module 13
2
1
2
1
3
-1
-4
_1 ; b = _1 ; h = _1 ; k = 2
(_ _ ) (_ _)
__
(h - 1, _ba + k) = (_12 - 1, ___ + 2) = (-_21 , _72 )
x
-6
⎫
⎭
2
3
Find reference points:
(h, a + k) = 12 , 12 + 2 = 12 , 52
y=2
0
1
g (x) = -3 _
2
Find the asymptote: y = 3
⎧
1
8.
1
4
+2
2
-1
1
⎧
⎫
Range: ⎨⎩y | y > 3⎬⎭
3
-2
(h - 1, _ba + k) = (-2 - 1, ___ + 3) = (-3, 7)
Domain: ⎨⎩x| -∞ < x < ∞⎬
4
y
4
Some students may think that a horizontal shift in
the graph of an exponential function affects the
domain. Demonstrate that the domain of all
exponential functions and their translations is the set
of all real numbers, just as with quadratic functions.
Go back to the definition of domain and point out
that the value of x can be any real number in any
exponential growth or decay function or any
translation of these functions. You might use a
graphing calculator demonstration to reinforce this
idea visually.
4
(h, a + k) = (-2, 1 + 3) = (-2, 4)
6
-4
_1 ; h = -2; k = 3
Find reference points:
8
-6
AVOID COMMON ERRORS
Lesson 2
10/06/15 2:34 AM
Exponential Decay Functions
662
Write the function represented by each graph and state the domain
and range using set notation.
Asymptote:
9.
8
y = -1
y
k = -1
(3, 7)
First reference point: (4, 3)
6
4
(4, 3) = (h, a + k)
h=4
(4, 3)
2
x
-1
0
2
4
6
8
y = -1
-2
10
a = 3 - (-1)
a=4
Second reference point: (3, 7)
a
+k
(3, 7) = h - 1, _
b
4
_-1= 7
b
4=8
_
b
4
b=_
8
1
b=_
2
x- 4
1
-1
g(x) = 4
2
⎧
⎫
Domain: ⎨xǀ-∞ < x < ∞⎬⎭
⎩
⎧
⎫
Range: ⎨yǀy > -1⎬⎭
⎩
(
)
(_)
Asymptote:
y
10.
y=3
k=3
4
y=3
First reference point: (-4, 1)
© Houghton Mifflin Harcourt Publishing Company
2
(-4, 1)
-6
-4
-2
x
0
-2
(-5, -3)
-4
2
(-4, 1) = (h, a + k)
h = -4
a=1- 3
a = -2
Second reference point: (-5, -3)
a
+k
(-5, -3) = h - 1, _
b
-2 + 3 = -3
_
b
-2 = -6
_
b
-2
b=_
-6
1
b=_
3
x+ 4
1
_
g(x) = -2
+3
3
⎧
⎫
Domain: ⎨xǀ-∞ < x < ∞⎬⎭
⎩
⎧
⎫
Range: ⎨yǀy < 3⎬⎭
⎩
(
)
()
Module 13
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Lesson 13.2
663
Lesson 2
3/31/14 5:07 PM
Write the exponential decay function described in the situation and
use a graphing calculator to answer each question asked.
PEER-TO-PEER DISCUSSION
Ask students to brainstorm and discuss two specific,
real-world situations, one of exponential growth and
one of exponential decay. For each situation, ask
students to discuss how the graphs should look and
tell how they would go about creating equations to
help them model real-world data.
11. Medicine A quantity of insulin used to regulate sugar in the bloodstream breaks
down by about 5% each minute after the injection. A bodyweight-adjusted dose is
generally 10 units. How long does it take for the remaining insulin to be half of the
original injection?
l(t) = 10(1 - 0.05)
t
= 10(0.95)
t
Half of the original injection (10) is 5. Use a graph to find the intersection
with l(t) = 5.
Intersection point is at (13.5, 5).
It takes about 13.5 minutes.
12. Paleontology Carbon-14 is a radioactive isotope of carbon
that is used to date fossils. There are about 1.5 atoms of
carbon-14 for every trillion atoms of carbon in the atmosphere,
which known as 1.5 ppt (parts per trillion). Carbon in a living
organism has the same concentration as carbon-14. When an
organism dies, the carbon-14 content decays at a rate of 11.4%
per millennium (1000 years). Write the equation for carbon-14
concentration (in ppt) as a function of time (in millennia)
and determine how old a fossil must be that has a measured
concentration of 0.2 ppt.
c(t) = 1.5(1 - 0.114)
t
= 1.5(0.886)
t
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Eliana
Aponte/Reuters/Corbis; (b) ©Joshua David Treisner/Shutterstock
Intersection point is at (16.65, 0.2).
The fossil is about 16.65 millennia, or 16,650 years old.
13. Music Stringed instruments like guitars and pianos create a
note when a string vibrates back and forth. The distance that
the middle of the string moves from the center is called the
amplitude (a), and for a guitar, it starts at 0.75 mm when a
note is first struck. Amplitude decays at a rate that depends
on the individual instrument and the note, but a decay rate of
about 25% per second is typical. Calculate the time it takes for
an amplitude of 0.75 mm to reach 0.1 mm.
a(t) = 0.75(1 - 0.25)
t
= 0.75(0.75)
t
Intersection point is at (7.004, 0.1).
The amplitude will reach 0.1 mm in about 7 seconds.
Module 13
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Lesson 2
3/31/14 5:07 PM
Exponential Decay Functions
664
JOURNAL
H.O.T. Focus on Higher Order Thinking
Have students write about the two types of
exponential models, and describe how they differ
from polynomial models such as quadratic and cubic.
()
x
1 and g (x) = x 2 .
