Chapter 10: Algebra: Nonlinear Functions and Polynomials

Chapter 10: Algebra: Nonlinear Functions and Polynomials
Algebra: Nonlinear
Functions and Polynomials
10
•
Standard 7AF2.0 Interpret
and evaluate expressions
involving integer powers
and simple roots.
•
Standard 7NS1.0 Use
exponents, powers, and
roots and use exponents in
working with fractions.
Key Vocabulary
cube root (p. 554)
nonlinear function (p. 522)
quadratic function (p. 528)
Real-World Link
Fountains Many real-world situations, such as this fountain
at Paramount’s Great America theme park in Santa Clara
California, cannot be modeled by linear functions. These
can be modeled using nonlinear functions.
Algebra: Nonlinear Functions and Polynomials Make this Foldable to help you organize your
notes. Begin with eight sheets of grid paper.
1 Cut off one section of
the grid paper along both
the long and short edges.
2 Cut off two sections
from the second sheet,
three sections from the
third sheet, and so on
to the 8th sheet.
3 Stack the sheets from
narrowest to widest.
4 Label each of the right
tabs with a lesson number.
520 Chapter 10 Algebra: Nonlinear Functions and Polynomials
Richard Cummins/SuperStock
GET READY for Chapter 10
Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2
Take the Online Readiness Quiz at ca.gr7math.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Graph each equation.
(Lesson 11-2)
Example 1
1. y = x – 4
Graph y = x + 1.
2. y = 2x
First, make a table of values. Then, graph
the ordered pairs and connect the points.
3. y = x + 2
x
y
(x, y)
y = 2.54x describes about how
many centimeters y are in
x inches. Graph the function.
0
1
(0, 1)
1
2
(1, 2)
2
3
(2, 3)
(Lessons 11-2)
3
4
(3, 4)
4. MEASUREMENT The equation
Write each expression using a
positive exponent. (Lesson 2-9)
5. a -9
6. 6 -4
-5
-2
7. x
8. 5
Write each expression using
exponents. (Lesson 2-9)
9. 6 · 6 · 6 · 6
10. 3 · 7 · 7 · 3 · 7
11. FUND-RAISER The students at
y
yx1
O
x
Example 2
Write n -3 using a positive exponent.
1
n -3 = _
3
n
definition of negative exponent
Example 3
Write 5 · 4 · 5 · 4 · 5 using exponents.
5 is multiplied by itself 3 times and 4 is
multiplied by itself 2 times.
So, 5 · 4 · 5 · 4 · 5 = 5 3 · 4 2.
Hampton Middle School raised
8 · 8 · 2 · 8 · 2 dollars to help
build a new community center.
How much money did they
raise? (Lesson 2-9)
Chapter 10 Get Ready for Chapter 10
521
Linear and
Nonlinear Functions
10-1
Main IDEA
ROCKETRY The tables
show the flight data
for a model rocket
launch. The first table
gives the rocket’s
height at each second
of its ascent, or
upward flight. The
second table gives its
height as it descends
back to Earth using
a parachute.
Determine whether a
function is linear or
nonlinear.
Preparation for
AF1.5 Represent
quantitative
relationships graphically and
interpret the meaning of a
specific part of a graph in the
situation represented by the
graph.
NEW Vocabulary
nonlinear function
Ascent
Descent
Time
(s)
Height
(m)
Time
(s)
Height
(m)
0
0
7
140
1
38
8
130
2
74
9
120
3
106
10
110
4
128
11
100
5
138
12
90
6
142
13
80
1. During its ascent, did the rocket travel the same distance each
second? Justify your answer.
2. During its descent, did the rocket travel the same distance each
second? Justify your answer.
3. Graph the ordered pairs (time, height) for the rocket’s ascent and
descent on separate axes. Connect the points with a straight line or
smooth curve. Then compare the graphs.
REVIEW Vocabulary
constant rate of change
occurs when the rate of
change between any two
data points is proportional.
(Lesson 4-10)
In Lesson 9-2, you learned that linear functions have graphs that
are straight lines. These graphs represent constant rates of change.
Nonlinear functions are functions that do not have constant rates of
change. Therefore, their graphs are not straight lines.
Identify Functions Using Tables
Determine whether each table represents a linear or nonlinear
function. Explain.
1
+2
+2
+2
x
y
2
50
4
35
6
20
8
5
2
-15
+3
-15
+3
-15
+3
As x increases by 2, y decreases
by 15 each time. The rate of
change is constant, so this
function is linear.
522 Chapter 10 Algebra: Nonlinear Functions and Polynomials
Doug Martin
x
y
1
1
4
16
7
49
10
100
+15
+33
+51
As x increases by 3, y increases
by a greater amount each time.
The rate of change is not
constant, so this function is
nonlinear.
Determine whether each table represents
a linear or nonlinear function. Explain.
a.
x
0
5
10
15
y
20
16
12
8
b.
x
0
2
4
6
y
0
2
8
18
Identify Functions Using Graphs
Determine whether each graph represents a linear or nonlinear
function. Explain.
y
3
y
4
x
y2 1
y 0.5x 2
x
O
x
O
The graph is a curve, not a
straight line. So, it represents
a nonlinear function.
This graph is also a curve.
So, it represents a nonlinear
function.
Determine whether each graph represents
a linear or nonlinear function. Explain.
c.
d.
y
O
e.
y
x
O
y
x
O
x
Recall that the equation for a linear function can be written in the form
y = mx + b, where m represents the constant rate of change.
Identifying Linear
Equations Always
examine an equation
after it has been
solved for y to see
that the power
of x is 1 or 0. Then
check to see that x
does not appear in
the denominator.
Identify Functions Using Equations
Determine whether each equation represents a linear or nonlinear
function. Explain.
_
6 y = 6x
5 y=x+4
Since the equation can be
written as y = 1x + 4, this
function is linear.
f. y = 2x 3 + 1
Extra Examples at ca.gr7math.com
g. y = 3x
The equation cannot be written
in the form y = mx + b. So, this
function is nonlinear.
h. y = _
x
5
Lesson 10-1 Linear and Nonlinear Functions
523
7 BASKETBALL Use the table to determine whether
Round(s)
of play
Teams
1
32
Examine the differences between the number of
teams for each round.
2
16
3
8
16 - 32 = -16
4 - 8 = -4
4
4
5
2
the number of teams is a linear function of the
number of rounds of play.
8 - 16 = -8
2 - 4 = -2
While there is a pattern in the differences, they are
not the same. Therefore, this function is nonlinear.
Graph the data to verify the
ordered pairs do not lie on
a straight line.
Real-World Link
The NCAA women’s
basketball tournament
begins with 64 teams
and consists of
6 rounds of play.
y
32
24
Teams
Check
16
8
0
2
4
6
8 x
Rounds of Play
i. TICKETS Tickets to the school dance
cost $5 per student. Are the ticket sales
a linear function of the number of
tickets sold? Explain.
Number of
Tickets Sold
1
Ticket Sales
$5
2
3
$10 $15
Personal Tutor at ca.gr7math.com
Determine whether each table, graph, or equation or represents a linear or
nonlinear function. Explain.
Examples 1–6
1.
(pp. 522–523)
x
0
1
2
3
y
1
3
6
10
3.
x
0
3
6
9
y
-3
9
21
33
4.
y
O
2.
y
x
O
5. y = _
x
3
Example 7
(p. 524)
Elise Amendola/AP/Wide World Photos
6. y = 2x 2
7. MEASUREMENT The table shows the measures
of the sides of several rectangles. Are the
widths of the rectangles a linear function
of the lengths? Explain.
524 Chapter 10 Algebra: Nonlinear Functions and Polynomials
x
Length (in.)
1
4
8
10
Width (in.)
64
16
8
6.4
(/-%7/2+ (%,0
For
Exercises
8–13
14–19
20–25
26–29
See
Examples
1, 2
3, 4
5, 6
7
Determine whether each table, graph, or equation or represents a linear or
nonlinear function. Explain.
8.
10.
12.
x
3
6
9
12
y
12
10
8
6
x
5
10
15
20
y
13
28
43
58
x
2
4
6
8
y
10
12
16
24
y
14.
9.
11.
13.
15.
x
1
2
3
4
y
1
4
9
16
x
1
y
-2
x
4
8
12
16
y
3
0
-3
-6
3
5
7
-18 -50 -98
16.
y
y
x
O
17.
18.
y
O
x
19.
y
y
x
O
x
O
20. y = x 3 - 1
21. y = 4x 2 + 9
22. y = 0.6x
23. y = _
24. y = _
x
25. y = _
x +5
3x
2
x
O
x
O
8
4
26. TRAVEL The Guzman family drove from Sacramento to Yreka. Use the table
to determine whether the distance driven is a linear function of the hours
traveled. Explain.
Time (h)
1
2
3
4
Distance (mi)
65
130
195
260
27. BUILDINGS The table shows the
height of several buildings in
Chicago, Illinois. Use the table
to determine whether the height
of the building is a linear function
of the number of stories. Explain.
Stories
Height
(ft)
Harris Bank III
35
510
One Financial Place
40
515
Kluczynski Federal Building
45
545
Mid Continental Plaza
50
582
North Harbor Tower
55
556
Building
Source: The World Almanac
Lesson 10-1 Linear and Nonlinear Functions
525
MEASUREMENT For Exercises 28 and 29, use the following information.
Recall that the circumference of a circle is equal to pi times its diameter and
that the area of a circle is equal to pi times the square of its radius.
28. Is the circumference of a circle a linear or nonlinear function of its
diameter? Explain your reasoning.
29. Is the area of a circle a linear or nonlinear function of its radius? Explain
your reasoning.
For Exercises 30–34, determine whether each equation or table represents a
linear or nonlinear function. Explain.
30. y - x = 1
33.
32. y = 2 x
31. xy = -9
x
0.5
1
1.5
2
y
15
8
1
-6
34.
x
-4
0
y
2
1
35. FOOTBALL The graphic shows
36. MEASUREMENT Make a graph
8
-1 -4
:fcc\^\9fnc>Xd\j
8m\iX^\8kk\e[XeZ\
6ISITORS
the decrease in the average
attendance at college bowl
games from 1983 to 2003.
Would you describe the decline
as linear or nonlinear? Explain.
4
showing the area of a square
as a function of its perimeter.
Explain whether the function
is linear.
9EAR
Source: USA Today
%842!02!#4)#% 37. GRAPHING Water is poured at a constant rate
into the vase at the right. Draw a graph of
See pages 702, 717.
the water level as a function of time. Is the
water level a linear or nonlinear function
Self-Check Quiz at
of time? Explain.
ca.gr7math.com
H.O.T. Problems
38. CHALLENGE True or false? All graphs
of straight lines are linear functions. Explain your reasoning or provide a
counterexample.
39. Which One Doesn’t Belong? Identify the function that is not linear. Explain
your reasoning.
y = 2x
y = x2
y -2 = x
x-y=2
40. OPEN ENDED Give an example of a nonlinear function using a table of
values.
