LESSON
6.1
Name
Graphing Cubic
Functions
Class
6.1
Date
Graphing Cubic Functions
(_
The student is expected to:
Explore 1
A2.2.A

Mathematical Processes
Graphing and Analyzing f(x) = x 3
Complete the table, graph the ordered pairs,
and then draw a smooth curve through the
plotted points to obtain the graph of ƒ(x) = x 3.
Communicate mathematical ideas, reasoning, and their implications
using multiple representations, including symbols, diagrams, graphs,
and language as appropriate.

1
1
2
8
© Houghton Mifflin Harcourt Publishing Company
ℝ
Range
ℝ
End behavior
As x → + ∞, f(x) → +∞ .
2
4
-2
x=0
Zeros of the function
Where the function has positive values
x>0
Where the function has negative values
x<0
Where the function is increasing
The function increases
throughout its domain.
Where the function is decreasing
The function never decreases.
Is the function even (f(-x) = f(x)), odd
(f(-x) = -f(x)), or neither?
Module 6
ges must
EDIT--Chan
DO NOT Key=TX-B
Correction
Odd , because (x) 3 = -x 3 .
be made
through
Lesson 1
315
"File info"
Date
Class
ns
bic Functio (_
Name
Cu
Graphing
)
3
1 (x - h) + k
f(x) = b
3
h) + k and
= a(x Resource
s of f(x)
3
Locker
are the graph of f(x) = x ?
ion: How
the graph
asymptotic
related to
symmetries,
intercepts,
domain, range, l. Also A2.6.A.
tes such as
an interva
3
key attribu
given
ze
um
A2.2.A ...analy
um and minim
rs
x) = x
and maxim
real numbe
lyzing f(
behavior,
and c are
b,
Ana
a,
are all
where
g and
2
b, c and d
+ bx + 2c
where a,
of cubic
(x) = ax 3
1 Graphin
+ cx + d
rd form ƒ
n for a family
Explore
ax + bx
functio
has the standa rd form ƒ(x) = 3
n
parent
x , as the
tic functio
standa
n, ƒ(x) =
on has the
that a quadra
You know
a cubic functi the basic cubic functio(x) = x3 .
Similarly,
use
of ƒ
y
and a ≠ 0.
0. You can
of the graph
8
rs and a ≠
rmations
real numbe
3
through transfo
x
related
=
y
functions
x
6
ordered pairs,
6.1
HARDCOVER PAGES 225234
Quest
Essential
A2_MTXESE353930_U3M06L1.indd 315
graph the
h the
the table,
Complete
h curve throug (x) = x3 .
of ƒ
draw a smoot
the graph
and then
to obtain
plotted points

-8
-2
0
0
2
x
1
1
8
2
Turn to these pages to
find this lesson in the
hardcover student
edition.
4
-1
-1
-4
-2
0
2
4
-2
-4
ete the table.
on and compl
e the functi
to analyz
x3
Use the graph
of f(x) =
Attributes
-6
-8
ℝ
y
g Compan
End
ℝ
.
) → +∞
+ ∞, f(x
As x →
.
f(x) → -∞
As x → -∞,
behavior
x=0
Publishin
Lesson 6.1
0
As x → -∞, f(x) → -∞ .
Range
315
x
-2
-8
Domain
View the Engage section online. Discuss the photo
and how the volume of a spherical aquarium can be
described by a cubic function. Then preview the
Lesson Performance Task.
2
-6
Domain

n Mifflin
Harcour t
function
Zeros of the
e values
has positiv
function
Where the
ive values
has negat
function
Where the
is increasing
function
Where the
© Houghto
PREVIEW: LESSON
PERFORMANCE TASK
4
-4
Attributes of f(x) = x 3
Explain to a partner how to predict transformations of a basic cubic
function.
Possible answer: Both graphs involve
transformations of the graph of f(x) = x 3. The first
involves vertically stretching or compressing the
graph of f(x) = x 3, reflecting it across the x-axis if
a < 0, and translating it horizontally and vertically.
The second involves horizontally stretching or
compressing the graph of f(x) = x 3, reflecting it
across the y-axis if b < 0, and translating it
horizontally and vertically.
0
Use the graph to analyze the function and complete the table.
2.C.4, 3.B.3, 3.C.4, 3.H.3
graph of f(x) = x 3?
0
y
6
-1
-4
Language Objective
(__b1 (x - h)) + k related to the
8
-8
-2
A2.1.D
f(x) =
y = x3
x
-1
3
Resource
Locker
You know that a quadratic function has the standard form ƒ(x) = ax 2 + bx + c where a, b, and c are real numbers
and a ≠ 0. Similarly, a cubic function has the standard form ƒ(x) = ax 3 + bx 2 + cx + d where a, b, c and d are all
real numbers and a ≠ 0. You can use the basic cubic function, ƒ(x) = x 3, as the parent function for a family of cubic
functions related through transformations of the graph of ƒ(x) = x 3.
1
3 ―
3
Graph the functions f(x ) = √―x , f(x ) = __
x , f(x ) = x , f(x) = √ x ,
f(x ) = bx, f(x ) = ⎜x⎟, and f(x ) = log b(x) where b is 2, 10, and e,
and, when applicable, analyze the key attributes such as domain, range,
intercepts, symmetries, asymptotic behavior, and maximum and
minimum given an interval. Also A2.6.A
Essential Question: How are the
3
graphs of f(x) = a(x - h) + k and
3
A2.2.A ...analyze key attributes such as domain, range, intercepts, symmetries, asymptotic
behavior, and maximum and minimum given an interval. Also A2.6.A.
Texas Math Standards
ENGAGE
) +k
3
1
Essential Question: How are the graphs of f(x) = a(x - h) + k and f(x) = (x - h)
b
3
)
(
related to the graph of f x = x ?
Where the
function
is
decreasing
(x))
(f(-x) = f
on even
Is the functi (x)), or neither?
