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Additional Topics
A-1
Introduction to Power Functions . . . . . . . . . . . . . . . . . . . . . . . . AT2
A-2
Piecewise and Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . AT5
A-3
Technology Lab: Graphing to Solve Equations . . . . . . . . . . . . . .AT9
A-4
Patterns and Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AT10
A-5
Linear and Nonlinear Rates of Change . . . . . . . . . . . . . . . . . . AT14
A-6
Reasoning and Counterexamples . . . . . . . . . . . . . . . . . . . . . . AT18
ADDITIONAL
TOPIC
A-1
Objective
Analyze the
characteristics of power
functions.
Introduction to Power
Functions
A power function is a function that can be written in the form f (x) = ax n,
where a and n are real numbers and a ≠ 0. This lesson will only address power
functions for which n is a positive integer.
Power Functions
Vocabulary
power function
n=1
n=2
y = ax
EXAMPLE
1
n=3
y = ax
2
y = ax
n=4
3
y = ax
n=5
4
y = ax
5
Graphing Power Functions
Graph the power function y = x 5, and then describe the graph.
Step 1 Make a table of values.
x
x5 = y
-2
(-2) 5 = -32
-1
(-1) 5 = -1
0
05 = 0
1
15 = 1
2
2 5 = 32
The ordered pairs (−2, −32), (−1, −1), (0, 0), (1, 1) and (2, 32) lie on the
graph.
Step 2 Sketch the graph by plotting the ordered pairs from the table.
40
y
20
-4
-2
0
2
4x
-20
-40
From left to right, the graph increases. It crosses the x-axis once, at
the origin.
1. Graph the power function y = x 4, and then describe the graph.
AT2
Additional Topics
Substituting any real value of x into a power function results in a real number.
Therefore, the domain of any power function is all real numbers. The range of a
power function depends on the degree of the function. The degree of the power
function f (x) = ax n is n.
The table summarizes the possibilities for the range and the maximum and
minimum values of a power function.
Range, Maximum Value, and Minimum Value
Odd Degree
The range is all real
numbers. There is
no maximum or
minimum value.
EXAMPLE
2
Even Degree, a > 0
Even Degree , a < 0
The graph opens
downward. The
maximum value is 0.
The range is y ≤ 0.
The graph opens
upward. The
minimum value is 0.
The range is y ≥ 0.
Finding Maximum or Minimum Values
Give the domain and range of each power function. Then give the minimum or
maximum value, if any.
A f (x) = -0.5x 7
The degree, 7, is odd.
The domain is all real numbers. The range is all real numbers. There is no
minimum or maximum value.
B f (x) = -2x 4
The degree, 4, is even, and a < 0.
The domain is all real numbers.
The range is y ≤ 0.
The maximum value is 0.
Check The graph supports the answer.
1 x6
C f (x) = _
20
The degree, 6, is even, and a > 0.
The domain is all real numbers.
The range is y ≥ 0.
The minimum value is 0.
Check The graph supports the answer.
Give the domain and range of each power function. Then give the
minimum or maximum value, if any.
1 x 10
2a. f (x) = -x 8
2b. f (x) = -2.5x 5 2c. f (x) = _
4
A-1 Introduction to Power Functions
AT3
A-1
Exercises
Graph each power function, and then describe the graph.
1. y = x 3
2. y = x 6
3. y = x 7
Give the domain and range of each power function. Then give the minimum or
maximum value, if any.
1 x 16
4. f(x) = -2x 11
5. f(x) = -_
6. f(x) = 0.05x 8
8
7. Geometry A cube has sides of length x.
a. Write a power function f that gives the area of each
face of the cube.
b. Write a power function g that gives the surface area
of the cube.
x
8. Geometry The volume V of a sphere of radius x is given by the
3
power function V(x) = __43 πx . Use a graphing calculator to determine
the radius of a sphere that has a volume of 100 cm 3. Round your answer
to the nearest tenth of a centimeter and explain your method.
Determine whether each statement is always, sometimes, or never true.
Explain.
9. A power function has two x-intercepts.
10. A power function has a minimum value of 1.
11. The graph of a power function that passes through the point (a, b) also passes
through the point (−a, −b).
12. A power function that has a minimum value has an even degree.
The end behavior of a function is a description of the function’s values as x
increases or decreases. For example, you can describe the end behavior of f(x) = x 3
as follows: As x increases, f(x) increases. As x decreases, f(x) decreases.
Describe the end behavior of each power function.
1 x7
13. f (x) = -4x 4
14. f (x) = _
15. f (x) = -2x 9
3
16. Physics A certain metal sphere sinks in a lake according to the function d(t) = 2.1t 2
where t is the time in seconds and d(t) is the depth of the sphere in meters. Sketch
a graph of the function. Then give a reasonable domain and range for the function
when the sphere is dropped in a lake with a maximum depth of 100 meters.
17. Determine whether the power function f(x) = -3x 3 is increasing or decreasing on the
interval -5 ≤ x ≤ -1.
18. Critical Thinking Is a direct variation a power function? Explain why or why not.
19. Graph f(x) = x 3, g(x) = x 3 + 1, and h(x) = x 3 - 2. Make a conjecture, in terms of a
transformation, about the effect of b on the graph of f(x) = ax 3 + b.
AT4
Additional Topics
ADDITIONAL
TOPIC
A-2
Objective
Recognize, identify, and
graph piecewise and step
functions.
Vocabulary
piecewise function
step function
greatest-integer function
Piecewise and Step Functions
A piecewise function is a function that is a combination of one or more
functions. The rule for a piecewise function may be different for different parts of
the function’s domain.
4
For example, the absolute value function y = ⎪x⎥
is a piecewise function and can be defined by the
following rule:
y
y = |x|
2
x
-4
0
-2
2
4
-2
⎧-x if x < 0
y=⎨
⎩ x if x ≥ 0
-4
To evaluate a piecewise function for a given input value, find the interval of the
domain that contains the input value and apply the rule for that interval.
EXAMPLE
1
Evaluating a Piecewise Function
⎧ x2 + 1
if x < 2
For f (x) = ⎨
, evaluate f (x) for x = −3 and x = 6.
⎩ -3x + 4 if x ≥ 2
2
f (-3) = (-3) + 1 = 10
Because -3 < 2 , use the rule for x < 2.
f (6) = -3(6) + 4 = -14
Because 6 ≥ 2 , use the rule for x ≥ 2.
⎧3x - 3 if x < -1
1. For g(x) = ⎨
, evaluate g(x) for x = –4 and x = –1.
⎩ -x 2
if x ≥ -1
EXAMPLE
2
Graphing a Piecewise Function
⎧ -3
Graph f (x) = ⎨
⎩ x2 + 1
if x < 0
if x ≥ 0
.
This function is composed of two pieces.
For x < 0, the function is the constant function
f (x) = –3. Draw a horizontal ray to the left of
(0, –3). Draw an open circle at (0, –3) to show
that this point is not part of the graph.
For x ≥ 0, the function is the quadratic function
2
f (x) = x + 1. Draw half of a parabola to the
right of (0, 1). Draw a solid circle at (0, 1) to
show that this point is part of the graph.
⎧ 2x
2. Graph g(x) = ⎨
⎩ x -1
4
y
2
x
-4
-2
0
2
4
-2
-4
if x < 1
.
if x ≥ 1
A-2 Piecewise and Step Functions
AT5
The graph and table at right give the price
of admission to a theme park. The function
is defined differently over different domain
intervals (age groups), so the function is a
piecewise function.
Regular admission
$25
Youth (under 15)
$10
Seniors (55 and older)
$20
Theme Park Admission Prices
30
Price ($)
The function that describes the theme park
admission prices is a step function. A step
function is a piecewise function that is
constant over each interval in its domain.
Theme Park Admission Prices
20
10
0
3
20
30 40
Age (yr)
50
60
10
12
Parking Fees
Step Functions
The graph shows parking fees at a garage.
Create a table and a verbal description to
represent the graph.
Step 1 Create a table.
y
6
Fee ($)
EXAMPLE
10
Use the endpoints of the segments
of the graph to identify the
intervals of the domain.
4
2
x
0
2
4
6
8
Hours
Parking Fees
Hours x
Fee y ($)
0<x≤2
3
2<x≤4
4
4<x≤6
5
x>6
6
On the graph, an open circle means that
a value is not included in the interval. A
solid circle means that a value is included
in the interval.
Step 2 Write a verbal description.
For any amount of time up to 2 hours, the parking fee is $3. For more
than 2 hours and no more than 4 hours, the fee is $4. For more than 4
hours and no more than 6 hours, the fee is $5. For more than 6 hours,
the fee is $6.
Shipping Costs
6
Cost ($)
3. The graph shows the shipping
costs for books that are ordered
online. Create a table and a
verbal description to represent
the graph.
y
4
2
x
0
AT6
Additional Topics
4
8
12 16
Weight (oz)
20
The greatest-integer function is a piecewise function,
written f (x) = x, in which the number x is rounded down
to the greatest integer that is less than or equal to x. For
example, f (3.14) = 3.14 = 3. The graph of the greatestinteger function shows that it is a step function.
4
f(x) = x
y
2
x
-4
-2
0
2
4
-2
-4
EXAMPLE
4
Using the Greatest-Integer Function
Write a function that gives the number of magazines that you can buy with x
dollars if a magazine costs $3.75. Then use the function to find the number of
magazines you can buy with $20.
x
The number of magazines you can buy with x dollars is ____
, rounded
3.75
down to the greatest integer that is less than or equal to this quotient.
x .
The function is f (x) = _
3.75
To find how many magazines you can buy with $20, evaluate f (x) for x = 20.
20 = 5.__
f (20) = _
3 = 5
3.75
You can buy 5 magazines with $20.
4. Write a function that gives the number of movie tickets that you
can buy with x dollars if a movie ticket costs $10.95. Then use the
function to find the number of movie tickets you can buy with $50.
A-2
Exercises
Evaluate each function for x = −2 and x = 3.
⎧ 2x + 1 if x < 0
⎧-x 2 if x ≤ -1
1. f (x) = ⎨
2. g (x) = ⎨
⎩ 5x if x > -1
⎩-x + 2 if x ≥ 0
⎧ x 2 + 1 if x < -3
3. h(x) = ⎨
⎩ 2x - 5 if x ≥ -3
Evaluate each function for x = −1 and x = 2.
⎧4x -1 if x < -1
5. f (x) = -x + 5 if -1 ≤ x < 2
⎩ 3x
if x ≥ 2
⎨
⎧ -5
4. f (x) = ⎨
if x < 3
⎩ 3x + 1 if x ≥ 3
2
⎧x 2
⎨
6. f (x) = 4x + 6
1
_
⎩ 2x
if x ≤ -2
if -2 ≤ x ≤ 1
if x ≥ 1
Graph each function.