41.
*/ -!4( Describe two methods for determining whether a
(*/
83 *5*/(
function is linear given its equation.
526 Chapter 10 Algebra: Nonlinear Functions and Polynomials
42. Which equation describes the data in
43. Which equation represents a nonlinear
the table?
function?
x
-7 -5 -3
0
4
y
50
1
17
A 5x + 1 = y
B xy = 68
26
10
F y = 3x + 1
x
G y=_
3
H 2xy = 10
C x2 + 1 = y
J
2
y = 3(x - 5)
D -2x + 8 = y
STATISTICS Determine whether a scatter plot of the data for the following
might show a positive, negative, or no relationship. (Lesson 9-8)
44. grade on a test and amount of time spent studying
45. age and number of siblings
46. number of Calories burned and length of time exercising
49. 7k + 12 = 8 - 9k
50. 13.4w + 17 = 5w - 4
51. 8.1a + 2.3 = 5.1a - 3.1
!R
AB
IC
48. 1 - 3c = 9c + 7
%N
GLI
SH
3P
AN
ISH
(Lesson 8-4)
(I
ND
I
Solve each equation. Check your solution.
,ANGUAGES3POKENBY.ATIVE3PEAKERS
-A
ND
ARI
N
languages spoken by at least 100 million
native speakers worldwide. What conclusions
can you make about the number of Mandarin
native speakers and the number of English
native speakers? (Lesson 9-7)
.ATIVE3PEAKERSMILLIONS
47. LANGUAGES The graph shows the top five
,ANGUAGES
52. 4.1x - 23 = -3.9x - 1 53. 3.2n + 3 = -4.8n - 29 Source: The World Almanac For Kids
54. PARKS A circular fountain in a park has a
diameter of 8 feet. The park director wants to build a fountain that has an
area four times that of the current fountain. What will be the diameter of
the new fountain? (Lesson 7-1)
55. MEASUREMENT The cylindrical air duct of a large furnace has a diameter
of 30 inches and a height of 120 feet. If it takes 15 minutes for the contents
of the duct to be expelled into the air, what is the volume of the
substances being expelled each hour? (Lesson 7-5)
PREREQUISITE SKILL Graph each equation.
56. y = 2x
57. y = x + 3
(Lesson 9-2)
58. y = 3x - 2
59. y = _x + 1
1
3
Lesson 10-1 Linear and Nonlinear Functions
527
10-2
Graphing Quadratic Functions
Main IDEA
Graph quadratic functions.
Standard 7AF1.5
Represent
quantitative
relationships graphically
and interpret the meaning
of a specific part of a graph
in the situation represented
by the graph.
Standard 7AF3.1 Graph
functions of the form
y = nx 2 and y = nx 3 and
use in solving problems.
You know that the area A of a square
is equal to the length of a side
s squared, A = s 2.
Copy and complete the table.
s
s2
(s, A)
0
0
(0, 0)
1
1
(1, 1)
2
3
Graph the ordered pairs from
the table. Connect them with
a smooth curve.
4
5
6
1. Is the relationship between the side length and
the area of a square linear or nonlinear? Explain.
2. Describe the shape of the graph.
NEW Vocabulary
quadratic function
A quadratic function, like A = s 2, is a function in which the greatest
power of the variable is 2. Its graph is U-shaped, opening upward or
downward. The graph opens upward if the number in front of the
variable that is squared is positive, downward if it is negative.
Graph Quadratic Functions
Quadratic Fuctions
The graph of a
quadratic function is
called a parabola.
1 Graph y = x 2.
To graph a quadratic function, make a table of values, plot the
ordered pairs, and connect the points with a smooth curve.
y
x
x2
y
(x, y)
-2
(-2) 2 = 4
4
(-2, 4)
-1
2
(-1) = 1
1
(-1, 1)
0
(0) 2 = 0
0
(0, 0)
2
1
(1) = 1
1
(1, 1)
2
(2) 2 = 4
4
(2, 4)
y x2
x
O
2 Graph y = -2x 2.
x
-2x 2
-2 -2(-2) 2 = -8
2
-1 -2(-1) = -2
0
-2(0) 2 = 0
1
-2(1) 2 = -2
2
-2(2) 2 = -8
y
(x, y)
-8
(-2, -8)
-2
(-1, -2)
4
O
⫺8
4
⫺4
(0, 0)
⫺4
-2
(1, -2)
⫺8
-8
(2, -8)
0
528 Chapter 10 Algebra: Nonlinear Functions and Polynomials
y
8x
y 2x 2
⫺12
Extra Examples at ca.gr7math.com
READING
in the Content Area
For more strategies in
reading this lesson, visit
ca.gr7math.com.
3 Graph y = x 2 + 2.
x
x2 + 2
-2 (-2) 2 + 2 = 6
-1
0
y
(x, y)
6
(-2, 6)
2
3
(-1, 3)
2
2
(0, 2)
2
(-1) + 2 = 3
(0) + 2 = 2
1
(1) + 2 = 3
3
(1, 3)
2
(2) 2 + 2 = 6
6
(2, 6)
y
y x2 2
x
O
4 Graph y = -x 2 + 4.
x
-x 2 + 4
y
(x, y)
-2
-(-2) 2 + 4 = 0
0
(-2, 0)
-1
-(-1) 2 + 4 = 3
3
(-1, 3)
2
0
-(0) + 4 = 4
4
(0, 4)
1
-(1) 2 + 4 = 3
3
(1, 3)
2
-(2) 2 + 4 = 0
0
(2, 0)
y
y x2 4
x
O
Graph each function.
a. y = 6x 2
b. y = x 2 - 2
c. y = -2x 2 - 1
5 MONUMENTS The function h = 0.66d 2 represents the distance d in
miles you can see from a height of h feet. Graph this function. Then
use your graph and the information at the left to estimate how far
you could see from the top of the Eiffel Tower.
Distance cannot be negative, so use only positive values of d.
(d, h)
0
0.66(0) 2 = 0
(0, 0)
10
0.66(10) 2 = 66
(10, 66)
Source: structurae.de
1,000
Height (ft)
Real-World Link
The Eiffel Tower in
Paris, France, opened
in 1889 as part of the
World Exposition. It is
about 986 feet tall.
h
h = 0.66d 2
d
800
20
2
0.66(20) = 264
(20, 264)
25
2
0.66(25) = 412.5
(25, 412.5)
30
0.66(30) 2 = 594
(30, 594)
400
35
0.66(35) 2 = 808.5
(35, 808.5)
200
40
2
(40, 1,056)
0.66(40) = 1,056
600
0
10
20
30
40
d
Distance (mi)
At a height of 986 feet, you could see approximately 39 miles.
d. TOWERS The outdoor observation deck of the Space Needle in
Seattle, Washington, is 520 feet above ground level. Estimate how
far you could see from the observation deck.
Personal Tutor at ca.gr7math.com
Lesson 10-2 Graphing Quadratic Functions
Lance Nelson/CORBIS
529
Examples 1–4
(pp. 528–529)
Example 5
(p. 529)
(/-%7/2+ (%,0
For
Exercises
8–11
12–19
20, 21
See
Examples
1, 2
3, 4
5
Graph each function.
1. y = 3x 2
2. y = -5x 2
3. y = -4x 2
4. y = -x 2 + 1
5. y = x 2 - 3
6. y = -x 2 + 2
7. CARS The function d = 0.006s 2 represents the braking distance d in meters
of a car traveling at a speed s in kilometers per second. Graph this function.
Then use your graph to estimate the speed of the car if its braking distance
is 12 meters.
Graph each function.
8. y = 4x 2
9. y = 5x 2
10. y = -3x 2
11. y = -6x 2
12. y = x 2 + 6
13. y = x 2 - 4
14. y = -x 2 + 2
15. y = -x 2 - 5
16. y = 2x 2 - 1
17. y = 2x 2 + 3
18. y = -4x 2 - 1
19. y = -3x 2 + 2
20. RACING The function d = _at 2 represents the distance d that a race car will
1
2
travel over an amount of time t given the rate of acceleration a. Suppose a
car is accelerating at a rate of 5 feet per second every second. Graph this
function. Then use your graph to find the time it would take the car to
travel 125 feet.
21. WATERFALLS The function d = -16t 2 + 182 models the distance d in feet a
drop of water falls t seconds after it begins its descent from the top of the
182-foot high American Falls in New York. Graph this function. Then use
your graph to estimate the time it will take the drop of water to reach the
river at the base of the falls.
Graph each function.
22. y = 0.5x 2 + 1
23. y = 1.5x 2
24. y = 4.5x 2 - 6
25. y = _x 2 - 2
26. y = _x 2
27. y = -_x 2 + 1
1
3
1
2
1
4
MEASUREMENT For Exercises 28 and 29, write a function for each of the
following. Then graph the function in the first quadrant.
%842!02!#4)#%
See pages 702, 717.
28. The surface area of a cube is a function of the edge length a. Use your
graph to estimate the edge length of a cube with a surface area of
54 square centimeters.
29. The volume V of a rectangular prism with a square base and a fixed height
Self-Check Quiz at
ca.gr7math.com
of 5 inches is a function of the base edge length s. Use your graph to
estimate the base edge length of a prism whose volume is 180 cubic inches.
530 Chapter 10 Algebra: Nonlinear Functions and Polynomials
H.O.T. Problems
CHALLENGE The graphs of quadratic functions may have exactly one highest
point, called a maximum, or exactly one lowest point, called a minimum. Graph
each quadratic equation. Determine whether each graph has a maximum or a
minimum. If so, give the coordinates of each point.
30. y = 2x 2 + 1
31. y = -x 2 + 5
32. y = x 2 - 3
33. OPEN ENDED Write and graph a quadratic function that opens upward and
has its minimum at (0, -3.5).
*/ -!4( Write a quadratic function of the form y = ax 2 + c and
(*/
83 *5*/(
34.
explain how to graph it.
35. Which graph represents the function y = -0.5x 2 - 2?
y
A
O
y
B
y
C
x
O
x
x
Determine whether each equation represents a linear or nonlinear function.
37. y = 3x 3 + 2
x
O
O
36. y = x - 5
y
D
38. x + y = -6
(Lesson 10-1)
39. y = -2x 2
STATISTICS For Exercises 40–42, use the information
at the right. (Lesson 9-8)
Year
Population
40. Draw a scatter plot of the data and draw a line of fit.
2000
172
41. Does the scatter plot show a positive, negative, or no
relationship?
42. Use your graph to estimate the population of the
whooping crane at the refuge in 2005.
Whooping Cranes
2001
171
2002
181
2003
194
2004
197
43. SAVINGS Anna’s parents put $750 into a college savings account. After
6 years, the investment had earned $540. Write an equation that you
could use to find the simple interest rate. Then find the simple interest
rate. (Lesson 5-9)
44. PREREQUISITE SKILL A section of a theater is arranged so that each row has
the same number of seats. You are seated in the 5th row from the front
and the 3rd row from the back. If your seat is 6th from the left and 2nd
from the right, how many seats are in this section of the theater? Use the
draw a diagram strategy. (Lesson 4-4)
Lesson 10-2 Graphing Quadratic Functions
531
10-3 Problem-Solving Investigation
MAIN IDEA: Solve problems by making a model.
Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to
explain mathematical reasoning. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an
equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a
number, half as large as area A.
e-Mail:
MAKE A MODEL
YOUR MISSION: Make a model to solve the problem.
THE PROBLEM: Determine if there are enough
tables to make a 10-by-10 square arrangement.
EXPLORE
PLAN
▲
Tonya: We have 35 square tables. We need to
arrange them into a square that is open in
the middle and has 10 tables on each side.
You know Tonya has 35 square tables.
Start by making models of a 4-by-4 square and of a 5-by-5 square.
Then look for a pattern.
SOLVE
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For a 10-by-10 square, Tonya needs 2 · 10 + 2 · 8 or 36 tables. She has 35 tables,
so she needs one more.
You can estimate that Tonya needs 4 × 10 or 40 tables. But each of the corner
tables is counted twice. So, she needs 40 - 4 or 36 tables.
1. Draw a diagram showing another way the students could have
grouped the tiles to solve this problem. Use a 4-by-4 square.
2.
*/ -!4( Write a problem that can be solved by making a
(*/
83 *5*/(
model. Describe the model. Then solve the problem.
532 Chapter 10 Algebra: Nonlinear Functions and Polynomials
Laura Sifferlin
For Exercises 3–5, solve by making a model.
3. STICKERS In how many different ways can
three rectangular stickers be torn from a
sheet of 3 × 3 stickers so that all three
stickers are still attached? Draw each
arrangement.
4. MEASUREMENT A 10-inch by 12-inch piece of
cardboard has a 2-inch square cut out of
each corner. Then the sides are folded up
and taped together to make an open box.
Find the volume of the box.
8. PETS Mrs. Harper owns both cats and
canaries. Altogether, her pets have thirty
heads and eighty legs. How many cats does
she have?
GEOMETRY For Exercises 9
and 10, use the figure at
the right.
9. How many cubes
would it take to build
this tower?
10. How many cubes would it take to build a
similar tower that is 12 cubes high?
5. GEOMETRY A computer game
requires players to stack arrangements
of five squares arranged to form a
single shape. One arrangement is
shown at the right. How many
different arrangements are there if touching
squares must border on a full side?
11. CARS Yesterday you noted that the mileage
on the family car read 60,094.8 miles. Today
it reads 60,099.1 miles. Was the car driven
about 4 or 40 miles?
Use any strategy to solve Exercises 6–11.
Some strategies are shown below.
For Exercises 12 and 13, select the appropriate
operation(s) to solve the problem. Justify your
selections(s) and solve the problem.
G STRATEGIES
PROBLEM-SOLVIN
tep plan.
• Use the four-s
m.
• Draw a diagra
eck.
• Guess and Ch
12. SCIENCE The light in the circuit will turn on
if one or more switches are closed. How
many combinations of open and closed
switches will result in the light being on?
l.
• Make a mode
6. CAMP The camp counselor lists 21 chores on
separate pieces of paper and places them in
a basket. The counselor takes one piece of
paper, and each camper takes one as the
basket is passed around the circle. There is
one piece of paper left when the basket
returns to the counselor. How many people
could be in the circle if the basket goes
around the circle more than once?
7. PARKING Parking space numbers consist of
3 digits. They are typed on a slip of paper
and given to students at orientation. Tara
accidentally read her number upside-down.
The number she read was 795 more than her
actual parking space number. What is Tara’s
parking space number?
a
b
c
d
e
13. HOBBIES Lorena says to Angela, “If you
give me one of your baseball cards, I will
have twice as many baseball cards as you
have.” Angela answers, “If you give me
one of your cards, we will have the same
number of cards.” How many cards does
each girl have?
Lesson 10-3 Problem-Solving Investigation: Make a Model
533
10-4
Graphing Cubic Functions
Main IDEA
Standard 7AF3.1
Graph functions of
the form y = nx 2 and
y = nx 3 and use in solving
problems.
Standard 7AF3.2 Plot the
values from the volumes of
three-dimensional shapes
for various values of the
edge lengths (e.g., cubes
with varying edge lengths or
a triangle prism with a fixed
height and an equilateral
triangle base of varying
lengths).
MEASUREMENT You can find the area A of a
square by squaring the length of a side s. This
relationship can be represented in different ways.
Words and Equation
Area
A
s2
=
s
Table
length of a
side squared.
equals
s
s2
(s, A)
2
0 =0
(0, 0)
2
s
0
Graph
1
1 =1
(1, 1)
2
22 = 4
(2, 4)
A
A s2
Area
Graph cubic functions.
s
O
Side
1. The volume V of a cube is found by cubing the length
of a side s. Write a formula to represent the
volume of a cube as a function of side length.
s
s
s
2. Graph the volume as a function of side length.
(Hint: Use values of s such as 0, 0.5, 1, 1.5, 2, and so on.)
3. Would it be reasonable to use negative numbers for x-values in this
situation? Explain.
You can graph cubic functions such as the formula for the volume of a
cube by making a table of values.
Graph a Cubic Function
1 Graph y = x 3.
x
y = x3
(x, y)
-1.5
(-1.5) 3 ≈ -3.4
(-1.5, -3.4)
-1
Graphing
It is often helpful to
substitute decimal
values of x in order
to graph points that
are closer together.
3
(-1, -1)
3
(-1) = -1
0
(0) = 0
(0, 0)
1
(1) 3 = 1
(1, 1)
1.5
(1.5) 3 ≈ 3.4
(1.5, 3.4)
y
O
Graph each function.
a. y = x 3 - 1
b. y = -4x 3
Personal Tutor at ca.gr7math.com
534 Chapter 10 Algebra: Nonlinear Functions and Polynomials
c. y = x 3 + 4
x
2 PACKAGING A packaging company wants to manufacture a
cardboard box with a square base of side length x inches and a
height of (x – 3) inches as shown.
Real-World Link
Packaging is the
nation’s third largest
industry, with over
$130 billion in sales
each year.
Source: San Jose State
University
(x 3) in.
x in.
x in.
Write the function for the volume V of the box. Graph the function.
Then estimate the dimensions of the box that would give a volume
of approximately 8 cubic inches.
V = lwh
Volume of a rectangular prism
V = x · x · (x – 3)
Replace l with x, w with x, and h with (x – 3).
V = x 2(x – 3)
x · x = x2
V = x 3 – 3x 2
Distributive Property
The function for the volume V of the box is V = x 3 – 3x 2. Make a table
of values to graph this function. You do not need to include negative
values of x since the side length of the box cannot be negative.
x
3
2
(x, V)
0
(0) – 3(0) = 0
(0, 0)
0.5
(0.5) 3 – 3(0.5) 2 ≈ –0.6
(0.5, –0.6)
3
2
1
(1) – 3(1) = –2
(1, –2)
1.5
(1.5) 3 – 3(1.5) 2 ≈ –3.8
(1.5, –3.8)
2
Analyze the Graph
Notice that the graph
is below the x-axis for
values of x < 3. This
means that the
“volume” of the box is
negative for x < 3. To
have a box with a
positive height and a
positive volume, x must
be greater than 3.
V = x 3 – 3x 2
3
2
(2) – 3(2) = –4
3
2
(2, –4)
2.5
(2.5) – 3(2.5) ≈ –3.1
(2.5, –3.1)
3
(3) 3 – 3(3) 2 = 0
(3, 0)
3
2
3.5
(3.5) – 3(3.5) ≈ 6.1
(3.5, 6.1)
4
(4) 3 – 3(4) 2 = 16
(4, 16)
20
y
18
16
14
12
10
8
6
4
2
0
2
4
y x 3 2x 2
x
2
4
6
8
10
12
Looking at the graph, we see that the volume of the box is
approximately 8 cubic inches when x is about 3.6 inches.
The dimensions of the box when the volume is about 8 cubic inches
are 3.6 inches, 3.6 inches, and 3.6 – 3 or 0.6 inch.
d. PACKAGING A packaging company wants to manufacture a
cardboard box with a square base of side length x feet and a height
of (x – 2) feet. Write the function for the volume V of the box.
Graph the function. Then estimate the dimensions of the box that
would give a volume of about 1 cubic foot.
Extra Examples at ca.gr7math.com
Getty Images
Lesson 10-4 Graphing Cubic Functions
535
Example 1
(p. 534)
Example 2
(p. 535)
(/-%7/2+ (%,0
For
Exercises
6–17
18, 19
See
Examples
1
2
Graph each function.
1. y = -x 3
2. y = 0.5x 3
3. y = x 3 – 2
4. y = 2x 3 + 1
5. MEASUREMENT A rectangular prism with a square base of side length x
centimeters has a height of (x + 1) centimeters. Write the function for the
volume V of the prism. Graph the function. Then estimate the dimensions
of the box that would give a volume of approximately 9 cubic centimeters.
Graph each function.
6. y = -2x 3
7. y = -3x 3
8. y = 0.2x 3
9. y = 0.1x 3
10. y = x 3 + 1
11. y = 2x 3 + 1
12. y = x 3 – 3
13. y = 2x 3 – 2
14. y = _ x 3
15. y = _ x 3 + 2
16. y = -x 3 – 2
17. y = -x 3 + 1
1
4
1
3
18. MEASUREMENT Jorge built a scale model of the Great
%842!02!#4)#%
See pages 703, 717.
Self-Check Quiz at
ca.gr7math.com
Pyramid. The base of the model is a square with
side length s and the model’s height is (s – 1) feet.
Write the function for the volume V of the model.
Graph this function. Then estimate the length of
one side of the square base of the model if the
model’s volume is approximately 8 cubic feet.
19. MEASUREMENT The formula for the volume V of a tennis ball is given by
4 3
the equation V = _
πr where r represents the radius of the ball. Graph
3
this function. Use 3.14 for π. Then estimate the length of the radius if the
volume of the tennis ball is approximately 11 cubic inches.
Graph each pair of equations on the same coordinate plane. Describe their
similarities and differences.
20. y = x 3
y = 3x
21. y = x 3
3
3
y=x –3
22. y = 0.5x 3
y = 2x
23. y = 2x 3
3
y = -2x 3
FARMING For Exercises 24 and 25, use the following
information.
A grain silo consists of a cylindrical main section and a
hemispherical roof. The cylindrical main section has a
radius of r units and a height h equivalent to the radius.
The volume V of a cylinder is given by the equation V = πr 2h.
24. Write the function for the volume V of the cylindrical main
section of the grain silo in terms of its radius r.
25. Graph this function. Use 3.14 for π. Then estimate the radius
and height in meters of the cylindrical main section of the
grain silo if the volume is approximately 15.5 cubic meters.
536 Chapter 10 Algebra: Nonlinear Functions and Polynomials
r
H.O.T. Problems
26. OPEN ENDED Write the equation of a cubic function whose graph in the first
quadrant shows faster growth than the function y = x 3.