(f(-x) = -f
Module 6
L1.indd
0_U3M06
SE35393
A2_MTXE
315
, odd
x>0
x<0
ases
ion incre
in.
The funct
t its doma
throughou
on never
The functi
decreases.
Odd , because (x)
3
3 .
= -x
Lesson 1
315
15-01-11
1:52 AM
15-01-11 1:52 AM
Reflect
1.
EXPLORE 1
How would you characterize the rate of change of the function on the intervals ⎡⎣-1, 0⎤⎦ and ⎡⎣0, 1⎤⎦
compared with the rate of change on the intervals ⎡⎣-2, -1⎤⎦ and ⎡⎣1, 2⎤⎦? Explain.
The rate of change on all four intervals is positive because the function is increasing.
Graphing and Analyzing f(x) = x 3
However, the rate of change on the intervals ⎡⎣-1, 0⎤⎦ and ⎡⎣0, 1⎤⎦ is much less than the rate
of change on the intervals ⎡⎣-2, -1⎤⎦ and ⎡⎣1, 2⎤⎦, because the graph rises more slowly on the
2.
intervals ⎡⎣-1, 0⎤⎦ and ⎡⎣0, 1⎤⎦ than it does on the intervals ⎡⎣-2, -1⎤⎦ and ⎡⎣1, 2⎤⎦.
INTEGRATE TECHNOLOGY
A graph is said to be symmetric about the origin (and the origin is called the graph’s point of symmetry)
if for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Is the graph of ƒ(x) = x 3
symmetric about the origin? Explain.
Yes, because for every point (x, y) = (x, x 3) on the graph of the function, the point
Students can use a graphing calculator to
graph and generate table values for the
function y = x 3.
(-x, (-x)3) = (-x, -x 3) = (-x, -y) is also on the graph.
Explore 2
QUESTIONING STRATEGIES
Predicting Transformations of the
Graph of f(x) = x 3
How do you verify that the graph of
f(x) = x 3 has symmetry about the
origin? Show that if (x, y) is a point on the graph,
then (-x, -y) is also a point on the graph.
You can apply familiar transformations to cubic functions. Check your predictions using a graphing calculator.
A
Predict the effect of the parameter h on the graph of g(x) = (x - h) for each function.
3
a. The graph of g(x) = (x - 2) is a
3
translation of the graph of ƒ(x)
How is the graph of f(x) = x 3 similar to the
graph of f(x) = x? The domain and range
for both functions are (-∞, ∞); the end behavior
for both functions is f(x) → -∞ as x → -∞, and
f(x) → ∞ as x → +∞.
[right/up/left/down] 2 units.
b. The graph of g(x) = (x + 2) is a
3
translation of the graph of ƒ(x)
[right/up/left/down] 2 units.
B
Predict the effect of the parameter k on the graph of g(x) = x 3 + k for each function.
translation of the graph of ƒ(x)
© Houghton Mifflin Harcourt Publishing Company
a. The graph of g(x) = x 3 + 2 is a
[right/up/left/down] 2 units.
b. The graph of g(x) = x 3 - 2 is a
translation of the graph of ƒ(x)
[right/up/left/down] 2 units.
C
Predict the effect of the parameter a on the graph of g(x) = ax 3 for each function.
vertical stretch
a. The graph of g(x) = 2x 3 is a
2
factor of
.
x-axis
Module 6
Predicting Transformations of the
Graph of f(x) = x 3
AVOID COMMON ERRORS
vertical compression of the graph of ƒ(x) by a
b. The graph of g(x) = -__12 x 3 is a
_1
2
factor of
the
of the graph of ƒ(x) by a
EXPLORE 2
as well as a
reflection
across
.
316
Lesson 1
PROFESSIONAL DEVELOPMENT
A2_MTXESE353930_U3M06L1 316
Math Background
A cubic function is a polynomial function of degree 3. The standard form of a
cubic function is f(x) = ax 3 + bx 2 + cx + d where a, b, c, and d are real numbers
and a ≠ 0. Each function of the family of cubic functions is increasing or
decreasing over its entire domain, depending on the sign of the leading coefficient
of x 3. There also may be turning points and relative maximum and relative
minimum points, depending on the transformations of the graph of f(x) = x 3. The
domain and range of cubic functions are the set of all real numbers. The inverse of
a cubic function may or may not be a function. If the graph has turning points, its
inverse is not a function.
2/22/14 4:45 AM
Students often assume that the graph of f(x - h)
is translated to the left due to the subtraction.
For example, if the function is f(x) = (x - 2) 3,
students may want to translate f(x) = x 3 two units to
the left. Point out that the value of h in f(x - h)
is positive, so the transformation is in the positive
direction, to the right. Have students graph the two
functions to verify this.
CONNECT VOCABULARY
Relate the term parent function to parents and
children. The children come from the parents;
likewise, cubic functions can all be traced back to a
parent function.
Graphing Cubic Functions 316
D
QUESTIONING STRATEGIES
How is the graph of g(x) = (x - h)3 + k
related to the parent graph f(x) = x 3? The
3
graph of g(x) = (x - h) + k is the graph of f(x) = x3
translated h units horizontally and k units vertically.
( )
3
Predict the effect of the parameter b on the graph of g(x) = __b1 x for the following
values of b.
of
horizontal stretch
2
of the graph of ƒ(x) by a factor
.
b. The graph of g(x) = (2x) is a horizontal compression of the graph of ƒ(x) by a
If a > 0, how is the graph of g(x) = ax related
to the graph of f(x) = x 3? If a > 1, the graph
of g(x) is the graph of f(x) = x 3 stretched a units
vertically. If 0 < a < 1, the graph of g(x) is the graph
1 units vertically.
of f(x) = x 3 compressed ____
|a|
3
__1
2
factor of
3
(
.
)
c. The graph of g(x) = -__12 x is a horizontal compression of the graph of ƒ(x) by
a factor of
the
2
3
y-axis
EXPLAIN 1
across
.