⎧ x -1 if x < 0
7. f (x) = ⎨
if x ≥ 0
⎩4
⎧ -1
if x ≤ -1
8. f (x) = 2x
if -1< x ≤ 2
⎩ x - 4 if x > 2
⎨
A-2 Piecewise and Step Functions
AT7
Create a table and a verbal description to represent each graph.
9.
10.
Admission Prices
y
6
Weeks
Price ($)
6
Weeks of Paid Vacation
4
2
y
4
2
x
0
4
8
12
Age (yr)
x
0
16
2
4
6
8
Years of employment
11. Write a function that gives the number of energy bars that you can buy with x dollars
if an energy bar costs $1.35. Then use the function to find the number of energy bars
you can buy with $10.
x
12. The function g (x) = ____
gives the number of gel pens you can buy with x dollars.
3.95
What is the price of one gel pen? How many gel pens can you buy with $15?
Write a piecewise function for each graph.
13.
4
14.
y
4
15.
y
-4
-2
2
4
x
x
x
-4
-2
0
y
2
2
2
0
4
2
4
-4
-2
0
-2
-2
-2
-4
-4
-4
2
4
16. Entertainment The cost of admission to a state fair is $4 for children less than
12 years old and $8 for everyone 12 or older.
a. Write a function that gives the cost of admission for a person who is x years old.
b. Graph the function.
17. Pet Care The table shows the
recommended amount of dog food
based on a dog’s weight.
a. Write a function that gives the
amount of dog food in cups for
a dog that weighs x pounds.
b. Graph the function.
Recommended Amount of Dog Food
Weight of dog (lb)
Amount of food (c)
Less than 10
1
__
At least 10 and no
more than 20
1
More than 20
1__12
2
18. Travel Carolyn leaves her house and drives for 3 hours at a constant rate of 60 mi/h.
Then she stops for 1 hour to have lunch. After lunch, she continues to drive away
from her house at a constant rate of 60 mi/h for another 2 hours. Graph a piecewise
function that gives Carolyn’s distance in miles from her house after x hours.
19. Critical Thinking What are the domain and range of the piecewise function
⎧ x if x < 0
f(x) = ⎨
?
⎩ x 2 if x ≥ 0
AT8
Additional Topics
A-3
Graphing to Solve Nonlinear
Equations
You can use a graphing calculator to solve nonlinear equations,
including quadratic and exponential equations.
Activity
Use a graph to solve x 2 – x – 5 = –3.
Enter the left side of the equation as Y1 and the right side as Y2.
Press
. The x-values of the points where the
graphs intersect (where Y1 = Y2) are the solutions of the
equation. Notice that there is more than one solution.
To find the coordinates of an intersection point:
Y1 = x2 - x - 5
and select 5:intersect.
Press
Press
to select Y1.
Press
again to select Y2.
Use
Press
Y2 = -3
to move the cursor close to the intersection point.
and
.
One point of intersection is (–1, –3), so one solution is x = –1.
Repeat these steps to find the coordinates of the second intersection
point. This point is (2, –3), so the second solution is x = 2.
The solutions of x 2 – x – 5 = –3 are –1 and 2.
Check
x 2 - x - 5 = -3
x 2 - x - 5 = -3
(-1) 2 - (-1) - 5 -3
(2)2 - 2 - 5 -3
1 + 1 - 5 -3
4 - 2 - 5 -3
-3 -3 -3 -3 Use a graph to solve each equation.
1. x 2 + 6x + 9 = 1
()
x
2. 1.5(2) x = 6
3. -x 2 + 3x - 4 = -4
()
x
1 =4
2 _
1 =6
4. 2 _
5. 2 x 2 + 2x - 12 = -8
6. _
2
3 3
7. Critical Thinking Explain how you could use a graphing calculator to show that
the equation x 2 - 2x + 4 = 1 has no real solutions.
8. Critical Thinking Could you use the method described in the activity to solve an
equation like x 2 + 4x = x + 4? Explain.
A- 3 Technology Lab
AT9
ADDITIONAL
TOPIC
A-4
Patterns and Recursion
Objective
Identify and extend
patterns using recursion.
In a recursive pattern or recursive sequence, each term is defined using one or
more previous terms. For example, the sequence 1, 4, 7, 10, 13, ... can be defined
recursively as follows: The first term is 1 and each term after the first is equal to
the preceding term plus 3.
Vocabulary
recursive pattern
You can use recursive techniques to identify patterns. The table summarizes the
characteristics of four types of patterns.
Using Recursive Techniques to Identify Patterns
Type of Pattern
EXAMPLE
1
Characteristics
Linear
First differences are constant.
Quadratic
Second differences are constant.
Cubic
Third differences are constant.
Exponential
Ratios between successive terms are constant.
Identifying and Extending a Pattern
Identify the type of pattern. Then find the next three numbers in the pattern.
A 4, 6, 10, 16, 24, ...
Find first, second, and, if necessary, third differences.
4
6
10
16
24
You may need to use
trial and error when
identifying a pattern.
If first, second, and
third differences are
not constant, check
for constant ratios.
+2
+4
+6
+8
+2
+2
+2
Second differences are constant, so the pattern is quadratic.
Extend the pattern by continuing the sequence of first and second
differences.
4
6
10
16
24
34
46
60
+2
+4
+6
+8
+ 10
+ 12
+ 14
+2
+2
+2
The next three numbers in the pattern are 34, 46, and 60.
B
__1 , __1 , 2, 8, 32
8 2
Find the ratio between successive terms.
1
1
__
__
2
8
32
8
2
×4
×4
×4
×4
Ratios between terms are constant, so the pattern is exponential.
Extend the pattern by continuing the sequence of ratios.
1
1
__
__
2
8
32
128
512
2048
8
2
×4
×4
×4
×4
×4
×4
×4
The next three numbers in the pattern are 128, 512, and 2048.
AT10
Additional Topics
Identify the type of pattern. Then find the next three numbers in the
pattern.
1a. 56, 47, 38, 29, 20, ...
1b. 1, 8, 27, 64, 125, ...
You can use a similar process to determine whether a function is linear, quadratic,
cubic, or exponential. Note that before comparing y-values, you must first make sure
there is a constant change in the corresponding x-values.
Using Recursive Techniques to Identify Functions
Type of Function
EXAMPLE
Characteristics
(Given a Constant Change in x-values)
Linear
First differences of y-values are constant.
Quadratic
Second differences of y-values are constant.
Cubic
Third differences of y-values are constant.
Exponential
Ratios between successive y-values are constant.
2
Identifying a Function
The ordered pairs {(–4, –4), (0, 0), (4, 4), (8, 32), (12, 108)} satisfy a function.
Determine whether the function is linear, quadratic, cubic, or exponential.
Then find three additional ordered pairs that satisfy the function.
Make a table. Check for a constant change in the x-values. Then find first,
second, and third differences of y-values.
+4
+4
+4
+4
x
-4
0
4
8
12
y
-4
0
4
32
108
+4
+4
+0
+ 28
+ 24
+ 24
+ 76
+ 48
+ 24
There is a constant change in the x-values. Third differences are constant. The
function is a cubic function.
To find additional ordered pairs, extend the pattern by working backward from
the constant third differences.
+4
+4
+4
+4
+4
+4
+4
In Example 2, the
constant third
differences are 24. To
extend the pattern,
first find each second
difference by adding
24 to the previous
second difference.
Then find each
first difference by
adding the second
difference below
to the previous first
difference.
x
-4
0
4
8
12
16
20
24
y
-4
0
4
32
108
256
500
864
+4
+4
+0
+ 24
+ 24
+ 28
+ 48
+ 24
+ 76
+ 148
+ 72
+ 24
+ 244 + 364
+ 96
+ 24
+ 120
+ 24
Three additional ordered pairs that satisfy this function are (16, 256),
(20, 500), and (24, 864).
A-4 Patterns and Recursion
AT11
Several ordered pairs that satisfy a function are given. Determine
whether the function is linear, quadratic, cubic, or exponential. Then
find three additional ordered pairs that satisfy the function.
2a. {(0, 1), (1, 3), (2, 9), (3, 19), (4, 33)}
⎧ 1
1 , 5, _
1 , 7, _
1 , 9, _
1 ⎫
2b. ⎨ 1, _
, 3, _
⎬
54
6
18
162 ⎭
⎩ 2
( )( )(
A-4
)(
)(
)
Exercises
Identify the type of pattern. Then find the next three numbers in the pattern.
1. 25, 28, 31, 34, 37, ...
2. 20, 45, 80, 125, 180, ...
3. 128, 64, 32, 16, 8, ...
3 , 1, 1_
1, _
1 , 1_
1 , ...
5. _
4 2
2 4
4. 4, 32, 108, 256, 500, ...
6. 0.3, 0.03, 0.003, 0.0003, 0.00003, ...
7. 127, 66, 29, 10, 3, ...
8. 2, 8, 18, 32, 50, ...
Several ordered pairs that satisfy a function are given. Determine whether the
function is linear, quadratic, cubic, or exponential. Then find three additional
ordered pairs that satisfy the function.
9. {(3, 1), (5, –3), (7, –7), (9, –11), (11, –15)}
10. {(–1, –2), (2, 7), (5, 124), (8, 511), (11, 1330)}
⎧ 1
1 , 4, _
1 , 5, _
1 , 6, _
1 ⎫
11. ⎨ 2, _
, 3, _
⎬
32
4
8
16
64
⎭
⎩
12. {(–3, –7), (0, 2), (3, –7), (6, –34), (9, –79)}
( )( )(
)(
)(
)
13. {(0, 600), (10, 480), (20, 384), (30, 307.2), (40, 245.76)}
14. {(–8, 2), (–5, 7), (–2, 12), (1, 17), (4, 22)}
15. Entertainment The table shows the cost of using
an online DVD rental service for different numbers
of months.
Online DVD Rentals
a. Determine whether the function that models
the data is linear, quadratic, cubic, or
exponential. Explain.
b. Graph the data in the table.
c. What do you notice about your graph? Why
does this make sense?
d. Predict the cost of the service for 18 months.
Months
Cost ($)
3
50
6
92
9
134
12
176
15
218
16. A student claimed that the function shown in the table is a quadratic function. Do you
agree or disagree? Explain.
x
3
7
10
14
17
y
2
6
12
20
30
+4
+6
+2
AT12
Additional Topics
+8
+2
+ 10
+2
17. Business The table shows the annual sales for a small
company.
a. Determine whether the function that models
the data is linear, quadratic, cubic, or
exponential. Explain.
b. Suppose sales continue to grow according to
the pattern in the table. Predict the annual sales
for 2011, 2012, and 2013.
c. If the pattern continues, in what year will annual
sales be $17,000 greater than the previous year’s sales?