CHALLENGE The zero of a cubic function is the x-coordinate at which the
function crosses the x-axis. Find the zeros of each function below.
27. y = x 3
29.
28. y = x 3 + 1
*/ -!4( The volume V of a cube with side length s is given by
(*/
83 *5*/(
the equation V = s 3. Explain why negative values are not necessary when
creating a table or a graph of this function.
30. Which equation could represent the
graph shown below?
31. Which equation could represent the
graph shown below?
y
O
y
O
x
A y = x3
F y = x3 – 2
B y = -x 3
G y = x3 + 2
C y = 2x 3
H y = -2x 3
D y = -2x 3
J
x
y = 2x 3 + 1
32. MANUFACTURING A company packages six small books for a children’s collection in a
decorated 4-inch cube. They are shipped to bookstores in cartons. Twenty cubes fit in a
carton with no extra space. What are the dimensions of the carton? Use the make a model
strategy. (Lesson 10-3)
Graph each function.
(Lesson 10-2)
33. y = -2x 2
34. y = x 2 + 3
35. y = -3x 2 + 1
Estimate each square root to the nearest whole number.
37. √
54
38. - √
126
(Lesson 3-2)
39. √
8.67
PREREQUISITE SKILL Write each expression using exponents.
41. 3 · 3 · 3 · 3 · 3
42. 5 · 4 · 5 · 5 · 4
43. 7 · (7 · 7)
36. y = 4x 2 + 3
40. - √
19.85
(Lesson 2-9)
44. (2 · 2) · (2 · 2 · 2)
Lesson 10-4 Graphing Cubic Functions
537
Extend
10-4
Main IDEA
Use a graphing calculator
to graph families of
nonlinear functions.
Standard 7AF3.1
Graph functions of
the form y = nx 2 and
y = nx 3 and use in solving
problems.
Standard 7MR3.3 Develop
generalizations of the results
obtained and the strategies
used and apply them to new
problem situations.
Graphing Calculator Lab
Families of Nonlinear Functions
Families of nonlinear functions share a common characteristic based on
a parent function. The parent function of a family of quadratic functions
is y = x 2. You can use a graphing calculator to investigate families of
quadratic functions.
Graph y = x 2, y = x 2 + 5, and y = x 2 - 3 on the same screen.
Clear any existing equations from the Y= list by pressing
CLEAR
.
Enter each equation. Press
X,T,␪,n
ENTER ,
X,T,␪,n
5 ENTER , and
X,T,␪,n
3 ENTER .
Graph the equations in the
standard viewing window.
Press ZOOM 6.
ANALYZE THE RESULTS
1. Compare and contrast the three equations you graphed.
2. Describe how the graphs of the three equations are related.
3. MAKE A CONJECTURE How does changing the value of c in the
equation y = x 2 + c affect the graph?
4. Use a graphing calculator to graph y = 0.5x 2, y = x 2, and y = 2x 2.
5. Compare and contrast the three equations you graphed in Exercise 4.
6. Describe how the graphs of the three equations are related.
7. MAKE A CONJECTURE How does changing the value of a in the
equation y = ax 2 affect the graph?
8. Use a graphing calculator to graph y = 0.5x 3, y = x 3, and y = 2x 3.
9. Compare and contrast the three equations you graphed in Exercise 8
to the equations you graphed in Exercise 4.
538 Chapter 10 Nonlinear Functions and Polynomials
Other Calculator Keystrokes at ca.gr7math.com
10-5
Multiplying Monomials
Main IDEA
Multiply monomials.
Standard
7NS2.3 Multiply,
divide, and simplify
rational numbers by using
exponent rules.
Standard 7AF2.1 Interpret
positive whole-number
powers as repeated
multiplication and negative
whole-number powers as
repeated division or
multiplication by the
multiplicative inverse.
Simplify and evaluate
expressions that include
exponents.
Standard 7AF2.2 Multiply
and divide monomials;
extend the process of taking
powers and extracting roots
to monomials when the latter
results in a monomial with an
integer exponent.
SCIENCE The pH of a solution describes its acidity. Neutral water has
a pH of 7. Lemon juice has a pH of 2. Each one-unit decrease in the
pH means that the solution is 10 times more acidic. So, a pH of 8 is
10 times more acidic than a pH of 9.
pH
Times More Acidic
Than a pH of 9
Written Using
Powers
8
10
10 1
7
10 × 10 = 100
10 1 × 10 1 = 10 2
6
10 × 10 × 10 = 1,000
10 1 × 10 2 = 10 3
5
10 × 10 × 10 × 10 = 10,000
10 1 × 10 3 = 10 4
4
10 × 10 × 10 × 10 × 10 = 100,000
10 1 × 10 4 = 10 5
1. Examine the exponents of the factors and the exponents of the
products in the last column. What do you observe?
A monomial is a number, a variable, or a product of a number and one
or more variables. Exponents are used to show repeated multiplication.
You can use this fact to find a rule for multiplying monomials.
2 factors
4 factors
NEW Vocabulary
3 2 · 3 4 = (3 · 3) · (3 · 3 · 3 · 3) or 3 6
monomial
6 factors
Notice that the sum of the original exponents is the exponent in the final
product. This relationship is stated in the following rule.
+%9#/.#%04
Product of Powers
To multiply powers with the same base, add their exponents.
Words
Examples
Numbers
4
3
2 ·2 =2
4+3
Algebra
or 2
7
m
a · an = am + n
Multiply Powers
1 Find 5 2 · 5. Express using exponents.
Common Error
When multiplying
powers, do not
multiply the bases.
4 5 · 4 2 = 4 7, not 16 7.
52 · 5 = 52 · 51
= 52 + 1
=5
3
5 = 51
Check
5 2 · 5 = (5 · 5) · 5
The common base is 5.
=5·5·5
Add the exponents.
= 53 Lesson 10-5 Multiplying Monomials
CORBIS
539
2 Find -3x 2(4x 5). Express using exponents.
-3x 2(4x 5) = (-3 · 4)(x 2 · x 5)
Commutative and Associative Properties
= (-12)(x 2 + 5)
= -12x
The common base is x.
7
Add the exponents.
Multiply. Express using exponents.
a. 9 3 · 9 2
b.
2
(_35 ) (_35 )
9
c. -2m(-8m 5)
3 The population of Groveton is 6 5. The population of Putnam is 6 3
times as great. How many people are in Putnam?
Real-World Link
A census is taken every
ten years by the U.S.
Census Bureau to
determine population.
The government uses
the data from the
census to make many
decisions.
The population of Putnam is 6 8 or 1,679,616 people.
Source: census.gov
d. RIVERS The Guadalupe River is 2 8 miles long. The Amazon River is
To find out the number of people, multiply 6 5 by 6 3.
6 5 · 6 3 = 6 5+3 or 6 8
Product of Powers
almost 2 4 times as long. Find the length of the Amazon River.
Personal Tutor at ca.gr7math.com
In Lesson 2-9, you learned to evaluate negative exponents. Remember
that any nonzero number to the negative n power is the multiplicative
inverse of that number to the n th power. The Product of Powers rule can
be used to multiply powers with negative exponents.
Multiply Negative Powers
4 Find x 4 · x -2. Express using exponents.
METHOD 1
METHOD 2
x 4 · x -2 = x 4 + (-2) The common
base is x.
=x
2
Add the
exponents.
x 4 · x -2
1 _
1
1
_
-2
=x·x·x·x·_
x · x x = x2
= x2
Simplify.
Simplify. Express using positive exponents.
d. 3 8 · 3 -2
e. n 9 · n -4
540 Chapter 10 Algebra: Nonlinear Functions and Polynomials
Prisma/SuperStock
f. 5 -1 · 5 -2
Extra Examples at ca.gr7math.com
Examples 1–4
(pp. 539–540)
Example 3
Simplify. Express using exponents.
1. 4 5 · 4 3
2. n 2 · n 9
3. -2a(3a 4)
4. 5 2x 2y 4 · 5 3xy 3
5. r 7 · r -3
6. 6m · 4m 2
7. AGE Angelina is 2 3 years old. Her grandfather is 2 3 times her age. How old
(p. 540)
(/-%7/2+ (%,0
For
Exercises
8–25
26–28
See
Examples
1, 2, 4
3
is her grandfather?
Simplify. Express using exponents.
8. 6 8 · 6 5
9. 2 9 · 2
10. n · n 7
11. b 13 · b
12. 2g · 7g 6
13. (3x 8)(5x)
14. -4a 5(6a 5)
15. (8w 4)(-w 7)
16. (-p)(-9p 2)
17. -5y 3(-8y 6)
18. 4m -2n 5(3m 4n -2)
19. (-7a 4bc 3)(5ab 4c 2)
20. x 6 · x -3
21. y -1 · y 4
22. z -2 · z -3
23. m 2n -1 · m -3n 3
24. 3f -4 · 5f 2
25. -3ab · 4a -3b
3
26. INSECTS The number of ants in a nest was 5 3. After the eggs hatched, the
number of ants increased 5 2 times. How many ants are there after the eggs
hatch?
27. COMPUTERS The processing speed of a certain computer is 10 11 insructions
per second. Another computer has a processing speed that is 10 3 times as
fast. How many instructions per second can the faster computer process?
28. LIFE SCIENCE A cell culture contains 2 6 cells. By the end of the day, there
are 2 10 times as many cells in the culture. How many cells are there in the
culture by the end of the day?
Simplify. Express using exponents.
%842!02!#4)#%
See pages 703, 717.
29. xy 2(x 3y)
3
32.
(_23 ) (_23 )
35.
(_14 ) (_14 )
Self-Check Quiz at
ca.gr7math.com
4
30. 2 6 · 2 · 2 3
-4
-5
33.
(_78 ) (_78 )
36.
(_25 ) (_25 )
3
31. 4a 2b 3(7ab 2)
13
-2
4
-7
(_25 ) (_25 ) (_25 )
2
37. (_)
(_72 )
7
34.
-2
6
-3
Lesson 10-5 Multiplying Monomials
541
38. CHALLENGE What is twice 2 30? Write using exponents.
H.O.T. Problems
39. OPEN ENDED Write a multiplication expression whose product is 4 15.
*/ -!4( Determine whether the following statement is true or
(*/
83 *5*/(
40.
false. Explain your reasoning or give a counterexample.
If you change the order in which you multiply two monomials,
the product will be different.
41. Which expression is equivalent to
2
42. Which expression describes the area in
2
square feet of the rectangle below?
2 2
A 64x y z
F 11x 10
B 64x 2 yz 2
G 30x 10
C 16x 2 y 2z 2
H 11x 16
D 384x 2 y 2z 2
J
8x y · 8yz ?
2
Graph each function.
2
5x ft
8
6x ft
30x 16
(Lessons 10-2 and 10-4)
43. y = -x 3
44. y = 0.5x 3
45. y = x 3 - 2
46. y = 5x 2
47. y = x 2 + 5
48. y = x 2 – 4
49. BIOLOGY
The table shows how long it took for the
first 400 bacteria cells to grow in a petri dish. Is the
growth of the bacteria a linear function of time?