__1
3
horizontal stretch
2
factor of
reflection
as well as a
d. The graph of g(x) = (-2x) is a
the
as well as a
y-axis
of the graph of ƒ(x) by a
reflection
across
.
Reflect
3.
Graphing Combined Transformations
of f(x) = x 3
The graph of g(x) = (-x) 3 is a reflection of the graph of ƒ(x) = x 3 across the y-axis, while the graph of
h(x) = -x3 is a reflection of the graph of ƒ(x) = x3 across the x-axis. If you graph g(x) and h(x) on a
graphing calculator, what do you notice? Explain why this happens.
The graphs of g(x) and h(x) are the same because f(x) is an odd function (that is,
f(-x) = -f(x)). Given that g(x) = f(-x) and h(x) = -f(x), you have g(x) = h(x).
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Patterns
Explain 1
© Houghton Mifflin Harcourt Publishing Company
Point out that to graph the function
3
f(x) = a(x - h) + k, you identify the point of
symmetry (h, k) and use the value of a to draw the
graph through two additional points
(-1 + h, –a + k) and (1 + h, a + k). So, to graph
3
f(x) = 2(x - 3) + 1, first identify the point of
symmetry, (3, 1). Then identify points
(-1 + 3, -2 + 1) = (2, -1), and
(1 + 3, 2 + 1) = (4, 3) as additional reference points.
A smooth curve through these three points is a good
beginning for the graph.
( )
3
a. The graph of g(x) = __12 x is a
Graphing Combined Transformations of f(x) = x 3
When graphing transformations of ƒ(x) = x 3, it helps to consider the effect of the transformations on the
three reference points on the graph of ƒ(x): (-1, -1), (0, 30), and (1,1). The table lists the three points and the
corresponding points on the graph of g(x) = a __b1 (x - h) + k. Notice that the point (0, 0), which is the point of
symmetry for the graph of ƒ(x), is affected only by the parameters h and k. The other two reference points are affected
by all four parameters.
(
)
(_
) +k
g(x) = a 1 (x - h)
b
f(x) = x 3
3
x
y
x
-1
-1
-b + h
-a + k
0
0
h
k
1
1
b+h
a+k
Module 6
317
y
Lesson 1
COLLABORATIVE LEARNING
A2_MTXESE353930_U3M06L1.indd 317
Whole Class Activity
Have groups of students create posters to describe the transformations to the
3
graph of f(x) = x3 when graphing f(x) = a(x - h) + k. Have them write an
example function in the top cell of a graphic organizer. Then, in each cell below
the example function, have them list the variable and describe the nature of the
transformation.
317
Lesson 6.1
2/21/14 3:30 AM
Example 1

Identify the transformations of the graph of ƒ(x) = x 3 that produce the
graph of the given function g(x). Then graph g(x) on the same coordinate
plane as the graph of ƒ (x) by applying the transformations to the
reference points (-1, -1), (0, 0), and (1, 1).
g(x) = 2(x - 1) - 1
3
4
The transformations of the graph of ƒ(x) that produce the graph of
g(x) are:
What is the point of symmetry for a graph of
3
the form f(x) = __b1 (x - h) + k? (h, k)
y
x
-4
-2
Note that the translation of 1 unit to the right affects only the
x-coordinates of points on the graph of ƒ(x), while the vertical stretch
by a factor of 2 and the translation of 1 unit down affect only the
y-coordinates.
0
2
(
)
(
)
How does a negative value for b affect the
3
graph of f(x) = __b1 (x - h) + k? The graph is
reflected across the y-axis.
2
• a vertical stretch by a factor of 2
• a translation of 1 unit to the right and 1 unit down
4
-4
g(x) = 2(x - 1) - 1
f(x) = x 3

QUESTIONING STRATEGIES
3
x
y
-1
-1
x
y
-1 + 1 = 0
2(-1) - 1 = -3
0
0
0+1=1
1
1
1+1=2
2(0) - 1 = -1
2(1) - 1 = 1
g(x) = (2(x + 3)) + 4
3
6
The transformations of the graph of ƒ(x) that produce the graph of
g(x) are:
y
4
• a horizontal compression by a factor of __12
2
• a translation of 3 units to the left and 4 units up
x
-4
-2
0
2
-2
© Houghton Mifflin Harcourt Publishing Company
Note that the horizontal compression by a factor of __12 and the translation
of 3 units to the left affect only the x-coordinates of points on the graph
of ƒ(x), while the translation of 4 units up affects only the y-coordinates.
g(x) = (2(x + 3)) + 4
3
f(x) = x 3
x
y
-1
-1
0
0
__1
1
1
__1
Module 6
x
__1
2
y
__
1
(-1) + -3 = -3 2
2 (0) + -3 = -3
__
-2 12
2 (1) + -3 =
318
-1 + 4 = 3
0+ 4 = 4
1+ 4 = 5
Lesson 1
DIFFERENTIATE INSTRUCTION
A2_MTXESE353930_U3M06L1.indd 318
Critical Thinking
2/21/14 3:30 AM
Have students form pairs. Give each pair of students a cubic function in general
form and a graphing calculator. Have them graph the parent function f(x) = x 3
and then discuss how their given cubic function will be transformed based the
graph of the parent function. Include choices such as reflection across the x-axis,
reflection across the y-axis, horizontal and vertical shifts, and horizontal and
vertical stretches. Have them graph their given functions to verify their
predictions.
Graphing Cubic Functions 318
Your Turn
EXPLAIN 2
Identify the transformations of the graph of ƒ(x) = x 3 that produce the graph of the
given function g(x). Then graph g(x) on the same coordinate plane as the graph of ƒ(x)
by applying the transformations to the reference points (-1, -1), (0, 0), and (1, 1).
Writing Equations for Combined
Transformations of f(x) = x 3
4.