Annual Sales
Year
Sales ($)
2006
513,000
2007
516,000
2008
521,000
2009
528,000
2010
537,000
18. Critical Thinking Use the table for the following problems.
a.
b.
c.
d.
x
0
1
y
3
6
2
3
4
Copy and complete the table so that the function is a linear function.
Copy and complete the table so that the function is a quadratic function.
Copy and complete the table so that the function is an exponential function.
For which of these three types of functions is there more than one correct way
to complete the table? Explain.
Use the description to write the first five terms in each numerical pattern.
19. The first term is 8. Each following term is 11 less than the term before it.
20. The first term is 1000. Each following term is 40% of the term before it.
21. The first two terms are 1 and 2. Each following term is the sum of the two terms
before it.
Make a table for a function that has the given characteristics. Include at least five
ordered pairs.
22. The function is linear. The first differences are -3.
23. The function is quadratic. The second differences are 6.
24. The function is cubic. The third differences are 1.
A recursive formula for a sequence shows how to find the value of a term from
one or more terms that come before it. For example, the recursive formula
a n = a n-1 + 3 tells you that each term is equal to the preceding term plus 3.
Given that a 1 = 5, you can use the formula to generate the sequence 5, 8, 11, 14, ... .
Write the first four terms of each sequence.
25. a n = a n-1 + 2; a 1 = 12
26. a n = a n-1 - 7; a 1 =16
27. a n = 2a n-1; a 1 = 4
28. a n = 0.6a n-1; a 1 = 100
29. a n = 5a n-1 - 2; a 1 = 0
30. a n = (a n-1)2; a 1 = -2
31. A recursive function defines a function for whole numbers by referring to the value
of the function at previous whole numbers. Consider the recursive function
f (n) = f (n – 1) + 5 with f (0) = 1.
a. According to the formula, f (1) = f (0) + 5. What is the value of f (1)?
b. Use the formula to find f (2), f (3), f (4), and f (5).
c. Graph f (n) by plotting points at x = 0, x = 1, x = 2, x = 3, x = 4, and x = 5.
d. What do you notice about your graph? What does this tell you about f (n)?
A-4 Patterns and Recursion
AT13
ADDITIONAL
TOPIC
A-5
Objectives
Identify linear and
nonlinear rates of
change.
Compare rates of
change.
EXAMPLE
Linear and Nonlinear
Rates of Change
Recall that a rate of change is a ratio that compares the amount of change in a
dependent variable to the amount of change in an independent variable.
change in dependent variable
rate of change = ___
change in independent variable
The table shows the price of one ounce of
gold in 2005 and 2008. The year is the
independent variable and the price is the
dependent variable. The rate of change is
870-513
357
________
= ___
= 119, or $119 per year.
2008-2005
3
1
Price of Gold
Year
Price ($/oz)
2005
513
2008
870
Identifying Constant and Variable Rates of Change
Determine whether each function has a constant or variable rate of change.
A {(0, 0), (1, 4), (3, 8), (6, 8), (8, 6)}
Find the ratio of the amount of change in the dependent variable y to
the corresponding amount of change in the independent variable x.
+1
+2
+3
+2
x
y
0
0
1
4
3
8
6
8
8
6
+4
+4
+0
The rates of change are
4
-2
__
= 4, __4 = 2, __0 = 0, and ___
= –1.
1
2
3
2
The function has a variable rate of
change.
-2
B {(0, 1), (1, 2), (4, 5), (6, 7), (7, 8)}
Find the ratio of the amount of change in the dependent variable y to
the corresponding amount of change in the independent variable x.
x
y
+1
0
1
+1
The rates of change are
+3
1
2
1
__
= 1, __3 = 1, __2 = 1, and __1 = 1.
4
5
+3
6
7
+2
The function has a constant rate
of change.
7
8
+2
+1
1
3
2
1
+1
Determine whether each function has a constant or variable rate
of change.
1a. {(–3, 10), (0, 7), (1, 6), (4, 3), (7, 0)}
1b. {(–2, –3), (2, 5), (3, 7), (5, 9), (8, 12)}
AT14
Additional Topics
The functions in Examples 1A and 1B are graphed below.
Example 1A (variable
rate of change)
Example 1B (constant
rate of change)
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
Recall from Chapter 5 that a function is a linear function if and only if the function
has a constant rate of change. The graph of such a function is a straight line and the
rate of change is the slope of the line, as in Example 1B.
A function with a variable rate of change, as in Example 1A, is a nonlinear function.
Examples of nonlinear functions include quadratic functions and exponential functions.
EXAMPLE
2
Identifying Linear and Nonlinear Functions
Use rates of change to determine whether each function is linear or nonlinear.
A
x
y
+2
-2
0
+1
0
1
1
1.5
4
3
10
6
+3
+6
B
+1
+4
+ 0.5
+4
+ 1.5
-2
+3
+4
x
y
-6
18
-2
- 16
2
2
2
+0
0
0
4
8
-2
+8
Find the rates of change.
Find the rates of change.
0.5 = _
1.5 = _
3 =_
1 _
1 _
1
1 _
_
2
1
2
3
2 6 2
There is a constant rate of
change, __12 , so this function
is linear.
0 =0 _
-16 = -4 _
-2 = 1
_
4
4
-2
The rates of change are not
constant, so this function is
nonlinear.
8 =_
1
_
4 2
Use rates of change to determine whether each function is linear or
nonlinear.
2a.
x
y
-2
1
_
4
-1
1
_
2
0
2b.
x
y
-5
3
-1
3
1
3
1
3
3
3
8
7
3
4
16
When you are given a verbal description of a function, you can determine whether
the function is linear or nonlinear by making a table of values and examining the
rates of change. You can compare two functions by comparing their rates of change.
A-5 Linear and Nonlinear Rates of Change
AT15
EXAMPLE
3
Physical Science Application
Two water tanks contain 512 gallons of water each. Tank A begins to drain,
losing half of its volume of water every hour. Tank B begins to drain at the same
time and loses 40 gallons of water every hour. Identify the function that gives
the volume of water in each tank as linear or nonlinear. Which tank loses water
more quickly between hour 4 and hour 5?
Use the verbal descriptions to make a table for the volume of water in each
tank.
Time (h)
0
1
2
3
4
5
Water in Tank A (gal)
512
256
128
64
32
16
Time (h)
0
1
2
3
4
5
Water in Tank B (gal)
512
472
432
392
352
312
For tank A, the rates of change are –256, –128, –64, –32, and –16, so the rate of
change is variable and the function is nonlinear.
For tank B, the rates of change are all –40, so the rate of change is constant
and the function is linear.
Between hours 4 and 5, the volume of water in tank A decreases at a rate of
16 gallons per hour. The volume of water in tank B decreases at a rate of
40 gallons per hour. Tank B loses water more quickly.
3. Reka and Charlotte each invest $500. Each month, Charlotte’s
investment grows by $25, while Reka’s investment grows by 5% of
the previous month’s amount. Identify the function that gives the
value of each investment as linear or nonlinear. Who is earning
money more quickly between month 3 and month 4?
A-5
Exercises
Use rates of change to determine whether each function is linear or nonlinear.
1.
2.
3.
4.
AT16
Additional Topics
x
4
5
7
10
12
y
–2
–1
1
4
6
x
–2
3
4
6
8
y
–4
6
8
14
20
x
0
3
9
12
18
y
14
12
8
6
2
x
–8
–6
–4
–2
0
y
–3
1
3
5
9
5. Hobbies Caitlin and Greg collect stamps. Each starts with a collection of 50 stamps.
Caitlin adds 15 stamps to her collection each week. Greg adds 1 stamp to his
collection the first week, 3 stamps the second week, 5 stamps the third week, and so
on. Identify the function that gives the number of stamps in each collection as linear
or nonlinear. Which collection is growing more quickly between week 5 and week 6?
Determine whether each function has a constant or variable rate of change.
6.
4
7.
y
4
8.
y
2
2
2
-2
9. y = 2x 2
1x
12. y = _
5
x
15. y = 3 √
0
2
x
x
x
-4
y
4
-4
4
0
-2
2
-4
4
0
-2
-2
-2
-2
-4
-4
-4
10. y + 1 = 3x
11. y = -7
13. y = 5 x
x-3
16. y = _
2x
14. y = x 2+ 1
2
4
17. x + y = 6.25
Determine whether each statement is sometimes, always, or never true.
18. A function whose graph is a straight line has a variable rate of change.
19. A quadratic function has a constant rate of change.
20. The rate of change of a linear function is negative.
21. The rate of change between two points on the graph of a nonlinear function is 0.
22. Critical Thinking The figure shows the graph of the
x
exponential function y = __12 .
()
8
A
6
a. Find the rates of change between points A and B,
between points B and C, and between points
A and D.
b. What do you notice about the rates of change you
found in part a? Do you think this would be true
for the rate of change between any two points on
the graph?
y
4
B
C
-4
-2
2
x
D
0
2
4
c. How do your findings about the rates of change relate
to the shape of the graph?
Model Rocket
a. Find the rates of change between points A and B and
between points B and C.
b. Which rate of change is greater? What does this tell
you about the motion of the rocket?
c. Find the rates of change between points C and D and
between points D and E.
d. What does the sign of the rates of change you found
in part c tell you about the motion of the rocket? Explain.
Height (ft)
23. A model rocket is launched from the ground. The graph
shows the height of the rocket at various times.
90
80
70
60
50
40
30
20
10
0
C(2, 64)
D(3, 48)
B(1, 48)
A(0, 0)
E(4, 0)
1 2 3 4 5
Time (s)
A-5 Linear and Nonlinear Rates of Change
AT17
ADDITIONAL
TOPIC
A-6
Objectives
Use inductive reasoning
to make conjectures.
Use deductive reasoning
to prove conjectures, and
find counterexamples to
disprove conjectures.
EXAMPLE
Vocabulary
inductive reasoning
conjecture
counterexample
deductive reasoning
Reasoning and
Counterexamples
Inductive reasoning is the process of
concluding that a general rule or statement
is true because specific cases are true. A
statement based on inductive reasoning is
called a conjecture.
Row 1
Row 2
Row 3
When several numbers or geometric figures
form a pattern and you assume the pattern will continue, you are
using inductive reasoning. For example, you might use inductive
reasoning to make the conjecture that the fourth row of the pattern
at right will contain seven stars.
1
Using Inductive Reasoning
Use inductive reasoning to make a conjecture about the value of the 100th
term in the sequence 5, 9, 13, 17, 21, ... .
