The graph of any cube root function is a transformation of the graph of the cube root
parent function f
(x) = 1
3
x.
ESSENTIAL UNDERSTANDING
TEKS (6)(A) Analyze the effect on the graphs of f
(x) = x
3
and f
(x) = 2x
3
when f (x) is replaced by af (x), f (bx),
f
(x - c), and f(x) + d for specific
positive and negative real values of a, b, c, and d.
TEKS (1)(F)
Analyze mathematical relationships to connect and
communicate mathematical ideas.
Additional TEKS (1)(D), (2)(A)
TEKS FOCUS
Ě
Analyze – closely examine objects,
ideas,or relationships to learn more
about their nature
VOCABULARY
Table Function Graph
f
(x) = 1
3
x
Key Concept The Cube Root Parent Function
x
-8
-1
0
1
8
-2
-1
0
1
2
3
y
= !
x
y
4
2
O
−4 −2
−4
−2
4
x
2
10-3
Transformations of Cube
Root Functions
429
PearsonTEXAS.com
Parent Function y = 2
3
x
Translation
y = 1
3
x + d y = 1
3
x - c
d 7 0 shifts up
0
d
0
units c 7 0 shifts to the right
0
c
0
units
d 6 0 shifts down
0
d
0
units c 6 0 shifts to the left
0
c
0
units
Stretch, Compression, and Reflection
y = a1
3
x y = 1
3
bx
0
a
0
7 1 vertical stretch
0
b
0
7 1 horizontal compression (shrink)
0 6 a 6 1 vertical compression (shrink) 0 6 b 6 1 horizontal stretch
a 6 0 reflection across the x-axis b 6 0 reflection across the y-axis
Concept Summary Cube Root Function Family
Problem 1
P
Analyzing y = f
(x) + d for f
(x) =
!
x
3
What is the graph of the functions g(x) = 2x
3
+ 2 and h(x) = 2x
3
− 2 on the
samesetof axes as the parent function f
(x) = 2x
3
?
Graph the parent function f
(x) = 2x
3
.
Each y-value of g(x) = 2x
3
+ 2 is 2 greater than the corresponding y-value of f (x).
So, g(x) shifts the graph of f (x) up by 2 units.
Each y-value of h
(x) = 2x
3
- 2 is 2 less than the corresponding y-value of f (x).
So, h(x) shifts the graph of f (x) down by 2 units.
TEKS Process Standard (1)(D)
x
O
y
2
-2
-4
-2-4
4
3
y = x − 2
!
3
y = x
!
3
y = !
x
+ 2
6
How can you check
that your graphs are
reasonable?
The graphs of g(x) and
h(x) should be 4 units
apart along any vertical
line.
430 Lesson 10-3 Transformations of Cube Root Functions
Problem 3
Analyzing y = af
(x) for f
(x) =
!
3
x
What is the graph of the given functions on the same
set of axes as the parent function f
(x) = 2x
3
?
A
g(x) = 32x
3
, h
(x) = - 3 2x
3
Graph the parent function f
(x) = 2x
3
.
Each y-value of g
(x) = 32x
3
is 3 times the
corresponding y-value of the parent function.
So,g(x) = 32x
3
stretches thegraph of f (x) by the
factor3.
The - 3 in h(x) =-32x
3
reflects the graph of f (x)
across the x-axis and stretches it by the factor 3.
B
g
(x) =
1
4
2x
3
, h
(x) =-
1
4
2x
3
Graph the parent function f
(x) = 2x
3
.
Each y-value of g
(x) =
1
4
2x
3
is one-fourth the
corresponding y-value of the parent function.
So,g
(x) =
1
4
2x
3
compresses thegraph of f (x) by the
factor
1
4
.
The -
1
4
in h(x) =-
1
4
2x
3
reflects the graph of f (x)
across the x-axis and compresses it by the factor
1
4
.
x
O
y
-4
-224-4
4
y = 3 x
3
!
y = −3 x
3
!
y = x
!
3
-6
x
O
y
-4
-2
4
2
y = x
1
4
−
3
!
y = x
1
4
3
!
y = x
!
3
bl 3
Problem 2
P
Analyzing y = f
(x − c) for f
(x) =
!