14. Analyze Relationships Compare the graphs of ƒ (x) = _
2
Which of the following properties are the same? Explain.
⎧
⎫ ⎧
⎫
a. Domain
⎨xǀ-∞ < x < ∞⎬; ⎨xǀx ≥ 0⎬
⎭ ⎩
⎭
⎩
b. Range
⎧
⎫ ⎧
⎫
⎨yǀy > 0⎬; ⎨yǀy ≥ 0⎬
⎭ ⎩
⎭
⎩
c. End behavior as x increases
f (x) → 0; g (x) → ∞
d. End behavior as x decreases
f (x) → ∞; g (x) is not defined for values less than 0.
None are the same.
1
_
15. Communicate Mathematical Ideas A quantity is reduced to half of its original
amount during each given time period. Another quantity is reduced to one quarter
of its original amount during the same given time period. Determine each decay rate,
state which is greater, and explain your results.
The decay rate of the first quantity is 50% because the decay factor is
1
1
, so the decay rate equals __
, or 50%. The decay rate of the
1 - r = __
2
2
1
second quantity is 75% because the decay factor is 1 - r = __
, so the
4
3
decay rate is __
, or 75%. The decay rate of the second quantity is greater.
4
16. Multiple Representations Exponential decay functions are written as
transformations of the function ƒ (x) = b x, where 0 < b < 1. However, it is also
possible to use negative exponents as the basis of an exponential decay function. Use
the properties of exponents to show why the function ƒ (x) = 2 -x is an exponential
decay function.
f (x) = 2 -x
© Houghton Mifflin Harcourt Publishing Company
Given
Power of a power property
= (2 -1)
Property of negative exponents
1
= _
2
()
x
x
The last result is in the form f (x) = b x where 0 < b < 1 and is therefore
an exponential decay function.
17. Represent Real-World Problems You buy a video game console for $500 and
sell it 5 years later for $100. The resale value decays exponentially over time. Write
a function that represents the resale value, R, in dollars over the time, t, in years.
Explain how you determined your function.
R(t) = 500(0.725) ; Sample answer: I used the general exponential decay
t
function f (x) = ab x and substituted 100 for f (x), 500 for a, and 5 for x,
resulting in 100 = 500(b 5). I then solved for b by dividing 100 by 500 to
get 0.2 and then took the fifth root of 0.2, resulting in 0.724779, which I
rounded to 0.725.
Module 13
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Lesson 13.2
665
Lesson 2
10/16/14 11:23 AM
Lesson Performance Task
AVOID COMMON ERRORS
Students may set r equal to -0.045 because this is a
decay situation. However, this would make the
t
term (1 - r) greater than one, and the function
m Na(t) would become a growth function. Explain to
t
students that the term (1 - r) already contains the
minus sign that turns m Na(t) into a decay function.
Sodium-24 is a radioactive isotope of sodium used as a diagnostic aid in medicine. It undergoes radioactive decay to
form the stable isotope magnesium-24 and has a half-life of about 15 hours. This means that, in this time, half the
amount of a sample mass of sodium-24 decays to magnesium-24. Suppose we start with an initial mass of 100 grams
sodium-24.
a. Use the half-life of sodium-24 to write an exponential decay function of the form m Na(t) = m 0(1 - r) ,
where m 0 is the initial mass of sodium-24, r is the decay rate, t is the time in hours, and m Na(t) is the mass
of sodium-24 at time t. What is the meaning of r?
t
Substitute 50 for m Na(t), 100 for a 0, and 15 for t in the function.
50 = 100(1 - r)
15
0.5 = (1 - r)
――
15 ―――
√(1 - r)15
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 Have students consider the graphs for m Na(t)
15
√0.5 =
15
0.955 ≈ 1 - r
0.045 ≈ r
m Na(t) = 100(0.955)
t
and m Mg(t), and have them explain which is
exponential decay and which is exponential growth,
based on the properties of the graphs. Have students
discuss whether they can determine from the graphs
the final values of the functions as t gets very large.
The value of r means that the mass of sodium-24 is reduced by 4.5% each hour.
b. The combined amounts of sodium-24 and magnesium-24 must equal m 0, or 100, for all possible values of t.
Show how to write a function for m Mg(t), the mass of magnesium-24 as a function of t.
The sum of the mass of magnesium-24 and sodium-24 is
equal to m 0 , which is 100.
m Mg(t) + m Na(t) = m 0
m Mg(t) + m Na(t) = 100
Solve for m Mg(t).
m Mg(t) = 100 - m Na(t)
Substitute 100(0.955) for m Na(t)
t
m Mg(t) = 100 - 100(0.955)
t
© Houghton Mifflin Harcourt Publishing Company
c. Use a graphing calculator to graph m Na(t) and m Mg(t). Describe the graph of m Mg(t) as a series of
transformations of m Na(t). What does the intersection of the graphs represent?
100
Mass of Na/Mg (g)
The graph of m Mg(t) is a
reflection of the graph
of m Na(t) across the
t-axis and a translation
of 100 units vertically.
The intersection of the
graphs represents the
point where the mass of
sodium-24 is equal to the
mass of magnesium-24,
which occurs at the first
half-life of sodium-24.
80
Na
60
40
Mg
20
0
5
10
15
20
Time in Minutes (t)
Module 13
666
Lesson 2
EXTENSION ACTIVITY
A2_MNLESE385900_U6M13L2 666
Have students research the half-life of technetium-99m, another radioactive
isotope widely used in medicine. Have students write an exponential decay
function for an initial mass of 100 grams. Then have students graph this function
and compare it to the one for sodium-24. Have students discuss the difference in
decay rates and how that might affect a real-world situation.
08/06/15 1:41 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.
Exponential Decay Functions
666