Make a table. Examine the values and look for a pattern.
Term
1st
2nd
3rd
4th
5th
Value
5
9
13
17
21
4(1) + 1
4(2) + 1
4(3) + 1
4(4) + 1
4(5) + 1
Pattern
Each value in the sequence is 4 times its position in the sequence, plus 1.
The rule 4n + 1 can be used to find the nth term. A reasonable conjecture
for the value of the 100th term is 4(100) + 1 = 401.
1. Use inductive reasoning to make a conjecture about the value
of the 45th term in the sequence 1, 4, 9, 16, 25, ... .
A counterexample is a case that proves that a conjecture or statement is false.
One counterexample is enough to disprove a statement.
Counterexamples
Statement
Counterexample
No month has fewer than 30
days.
February has fewer than 30 days, so
the statement is false.
All prime numbers are odd.
The number 2 is a prime number,
but it is not odd. This example shows
that the statement is false.
Every integer that is divisible
by 2 is also divisible by 4.
The integer 18 is divisible by 2, but
it is not divisible by 4. This example
shows that the statement is false.
Recall that the Commutative and Associative Properties are true for addition and
multiplication. The following example shows how you can use a counterexample
to demonstrate that those properties are not true for other operations.
AT18
Additional Topics
EXAMPLE
2
Finding a Counterexample
Find a counterexample to disprove the statement “The Associative Property is
true for subtraction.”
Find three real numbers a, b, and c such that a - (b - c) ≠ (a - b) - c.
Try a = 10, b = 7, and c = 2.
a - (b - c)
(a - b) - c
10 - (7 - 2)
(10 - 7) - 2
10 - 5 = 5
3-2=1
Since 10 - (7 - 2) ≠ (10 - 7) - 2, this is a counterexample that shows that the
statement is false.
2. Find a counterexample to disprove the statement “The
Commutative Property is true for division.”
Inductive reasoning can be used to make conjectures, but it cannot be used to prove
a statement. To prove a statement, you must use deductive reasoning. Deductive
reasoning is the process of drawing conclusions from given facts, definitions, and
properties.
You may not realize it, but you use deductive reasoning every time you solve an
equation. In fact, solving an equation can be thought of as a proof.
EXAMPLE
3
Using Deductive Reasoning
Use deductive reasoning to prove each statement.
A If 3x + 4 = 19, then x = 5.
Notice that each step
of a proof is justified
with a definition,
a property, an
operation, or a piece
of given information.
Statements
Reasons
1. 3x + 4 = 19
Given
2. 3x + 4 - 4 = 19 - 4
Subtraction Property of Equality
3. 3x = 15
Subtraction
3x
15
4. __
= __
Division Property of Equality
5. x = 5
Division
3
3
B If 5(x – 3) = –20, then x = –1.
Statements
Reasons
1. 5(x - 3) = -20
Given
2. 5x - 5(3) = -20
Distributive Property
3. 5x - 15 = -20
Multiplication
4. 5x - 15 + 15 = -20 + 15
Addition Property of Equality
5. 5x = -5
Addition
5x
-5
6. __
= ___
5
5
Division Property of Equality
7. x = -1
Division
Use deductive reasoning to prove each statement.
3a. If __2x -1 = 7, then x = 16.
3b. If 4(x + 2) = 8, then x = 0.
A-6 Reasoning and Counterexamples
AT19
A-6
Exercises
Use inductive reasoning to make a conjecture about the value of the 40th term in
each sequence.
1. 7, 8, 9, 10, 11, ...
2. 4, 7, 10, 13, 16, ...
3. 5, 10, 15, 20, 25, 30, ...
4. 0, 3, 6, 9, 12, ...
5. 3, 5, 3, 5, 3, ...
6. 1, 2, 3, 1, 2, 3, 1, ...
Use inductive reasoning to make a conjecture about the next item in each pattern.
3, _
15 , ...
7, _
1, _
7. d, e, f, d, e, f, ...
8. _
9. 0.1, 0.02, 0.003, ...
2 4 8 16
, ,
, ...
10. 66, 57, 48, 39, ...
11. −1, 5, −3, 5, −5, 5, ...
12.
Find a counterexample to disprove each statement.
13. The Associative Property is true for division.
14. The Commutative Property is true for subtraction.
Use deductive reasoning to fill in each missing term.
15. The Red-Q company makes only red clothes. This shirt is made by Red-Q. Therefore,
this shirt is
?
.
16. Lance goes to the library every
went to the library today.
?
. Today is Wednesday. Therefore, Lance
Tell whether each statement is true or false. If the statement is false, give a
counterexample to disprove the statement.
17. If n is an even number, then 3n is also an even number.
18. The sum of two odd numbers is also an odd number.
19. If a number is divisible by 6, then it is divisible by 12.
20. The product of two odd numbers is an odd number.
21. Every number that is a multiple of 4 is a multiple of 2.
22. If m is a multiple of 3 and n is a multiple of 4, then m + n is a multiple of 7.
23. Fill in the missing statement or reason to complete the proof of the following
statement: If –2(x + 6) = –18, then x = 3.
Statements
Reasons
1. -2(x + 6) = -18
Given
2. (-2 )x + (-2)6 = -18
Distributive Property
?
3. a.
Multiplication
4. -2x - 12 + 12 = -18 + 12
b.
5. -2x = -6
Addition
-2x
-6
6. ____
= ___
c.
7. x = 3
Division
-2
-2
?
?
Use deductive reasoning to prove each statement.
AT20
24. If x – 16 = 41, then x = 57.
25. If 3x – 17 = 13, then x = 10.
26. If 7(x – 2) = 21, then x = 5.
27. If 3(x + 1) = –9, then x = –4.
Additional Topics
28. Geometry A student drew the set of figures shown below. Then the student made
the following conjecture: Given three points in a plane, it is possible to draw exactly
three distinct lines through the points. Do you agree or disagree? Explain.
29. Geometry An exterior angle of a triangle is
Remote
1
the angle formed by extending one side of the
Exterior angle
triangle. An exterior angle of a triangle has two interior
angles
remote interior angles. In the figure, ∠4 is an
2
3 4
exterior angle. Its remote interior angles are
∠1 and ∠2 .
a. Draw a triangle and an exterior angle. Use a protractor to measure the exterior
angle and its remote interior angles. What do you notice about the sum of the
measures of the remote interior angles?
b. Repeat the process in part a several more times. Make a conjecture.
c. What type of reasoning did you use to make your conjecture? Explain.
Determine whether inductive or deductive reasoning was used in each situation.
Explain.
30. Christine visited San Diego four times. Each time it was raining. Christine concludes,
“San Diego is a very rainy city.”
31. Stephanie knows that her cousin’s new pet is a lizard. She also knows that every lizard
is a reptile and that every reptile has scales. Stephanie concludes that her cousin’s new
pet has scales.
32. According to Aaron’s textbook, the sum of the angle measures in any triangle is 180°.
Aaron finds that two angles of a triangle measure 50° and 60°. Aaron concludes that
the third angle must measure 70°.
33. Looking at the sequence 3, 9, 27, 81, ..., David concludes that the next number must
be 243 because each term is 3 times the previous term.
A conditional statement is a statement that can be written in the form “If p, then q.”
The contrapositive of a conditional statement is the statement “If not q, then not p.”
The contrapositive of a true conditional statement is also true. Use the contrapositive
to make a true conjecture based on each conditional statement.
34. If today is Tuesday, then tomorrow is Wednesday.
35. If Sam lives in Detroit, then Sam lives in Michigan.
36. If n is an even number, then n + 1 is an odd number.
37. If 8x + 3 = 19, then x = 2.
38. Logic The mockingbird, the cardinal, the goldfinch, and the grouse are the state birds
of Illinois, New Jersey, Pennsylvania, and Tennessee, but not necessarily in that order.
Use deductive reasoning and the following clues to match each bird with its state.
• The cardinal is not the state bird of Tennessee.
• The state bird of Pennsylvania begins with the same letter as the state bird of New
Jersey.
• The state whose capital is Trenton has the goldfinch as its state bird.
A-6 Reasoning and Counterexamples
AT21
Student
Handbook
Extra Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4
Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4
Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S6
Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S8
Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S10
Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S12
Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S14
Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S16
Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S18
Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S20
Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S22
Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S24
Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S26
Application Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S28
Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S28
Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S29
Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S30
Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S31
Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S32
Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S33
Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S34
Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S35
Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S36
Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S37
Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S38
Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S39
S2
Student Handbook
Problem-Solving Handbook . . . . . . . . . . . . . . . . . . . . . . . PS2
Draw a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS2
Make a Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS3
Guess and Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS4
Work Backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS5
Find a Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS6
Make a Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS7
Solve a Simpler Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS8
Use Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS9
Use a Venn Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS10
Make an Organized List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS11
Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SA2
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G1
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IN1
Symbols and Formulas
. . . . . . . . . . . . . . inside back cover
S3
Extra Practice
Chapter 1
Lesson
1-1
Skills Practice
Give two ways to write each algebraic expression in words.
1. x + 8
2. 6(y)
3. g - 4
12
4. _
h
Evaluate each expression for a = 4, b = 2, and c = 5.
a
5. b + c
6. _
7. c - a
8. ab
b
Write an algebraic expression for each verbal expression. Then evaluate the
algebraic expression for the given values of y.
Verbal
9.
Lesson
1-2
Lesson
1-3
Algebraic
y=9
y=6
y reduced by 4
10.
the quotient of y and 3
11.
5 more than y
12.
the sum of y and 2
Add or subtract using a number line.
13. -7 - 9
14. -2.2 + 4.3
1 - 2_
1
15. -5 _
2
2
Subtract.
17. 12 - 47
18. 1.3 - 9.2
Compare. Write <, >, or =.
20. -5 - (-8)
-4 - 9
21. ⎪-6 - (-2)⎥
1 - 4_
2
19. 1_
3
3
7-4
Evaluate the expression g - (-7) for each value of g.
2
23. g = 121
24. g = 1.25
25. g = - _
5
Find the value of each expression.
27. -24 ÷ (-8)
28. 5(-9)
( )
6
2 ÷ -_
30. _
7
7
16. 3.4 - 6.5
( )
22. -2 - 5
7 - 14
1
26. g = -8 _
3
29. -5.2 ÷ (-1.3)
9 ÷0
32. _
10
4
31. 0 ÷ - _
5
Evaluate each expression for x = -8, y = 6, and z = -4.
y
33. xy
34. yz
35. _
z
z
36. _
x
Let a represent a positive number, b represent a negative number, and z represent
zero. Tell whether each expression is positive, negative, zero, or undefined.
a
ab
37. ab
38. -bz
39. - _
40. _
z
b
Lesson
1-4
Write each expression as repeated multiplication. Then simplify the expression.