3
x
What is the graph of the functions g(x) = 2x − 3
3
and
h(x) = 2x + 3
3
on the sameset of axes as the parent
function f
(x) = 2x
3
?
Graph the parent function f
(x) = 2x
3
.
A function of the form f
(x) = 2x - c
3
, where c 7 0, is a
horizontal translation of the graph of f (x) by
0
c
0
units to
theright. So, g(x) = 2x - 3
3
shifts the graph of f (x) to the
right by 3 units.
A function of the form f
(x) = 2x - c
3
, where c 6 0, is a
horizontal translation of the graph of f (x) by c units to
theleft. So h
(x) = 2x + 3
3
shifts the graph of f (x) to the
leftby 3 units.
x
O
y
2
-2
-4
-224-4
4
y = x − 3
3
!
y = x + 3
3
!
y = x
!
3
- 6
h
t
r
A
h
t
l
How can you check
that your graphs are
reasonable?
The graphs of g(x)
and h(x) should be
6 units apart along any
horizontal line.
How are the graphs
of g(x) and h(x)
related?
The graphs are reflections
of each other in the x-axis.
431
PearsonTEXAS.com
Problem 4
P
Analyzing y = f
(bx) for f
(x) =
!
3
What is the graph of the given functions on the same set of axes as the parent
function f
(x) = 2x
3
?
A
g
(x) = 20.5x
3
, h
(x) = 2- 0.5x
3
Graph the parent function f
(x) = 2x
3
.
A function of the form f
(x) = 2bx
3
where 0 6
0
b
0
6 1 is a horizontal stretch
ofthegraph of f (x). So, g
(x) = 20.5x
3
stretches the graph of f (x) horizontally.
The - 0.5 in h
(x) = 2- 0.5x
3
reflects the graph of f (x) across the y-axis and
stretchesithorizontally.
B
g
(x) = 26x
3
, h
(x) = 2- 6x
3
Graph the parent function f
(x) = 2x
3
.
A function of the form f
(x) = 2
3
bx, where b 7 1, is a horizontal compression
ofthegraph of f (x). So, g(x) = 26x
3
compresses the graph of f (x) horizontally.
The - 6 in h(x) = 2- 6x
3
reflects the graph of f (x) across the y-axis andcompresses it
horizontally.
x
y
-4
-2
-2-424
4
2
y = −0.5x
3
!
y = 0.5x
3
!
y = x
!
3
6
x
O
y
-4
-2
-2-424
4
2
y = −6x
3
!
y = 6x
3
!
y = x
!
3
What is the amount
of the horizontal
stretch?
Since
b = 0.5 and
0
6
0
b
0
6 1,
f
(
x
) is
horizontally stretched.
Notice that
f(x) = 2
when
x = 8, but g(x) = 2
when
x = 16. So,
g
(
x
) is a
horizontal stretch of
f
(
x
)
by a factor of 2.
432 Lesson 10-3 Transformations of Cube Root Functions
Problem 5
Graphing a Cube Root Function
The function y = 22x + 1 + 4
3
models the monthly profit y, in thousands of
dollars, x months after a café opens for business. What is the graph of the function?
Step 1 Graph the parent function y = 2x
3
.
Step 2 Multiply the y-coordinates by 2. This
stretches the graph vertically.
Step 3 Translate the graph 1 unit to the left
and 4 units up.
Step 4 In this situation, the domain is
x Ú 0. Redraw the graph in the first
quadrant only.
TEKS Process Standard (1)(F)
x
O
y
-2-424
4
6
2
Step 3
Step 3
Step 1
Step 3
Step 2
y = 2 x + 1 + 4
3
!
y = 2 x
3
!
y = x
!
3
6
Café Profits
Month
Profit (thousands of dollars)
0
0246810
2
4
6
8
y
x
10
C
O
F
F
E
E
C
A
F
É
S
S
S
S
How is
y = a
3
2x − c + d
related to its parent
function?
The factor a vertically
stretches or compresses
the parent function
while c and d translate it
horizontally and vertically.
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PearsonTEXAS.com
PRACTICE and APPLICATION EXERCISES
O
N
L
I
N
E
H
O
M
E
W
O
R
K
For additional support when
completing your homework,
go to
PearsonTEXAS.com
.