(
4
y
2
1
• a vertical compression by a factor of __
2
For students having difficulty identifying the
parameters in the graph of g(x) = a
3
The transformations of the graph of f(x)
that produce the graph of g(x) are:
AVOID COMMON ERRORS
__1 (x
b
1 (x - 3)
g(x) = -_
2
x
-4
• a reflection across the x-axis
-2
0
2
4
• a translation of 3 units to the right
)
- h) + k,
3
-4
Note that the translation of 3 units to the right affects
have them start with identifying the images of
the reference points (-1, -1 ), (0, 0), (1, 1) from the
graph of f(x) = x 3. Then have them solve for the
parameters a, b, h, and k using the image points and
(-b + h, -a + k), (h, k), or (b + h, a + k),
respectively.
only the x-coordinates of points on the graph of f(x),
1
while the vertical compression by a factor of and the reflection across the x-axis affect
2
1
only the y-coordinates. Points: (2, ), (3, 0), and (4, - 1 )
2
2
_
_
_
Writing Equations for Combined Transformations
of f(x) = x 3
Explain 2
(
)
Given the graph of the transformed function g(x) = a __b1 (x - h) + k, you can determine the values of the
parameters by using the same reference points that you used to graph g(x) in the previous example.
Example 2
© Houghton Mifflin Harcourt Publishing Company

3
A general equation for a cubic function g(x) is given along with the
function’s graph. Write a specific equation by identifying the values of the
parameters from the reference points shown on the graph.
g(x) = a(x - h) + k
3
y
Identify the values of h and k from the point of symmetry.
4
(h, k) = (2, 1), so h = 2 and k = 1.
2
(3, 4)
(2, 1)
x
Identify the value of a from either of the other two reference points.
The rightmost reference point has general coordinates (h + 1, a + k).
Substituting 2 for h and 1 for k and setting the general coordinates equal
to the actual coordinates gives this result:
-2
0
-2
2
4
(1, -2)
6
(h + 1, a + k) = (3, a + 1) = (3, 4), so a = 3.
Write the function using the values of the parameters: g(x) = 3(x - 2) + 1
3
Module 6
319
Lesson 1
LANGUAGE SUPPORT
A2_MTXESE353930_U3M06L1.indd 319
Connect Vocabulary
Have students work in pairs. Instruct one student to explain the effect of any
transformation of the parent function f(x) = x 3. While the first student explains,
the second student writes down the steps in the explanation. They switch roles
and repeat the procedure for another change, such as the effect of b on the graph
of g(x) = __b1 x 3. The goal is to support students as they learn to articulate their
understanding—at first, using their own words, then moving towards more
mathematically precise explanations.
319
Lesson 6.1
2/21/14 3:30 AM
B
(b
)
3
g(x) = __1 x - h + k
4
Identify the values of h and k from the point of symmetry.
(
)
(-4.5, 0)
-7
Identify the value of b from either of the other two reference points.
-5
The rightmost reference point has general coordinates
(-4, 1)
-3
Why does the general form
3
f(x) = a(x - h) + k or
3
f(x) = __b1 (x - h) + k need to be specified before
finding the equation of the transformed
function? The general form must be specified so
that a unique equation can be found.
x
-1 0
-2
(
1
-4
1 for k and setting the
general coordinates equal to the actual coordinates gives this result:
(b + h, 1 + k). Substituting -4 for h and
(b + h, 1 +
QUESTIONING STRATEGIES
2
(-3.5, 2)
(h, k) = -4, 1 , so h = -4 and k = 1 .
y
) = (b - 4, 2 ) = (-3.5, 2), so b = 0.5 .
1
)
Write the function using the values of the parameters, and then simplify.
(
or
g(x) =
(
2
)
(
)
(
)) +
1
x - -4
g(x) = _____
0.5
3
+
1
3
x+ 4
1
Your Turn
A general equation for a cubic function g(x) is given along with the function’s graph.
Write a specific equation by identifying the values of the parameters from the reference
points shown on the graph.
5.
g(x) = a(x - h) + k
3
2
6.
y
(-3, -5)
3
x
y
2
2
(5, 0)
-2
0
-2
(-3, -2)
-4
-4
-6
2
(1, -1)
x
(h, k) = (-2, -2), so h = -2 and k = -2.
(h, k) = (1, -1), so h = 1 and k = -1.
g(x) = 3(x + 2) - 2
1(
g(x) = _
x - 1)
4
(h + 1, a + k) = (-2 + 1, a - 2) = (-1, 1), so a = 3. (b + h, 1 + k) = (b + 1, 1 - 1) = (5, 0), so b = 4.
(
3
Module 6
A2_MTXESE353930_U3M06L1.indd 320
320
)
3
-1
© Houghton Mifflin Harcourt Publishing Company
-4 -2 0
(-2, -2) -2
)
4
(-1, 1)
-6
(
1 (x - h) + k
g(x) = _
b
Lesson 1
2/21/14 3:30 AM
Graphing Cubic Functions 320
Explain 3
EXPLAIN 3
Modeling with a Transformation of f(x) = x 3
You may be able to model a real-world situation that involves volume with a cubic function. Sometimes mass may
also be involved in the problem. Mass and volume are related through density, which is defined as an object’s mass per
m
unit volume. If an object has mass m and volume V, then its density d is d = __
. You can rewrite the formula as m =
V
dV to express mass in terms of density and volume.
Modeling with a Transformation of
f (x ) = x 3
Example 3
QUESTIONING STRATEGIES

How is the density formula for a sphere of
radius r, m(r) = DV(r) with mass m, density
D, and volume V related to the volume of a sphere,
V(r) = __43 πr 3, with radius r? The mass of the sphere
is the product of the density and the volume, so
4
m(r) = __
πDr 3.
3
Use a cubic function to model the situation, and graph the function
using calculated values of the function. Then use the graph to obtain the
indicated estimate.
Estimate the length of an edge of a child’s alphabet
block (a cube) that has a mass of 23 g and is made from oak
with a density of 0.72 g/cm 3.