Explain. (Lesson 10-1)
Express each number in scientific notation.
Time
(min)
46
53
57
60
Number
of cells
100
200
300
400
(Lesson 2-10)
50. The flow rate of some Antarctic glaciers is 0.00031 mile per hour.
51. A human blinks about 6.25 million times a year.
ALGEBRA Solve each equation. Check your solution.
52. k - 4.1 = -9.38
Find each sum or difference. Write in simplest form.
3
7
55. _ - _
8
10
PREREQUISITE SKILL
59. 3 · 3 · 3 · 3
(Lesson 2-7)
3
1
53. 1_ + p = -6_
2
4
5
1
56. -_ + _
12
5
54.
61. 7 · (7 · 7)
542 Chapter 10 Algebra: Nonlinear Functions and Polynomials
10
(Lesson 2-6)
2
1
57. 9_ + _
3
6
Write each expression using exponents.
60. 5 · 4 · 5 · 5 · 4
c
_
= 0.845
58. -2_ - 1_
3
4
1
8
(Lesson 2-9)
62. (2 · 2) · (2 · 2 · 2)
CH
APTER
Mid-Chapter Quiz
10
Lessons 10-1 through 10-5
Determine whether each equation or table
represents a linear or nonlinear function.
Explain. (Lesson 10-1)
STANDARDS PRACTICE Which graph
shows y = x 2 + 1? (Lesson 10-2)
11.
A
C
y
y
1. 3y = x
2. y = 5x 3 + 2
3.
4.
1
x
3
5
7
y
-5 -6 -7 -8
x
-1
0
1
2
y
1
0
1
4
O
O
B
x
x
D
y
O
y
O
x
x
5. LONG DISTANCE The graph shows the
amount of data transferred as a function of
time. Is this a linear or nonlinear function?
Explain your reasoning. (Lesson 10-1)
12. MEASUREMENT Brenda has a photograph
that is 10 inches by 13 inches. She decides
$ATA4RANSFER
1
to frame it, using a frame that is 2_
inches
4
wide on each side. Find the total area of the
framed photograph. Use the make a model
strategy. (Lesson 10-3)
'IGABYTES
Graph each function.
(Lesson 10-4)
13. y = -2x 3
4IMEMIN
Graph each function.
6. y = 2x
(Lesson 10-2)
2
2
7. y = -x + 3
2
14. y = 3x 3
15. y = 2x 3
16. y = 0.1x 3
Simplify. Express using exponents.
4
17. 10 · 10
(Lesson 10-4)
7
8. y = 4x - 1
18. 3 -3 · 3 5 · 3 2
9. y = -3x 2 + 1
19. 2 3a 7 · 2a -3
20. (3 2xy 4z 2)(3 5x 3y -2z 3)
10. AMUSEMENT PARK RIDES Your height h feet
above the ground t seconds after being
released at the top of a free-fall ride is given
by the function h = -16t 2 + 200. Graph this
function. After about how many seconds
will the ride be 60 feet above the
ground? (Lesson 10-2)
21.
STANDARDS PRACTICE Which
expression below has the same value
as 5m 2? (Lesson 10-5)
F 5m
H 5·5·m·m
G 5·m·m
J
5·m·m·m
10-6
Dividing Monomials
Main IDEA
Divide monomials.
Standard
7NS2.3 Multiply,
divide, and simplify
rational numbers by using
exponent rules.
Standard 7AF2.1 Interpret
positive whole-number
powers as repeated
multiplication and negative
whole-number powers as
repeated division or
multiplication by the
multiplicative inverse.
Simplify and evaluate
expressions that include
exponents.
Standard 7AF2.2 Multiply
and divide monomials;
extend the process of taking
powers and extracting roots
to monomials when the latter
results in a monomial with an
integer exponent.
NUMBER SENSE Refer to the table shown
that relates division sentences using
the numbers 2, 4, 8, and 16, and the
same sentences written using powers
of 2.
1. Examine the exponents of the divisors
Division
Sentence
Written Using
Powers of 2
4÷2=2
22 ÷ 21 = 21
8÷2=4
23 ÷ 21 = 22
8÷4=2
23 ÷ 22 = 21
16 ÷ 2 = 8
24 ÷ 21 = 23
16 ÷ 4 = 4
24 ÷ 22 = 22
16 ÷ 8 = 2
24 ÷ 23 = 21
and dividends. Compare them to the
exponents of the quotients. What do you notice?
2. MAKE A CONJECTURE Write the quotient of 2 5 and 2 2 using
powers of 2.
As you learned in Lesson 10-5, exponents are used to show repeated
multiplication. You can use this fact to find a rule for dividing powers
with the same base.
7 factors
Notice that the difference of the
original exponents is the exponent
in the final quotient. This relationship
is stated in the following rule.
57
5·5·5·5·5·5·5
_
= __
or 5 3
5·5·5·5
54
4 factors
+%9#/.#%04
Quotient of Powers
To divide powers with the same base, subtract their exponents.
Words
Examples
Numbers
Algebra
37
_
= 3 7 – 3 or 3 4
am
_
= a m – n, where a ≠ 0
an
33
Divide Powers
Simplify. Express using exponents.
n9
2 _
4
48
1 _
2
4
Common Error
When dividing
powers, do not divide
n
48
_
= 48 – 2
42
=4
6
The common base is 4.
Simplify.
n9
_
= n9 – 4
n4
=n
The common base is n.
5
Simplify.
48
4
the bases. _2 = 4 6,
not 1 6.
Simplify. Express using exponents.
a.
57
_
54
b.
x 10
_
544 Chapter 10 Algebra: Nonlinear Functions and Polynomials
x3
c.
12w 5
_
2w
The Quotient of Powers rule can also be used to divide powers with
negative exponents. It is customary to write final answers using positive
exponents.
Look Back
To review adding
and subtracting
integers, see Lessons
1-4 and 1-5.
Use Negative Exponents
Simplify. Express using positive exponents.
69
3 _
-3
6
69
_
= 6 9 – (-3)
Quotient of Powers
6 -3
= 6 9 + 3 or 6 12
Simplify.
w -1
4 _
-4
w
w -1
_
= w -1 – (-4)
Quotient of Powers
w -4
= w -1 + 4 or w 3
Simplify.
Simplify. Express using positive exponents.
d.
11 -8
_
11
e.
2
b -4
_
f.
b -7
6h 5
_
3h -5
22 · 45 · 52
5 _
=
5
4
2
2 ·4 ·5
A 2
Remember that the
Quotient of Powers
Rule allows you to
Read the Item
52
_
= 5 2 - 2 = 5 0 = 1.
Solve the Item
52
5
simplify _2 .
52
1
C _
B 1
D 0
2
You are asked to divide one monomial by another.
( )( )( )
22 · 45 · 52
52
22 _
45 _
_
= _
25 · 44 · 52
25
44
52
Group by common base.
= 2 -3 · 4 1 · 5 0
Subtract the exponents.
1
=_
·4·1
3
2 -3 = _3
4
1
=_
or _
Simplify.
2
8
2
1
2
The answer is C.
Extra Examples at ca.gr7math.com
Lesson 10-6 Dividing Monomials
545
(_16 ) × (_16 )
__
g. Simplify
.
1
_
(6)
-12
4
-3
1
F _
(6)
5
1
G _
H 64
6
J 65
Personal Tutor at ca.gr7math.com
6 SOUND The loudness of a conversation is 10 6 times as intense as the
loudness of a pin dropping, while the loudness of a jet engine is
10 12 times as intense. How many times more intense is the loudness
of a jet engine than the loudness of a conversation?
Real-World Link
The decibel measure
of the loudness of a
sound is the exponent
of its relative intensity
multiplied by 10. A jet
engine has a loudness
of 120 decibels.
To find how many times more intense, divide 10 12 by 10 6.
10 12
_
= 10 12 – 6 or 10 6
10 6
Quotient of Powers
The loudness of a jet engine is 10 6 or 1,000,000 times as intense as the
loudness of a conversation.
h. SOUND The loudness of a vacuum cleaner is 10 4 times as intense as
the loudness of a mosquito buzzing, while the loudness of a jack
hammer is 10 9 times as intense. How many times more intense is
the loudness of a jack hammer than that of a vacuum cleaner?
Personal Tutor at ca.gr7math.com
Examples 1–4
Simplify. Express using positive exponents.
(pp. 544-545)
Example 5
(p. 545)
1.
76
_
5.
9c 7
_
7
3c 2
(p. 546)
29
_
6.
24k 9
_
2
13
6k 6
y
_
7.
15 -6
_
y
5
15 2
4.
z
_
8.
35p
_
z2
1
5p -4
22 · 33 · 44
2·3 ·4
B 2
1
C _
2
1
D _
(2)
2
10. ASTRONOMY Venus is approximately 10 8 kilometers from the Sun. Saturn is
more than 10 9 kilometers from the Sun. About how many times farther
away from the Sun is Saturn than Venus?
546 Chapter 10 Algebra: Nonlinear Functions and Polynomials
Mug Shots/Corbis
8
3.
9. Simplify _
.
3
5
A 22
Example 6
2.
(/-%7/2+ (%,0
For
Exercises
11–26
27–30
31–34
See
Examples
1–4
5
6
Simplify. Express using positive exponents.
11.
8 15
_
12.
29
_
15.
h7
_
16.
g
_
19.
36d 10
_
20.
23.
22 -9
_
27.
x y
_
84
h6
6d 5
22 4
13.
43
_
14.
13 2
_
17.
x8
_
18.
n
_
16t 4
_
21.
20m 7
_
22.
75r 6
_
24.
3 -1
_
25.
42w -6
_
26.
12y
_
28.
63 · 66 · 64
_
=
29.
(5) (5)
__
30.
3x 4
_
18
6 14
x 4y 9
2
g
6
8t
3 -5
62 · 63 · 64
47
x 11
5m 5
n8
25r 5
-6
7w -2
_1 2 × _1
(_15 )
13 5
2y -10
-6
2
3 4x -2
31. POPULATION The continent of North America contains approximately 10 7
square miles of land. If the population doubles, there will be about 10 9
people on the continent. At that point, on average, how many people will
occupy each square mile of land?
32. FOOD An apple is 10 3 times as acidic as milk, while a lemon is
10 4 times as acidic. How many times more acidic is a lemon than
an apple?
33. ANIMALS A common flea 2 -4 inch long can jump about 2 3 inches high.
About how many times its body size can a flea jump?
34. MEDICINE The mass of a molecule of penicillin is 10 -18 kilograms
and the mass of a molecule of insulin is 10 -23 kilograms. How
many times greater is a molecule of penicillin than a molecule of
insulin?
Find each missing exponent.
35.
17 _
= 17 8
17 4
36.
k6
_
= k2
k
37.
5
_
= 53
5 -9
ANALYZE TABLES For Exercises 39 and 40, use
the information below and in the table.