41. 3 3
42. -2 4
43. (-5)3
44. (-1) 5
Write each expression using a base and an exponent.
45. 5 · 5 · 5 · 5 · 5
46. 4 · 4 · 4
47. 2 · 2 · 2 · 2
Write the exponent that makes each equation true.
48. 2 ■ = 16
S4
Extra Practice
49. 4 ■ = 256
50. (-3)■ = 81
51. -5 ■ = -125
Chapter 1
Lesson
1-5
Skills Practice
Find each root.
52. - √
64
53. √
144
Compare. Write <, >, or =.
55. √
118
11
56. 6
√
35
54.
57. 14
8
_
√
125
3
√
196
58. √
50
Write all classifications that apply to each real number.
59. -44
60. √
49
61. 15.982
Lesson
1-6
1
62. _
9
Evaluate each expression for the given value of the variable.
63. 22 - 3g + 5 for g = 4
64. 12 - 30 ÷ h for h = 6
65.
Simplify each expression.
66. 4 + 12 ÷ ⎪3 - 9⎥
67. -36 - √
4 + 15 ÷ 3
Translate each word phrase into an algebraic expression.
69. the quotient of 8 and the difference of a and 5
7
(11j + j) + 6 for j = 3
√
5 - √
12(3)
68. __
-4 + √
2(8)
70. the sum of -9 and the square root of the product of 7 and c
Lesson
1-7
Simplify each expression.
71. -5 + 38 + 5 + 62
1 - 42 + 7 _
2
1 · 4 · 25
72. 2 _
73. _
5
3
3
Write each product using the Distributive Property. Then simplify.
74. 12(108)
75. 7(89)
76. 11(33)
Simplify each expression by combining like terms.
77. 7a - 3a
78. -2b - 12b
79. 4c + 5c 2 - c
Simplify each expression. Justify each step with an operation or property.
80. 6(p - 2) + 3p
81. 8q - 3 + 5q(2 + q)
82. -4 + 3r - 7(2s - r)
Lesson
1-8
Graph each point.
83. A(2, 3)
84. B(-4, 1)
85. C (0, -2)
86. D (-4, -1)
Name the quadrant in which each point lies.
87. J
88. K
89. L
90. M
91. N
92. P
*
Ó
{
Þ
{ Ý
Ó
ä
Ó
Ó
{
{
Generate ordered pairs for each function for x = -2, -1, 0, 1, and 2.
Graph the ordered pairs and describe the pattern.
93. y = x - 3
94. y = -2x
95. y = -x 2
96. y = ⎪3x⎥
Write an equation for each rule. Use the given values for x to generate ordered
pairs. Graph the ordered pairs and describe the pattern.
97. y is equal to the sum of one-third of x and -2; x = -6, -3, 0, 3, and 6.
98. y is equal to 4 less than x squared; x = -2, -1, 0, 1, and 2.
Extra Practice
S5
Chapter 2
Lesson
2-1
Skills Practice
Solve each equation. Check your answer.
1. x - 9 = 5
2. 4 = y - 12
3 =7
3. a + _
5
4. 7.3 = b + 3.4
5. -6 + j = 5
6. -1.7 = -6.1 + k
Write an equation to represent each relationship. Then solve the equation.
7. A number decreased by 7 is equal to 10. 8. The sum of 6 and a number is -3.
Lesson
2-2
Solve each equation. Check your answer.
n = 15
9. _
5
k
10. -6 = _
4
r =5
11. _
2.6
12. 3b = 27
13. 56 = -7d
14. -3.6 = -2f
1z=3
15. _
4
4g
16. 12 = _
5
1 a = -5
17. _
3
Write an equation to represent each relationship. Then solve the equation.
18. A number multiplied by 4 is -20.
19. The quotient of a number and 5 is 7.
Lesson
2-3
Solve each equation. Check your answer.
20. 2k + 7 = 15
21. 11 - 5m = -4
2 b + 6 = 10
23. _
5
f
24. _ - 4 = 2
3
22. 23 = 9 - 2d
25. 6n + 4 = 22
Write an equation to represent each relationship. Solve each equation.
26. The difference of 11 and 4 times a number equals 3.
27. Thirteen less than 5 times a number is equal to 7.
Lesson
2-4
Solve each equation. Check your answer.
28. 5b - 3 = 4b + 1
29. 3g + 7 = 11g - 17
30. -8 + 4y = y - 6 + 3y - 2
31. 7 + 3d - 5 = -1 + 2d - 12 + d
Write an equation to represent each relationship. Then solve the equation.
32. Three more than one-half a number is the same as 17 minus three times the number.
33. Two times the difference of a number and 4 is the same as 5 less than the number.
Lesson
2-5
Lesson
2-6
Solve each equation for the indicated variable.
5 - c = d - 7 for c
34. q - 3r = 2 for r
35. _
6
y
_
36. 2x + 3 = 5 for y
37. 2fgh - 3g = 10 for h
4
Solve each equation. Check your answer.
38. ⎪a⎥ = 13
39. ⎪x⎥ - 16 = 3
f
41. ⎪7s ⎪ - 6 = 8
42. _ + 1 = 15
2
⎪
44. 500 = 25 ⎪z ⎪+ 200
S6
Extra Practice
⎥
45. ⎪7j + 14⎥ - 5 = 16
40. ⎪g + 5⎥ = 11
43. ⎪p - 5⎥ - 12 = -9
⎪p - 2⎥ - 15
46. __ = -1
5
Chapter 2
Lesson
2-7
Skills Practice
47. A long-distance runner ran 9000 meters in 30 minutes. Find the unit rate in meters
per minute.
48. A hummingbird flapped its wings 770 times in 14 seconds. Find the unit rate in flaps
per second.
49. A car traveled 210 miles in 3 hours. Find the unit rate in miles per hour.
50. A printer printed 60 pages in 5 minutes. Find the unit rate in pages per minute.
Lesson
2-8
Solve each proportion.
h =_
5
51. _
4
6
5
2
_
52. _
m=5
r =_
10
53. _
7
3
x
2=2
_
54. _
3
8
5 =_
3
55. _
x - 3 10
b-2 =_
7
56. _
4
12
Find the value of x in each diagram.
57. ABCD ∼ EFGH
58. JKL ∼ MNO
ÝÊvÌ
nʓ
£äÊvÌ
£äÊvÌ
{ÊvÌ
Èʓ
Ýʓ
£{ʓ
"
Lesson
59. Find 25% of 60.
60. Find 40% of 95.
2-9
61. What percent of 75 is 15?
62. What percent of 60 is 33?
63. 91 is what percent of 65?
64. 35% of what number is 24.5?
Write each decimal or fraction as a percent.
9
4
65. _
66. 0.55
67. _
5
6
Write each percent as a decimal and as a fraction.
69. 32%
70. 24%
71. 37.5%
68. 0.0375
72. 118.75%
Write each list in order from least to greatest.
6 , 0.19
5 , 9.2, 117%, _
9 , 8.8%
2 , 0.28, 1.9%, _
74. _
73. _
17
25
11
3
Lesson
75. Estimate the tax on a $2438.00 clarinet when the sales tax is 7.9%.
2-10
76. Estimate the tip on a $21.32 check using a tip rate of 20%.
Lesson
Find each percent change. Tell whether it is a percent increase or decrease.
77. 10 to 25
78. 40 to 2
79. 800 to 160
2-11
80. 4 to 14
81. 8 to 30
82. 60 to 36
83. Find the result when 45 is increased by 40%.
84. Find the result when 120 is decreased by 70%.
Extra Practice
S7
Chapter 3
Lesson
3-1
Skills Practice
Describe the solutions of each inequality in words.
1. 3 + v < -2
2. 15 ≤ k + 4
3. -3 + n > 6
4. 1 - 4x ≥ -2
Graph each inequality.
5. f ≥ 2
6. m < -1
8. (-1 - 1)2 ≤ p
2
7. √4
+ 32 > c
Write the inequality shown by each graph.
9.
10.
ä
£
Ó
Î
{
x
È
ä
Ó
{
È
n £ä £Ó
Î Ó £
ä
£
11.
12.
13.
Î Ó £
ä
£
Ó
Î
-6 -4 -2
0
2
4
6
Î
{
x
È
14.
Ó
Î
ä
£
Ó
Write each inequality with the variable on the left. Graph the solutions.
15. 14 > b
16. 9 ≤ g
17. -2 < x
18. -4 ≥ k
Lesson
3-2
Solve each inequality and graph the solutions.
19. 8 ≥ d - 4
20. -5 < 10 + w
21. a + 4 ≤ 7
22. 9 + j > 2
Write an inequality to represent each statement. Solve the inequality and graph
the solutions.
23. Five more than a number v is less than or equal to 9.
24. A number t decreased by 2 is at least 7.
25. Three less than a number r is less than -1.
26. A number k increased by 1 is at most -2.
Use the inequality 4 + z ≤ 11 to fill in the missing numbers.
27. z ≤
28. z ≤4
29. z - 3 ≤
Lesson
Solve each inequality and graph the solutions.
3-3
30. 24 > 4b
31. 27g ≤ 81
34. 4p < -2
3s > 3
35. _
8
-2e ≥ 4
39. _
5
38. -3k ≤ -12
h
42. 9 > _
-2
43. 49 > -7m
x <3
32. _
5
3d
36. 0 ≥ _
7
33. 10y ≥ 2
40. 8 < -12y
41. -3.5 > 14c
44. 60 ≤ -12c
1 q < -6
45. - _
3
a ≥_
3
37. _
4
8
Write an inequality for each statement. Solve the inequality and graph the
solutions.
1 and a number is not more than 6.
46. The product of _
2
47. The quotient of r and -5 is greater than 3.
48. The product of -11 and a number is greater than -33.
49. The quotient of w and -4 is less than or equal to -6.
S8
Extra Practice
Chapter 3
Lesson
3-4
Skills Practice
Solve each inequality and graph the solutions.
50. 3t - 2 < 5
51. -6 < 5b - 4
2f + 3
52. 4 < _
2
3
2
4
1
_
_
_
_
53. 10 ≤ 3(4 - r)
54. + h <
55. (10k - 2) > 1
5
3 4
3
3 8q - 2 2 < -3
3
2
2
56. -n - 3 < -2
57. 37 - 4d ≤ √
3 +4
58. - _
)
(
4
Use the inequality -6 - 2w ≥ 10 to fill in the missing numbers.
59. w ≤
60. w - 3 ≤
61.
+w≤1
Write an inequality for each statement. Solve the inequality and graph the solutions.