Let ℓ represent the length (in centimeters) of an edge of
the block. Since the block is a cube, the volume V (in cubic
centimeters) is V(ℓ) = ℓ 3. The mass m (in grams) of the block
is m(ℓ) = 0.72 · V(ℓ) = 0.72ℓ 3. Make a table of values for
this function.
Length (cm)
What kind of transformation does the
equation m(r) = __43 πDr 3 show? a vertical
4
stretch of the function m(r) = r 3 by a factor of __
πD
3
Mass (g)
0
0
1
0.72
2
5.76
3
19.44
4
46.08
Mass (g)
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©MidoSemsem/Shutterstock
Draw the graph of the mass function, recognizing that the graph is a vertical compression
of the graph of the parent cubic function by a factor of 0.72. Then draw the horizontal line
m = 23 and estimate the value of ℓ where the graphs intersect.
45
40
35
30
25
20
15
10
5
0
m
ℓ
0.5
1.5
2.5
3.5
4.5
Length (cm)
The graphs intersect where ℓ ≈ 3.2, so the edge length of the child’s block is about 3.2 cm.
Module 6
A2_MTXESE353930_U3M06L1 321
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Lesson 6.1
321
Lesson 1
2/22/14 4:46 AM
B
Estimate the radius of a steel ball bearing with a mass of 75 grams and a density of 7.82 g/cm 3.
Let r represent the radius (in centimeters) of the ball bearing.
The volume V (in cubic centimeters) of the ball bearing is
0.5
4.10
1
32.76
0
1.5
110.57
2
262.08
3
r . The mass m (in grams) of the ball
3
bearing is m(r) = 7.82 · V(r) =
Radius (cm)
0
32.76
Mass (g)
V(r) =
270
240
210
180
150
120
90
60
30
_4π
r 3.
Mass (g)
0
m
r
0.5
1
1.5
2
Radius (cm)
Draw the graph of the mass function, recognizing that the graph is a vertical
of the graph of the parent cubic function by a factor of
7.82
stretch
. Then draw the
horizontal line m = 75 and estimate the value of r where the graphs intersect.
The graphs intersect where r ≈ 1.3 , so the radius of the steel ball bearing is about
1.3
cm.
Reflect
7.
Discussion Why is it important to plot multiple points on the graph of the volume function.
Because you are using the graph to estimate the value of the independent variable from
a given value of the dependent variable, the graph should be as accurate as possible.
© Houghton Mifflin Harcourt Publishing Company
Module 6
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Lesson 1
2/21/14 3:30 AM
Graphing Cubic Functions 322
Your Turn
ELABORATE
8.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Critical Thinking
Let ℓ represent the length (in centimeters) of an inner
edge of the box. Since the box is a cube, the
volume V (in cubic centimeters) is V(ℓ) = ℓ 3. The mass
m (in kilograms) of the polystyrene beads in the box is
m(ℓ) = 0.00076 · V(ℓ) = 0.00076ℓ 3. Make a table of values
for this function.
Before students graph any cubic function, encourage
them to predict how the graph will look based on the
leading coefficient of x when the function is written
3
in general form g(x) = a __b1 (x - h) + k, and on the
values of h and k.
(
Polystyrene beads fill a cube-shaped box with an effective density of
0.00076 kg/cm 3 (which accounts for the space between the beads).
The filled box weighs 6 kilograms while the empty box had weighed
1.5 kilograms. Estimate the inner edge length of the box.
)
Length (cm)
SUMMARIZE THE LESSON
9
8
7
6
5
4
3
2
1
0
0
5
0.095
10
0.76
15
2.565
20
6.08
ℓ
5
10
15
20
Length (cm)
Mass (kg)
0
m
Draw the graph of the mass function, recognizing that the graph is a vertical compression
of the graph of the parent cubic function by a factor of 0.00076. Then draw the horizontal
line m = 6 - 1.5 = 4.5 and estimate the value of ℓ where the graphs intersect.
The graphs intersect where ℓ ≈ 18, so the inner edge length of the box is about 18 cm.
Elaborate
9.
© Houghton Mifflin Harcourt Publishing Company
How do you use reference points to graph or
write equations for combined transformations
of f(x) = x 3? Use the reference points (-1, -1),
(0, 0), and (1, 1). Under a combined transformation
(-1, -1) becomes (-b + h, -a + k), (0, 0) becomes
(h, k), and (1, 1) becomes (b + h, a + k). To graph the
combined transformation, you apply the
transformations to the reference points and use
those transformed points to graph the transformed
function. To write an equation, you use the
transformations of the reference points to solve for
a, b, h, and k and use those values to write the
equation of the transformed function.
Mass (kg)
Use a cubic function to model the situation, and graph the function using calculated
values of the function. Then use the graph to obtain the indicated estimate.
Identify which transformations (stretches or compressions, reflections, and translations) of ƒ(x) = x 3
change the following attributes of the function.
a. End behavior
Reflections across the x-axis (a < 0) and reflections across the y-axis (b < 0) change
the end behavior.
b. Location of the point of symmetry
Vertical translations (k ≠ 0) and horizontal translations (h ≠ 0) change the location of
the point of symmetry.
c.
Symmetry about a point
No transformations change the function’s symmetry about a point.
10. Essential Question Check-In Describe the transformations you must perform on the graph of ƒ(x) = x 3
3
to obtain the graph of g(x) = a(x - h) + k.
If a < 0, reflect the parent graph across the x-axis. Then either stretch the graph vertically
by a factor of a if ⎜a⎟ > 1 or compress the graph vertically by a factor of a if ⎜a⎟ < 1. Finally,
translate the graph h units horizontally and k units vertically.
Module 6
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Lesson 6.1
323
Lesson 1
1/12/15 9:39 PM
Evaluate: Homework and Practice
1.
EVALUATE
• Online Homework
• Hints and Help
• Extra Practice
Graph the parent cubic function ƒ(x) = x 3 and use the graph to answer each question.
a. State the function’s domain and range.
The domain is ℝ, and the range is ℝ.
8
b. Identify the function’s end behavior.
6
As x → +∞, f(x) → +∞. As x → -∞, f(x) → -∞.