%842!02!#4)#%
See pages 703, 717.
Self-Check Quiz at
ca.gr7math.com
For each increase of one on the Richter scale,
an earthquake’s vibrations, or seismic waves,
are 10 times greater.
-1
38.
Earthquake
p
_
= p 10
p
Richter Scale
Magnitude
San Francisco, 1906
8.3
Adana, Turkey, 1998
6.3
Source: usgs.gov
39. How many times greater are the seismic waves of an earthquake with a
magnitude of 6 than an aftershock with a magnitude of 3?
40. How many times greater were the seismic waves of the 1906 San Francisco
earthquake than the 1998 Adana earthquake?
Lesson 10-6 Dividing Monomials
547
3 100
3
41. NUMBER SENSE Is _
greater than, less than, or equal to 3? Explain your
99
H.O.T. Problems
reasoning.
42. OPEN ENDED Write a division expression with a quotient of 4 15.
43. CHALLENGE What is half of 2 30? Write using exponents.
44.
*/ -!4( Explain why the Quotient of Powers Rule cannot
(*/
83 *5*/(
5
x
be used to simplify the expression _
.
2
y
45. Which expression below is equivalent
8
47. One meter is 10 3 times longer than one
millimeter. One kilometer is 10 6 times
longer than one millimeter. How many
times longer is one kilometer than one
meter?
9m
to _
?
2
3m
A 6m 4
C 3m 4
B 6m 6
D 3m 6
A 10 9
46. The area of a rectangle is 2 6 square
B 10 6
feet. If the length is 2 3 feet, find the
width of the rectangle.
F 2 feet
H 2 3 feet
G 2 2 feet
J
D 10
2 9 feet
Simplify. Express using positive exponents.
4
48. 6 · 6
7
Graph each function.
3
52. y = x + 2
C 10 3
3
49. 18 · 18
-5
(Lesson 10-5)
50. (-3x 11)(-6x 3)
51. (-9a 4)(2a -7) –
54. y = -2x 3
55. y = -0.1x 3
(Lesson 10-4)
53. y = _ x 3
1
3
State the slope and the y-intercept for the graph of each equation.
56. y = x – 3
57.
2
y=_
x+7
3
58. 3x + 4y = 12
(Lesson 9-5)
59. x + 2y = 10
60. COIN COLLECTING Jada has 156 coins in her collection. This is 12 more than 8 times the
number of nickels in the collection. How many nickels does Jada have in her
collection? (Lesson 8-3)
Simplify. Express using positive exponents.
(Lesson 10-5)
61. 5n · 3n 4
63. (-5b 7)(-2b 4)
62. (-x)(-8x 3)
548 Chapter 10 Algebra: Nonlinear Functions and Polynomials
64. (-4w)(6w -2)
10-7
Powers of Monomials
Main IDEA
Find powers of
monomials.
MEASUREMENT Suppose the side
length of a cube is 2 2 centimeters.
Standard 7AF2.2
Multiply and divide
monomials; extend
the process of taking
powers and extracting roots
to monomials when the
latter results in a monomial
with an integer exponent.
1. Write a multiplication expression
for the volume of the cube.
2 2 cm
2. Simplify the expression. Write as a single power of 2.
3. Using 2 2 as the base, write the multiplication expression
2 2 · 2 2 · 2 2 using an exponent.
3
4. Explain why (2 2) = 2 6.
You can use the rule for finding the product of a power to discover the
rule for finding the power of a power.
5 factors
(6 4) 5 = (6 4) (6 4) (6 4) (6 4) (6 4)
Apply the rule for the product of powers.
= 64 + 4 + 4 + 4 + 4
= 6 20
Notice that the product of the original exponents, 4 and 5, is the final
power 20. This relationship is stated in the following rule.
+%9#/.#%04
Power of a Power
To find the power of a power, multiply the exponents.
Words
Examples
Numbers
2 3
(5 ) = 5
2·3
Algebra
or 5
6
m n
(a ) = a m · n
Find the Power of a Power
Common Error
When finding the
power of a power,
do not add the
exponents.
1 Simplify (8 4) 3.
(8 4) 3 = 8 4 · 3
=8
12
2 Simplify (k 7) 5.
Power of a Power
(k 7) 5 = k 7 · 5
= k 35
Simplify.
Power of a Power
Simplify.
(8 4) 3 = 8 12, not 8 7.
Simplify. Express using exponents.
a. (2 5) 2
Extra Examples at ca.gr7math.com
b. (w 4) 6
c. [(3 2) 3] 2
Lesson 10-7 Powers of Monomials
549
Extend the power of a power rule to find the power of a product.
5 factors
(3a 4) 5 = (3a 4) (3a 4) (3a 4) (3a 4) (3a 4)
ssociative Property of
= 3 · 3 · 3 · 3 · 3 · a 4 · a 4 · a 4 · a 4 · a 4 AMultiplication
= 3 5 · ( a 4) 5
Write using powers.
Apply the rule for power of
= 243 · a 20 or 243a 20
a power.
This example suggests the following rule.
+%9#/.#%04
Power of a Product
To find the power of a product, find the power of each factor
and multiply.
Words
Examples
Numbers
2 3
3
Algebra
2 3
(6x ) = (6) • (x ) or 216x
6
(ab) m = a mb m
Power of a Product
3 Simplify (4p 3) 4.
4 Simplify (-2m 7n 6) 5.
(4p 3) 4 = 4 4 · p 3 · 4
Alternative Method
(4p 3) 4 can also be
expressed as
(4p 3)(4p 3)(4p 3)(4p 3)
or (4 · 4 · 4 · 4)
(p · p · p)(p · p · p)
(p · p · p)(p · p · p)
which is 256p 12.
= 256p 12
(-2m 7n 6) 5 = (-2) 5m 7 · 5n 6 · 5
= -32m 35n 30 Simplify.
Simplify.
Simplify.
d. (8b 9) 2
e. (6x 5y 11)
4
f. (-5w 2z 8) 3
5 GEOMETRY Express the area of the square
as a monomial.
A = s2
Area of a square
4
A = (7a b)
2
Replace s with 7a 4b.
A = 7 2(a 4) 2(b 1) 2 Power of a Product
A = 49a 8b 2
7a 4b
Simplify.
The area of the square is 49a 8b 2 square units.
g. GEOMETRY Find the volume of a cube with sides of length 8x 3y 5.
Express as a monomial.
Personal Tutor at ca.gr7math.com
550 Chapter 10 Algebra: Nonlinear Functions and Polynomials
Examples 1–4
Simplify.
(pp. 549-550)
1. (3 2)
Example 5
4. (7w 7)
5
2. (h 6) 4
3
3. [(2 3) 2] 3
12
5. (5g 8k ) 4
6. (-6r 5s 9) 2
(p. 550)
7. MEASUREMENT Express the volume of the cube
at the right as a monomial.
(/-%7/2+ (%,0
For
Exercises
8–27
28–31
See
Examples
1–4
3
3c 3d 2
Simplify.
8. (4 2) 3
9. (2 2) 7
10. (5 3) 3
11. (3 4) 2
12. (d 7) 6
13. (m 8) 5
14. (h 4) 9
15. (z 11) 5
16. [(3 2) 2] 2
17. [(4 3) 2] 2
18. [(5 2) 2] 2
19. [(2 3) 3] 2
20. (5j 6) 4
21. (8v 9) 5
22. (11c 4) 3
23. (14y) 4
24. (6a 2b 6) 3
25. (2m 5n 11) 6
26. (-3w 3z 8) 5
27. (-5r 4s 12) 4
GEOMETRY Express the area of each square below as a monomial.
28.
29.
8g 3h
12d 6e 7
GEOMETRY Express the volume of each cube below as a monomial.
30.
31.
5r 2s 3
7m 6n 9
Simplify.
32. (0.5k 5) 2
35.
3 -6 9 2
(_
a b )
5
1
3
34. (_w 5z ) 2
33. (0.3p 7) 3
36. (3x
-2 4
4
6 2
) (5x )
37. (-2v 7) 3(-4v -2) 4
38. PHYSICS A ball is dropped from the top of a building. The expression 4.9x 2
%842!02!#4)#%
gives the distance in meters the ball has fallen after x seconds. Write and
simplify an expression that gives the distance in meters the ball has fallen
after x 2 seconds. Then write and simplify an expression that gives the
distance the ball has fallen after x 3 seconds.
See pages 703, 717.
39. BACTERIA A certain culture of bacteria doubles in population every hour. At
Self-Check Quiz at
1 P.M., there are 5 cells. The expression 5(2 x)gives the number of bacteria
that are present x hours after 1 P.M. Simplify the expressions [5(2 x)] 2 and
[5(2 x)] 3 and describe what they each represent.
ca.gr7math.com
Lesson 10-7 Powers of Monomials
551
MEASUREMENT For Exercises 40-42, use the
table that gives the area and volume of
a square and cube, respectively, with side
lengths shown.
Side
Length
(units)
Area of
Square
(units 2)
Volume of
Cube
(units 3)
x
x2
x3
40. Copy and complete the table.
2x
41. Describe how the area and volume are
3x
each affected if the side length is doubled.
Then describe how they are each affected
if the side length is tripled.
x2
x3
42. Describe how the area and volume are each affected if the side length is
squared. Describe how they are each affected if the side length is cubed.
H.O.T. Problems
43. OPEN ENDED A googol is 10 100. Use the Power of a Power rule to write three
different expressions that are equivalent to a googol where each expression
uses exponents.
CHALLENGE Solve each equation for x.
44. (7 x) 3 = 7 15
5
45. (-2m 3n 4) x = -8m 9n 12 3
*/ -!4( Compare and contrast how you would correctly
(*/
46.
83 *5*/(
simplify the expressions (2a 3)(4a 6) and (2a 3) 6.
47. Which expression is equivalent to
49. Which of the following has the same
4 8
value as 64m 6?
(10 ) ?
A 10 2
C 10 12
B 10 4
D 10 32
A the area in square units of a square
whose side length is 8m 2
B the expression (32m 3) 2
48. Which expression has the same value
C the expression (8m 3) 2
as 81h 8k 6?
F (9h 6k 4) 2
H (6h 5k 3) 3
G (9h 4k 3) 2
J
(3h 2k) 6
Simplify. Express using positive exponents.
15
_
15 4
(Lesson 10-6)
10
7
50.
D the volume in cubic units of a cube
whose side length is 4m 3
51.
y
_
y2
52.
18m 9
_
6m 4
3
53.
24g
_
3g 8
54. MEASUREMENT Find the area of a rectangle with a length of 9xy 2 and a width of 4x 2y.
(Lesson 10-5)
Find each square root.
55. √
49
(Lesson 3-1)
56. √
121
57. √
225
552 Chapter 10 Algebra: Nonlinear Functions and Polynomials
58. √
400
10-8
Roots of Monomials
Main IDEA
Find roots of monomials.