62. Twelve is less than or equal to the product of 6 and the difference of 5 and a number.
63. The difference of one-third a number and 8 is more than -4.
64. One-fourth of the sum of 2x and 4 is more than 5.
Lesson
Solve each inequality and graph the solutions.
3-5
65. 4v - 2 ≤ 3v
66. 2(7 - s) > 4(s + 2)
5 ≥_
1u-_
1u
67. _
3
2 6
69. 4(k + 2) ≥ 4k + 5
70. 2(5 - b) ≤ 3 - 2b
Solve each inequality.
68. 3 + 3c < 6 + 3c
Write an inequality to represent each relationship. Solve your inequality.
71. The difference of three times a number and 5 is more than the number times 4.
72. One less than a number is greater than the product of 3 and the difference of 5 and
the number.
Lesson
3-6
Solve each compound inequality and graph the solutions.
73. 6 < 3 + x < 8
74. -1 ≤ b + 4 ≤ 3
75. k + 5 ≤ -3 OR k + 5 ≥ 1
76. r - 3 > 2 OR r + 1 < 4
Write the compound inequality shown by each graph.
77.
78.
Î Ó £
ä
£
Ó
È { Ó
Î
ä
Ó
{
È
Write and graph a compound inequality for the numbers described.
79. all real numbers less than 2 and greater than or equal to -1
80. all real numbers between -3 and 1
Lesson
Solve each inequality and graph the solutions.
3-7
81. ⎪n + 5⎥ ≤ 26
82. ⎪x⎥ + 6 < 13
83. 4⎪k⎥ ≤ 12
84. ⎪c - 8⎥ > 18
85. 6⎪p⎥ ≥ 48
86. ⎪3 + t⎥ - 1 ≥ 5
88. 2⎪w⎥ + 5 < 3
89. ⎪s⎥ + 12 > 8
Solve each inequality.
87. ⎪a⎥ -2 ≤ -5
Write and solve an absolute-value inequality for each expression. Graph the
solutions on a number line.
90. All numbers whose absolute value is greater than 14.
91. All numbers whose absolute value multiplied by 3 is less than 27.
Extra Practice
S9
Chapter 4
Lesson
4-1
Skills Practice
Choose the graph that best represents each situation.
1. A person blows up a balloon with a steady airstream.
2. A person blows up a balloon and then lets it deflate.
3. A person blows up a balloon slowly at first and then uses more and more air.
À>«…Ê
6œÕ“i
/ˆ“i
Lesson
4-2
À>«…Ê
6œÕ“i
6œÕ“i
À>«…Ê
/ˆ“i
/ˆ“i
Express each relation as a table, as a graph, and as a mapping diagram.
⎧
⎫
⎫
⎧
4. ⎨(0, 2), (-1, 3), (-2, 5)⎬
5. ⎨(2, 8), (4, 6), (6, 4), (8, 2)⎬
⎩
⎭
⎩
⎭
Give the domain and range of each relation. Tell whether the relation is a function.
Explain.
⎫
⎧
⎫
⎧
7. ⎨(5, 4), (0, 2), (5, -3), (0, 1)⎬
6. ⎨(3, 4), (-1, 2), (2, -3), (5, 0)⎬
⎭
⎩
⎭
⎩
y
8.
9.
x
2
0
1
2
-1
y
1
0
-1
-2
-3
8
6
4
2
0
Lesson
4-3
2
4
6
8
x
Determine a relationship between the x- and y-variables. Write an equation.
⎧
⎫
10. ⎨(1, 3), (2, 6), (3, 9), (4, 12)⎬
11.
x
1
2
3
4
⎩
⎭
y
1
4
9
16
Identify the independent and dependent variables. Write an equation in function
notation for each situation.
12. A science tutor charges students $15 per hour.
13. A circus charges a $10 entry fee and $1.50 for each pony ride.
14. For f (a) = 6 - 4a, find f (a) when a = 2 and when a = -3.
2 d + 3, find g (d) when d = 10 and when d = -5.
15. For g (d) = _
5
16. For h (w) = 2 - w 2, find h (w) when w = -1 and when w = -2.
17. Complete the table for f (t ) = 7 + 3t.
t
f(t)
S10
Extra Practice
0
1
2
3
18. Complete the table for h(s) = 2s + s 3 - 6.
s
h(s)
-1
0
1
2
Chapter 4
Lesson
4-4
Skills Practice
Graph each function for the given domain.
⎧
⎫
⎧
⎫
19. 2x - y = 2; D: ⎨-2, -1, 0, 1⎬
20. f(x) = x 2 - 1; D: ⎨-3, -1, 0, 2⎬
⎩
⎭
⎩
⎭
Graph each function.
21. f(x) = 4 - 2x
22. y + 3 = 2x
23. y = -5 + x 2
5 - 2x to find the value of y when x = _
1.
24. Use a graph of the function y = _
2
2
Check your answer.
25. Find the value of x so that (x, 4) satisfies y = -x + 8.
26. Find the value of y so that (-3, y) satisfies y = 15 - 2x 2.
Lesson
For each function, determine whether the given points are on the graph.
x + 4; -3, 3 and 3, 5
27. y = _
28. y = x 2 - 1; (-2, 3) and (2, 5)
)
(
( )
3
Describe the correlation illustrated by each scatter plot.
4-5
29.
30.
Þ
31.
Þ
Ý
Þ
Ý
Ý
Identify the correlation you would expect to see between each pair of data sets. Explain.
32. the number of chess pieces captured and the number of pieces still on the board
33. a person’s height and the color of the person’s eyes
Choose the scatter plot that best represents the described relationship. Explain.
À>«…Ê
34. the number of students in a class and the
À>«…Ê
Þ
Þ
grades on a test
35. the number of students in a class and the
number of empty desks
Ý
Ý
Lesson
4-6
Determine whether each sequence appears to be an arithmetic sequence. If so, find
the common difference and the next three terms.
36. -10, -7, -4, -1, …
37. 8, 5, 1, -4, …
38. 1, -2, 3, -4, …
39. -19, -9, 1, 11, …
Find the indicated term of each arithmetic sequence.
40. 15th term: -5, -1, 3, 7, …
41. 20th term: a 1 = 2; d = -5
42. 12th term: 8, 16, 24, 32, …
43. 21st term: 5.2, 5.17, 5.14, 5.11, …
Find the common difference for each arithmetic sequence.
7, _
10 , …
1 , 1, _
44. 0, 7, 14, 21, …
45. 132, 121, 110, 99, …
46. _
4
4 4
47. 1.4, 2.2, 3, 3.8, …
48. -7, -2, 3, 8, …
49. 7.28, 7.21, 7.14, 7.07, …
Find the next four terms in each arithmetic sequence.
50. -3, -6, -9, -12, …
51. 2, 9, 16, 23, …
5, …
1, _
1 , 1, _
52. - _
53. -4.3, -3.2, -2.1, -1, …
3 3
3
Extra Practice
S11
Chapter 5
Lesson
5-1
Skills Practice
Identify whether each graph represents a function. Explain. If the graph does
represent a function, is the function linear?
1.
{
2.
Þ
3.
Þ
{
Þ
È
Ó
Ó
Ý
È
{
ä
Ó
{
Ý
{
Ó
Ó
ä
Ó
{
Ó
Ó
{
Ý
{
{
Ó
ä
Tell whether the given ordered pairs satisfy a linear function. Explain.
4.
5.
x
2
5
8
x
-4
-2
0
2
4
y
7
6
5
4
y
3
12
8
7
Ó
{
11
14
3
3
Lesson
Tell whether each equation is linear. If so, write the equation in standard form and
give the values of A, B, and C.
x = 4 - 2y
6. y = 8 - 3x
7. _
8. -3 + xy = 2
9. 4x = -3 - 3y
3
Find the x- and y-intercepts.
5-2
10. -4x = 2y - 1
11. x - y = 3
12. 2x - 3y = 12
13. 2.5x + 2.5y = 5
Use intercepts to graph the line described by each equation.
14. 15 = -3x - 5y
15. 4y = 2x + 8
16. y = 6 - 3x
Lesson
Find the slope of each line.
5-3
18.
{
19.
Þ
Þ
n
Ó
17. -2y = x + 2
{
Ý
{
Lesson
5-4
Ó
ä
Ó
{
Ý
n
{
Ó
{
{
n
5-5
{
n
Find the slope of the line that contains each pair of points.
20. (-1, 2) and (-4, 8)
21. (2, 6) and (0, 1)
22. (-2, 3) and (4, 0)
Find the slope of the line described by each equation.
23. 2y = 42 - 6x
24. 3x + 4y = 12
Lesson
ä
25. 3x = 15 + 5y
Find the coordinates of the midpoint of each segment.
−−
26. AB with endpoints A(-3, 4) and B(1, 5)
−−
27. CD with endpoints C(9, -8) and D(9, -2)
Find the distance, to the nearest hundredth, between each pair of points.
28. A(0, 6) and B(4, 8)
29. J(-1, 7) and K(-4, 5)
30. S(-3, -9) and T(2, 3)
S12
Extra Practice
Chapter 5
Lesson
5-6
Skills Practice
Tell whether each equation represents a direct variation. If so, identify the constant
of variation.
31. x - 2y = 0
32. x - y = 3
33. 3y = 2x
34. The value of y varies directly with x, and y = 2 when x = -3. Find y when x = 6.
35. The value of y varies directly with x, and y = -3 when x = 9. Find y when x = 12.
Lesson
Write the equation that describes each line in slope-intercept form.
5-7
36. slope = 2, y-intercept = -2
38. slope = -2, (5, 4) is on the line.
40.
y
(-3, 2)
2
-2
0
(3, 0)
x
-2
5-8
Lesson
5-9
0
-2
2
-2
Lesson
37. slope = 0.25, y-intercept = 4
1 , (-8, 0) is on the line.
39. slope = _
3
41.
y
x
(2, -1)
(-2, -5)
Write each equation in slope-intercept form. Then graph the line described by the
equation.
1 x=2
42. 2y = x - 3
43. -3x - 2y = 1
44. 2y - _
2
Write an equation in point-slope form for the line with the given slope that
contains the given point.
1 ; (2, 4)
45. slope = 2; (0, 3)
46. slope = -1; (1, -1)
47. slope = _
2
Write the equation that describes each line in slope-intercept form.
48. slope = 3, (-2, -5) is on the line.
49. (-1, 1) and (1, -2) are on the line.
50. (3, 1) and (2, -3) are on the line.
51. x-intercept = 4, y-intercept = -5
Write an equation in slope-intercept form for the line that is parallel to the given
line and that passes through the given point.
52. y = -2x + 3; (1, 4)
53. y = x - 5; (2, -4)
54. y = 3x; (-1, 5)
Write an equation in slope-intercept form for the line that is perpendicular to the
given line and that passes through the given point.