4
Identify the graph’s x- and y-intercepts.
c.
y
ASSIGNMENT GUIDE
2
The graph’s only x-intercept is 0. The graph’s only
y-intercept is 0.
x
-4
d. Identify the intervals where the function has positive values
and where it has negative values.
-2
0
2
4
-2
-4
The function has positive values on the interval
(0, +∞) and negative values on the interval (-∞, 0).
-6
-8
e. Identify the intervals where the function is increasing and
where it is decreasing.
The function is increasing throughout its domain.
The function never decreases.
Tell whether the function is even, odd, or neither. Explain.
f.
The function is odd because f(-x) = (-x) 3 = -x 3 = -f(x).
g. Describe the graph’s symmetry.
The graph is symmetric about the origin.
2.
g(x) = (x - 4)
3
3.
translation of the graph of f(x) right
4 units.
4.
g(x) = x 3 + 2
vertical stretch of the graph of f(x) by
a factor of 5 and a reflection across the
x-axis.
5.
translation of the graph of
f(x) up 2 units.
Module 6
g(x) = -5x 3
g(x) = (3x)
3
horizontal compression of the
1
.
graph of f(x) by a factor of _
3
Exercise
Depth of Knowledge (D.O.K.)
Practice
Explore 1
Graphing and Analyzing f(x) = x 3
Exercise 1
Explore 2
Predicting Transformations of the
Graph of f(x) = x 3
Exercises 2–9
Example 1
Graphing Combined
Transformations of f(x) = x 3
Exercises 10–17
Example 2
Writing Equations for Combined
Transformations of f(x) = x 3
Exercises 18–21
Example 3
Modeling with a Transformation of
f(x) = x 3
Exercises 22–23
INTEGRATE TECHNOLOGY
When graphing a cubic function or any other
function on a graphing calculator, students
should choose a good viewing window: one that
displays the important characteristics of the graph. As
a class, have students brainstorm what should show
on the graphing screen. Sample answer: the end
behavior; the turning point, or point of symmetry of
the graph; the x-axis of the graph
Lesson 1
324
A2_MTXESE353930_U3M06L1.indd 324
© Houghton Mifflin Harcourt Publishing Company
Describe how the graph of g(x) is related to the graph of ƒ(x) = x 3.
Concepts and Skills
Mathematical Processes
1–10
1 Recall of Information
1.F Analyze relationships
11–21
2 Skills/Concepts
1.F Analyze relationships
22–24
3 Strategic Thinking
1.A Everyday life
25
2 Skills/Concepts
1.F Analyze relationships
26
3 Strategic Thinking
1.G Explain and justify arguments
2/21/14 3:30 AM
Graphing Cubic Functions 324
6.
AVOID COMMON ERRORS
3
7.
translation of the graph of f(x) left
1 unit.
Students may be confused about where to start when
finding the equation for a combined transformation
of the graph of f(x) = x 3. Explain that they should
compare the corresponding points of the graph to the
reference points of the graph of f(x) = x 3. If the graph
is stretched or compressed horizontally, then they
3
may use the form f(x) = __b1 (x - h) + k. If the
graph is stretched or compressed vertically, then they
3
may use the form f(x) = a(x - h) + k.
(
g(x) = (x + 1)
8.
g(x) = x 3 - 3
1 x3
g(x) = _
4
vertical compression of the graph of
1
f(x) by a factor of _
.
4
9.
translation of the graph of f(x) down
3 units.
( )
2x
g(x) = -_
3
horizontal stretch of the graph of f(x) by
3
a factor of _
as well as a reflection across
2
the y-axis.
)
Identify the transformations of the graph of ƒ(x) = x 3 that
produce the graph of the given function g(x). Then graph g(x) on
the same coordinate plane as the graph of ƒ(x) by applying the
transformations to the reference points (-1, -1), (0, 0), and (1, 1).
3
1x
1 x3
10. g(x) = _
11. g(x) = _
3
3
( )
y
4
4
x
x
-4
0
-2
2
-4
4
-2
© Houghton Mifflin Harcourt Publishing Company
Reference points on the graph of g(x) are
(-1, -_13), (0, 0) and (1, _13).
3
3
y
4
2
2
x
2
4
x
6
-4
-4
Lesson 6.1
-2
0
2
4
-4
a translation of 1 unit to the left and
2 units up. Reference points on the graph
of g(x) are (-2, 1), (-1, 2), and (0, 3).
translation of 4 units to the right
and 3 units down. Reference
points on the graph of g(x) are
(3, -4), (4, -3), and (5, -2).
A2_MTXESE353930_U3M06L1.indd 325
4
13. g(x) = (x + 1) + 2
12. g(x) = (x - 4) - 3
Module 6
2
1
.
a vertical compression by a factor of _
3
a horizontal stretch by a factor of 3.
Reference points on the graph of g(x)
are (-3, -1), (0, 0), and (3, 1).
0
0
-4
-4
-2
y
2
2
325
3
325
Lesson 1
15-01-11 1:52 AM
15. g(x) = (-2(x - 2))
14. g(x) = -2x 3 + 3
6
y
4
x
2
-4
x
-2
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
y
2
4
-4
3
0
2
0
-2
4
2
-4
-2
1
•ahorizontalcompressionbyafactorof_
2
•averticalstretchbyafactorof2
•areflectionacrossthex-axis
•areflectionacrossthey-axis
•atranslationof3unitsup
•atranslationof2unitstotheright
Referencepointsonthegraphofg(x)are
(-1,5),(0,3),and(1,1).
Referencepointsonthegraphofg(x)are
(1_12, 1),(2,0),and(2_12, -1).
(
3
y
2
Horizontal shift
3
Vertical stretch
y
Horizontal
compression
x
4
-2
-2
-4
0
2
4
-4
•ahorizontalstretchbyafactorof4
•areflectionacrossthey–axis
•atranslationof1unittotheleftand
1unitdown
•atranslationof1unittotherightand
2unitsup
Referencepointsonthegraphofg(x)are
(-2,2),(-1,-1),and(0,-4).