Standard 7AF2.2
Multiply and divide
monomials; extend
the process of taking powers
and extracting roots to
monomials when the latter
results in a monomial with
an integer exponent.
NEW Vocabulary
cube root
REVIEW Vocabulary
square root: the opposite of
squaring a number (Lesson 3-1)
NUMBER THEORY The square root of a number is one of the two equal
factors of the number. Some perfect squares can be factored into the
product of two other perfect squares.
1. Find two factors of 100 that are also perfect squares.
2. Find the square root of 4 and 25. Then find their product.
3. How does the product relate to 100?
4. Repeat Questions 1–3 using 144.
The pattern you discovered about the factors of a perfect square is true
for any number.
+%9#/.#%04
Product Property of Square Roots
For any numbers a and b, where a ≥ 0 and b ≥ 0, the square
root of the product ab is equal to the product of each square
root.
Words
Examples
Numbers
√
9 · 16 = √
9 · √
16
Algebra
√
ab = √
a · √
b
= 3 · 4 or 12
The square root of a monomial is also one of its two equal factors. You
can use the product property of square roots to find the square roots of
monomials.
√
x 2 = √x
· x = ⎪x⎥
Since x represents an unknown value,
absolute value is used to indicate the
positive value of x.
√
x 4 = √
x2 · x2 = x2
Absolute value is not necessary since the
value of x 2 will never be negative.
Simplify Square Roots
4y 2 .
√
4y 2 = √4 · √
y2
√
36q 6 .
√
36q 6 = √
36 · √
q6
√
= √
6 · 6 · √
q3 · q3
1 Simplify
Absolute Value
Use absolute value to
indicate the positive
value of y and q 3.
2 Simplify
= √
2 · 2 · √
y·y
= 2⎪y⎥
= 6 ⎪q 3 ⎥
Simplify.
a.
√
v2
Extra Examples at ca.gr7math.com
b.
√
c 6d 8
c.
√
121x 4z 10
Lesson 10-8 Roots of Monomials
553
READING Math
Cube Root Symbol The
cube root of a is shown by
3
the symbol √
a.
The process of simplifying expressions involving square roots can be
extended to cube roots. The cube root of a monomial is one of the three
equal factors of the monomial.
3
3
√
8 = √
2·2·2=2
3
3
√
a3 = √
a
·a·a=a
+%9#/.#%04
Product Property of Cube Roots
For any numbers a and b, the cube root of the product ab is
equal to the product of each cube root.
Words
Examples
Numbers
3
Algebra
3
3
= √
√ab
a · √
b
3
3
3
√
216 = √
8 · √
27
= 2 · 3 or 6
Simplify Cube Roots
3
3 Simplify √
c3.
3
√
c3 = c
(c) 3 = c 3
3
64g 6 .
√
64g 6 = √
64 · √
g6
√
4 Simplify
3
3
3
3
= √
4·4·4·
Product Property of Cube Roots
3
g2 · g2 · g2
√
= 4 · g 2 or 4g 2
Absolute Value
Because a cube root
can be negative,
absolute value is not
necessary when
simplifying cube
roots.
Simplify.
Simplify.
d.
3
√
s3
e.
3
27y 3
√
f.
3
√
216k 12
5 GEOMETRY Express the length of one side of the square whose area
is 81y 2z 6 square units as a monomial.
A = s 2 Area of a square
81y 2z 6 = s 2 Replace A with 81y 2z 6.
y 2z 6 = s Definition of square root.
√81
√
81 · √
y 2 · √
z 6 = s Product Property of Square Roots.
9⎪yz 3⎥ = s
Simplify. Add absolute value.
The length of one side of the square is 9⎪yz 3⎥ units.
g. GEOMETRY Find the length of one side of a square whose volume is
is 125a 15 cubic units.
Personal Tutor at ca.gr7math.com
554 Chapter 10 Algebra: Nonlinear Functions and Polynomials
Examples 1–2
(p. 553)
Example 3–4
Simplify.
1. √
d2
5.
3
√
m3
2.
√
25a 2
6.
8p 3
√
(p. 554)
Example 5
3
3.
49x 6y 2
√
7.
√
125r 6s 9
3
4.
√
121h 8k 10
8.
64 x 12y 3
√
3
9. GEOMETRY Express the length of one side of the square whose area is
256u 2v 6 square units as a monomial.
(p. 554)
10. GEOMETRY Express the length of one side of a cube whose volume is
27b 3c 12 cubic units as a monomial.
(/-%7/2+ (%,0
For
Exercises
11–18
14–26
27–34
See
Examples
1–2
3
5
Simplify.
2
11. √n
12.
y4
√
13.
g 8k 14
√
14.
√
64a 2
17.
9p 8q 4
√
18.
225x 4y 6
√
22.
√
64k 3
26.
√
216x 12w 15
15.
√
36z 12
16.
√
144k 4m 6
19.
3
√
h3
20.
√
v3
21.
√
27b 3
23.
√
125d 9e 3
24.
8q 9r 18
√
25.
√
343m 3n 21
3
3
3
3
3
3
3
GEOMETRY Express the length of one side of each square whose area is given
as a monomial.
27.
28.
29.
30.
A 36m 6n 8
A 400x 2y 10
A 121a 2b 2
A 49p 4q 6
GEOMETRY Express the length of one side of each cube whose volume is
given as a monomial.
31.
32.
33.
34.
V 125k 9m 18
V 27g 24h 3
V 64w 3z 3
V 343c 6d 12
Simplify.
%842!02!#4)#%
See pages 704, 717.
Self-Check Quiz at
ca.gr7math.com
35.
√
0.25x 2
36.
Simplify each expression if
x
_
√
16
2
38.
39.
3
0.08p 9
√
37.
8 3 6
w x
√_
27
40.
121
√_
h k
3
√a
.
√_ab = _
√b
81
√_
m
4
8 6
Lesson 10-8 Roots of Monomials
555
H.O.T. Problems
41. OPEN ENDED Write a monomial and its square root.
CHALLENGE Solve each equation for x.
42.
√
25a x = 5 ⎪a 3⎥
43.
3
√
64a 3b x = 4ab 7
simplifying the expression
y 2 and not necessary when simplifying √
y4.
√
46. Which expression is equivalent
48. Which of the following has the same
3
value as √
27m 3n 6 ?
to √
144g 2 ?
A 12g
C 12g 2
B 12⎪g⎥
D 12⎪g
2
A the length of the side of a square
whose area is 27m 3n 6
⎥
B the expression 9mn 3
47. Which expression has the same value
F 20hk 2
H 20h 2k 4
G 20 ⎪h⎥ k 2
J
49. (6 3) 5
(Lesson 10-7)
50. (n 7) 2
5
9
_
93
D the length of the side of a cube
whose volume is 3mn 2
200 ⎪h⎥ k 2
Simplify. Express using positive exponents.
53.
C the expression 3mn 2
√
400h 2k 4 ?
Simplify.
√
81a 4b x = 9a 4 ⎪b 5⎥
*/ -!4( Explain why absolute value is necessary when
(*/
83 *5*/(
45.
as
44.
54.
k 15
_
k6
51. (2a 3b 2) 4
52. (-4p 11q) 3
(Lesson 10-6)
4
55.
24y
_
4y 2
3
56.
45g
_
3g 7
57. RETAIL Find the discount to the nearest cent for a flat-screen television that costs $999
and is on sale at 15% off.
(Lesson 5-8)
Math and Economics
Getting Down to Business It’s time to complete your project. Use the information and
data you have gathered about the cost of materials and the feedback from your peers to
prepare a video or brochure. Be sure to include a scatter plot with your project.
Cross-Curricular Project at ca.gr7math.com
556 Chapter 10 Algebra: Nonlinear Functions and Polynomials
CH
APTER
10
Study Guide
and Review
Download Vocabulary
Review from ca.gr7math.com
Key Vocabulary
cube root (p. 554)
monomial (p. 539)
Be sure the following
Key Concepts are noted
in your Foldable.
Key Concepts
Functions
(Lessons 10-1, 10-2, and 10-3)
• Linear functions have constant rates of change.
• Nonlinear functions do not have constant rates of
change.
• Quadratic functions are functions in which the
greatest power of the variable is 2.
• Cubic functions are functions in which the
greatest power of the variable is 3.
Monomials
nonlinear function (p. 522)
quadratic function (p. 528)
Vocabulary Check
State whether each sentence is true or false.
If false, replace the underlined word or
number to make a true sentence.
1. The expression y = x 2 - 3x is an example
of a monomial.
2. A nonlinear function has a constant rate of
change.
(Lessons 10-5 through 10-8)
• To multiply powers with the same base, add their
exponents.
• To divide powers with the same base, subtract
their exponents.
3. A quadratic function is a function whose
greatest power is 2.
4. The product of 3x and x 2 + 3x will have 3
terms.
• To find the power of a power, multiply the
exponents.
5. A quadratic function is a nonlinear
• To find the power of a product, find the power of
each factor and multiply.
6. The graph of a linear function is a curve.
function.
7. To divide powers with the same base,
subtract the exponents.
8. The Quotient of Powers states when
dividing powers with the same base,
subtract their exponents.
9. The graph of a cubic function is a straight
line.
Vocabulary Review at ca.gr7math.com
Chapter 10 Study Guide and Review
557
CH
APTER
10
Study Guide and Review
Lesson-by-Lesson Review
10-1
Linear and Nonlinear Functions
(pp. 522–527)
Determine whether each equation or
table represents a linear or nonlinear
function. Explain.
10. y - 4x = 1
12.
10-2
11. y = x 2 + 3
Time (h)
2
Number of Pages
98
3
4
5
147 199 248
Graphing Quadratic Functions
Example 1 Determine
whether the table
represents a linear or
nonlinear function.
y
-2
-3
-1
-1
0
1
1
3
As x increases by 1,
y increases by 2. The rate
of change is constant, so this
function is linear.
(pp. 528–531)
Graph y = -x 2 - 1.
Graph each function.
Example 2
13. -4x 2
14. y = x 2 + 4
15. y = -2x 2 + 1
16. y = 3x 2 - 1
Make a table of values. Then plot and
connect the ordered pairs with a smooth
curve.
17. SCIENCE A ball is dropped from the
top of a 36-foot tall building. The
quadratic equation d = -16t 2 + 36
models the distance d in feet the ball
is from the ground at time t seconds.
Graph the function. Then use your
graph to find how long it takes for the
ball to reach the ground.
10-3
x
PSI: Make a Model
x
y = -x 2 - 1
(x, y)
-2
-(-2) 2 - 1
(-2, -5)
-1
-(-1) 2 - 1
(-1, -2)
0
2
-(0) - 1
(0, -1)
1
2
-(1) - 1
(1, -2)
2
2
(2, -5)
-(2) - 1
y
O
x
y x 2 1
(pp. 532-533)
Solve the problem by using the make a
model strategy.