55. y = x + 1; (3, -2)
Lesson
5-10
56. y = -4x - 1; (-1, 0)
57. y = 4x + 5; (2, -1)
Graph f (x) and g (x). Then describe the transformation(s) from the graph of f (x) to
the graph of g (x).
1
58. f (x) = x, g(x) = x + 2
59. f (x) = x, g (x) = x - _
2
60. f (x) = 6x + 1, g(x) = 2 x + 1
61. f (x) = 3x - 1, g (x) = 9x - 1
1x
62. f (x) = x, g(x) = 2x - 1
63. f (x) = x + 1, g (x) = - _
2
Extra Practice
S13
Chapter 6
Lesson
6-1
Skills Practice
Tell whether the ordered pair is a solution of the given system.
⎧ 2x - 3y = -7
⎧4x + 3y = -2
⎧ -2x - 3y = 1
1. (1, 3); ⎨
2. (-2, 2); ⎨
3. (4, -3); ⎨
⎩ -5x + 3y = 4
⎩ -2x - 2y = 2
⎩ x + 2y = -2
Use the given graph to find the solution of each system.
⎧
1
_
y = 2 x - 1
⎧y = x + 1
4. ⎨
5. ⎨
1x+3
⎩ y = -x + 1
y = - _
2
⎩
{
Þ
Þ
{
Ó
Ó
Ý
{
Ó
ä
Ó
{Ý
{
Ó
ä
Ó
Ó
{
{
Ó
{
Solve each system by graphing. Check your answer.
⎧y = x + 1
6. ⎨
⎩ y = -2x - 2
Lesson
6-2
⎧3x + y = -8
7. ⎨
1
⎩ 3y = _ x - 5
2
Solve each system by substitution.
⎧y = 12 - 3x
⎧2x + y = -6
9. ⎨
10. ⎨
⎩ y = 2x - 3
⎩ -5x + y = 1
⎧2x + 3y = 2
12. ⎨
1
⎩ - _ x + 2y = -6
2
⎧3x - 2y = -3
13. ⎨
⎩ y = 7 - 4x
⎧x = 2 - 2y
8. ⎨
⎩ -1 = -2x - 3y
⎧y = 11 - 3x
11. ⎨
⎩ -2x + y = 1
⎧4y - 2x = -2
14. ⎨
⎩ x + 3y = -4
Two angles whose measures have a sum of 90° are called complementary angles.
For Exercises 15–17, x and y represent the measures of complementary angles. Use
this information and the equation given in each exercise to find the measure of
each angle.
15. y = 9x - 10
16. y - 4x = 15
17. y = 2x + 15
Lesson
6-3
S14
Solve each system by elimination.
⎧x - 3y = -1
⎧-3x - y = 1
18. ⎨
19. ⎨
⎩ -x + 2y = -2
⎩ 5x + y = -5
⎧-x - 3y = -1
20. ⎨
⎩ 3x + 3y = 9
⎧3x - 2y = 2
21. ⎨
⎩ 3x + y = 8
⎧5x - 2y = -15
22. ⎨
⎩ 2x - 2y = -12
⎧-4x - 2y = -4
23. ⎨
⎩ -4x + 3y = -24
⎧-3x - 3y = 3
24. ⎨
⎩ 2x + y = -4
⎧4x - 3y = -1
25. ⎨
⎩ 2x - 2y = -4
⎧3x + 6y = 0
26. ⎨
⎩ 7x + 4y = 20
Extra Practice
Chapter 6
Lesson
6-4
Skills Practice
Solve each system of linear equations.
⎧y = 2x + 4
⎧-y = 3 - 5x
27. ⎨
28. ⎨
⎩ -2x + y = 6
⎩ y - 5x = 6
⎧y + 2 = 3x
29. ⎨
⎩ 3x - y = -1
⎧y - 1 = -3x
31. ⎨
⎩ 12x + 4y = 4
⎧2y = 6 - 6x
30. ⎨
⎩ 3y + 9x = 9
⎧4x - 2y = 4
32. ⎨
⎩ 3y = 6 (x - 1)
Classify each system. Give the number of solutions.
⎧2y = 2 (4x - 3)
33. ⎨
⎩ y - 1 = 4x
⎧3y + 6x = 9
34. ⎨
⎩ 2(y - 3) = -4x
⎧3x - 13 = 2y
35. ⎨
⎩ -3y = 2x
Lesson
Tell whether the ordered pair is a solution of the given inequality.
6-5
36. (3, 6); y > 2x + 4
37. (-2, -8); y ≤ 3x - 2
38. (-3, 3); y ≥ -2x + 5
Graph the solutions of each linear inequality.
39. y > 2x
40. y ≤ -3x + 2
41. y ≥ 2x - 1
42. -y < -x + 4
43. y ≥ -2x + 4
44. y > -x - 3
1 x + 1_
1
45. y < _
2
2
46. y ≤ 4x - (-1)
Write an inequality to represent each graph.
47.
n
48.
Þ
Þ
n
{
{
Ý
n
Lesson
6-6
{
ä
{
Ý
n
n
{
ä
{
{
n
n
{
n
Tell whether the ordered pair is a solution of the given system.
⎧y > 3x - 3
⎧y > -3x - 2
⎧y > 2x
49. (2, 5); ⎨
50. (3, 9); ⎨
51. (2, 3); ⎨
⎩y ≥ x + 1
⎩ y < 2x + 3
⎩y ≤ x - 3
Graph each system of linear inequalities. Give two ordered pairs that are solutions
and two that are not solutions.
⎧x + 4y < 2
52. ⎨
⎩ 2y > 3x + 8
⎧y ≤ 6 - 2x
53. ⎨
⎩ x - 2y < -2
⎧2x - 2 > -3y
54. ⎨
⎩ -x + 3y ≥ -10
Graph each system of linear inequalities. Describe the solutions.
⎧y > 2x + 1
55. ⎨
⎩ y < 2x - 2
⎧y < 3x - 1
56. ⎨
⎩ y > 3x - 4
⎧y ≥ -x + 2
57. ⎨
⎩ y ≥ -x + 5
⎧y ≥ 2x - 3
58. ⎨
⎩ y ≥ 2x + 3
⎧y > -4x - 2
59. ⎨
⎩ y ≤ -4x - 5
⎧y ≥ -2x + 1
60. ⎨
⎩ y < -2x + 6
Extra Practice
S15
Chapter 7
Skills Practice
Lesson
Simplify.
7-1
1. 3 -4
2. 5 -3
3. -4 0
4. -2 -5
6. (-2)-4
7. 1-7
8. (-4)-3
9. (-5)0
5. 6 -3
10. (-1)-5
Evaluate each expression for the given value(s) of the variable(s).
11. x -4 for x = 2
12. (c + 3)-3 for c = -6
13. 3j -7k -1 for j = -2 and k = 3
14. (2n - 2)-4 for n = 3
Simplify.
15. b 4g -5
k -3
16. _
r5
17. 5s -3c 0
z -4
18. _
5t -2
f2
19. _
3a -4
-3t 4
20. _
q -5
a 0k -4
21. _
p2
22. 3f -1y -5
25. 10 6
26. 10 -8
Lesson
Find the value of each power of 10.
7-2
23. 10 -7
24. 10 9
Write each number as a power of 10.
27. 10,000,000
28. 0.00001
29. 10,000,000,000,000
Find the value of each expression.
30. 72.19 × 10 -2
31. 0.096 × 10 -7
32. 7384.5 × 10 6
Write each number in scientific notation.
33. 3,605,000
34. 0.0063
35. 100,500,000
38. (k 4)
Lesson
Simplify.
7-3
36. 3 4 · 3 2
37. r 7 · r 0
39. (b 4)
40. (c 3d 2) · (c d 2)
4
41. (-3q 3)
-2
3
3
-2
Find the missing exponent in each expression.
a
43. (a 3b ■) = _
b6
3
42. a ■a 6 = a 9
Lesson
Simplify.
7-4
3 11
45. _
38
44 · 53
46. _
2
3 · 43 · 53
b
44. (a 4b -2) · a 3 = _
a5
9
4
■
6h 4
47. _
12h 3
r 6s 5
48. _
r 5s 6
Simplify each quotient and write the answer in scientific notation.
49. (4 × 10 7) ÷ (1.6 × 10 5)
Simplify.
()
2
52. _
3
S16
Extra Practice
4
50. (10 × 10 4) ÷ (2 × 10 7)
( )
x 2y 2
53. _
y3
2
()
4
54. _
5
-3
51. (2.5 × 10 8) ÷ (5 × 10 3)
55.
( )
2xy 2
_
3(xy)2
-3
Chapter 7
Skills Practice
Lesson
Simplify each expression.
7-5
1
_
1
_
1
_
56. 27 3
57. 256 4
58. 169 2
59. 0 5
60. 4 2
61. 49 2
62. 36 2
63. 16 4
1
_
3
_
3
_
3
_
5
_
Simplify. All variables represent nonnegative numbers.
1
_
64. √
x2 y6
Lesson
7-6
65. √
a 9 b 15
3
Find the degree of each monomial.
68. 4 7
69. x 3 y
Find the degree of each polynomial.
72. a 2 b + b - 2 2
73. 5x 4 y 2 - y 5 z 2
g )
( √
1
_
7
3
(m 8) 2
66. _
√
m4
67.
r 6 st 2
70. _
2
71. 9 0
74. 3g 4 h + h 2 + 4j 6
75. 4nm 7 - m 6 p3 + p
5
60
√
t 14
Write each polynomial in standard form. Then give the leading coefficient.
1 t3 + t - _
1 t5 + 4
76. 4r - 5r 3 + 2r 2
77. -3b 2 + 7b 6 + 4 - b
78. _
2
3
Classify each polynomial according to its degree and number of terms.
80. -4x 2 + x 6 - 4 + x 3
81. x 3 - 7 2
79. 3x 2 + 4x - 5
Lesson
7-7
Lesson
7-7
Add or subtract.
82. 4y 3 - 2y + 3y 3
83. 9k 2 + 5 - 10k 2 - 6
84. 7 - 3n 2 + 4 + 2n 2
85. (9x 6 - 5x 2 + 3) + (6 x 2 - 5)
86. (2y 5 - 5y 2) + (3y 5 - y 3 + 2y 2)
87. (r 3 + 2r + 1) - (2r 3 - 4)
88. (10s 2 + 5) - (5s 2 + 3s - 2)
89. (2s 7 - 6s 3 + 2) - (3s 7 + 2)
Multiply.