Referencepointsonthegraphofg(x)are
(-3,3),(1,2),and(5,1).
A2_MTXESE353930_U3M06L1.indd 326
326
© Houghton Mifflin Harcourt Publishing Company
•averticalstretchbyafactorof3
•areflectionacrossthex-axis
Module 6
Example
Vertical shift
2
x
0
)
4
2
-2
Transformation
1 (x - 1) + 2
17. g(x) = -_
4
16. g(x) = -3(x + 1) - 1
-4
Suggest that students make a graphic organizer
similar to the one below to help them remember
the differences in the types of transformations of
f(x) = x 3. Suggest students choose one or more cubic
functions from this exercise set and give an example
of each type of transformation.
4
Lesson 1
15-01-11 1:52 AM
Graphing Cubic Functions 326
A general equation for a cubic function g(x) is given along with the function’s graph.
Write a specific equation by identifying the values of the parameters from the reference
points shown on the graph.
3
3
1 (x - h) + k
18. g(x) = _
19. g(x) = a(x - h) + k
b
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
(
Point out that another name for the point of
symmetry of a cubic function in this lesson is the
inflection point. This is the point for which the
“curvature” of the graph changes. For a cubic
equation with a positive leading coefficient, the rate
of change in the y-values of the graph decreases as x
approaches this point from below and then begins to
increase again as x increases past the point. In later
mathematics classes, students will find that the slope
of the curve at this point is 0.
-6
)
-4
y
(-2, -3) -2
(-5, -4)
-4
6
x
0
-2
4
(1, -2)
-2
(h, k) = (-2, -3), so h = -2 and k = -3.
)
(
3
(2, 1)
0
4
(h + 1, a + k) = (1 + 1, a + 4) = (2, 1), so a = -3.
g(x) = -3(x - 1) + 4
-3
3
)
3
1 (x - h) + k
20. g(x) = _
b
21. g(x) = a(x - h) + k
3
y
6
2
(1.5, 0)
-2
0
(2, -1)
-2
(-1, 2)
4
6
(2.5, -2)
© Houghton Mifflin Harcourt Publishing Company
(0, 2.5)
(-2, 1.5)
-4
-2
0
2
x
-2
(h, k) = (2, -1), so h = 2 and k = -1.
(h, k) = (-1, 2), so h = -1 and k = 2.
so b = -0.5.
so a = 0.5.
(h + 1, a + k) = (-1 + 1, a + 2) = (0, 2.5),
(b + h, 1 + k) = (b + 2, 1 - 1) = (1.5, 0),
3
3
1
g(x) = (____
(x - 2)) - 1 = (-2(x - 2)) - 1
-0.5
Module 6
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Lesson 6.1
y
4
x
-4
327
x
(h, k) = (1, 4), so h = 1 and k = 4.
(b + h, 1 + k) = (b - 2, 1 - 3) = (1, -2), so b = 3.
(
(1, 4)
2
-6
1(
g(x) = _
x + 2)
3
y
(0, 7)
327
g(x) = 0.5(x + 1) + 2
3
Lesson 1
15-01-11 1:52 AM
Use a cubic function to model the situation, and graph the function
using calculated values of the function. Then use the graph to
obtain the indicated estimate.
22. Estimate the edge length of a cube of gold with a mass of 1 kg.
The density of gold is 0.019 kg/cm 3.
Length (cm)
0
Mass (kg)
m
2
Mass (kg)
The volume V (in cubic centimeters) of the cube is
V(ℓ) = ℓ 3. The mass m (in kilograms) of the cube is
m(ℓ) = 0.019 · V(ℓ) = 0.019ℓ 3.
0.019
2
0.152
3
0.513
4
1.216
For modeling problems involving geometry, review
the formulas for the volumes of a sphere and
rectangular prism. Students should recall that the
volume V of a sphere can be found with the formula
V = __43 πr 3, where r is the radius, and the volume of a
rectangular prism is V = lwh, where l is the length of
the rectangular base, w is the width of the base, and h
is the height of the prism.
1.5
1
0.5
0
1
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
ℓ
0
0.5
1.5
2.5
3.5
Length (cm)
The graphs intersect where ℓ ≈ 3.75, so the edge
length of the cube of gold is about 3.75 centimeters.
23. A proposed design for a habitable Mars colony is a hemispherical
biodome used to maintain a breathable atmosphere for the colonists.
Estimate the radius of the biodome if it is required to contain 5.5
billion cubic feet of air.
Since the biodome is a hemisphere,
the volume V (in billions of cubic feet) of
(
)
Volume
(billions of cubic feet)
0
0.26
2.09
7.05
16.72
The graphs intersect where r ≈ 1.4, so the radius of the
biodome is about 1.4 thousand (1400) ft.
V
9
8
7
6
5
4
3
2
1
0
r
0.5
1
1.5
2
Radius (thousands of feet)
Module 6
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328
© Houghton Mifflin Harcourt Publishing Company
Radius
(thousands of feet)
0
0.5
1
1.5
2
Volume (billions of cubic feet)
1 _
4 3
the biodome is V(r) = _
πr ≈ 2.09r 3.
2 3
Lesson 1
2/14/15 10:13 PM
Graphing Cubic Functions 328
24. Multiple Response Select the transformations of the graph of the parent cubic
3
function that result in the graph of g(x) = (3(x - 2)) + 1.
CONNECT VOCABULARY
To help students remember how the words shift,
stretch, and compression are used in the context of this
lesson, remind students that a shift is a translation, a
transformation that does not change the shape or size
of the original figure as the figure moves on the
coordinate plane. A stretch (or compression), on the
other hand, represents what happens to the graph of
the parent figure if the leading coefficient of x is not
equal to one. Reference points from the transformed
graph will have more space between the
corresponding y-values (“stretched”) than the parent
graph, or less space between the y-values
(“compressed”) than points on the parent graph.