18. MEASUREMENT Sydney has a postcard
that measures 5 inches by 3 inches. She
decides to frame it, using a frame that
3
is 1_
inches wide. What is the
4
Example 3
DISPLAYS Cans of oil are displayed in the
shape of a pyramid. The top layer has 2
cans in it. One more can is added to each
layer, and there are 4 layers in the
pyramid. How many cans are there in
the display?
perimeter of the framed postcard?
19. MAGAZINES A book store arranges it
best-seller magazines in the front
window. In how many different ways
can five best-seller magazines be
arranged in a row?
So, based on the model there are 14 cans.
558 Chapter 10 Algebra: Nonlinear Functions and Polynomials
Mixed Problem Solving
For mixed problem-solving practice,
see page 717.
10-4
Graphing Cubic Functions
(pp. 534-537)
20. y = 2x 3 – 4
21. y = 0.25x 3 - 2
x
y = -x 3
(x, y)
22. y = 2x 3 + 4
23. y = 0.25x 3 + 2
-2
-(-2) 3
(-2, 8)
-1
-(-1)
3
(-1, 1)
0
-(0) 3
1
-(1)
3
(1, -1)
2
-(2) 3
(2, -8)
24. MEASUREMENT A rectangular prism
with a square base of side length x
inches has a height of (x - 1) inches.
Write the function for the volume V of
the prism. Graph the function. Then
estimate the dimensions of the box that
would give a volume of approximately
18 cubic inches.
10-5
Multiplying Monomials
25. 4 · 4 5
26. x 6 · x 2
27. -9y 2(-4y 9)
28.
_3
-4
3
· _
(7) (7)
2
29. LIFE SCIENCE The number of bacteria
after t cycles of reproduction is 2 t.
Suppose a bacteria reproduces every 30
minutes. If there are 1,000 bacteria in a
dish now, how many will there be in 1
hour?
Dividing Monomials
n5
31. _
n
21c
_
-7c 8
(0, 0)
x
Example 5 Find 4 · 4 3. Express using
exponents.
4 · 43 = 41 · 43 4 = 41
= 41 + 3
The common base is 4.
4
=4
Add the exponents.
Example 6 Find 3a 3 · 4a 7.
3a 3 · 4a 7 = (3 · 4)a 3 + 7 Commutative and
Associative Properties
= 12a
10
33.
Example 7
68
6
Simplify_
. Express using exponents.
3
-1
3
11
32.
y x 3
(pp. 544-548)
Simplify. Express using exponents.
59
30. _
52
y
(pp. 539-542)
Simplify. Express using exponents.
10-6
Graph y = -x 3.
Example 4
Graph each function.
(_47 ) × (_47 )
__
_4
7
34. MEASUREMENT The area of the family
room is 3 4 square feet. The area of the
kitchen is 4 3 square feet. What is the
difference in area between the two
rooms?
68
_
= 68 - 3
63
The common base is 6.
5
=6
Example 8
Simplify.
-8
s
. Express using exponents.
Simplify _
-4
s
s -8
=_
= s -8 - (-4)
s –4
= s -8 + 4 or s -4
Quotient of
Powers
Simplify.
Chapter 10 Study Guide and Review
559
CH
APTER
10
Study Guide and Review
10-7
Powers of Monomials
(pp. 549-552)
Example 9
Simplify.
35. (9 2) 3
36. (d 6f 3) 4
Simplify (7 3) 5.
37. (5y 5) 4
38. (6z 4x 3) 5
39. (_n -1) 2
40. [(p 2) 3] 2
41. (5 -1) 2
42. (-3k 2) 2(4k -3) 2
(7 3) 5 = 7 3 · 5 Power of a Power
= 7 15 Simplify.
Example 10
3
4
43. GEOMETRY Find the volume of a cube
with sides of length 5s 2t 4 as a
monomial.
Simplify (2x 2y 3) 3.
(2x 2y 3) 3 = 2 3 · x 2 · 3 y 3 · 3 Power of a Product
= 8 x 6y 9
Simplify.
44. GEOMETRY Find the area of a square
with sides of length 6a 3b 5 as a
monomial.
10-8
Roots of Monomials
Simplify.
2
45. √a
47.
49.
36x 2y 6
√
3
6
√p
3
51. √
64c 6d 21
(pp. 553-556)
46.
√4
48.
81q 14
√
50.
49n
3
18
√
8m
3
52. √
125r 9s 15
Example 11
16f 8g 6 .
Simplify √
16f 8g 6 = √
16 · √
f 8 · √
g 6 Product
√
Property of
Square Roots
= 4 · f 4 · ⎪g 3⎥ or 4f 4 ⎪g 3⎥
53. GEOMETRY Express the length of one
side of the square whose area is 64b 16
square units as a monomial.
54. GEOMETRY Express the length of one
side of a cube whose volume is
216a 9c 3 cubic units as a monomial.
Example 12
3
Simplify √
x9.
3
√
x9 = x3
560 Chapter 10 Algebra: Nonlinear Functions and Polynomials
(x) 9 = x 3
Use absolute
value to
indicate the
positive value
of g 3.
CH
APTER
Practice Test
10
Determine whether each graph, equation, or
table represents a linear or nonlinear function.
Explain.
1.
12. CRAFTS Martina is making cube-shaped gift
boxes from decorative cardboard. Each side
of the cube is to be 6 inches long, and there
1
is a _
-inch overlap on each side. How much
2. 2x = y
y
2
cardboard does Martina need to make each
box?
x
O
Simplify. Express using exponents.
3.
x
-3 -1
2
y
10
1
3
18
26
13. 15 3 15 5
15.
3 15
_
14. -5m 6(-9m 8)
16.
37
-40w 8
_
8w
Graph each function.
4. y = _x 2
1
2
5. y = -2x 2 + 3
6. BUSINESS The function p = 60 + 2d 2 models
the profit made by a manufacturer of digital
audio players. Graph this function. Then use
your graph to estimate the profit earned
after making 20 players.
Simplify.
17. √
m2
18.
√
144a 2b 6
19.
64x 3y 15
√
20.
STANDARDS PRACTICE Simplify the
algebraic expression (3x 3y 2)(7x 3y).
7.
A 21x 9y 2
B 21x 6y 2
C 21x 6y 3
D 21x 6y 6
Graph each function.
3
STANDARDS PRACTICE Which
(12x 4)(4x 3)
expression is equivalent to _
?
5
F 12x 7
H 6x 4
G 12x 2
J
8x
6x 2
21. MEASUREMENT Find the
area of the rectangle at
the right.
4s 2t 2
3st 3
3
8. y = x + 4
9. y = x 3 - 4
Simplify.
10. y = _x 3
22. [(x 2) 4] 3
11. MEASUREMENT A neighborhood group
24. (3 -3) 2
1
3
23. (-2b 3) 2(4b 2) 2
would like Jacob to fertilize their lawns. The
average area of each lawn is 6 4 square feet.
If there are 6 2 lawns in this neighborhood,
how many total square feet of lawn does
Jacob need to fertilize?
Chapter Test at ca.gr7math.com
25. GEOMETRY Express the length of one side of
a square with an area of 121x 4y 10 square
units as a monomial.
Chapter 10 Practice Test
561
CH
APTER
10
California
Standards Practice
Cumulative, Chapters 1–10
Read each question. Then fill in the
correct answer on the answer
document provided by your teacher or
on a sheet of paper.
1
3
The equation c = 0.8t represents c, the cost
of t tickets on a ferry. Which table contains
values that satisfy this equation?
A
A car used 4.2 gallons of gasoline to travel
126 miles. How many gallons of gasoline
would it need to travel 195 miles?
t
c
Cost of Ferry Tickets
1
2
3
$0.80
$1.00
$1.20
4
$1.40
t
c
Cost of Ferry Tickets
1
2
3
$0.80
$1.60
$2.40
4
$3.20
t
c
Cost of Ferry Tickets
1
2
3
$0.75
$1.50
$2.25
4
$3.00
t
c
Cost of Ferry Tickets
1
2
3
$1.80
$2.60
$3.40
4
$4.20
B
A 2.7
B 5.0
C 6.5
C
D 7.6
2
The scatter plot below shows the cost of
computer repairs in relation to the number
of hours the repair takes. Based on the
information in the scatter plot, which
statement is a valid conclusion?
Total Cost ($)
Cost of
Computer Repairs
55
50
45
40
35
30
25
20
15
10
5
y
D
4
Shanelle purchased a new computer for
$1,099 and a computer desk for $699
including tax. She plans to pay the total
amount in 24 equal monthly payments.
What is a reasonable amount for each
monthly payment?
F $50
G $75
1 2 3 4 5 6 7 x
Number of Hours
H $150
J $1,800
F As the length of time increases, the cost
of the repair increases.
G As the length of time increases, the cost
of the repair stays the same.
H As the length of time decreases, the cost
of the repair increases.
J As the length of time increases, the cost
of the repair decreases.
Question 4 You can often use
estimation to eliminate incorrect
answers. In this question, Shanelle’s
total spent can be estimated by adding
$1,100 and $700, then dividing by 24.
The sum of $1,100 and $700 is $1,800
before dividing by 24, so choice J can
be eliminated.
562 Chapter 10 Algebra: Nonlinear Functions and Polynomials
More California
Standards Practice
For practice by standard,
see pages CA1–CA39.
5
Which of the following is the graph of
2 x 2?
y=_
3
y
y
C
A
O
O
x
y
B
D
O
8
9
x
The area of a rectangle is 30m 11 square feet.
If the length of the rectangle is 6m 4 feet,
what is the width of the rectangle?
F 5m 7 ft
H 36m 15 ft
G 24m 7 ft
J 180m 15 ft
Which expression is equivalent to 5 4 × 5 6?
A 5 10
C 25 10
B 5 24
D 25 24
y
x
Pre-AP
O
6
Record your answers on a sheet of paper.
Show your work.
x
10 An electronics store is having a sale on
Simplify the expression shown below.
certain models of televisions. Mr. Castillo
would like to buy a television that is on sale.
This television normally costs $679.
(3m 3n 2)(6m 4n)
F 18m 12n 2
H 18 m 7n 3
G 18 m 7n 2
J 18 m 7n 6
Last Year’s Models
7
40% off
What is the height h of the gutter in the
figure below?
Wednesday
Only
Television
Sale!
Take an additional
10% off
a. What price, not including tax, will Mr.
Castillo pay if he buys the television on
Saturday?
20 ft
h
b. What price, not including tax, will Mr.
Castillo pay if he buys the television on
Wednesday?
12 ft
A 10 ft
C 16 ft
B 14 ft
D 18 ft
c. How much money will Mr. Castillo save
if he buys the television on Saturday?
NEED EXTRA HELP?
If You Missed Question...
1
2
3
4
5
6
7
8
9
10
Go to Lesson...
4-3
9-8
9-1
5-5
10-2
10-4
10-4
10-5
10-5
5-8
For Help with Standard...
AF4.2
PS1.2
MR2.5
MR3.1
AF3.1
AF2.2
MG3.3
NS2.3
NS2.3
NS1.7
California Standards Practice at ca.gr7math.com
Chapter 10 California Standards Practice
563
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