Graph each transformation of the parent function f
(x) = 2x
3
on the same set
of axes as the parent function. Analyze the effect of the transformation on the
graph of the parent function.
1. g
(x) = 2x
3
+ 4 2. g
(x) = 2- 3x
3
3. g
(x) =-22x
3
4. g
(x) = 2x
3
- 1 5. g
(x) =
5
1
4
x
3
6. g
(x) =
1
3
2x
3
7. g
(x) =
5
-
1
10
x
3
8. g
(x) = 2x + 3.5
3
9. g
(x) = 2x - 1.5
3
10. The function y =-22x + 4 + 8
3
models the price y of one share of a company’s
stock, in dollars, x months after the stock became available.
a. Describe a sequence of transformations you can use to graph the function if
youstart with the graph of the parent function f
(x) = 1x.
3
Graph the function.
b. What does the graph tell you about the price of the company’s stock?
c. Do you think the function is a good model of the stock’s price for any number
of months? Explain.
11.
Explain Mathematical Ideas (1)(G) Your friend said the graph of g(x) = 2- x
3
isareflection of the graph of the parent function f
(x) = 1x.
3
in the x-axis. You
saidthe graph of g(x) is a reflection of the graph of the parent function in the
y-axis.Who is correct? Explain.
The graph of g(x) can be obtained from the graph of the parent function f
(x) = 2x
3
by using the given transformations. Write an equation for the function g(x).
12. Reflect the graph in the x-axis, then translate it 2 units right.
13. Vertically compress the graph by a factor of
1
3
, then translate it 4 units left and
1 unit up.
14. Vertically stretch the graph by a factor of 6, then translate it 1 unit right and
7 units up.
15. Horizontally stretch the graph by a factor of 2, then translate it 2 units down.
Match each function with the correct graph.
16. y = 2- 2x
3
+ 2 17. y = 22x + 2
3
18. y =-2x
3
- 2
A. B. C.
x
O
2-2
-2
-4
y
2
4
x
O
2
-2
-4
y
2
4
x
O
2
-2
-4
y
4
Scan page for a Virtual Nerd™ tutorial video.
434 Lesson 10-3 Transformations of Cube Root Functions
19. Analyze Mathematical Relationships (1)(F) The function y = 0.52x - 2
3
+ 3
models the annual revenue y of a software company, in millions of dollars,
xyearsafter 2000.
a. Describe a sequence of transformations you can use to graph the function if
youstart with the graph of the parent function y = 2x
3
.
b. Graph the function.
c. What does the graph tell you about the company’s annual revenue since 2000?
Determine whether each statement is always, sometimes, or never true.
20. For a not equal to zero, the graph of f
(x) = a2x
3
rises as x increases.
21. The graph of f
(x) = 2x
3
+ d has 180° rotational symmetry around a point on
they-axis.
22. You can draw the graph of g
(x) = 2x - c
3
by translating the graph of the parent
function f
(x) = 2x
3
.
23. The graph of g
(x) =-2x
3
- 1 passes through the point (a, 0), where a 7 0.
24. For real numbers m and n, with m ≠ n, the graph of f
(x) = 2x - m
3
intersects
thegraph of g
(x) = 2x - n
3
.
TEXAS Test Practice
T
25. Which function has a graph that is not a translation of the graph of the parent
function f
(x) = 2x
3
?
A. g
(x) = 2x - 3.7
3
C. g (x) = 23.7x
3
B. g (x) = 2x
3
+ 3.7 D. g (x) = 2x + 3.7
3
26. Which function has a graph that intersects the negative x-axis?
F. f
(x) = 2x
3
+ 8 H. f
(x) = 28x
3
G. f
(x) =-2x
3
+ 8 J. f
(x) = 2x - 8
3
27. You graph the function f
(x) = 1x.
3
Then you reflect the graph across the x-axis,
stretch the graph vertically by a factor of 2, and then translate the graph 2 units
tothe right. Which of the following is an equation for the resulting graph?
A. y =-2
3
2x + 2 C. y =-2
3
2x - 2
B. y =-2
3
2x - 2 D. y = 2- 2x
3
+ 2
28. Explain how the graph of g(x) = 4
3
2x - 1 is related to the graph of h(x) = 4
3
2x + 4.
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