You can see these changes on the graph for small
values of x.
A. Horizontal stretch by a factor of 3
E. Translation 1 unit up
1
B. Horizontal compression by a factor of _
3
F. Translation 1 unit down
C. Vertical stretch by a factor of 3
G. Translation 2 units left
_
D. Vertical compression by a factor of 1
3
H. Translation 2 units right
H.O.T. Focus on Higher Order Thinking
25. Justify Reasoning Explain how horizontally stretching (or compressing) the
graph of ƒ(x) = x 3 by a factor of b can be equivalent to vertically compressing (or
stretching) the graph of ƒ(x) = x 3 by a factor of a.
Horizontally stretching or compressing the graph of the parent cubic function by a factor
( )
1 3
of b results in the graph of the function g(x) = _
x . You can rewrite g(x) as follows:
b
( )
1
·x
= (_
b)
1
x
g(x) = _
b
3
3
3
1
3
= ___
3 · x
b
1
⎜ ⎟
The function is now in the form g(x) = ax 3 where a = __
3 . Notice that when b > 1, ⎜a⎟ < 1,
b
which means that a horizontal stretch is equivalent to a vertical compression. Similarly,
PEERTOPEER DISCUSSION
JOURNAL
when ⎜b⎟ < 1, ⎜a⎟ > 1, which means that a horizontal compression is equivalent to a
vertical stretch.
26. Critique Reasoning A student reasoned that g(x) = (x - h) can be rewritten
as g(x) = x 3 - h 3, so a horizontal translation of h units is equivalent to a vertical
translation of -h 3 units. Is the student correct? Explain.
3
© Houghton Mifflin Harcourt Publishing Company
Have students work in pairs. Instruct one student in
each pair to sketch the graph of a cubic function of
3
the form f(x) = a(x - h) + k, while the other gives
verbal instructions for each step. Then have the
student who sketched the graph write the steps that
were followed. Have students switch roles and repeat
the exercise using a cubic function of the form
f(x) = __b1 (x - h) 3 + k.
No, the student isn’t correct. There is no property of exponents that
allows you to distribute an exponent to a sum or difference, so
3
g(x) = (x - h) cannot be rewritten in the form g(x) = x 3 + k. From a
geometric perspective, it doesn’t make sense to think that there can
be a vertical translation of a graph that is equivalent to a horizontal
translation of the graph except in the trivial case where the graph is not
moved at all (h = k = 0).
Have students compare and contrast algebraic
representations of vertical and horizontal
transformations of cubic functions.
Module 6
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Lesson 6.1
329
Lesson 1
2/25/14 12:35 PM
Lesson Performance Task
AVOID COMMON ERRORS
Make sure students are multiplying the volume by the
density instead of dividing by the density. Have
students write the units next to the values as they do
the calculation. If the units cancel properly, the final
units should be mass. Ask students what the real-life
consequence of dividing by the density would be.
Julio wants to purchase a spherical aquarium and fill it with salt water, which has an average
density of 1.027 g/cm 3. He has found a company that sells four sizes of spherical aquariums.
Aquarium Size
Diameter (cm)
Small
15
Medium
30
Large
45
Extra large
60
QUESTIONING STRATEGIES
a. If the stand for Julio’s aquarium will support a maximum of 50 kg, what is the
largest size tank that he should buy? Explain your reasoning.
Why would using fresh water allow Julio to
buy a larger tank? Fresh water has a lower
density and therefore has a lower mass for the same
volume.
b. Julio’s friend suggests that he could buy a larger tank if he uses fresh water, which
has a density of 1.0 g/cm 3. Do you agree with the friend? Why or why not?
a. The volume V (in cubic centimeters) of a sphere with diameter d (in centimeters) is
()
3
4
d
π 3
V(d) = _
π _
= __
d . If the sphere is filled with salt water having a density of 1.027 g/cm3,
3
6
2
1.027
then the mass m (in kilograms) of the salt water is m(V) = ____
V = 0.001027V. The table
1000
If the diameter of the Medium tank is twice
the diameter of the Small tank, why isn’t the
volume of the Medium tank twice that of the Small
tank? The volume is described by a cubic function.
The ratio of the volumes is the ratio of their
diameters cubed.
summarizes the volume and mass of salt water for each size of tank.
Aquarium Size
Small
Volume (cm 3)
1,760
Medium
14,100
Large
47,700
Extra large
113,000
Mass of Water (kg)
1.81
14.5
49.0
Why is it important to know the density of the
water? The density is the mass per volume. If
the volume is known, then the density will give the
mass that the volume will hold.
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49.0 kg, is less than 50 kg. Other students may say that he should buy the medium tank,
because he has to consider the mass of the empty tank, which will likely make the total mass
greater than 50 kg.
b. Students’ answers will depend on how they answered Part a. If they answered that Julio
should buy the large tank, then the minor difference in density will have no effect on his
choice. If they said he should buy the medium tank, some students may decide that the
© Houghton Mifflin Harcourt Publishing Company
Some students may say that Julio should buy the large tank, because the mass of the water,
lesser density of fresh water might reduce the mass of the water just enough to allow him to
buy the large tank.
Module 6
330
Lesson 1
EXTENSION ACTIVITY
A2_MTXESE353930_U3M06L1.indd 330
Julio decides he wants to fill one-third of his aquarium with sand that has a
density of 1.6 g/cm 3. The remaining two-thirds of the aquarium will be filled with
water. Have students answer the questions in the Performance Task using this new
information. Then have them graph the mass in the aquarium compared with the
aquarium size, both with and without the sand. Ask them how the addition of the
sand affects the total mass in the aquarium and if this information will help Julio
decide between a Medium and a Large tank.
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Scoring Rubric
2 points: The student’s answer is an accurate and complete execution of the
task or tasks.
1 point: The student’s answer contains attributes of an appropriate response
but is flawed.
0 points: The student’s answer contains no attributes of an appropriate response.
Graphing Cubic Functions 330