90. (3a 7)(2a 4)
91. (-3xy 3)(2x 2z)(yz 4)
92. (4k 3m)(-2k 2m 2)
93. 3jk 2(2j 2 + k)
94. 4q 3r 2 (2qr 2 + 3q)
95. 3xy 2(2x 2y - 3y)
96. (x - 3)(x + 1)
97. (x - 2)(x - 3)
98. (x 2 + 2xy)(3x 2y - 2)
99. (x 2 - 3x)(2xy - 3y)
102. (x + 3)(2x 4 - 3x 2 - 5)
100. (x - 2)(x 2 + 3x - 4)
101. (2x - 1)(-2x 2 - 3x + 4)
103. (3a + b)(2a 2 + ab - 2b 2) 104. (a 2 - b)(3a 2 - 2ab + 3b 2)
Lesson
Multiply.
7-8
105. (x + 3) 2
106. (3 + 2x) 2
107. (4x + 2y)2
108. (3x - 2)2
109. (5 - 2x) 2
110. (3x - 5y)2
111. (3 + x)(3 - x)
112. (x - 5)(x + 5)
113. (2x + 1)(2x - 1)
114. (x 2 + 4)(x 2 - 4)
115. (2 + 3x 3)(2 - 3x 3)
116. (4x 3 - 3y)(4x 3 + 3y)
Extra Practice
S17
Chapter 8
Lesson
8-1
Skills Practice
Write the prime factorization of each number.
1. 24
2. 78
3. 88
4. 63
5. 128
6. 102
7. 71
8. 125
Find the GCF of each pair of numbers.
9. 18 and 66
10. 24 and 104
11. 30 and 75
12. 24 and 120
13. 36 and 99
14. 42 and 72
Find the GCF of each pair of monomials.
15. 4a 3 and 9a 4
16. 6q 2 and 15q 5
17. 6x 2 and 14y 3
18. 4z 2 and 10z 5
19. 5g 3 and 9g
20. 12x 2 and 21y 2
Lesson
Factor each polynomial. Check your answer.
8-2
21. 6b 2 - 15b 3
22. 11t 4 - 9t 3
23. 10v 3 - 25v
24. 12r + 16r 3
25. 17a 4 - 35a 2
26. 9f + 18f 5 + 12f 2
27. 3(a + 3) + 4a(a + 3)
28. 5(k - 4) - 2k (k - 4)
29. 5(c - 3) + 4c 2(c - 3)
30. 3(t - 4) + t (t - 4)
31. 5(2r - 1) - s(2r - 1)
32. 7(3d + 4) - 2e(3d + 4)
Factor each expression.
Factor each polynomial by grouping. Check your answer.
33. x 3 + 3x 2 - 2x - 6
34. 2m 3 - 3m 2 + 8m - 12
35. 3k 3 - k 2 + 15k - 5
36. 15r 3 + 25r 2 - 6r - 10
37. 12n 3 - 6n 2 - 10n + 5
38. 4z 3 - 3z 2 + 4z - 3
39. 2k 2 - 3k + 12 - 8k
40. 3p 2 - 2p + 8 - 12p
41. 10d 2 - 6d + 9 - 15d
42. 6a 3 - 4a 2 + 10 - 15a
43. 12s 3 - 2s 2 + 3 - 18s
44. 4c 3 - 3c 2 + 15 - 20c
Lesson
Factor each trinomial. Check your answer.
8-3
45. x 2 + 15x + 36
46. x 2 + 13x + 40
47. x 2 + 10x + 16
48. x 2 - 9x + 18
49. x 2 - 11x + 28
50. x 2 - 13x + 42
51. x 2 + 4x - 21
52. x 2 - 5x - 36
53. x 2 - 7x - 30
54. Factor c 2 - 2c - 48. Show that the original polynomial and the factored form
describe the same sequence of values for c = 0, 1, 2, 3, and 4.
Copy and complete the table.
S18
x 2 + bx + c
Sign of c
Binomial factors
Sign of Numbers
in Binomials
x 2 + 9x + 20
Positive
(x + 4)(x + 5)
Both positive
55.
x - x - 20
?
?
?
56.
x - 2x - 8
?
?
?
57.
x - 6x + 8
?
?
?
Extra Practice
2
2
2
Chapter 8
Skills Practice
Lesson
Factor each trinomial. Check your answer.
8-4
58. 2x 2 + 13x + 15
59. 3x 2 + 14x + 16
60. 8x 2 - 16x + 6
61. 6x 2 + 11x + 4
62. 3x 2 - 11x + 6
63. 10x 2 - 31x + 15
64. 6x 2 - 5x - 4
65. 8x 2 - 14x - 15
66. 4x 2 - 11x + 6
67. 12x 2 - 13x + 3
68. 6x 2 - 7x - 10
69. 6x 2 + 7x - 3
70. 2x 2 + 5x - 12
71. 6x 2 - 5x - 6
72. 8x 2 + 10x - 3
73. 10x 2 - 11x - 6
74. 4x 2 - x - 5
75. 6x 2 - 7x - 20
76. -2x 2 + 11x - 5
77. -6x 2 - x + 12
78. -8x 2 - 10x - 3
79. -4x 2 + 16x - 15
80. -10x 2 + 21x + 10
81. -3x 2 + 13x - 14
Lesson
8-5
Determine whether each trinomial is a perfect square. If so, factor. If not,
explain why.
82. x 2 - 8x + 16
83. 4x 2 - 4x + 1
84. x 2 - 8x + 9
85. 9x 2 - 14x + 4
86. 4x 2 + 12x + 9
87. x 2 + 8x - 16
88. 9x 2 - 42x + 49
89. 4x 2 + 18x + 25
90. 16x 2 - 24x + 9
Determine whether each trinomial is the difference of two squares. If so, factor. If
not, explain why.
91. 4 - 16x 4
92. -t 2 - 35
93. c 2 - 25
94. g 5 - 9
95. v 4 - 64
96. x 2 - 120
97. x 2 - 36
98. 9m 2 - 15
99. 25c 2 - 16
Find the missing term in each perfect-square trinomial.
100. 4x 2 - 20x +
101. 9x 2 +
103. 9b 2 -
104.
+ 25
+1
+ 28a + 49
102.
- 56x + 49
105. 4a 2 + 4a +
Lesson
Tell whether each expression is completely factored. If not, factor.
8-6
106. 5(16x 2 + 4)
107. 3r (4x - 9)
108. (9d - 6)(2d - 7)
109. (5 - h)(6 - 5h)
110. 12y 2 - 2y - 24
111. 3f (2f 2 + 5fg + 2g 2)
Factor each polynomial completely. Check your answer.
112. 12b 3 - 48b
113. 24w 4 - 20w 3 - 16w 2
114. 18k 3 - 32k
115. 4a 3 + 12a 2 - a 2b - 3ab 116. 3x 3y - 6x 2y 2 + 3xy 3
117. 36p 2q - 64q 3
118. 32a 4 - 8a 2
119. m 3 + 5m 2n + 6mn 2
120. 4x 2 - 3x 2 - 16x + 48x
121. 18d 2 + 3d - 6
122. 2r 2 - 9r - 18
123. 8y 2 + 4y - 4
124. 81 - 36u 2
125. 8x 4 + 12x 2 - 20
126. 10j 3 + 15j 2 - 70j
127. 27z 3 - 18z 2 + 3z
128. 4b 2 + 2b - 72
129. 3f 2 - 3g 2
Extra Practice
S19
Chapter 9
Lesson
9-1
Skills Practice
Tell whether each function is quadratic. Explain.
1. y + 4x 2 = 2x - 3
2. 4x - y = 3
4.
5.
x
-6
-4
-2
0
2
y
-5
-6
-4
2
11
3. 3x 2 - 4 = y + x
x
0
1
2
3
4
y
-5
-5
-3
1
7
Tell whether the graph of each quadratic function opens upward or downward.
Then use a table of values to graph each function.
2 x2
6. y = -3x 2
7. y = _
8. y = x 2 + 2
9. y = -4x 2 + 2x
3
Identify the vertex of each parabola. Then find the domain and range.
10.
11.
y
12.
y
2
2
x
x
-2
-4
-2
y
6
2
2
8
-2
4
-4
2
-2
x
2
Lesson
9-2
4
6
Find the zeros of each quadratic function and the axis of symmetry of each
parabola from the graph.
13.
8
14.
y
2
15.
y
2
y
x
x
6
-2
0
2
4
-2
0
4
-2
-2
2
-4
-4
2
4
x
-4
-2
0
Find the vertex.
16. y = 3x 2 - 6x + 2
Lesson
9-3
2
4
17. y = -2x 2 + 8x - 3
18. y = x 2 + 2x - 4
Graph each quadratic function.
19. y = x 2 - 4x + 1
20. y = -x 2 - x + 4
21. y = 3x 2 - 3x + 1
22. y - 2 = 2x 2
24. y - 4 = x 2 + 2x
23. y + 3x 2 = 3x - 1
Lesson
Order the functions from narrowest to widest.
9-4
25. f (x) = 2x 2, g(x) = -4x 2, h(x) = -x 2
1 x 2, h(x) = -2x 2
26. f (x) = 3x 2, g(x) = _
2
1 x2
27. f (x) = 4x 2, g(x) = x 2, h(x) = - _
28. f (x) = 2x 2, g(x) = 5x 2, h(x) = -3x 2
4
Compare the graph of each function with the graph of f (x) = x 2.
1 x2
29. g(x) = 2x 2 - 2
30. g(x) = - _
31. g(x) = -3x 2 + 1
2
S20
8
Extra Practice
Chapter 9
Lesson
9-5
Lesson
9-6
Skills Practice
Solve each quadratic equation by graphing the related function.
32. x 2 - x - 2 = 0
33. x 2 - 2x + 8 = 0
34. 2x 2 + 4x - 6 = 0
35. 2x 2 + 9x = -4
36. 2x 2 + 3 = 0
37. 2x 2 - 2x - 12 = 0
38. 3x 2 = -3x + 6
39. x 2 = 4
40. 2x 2 + 6x - 20 = 0
41. -3x 2 - 2 = 0
42. x 2 = -2x + 8
43. x 2 - 2x = 15
Use the Zero Product Property to solve each equation. Check your answer.
44. (x + 3)(x - 2) = 0
45. (x - 4)(x + 2) = 0
46. (x)(x - 4) = 0
47. (2x + 6)(x - 2) = 0
48. (3x - 1)(x + 3) = 0
49. (x)(2x - 4) = 0
Solve each quadratic equation by factoring. Check your answer.
50. x 2 + 5x + 6 = 0
51. x 2 - 3x - 4 = 0
52. x 2 + x - 12 = 0
Lesson
9-7
Lesson
9-8
Lesson
9-9
53. x 2 + x - 6 = 0
54. x 2 - 6x + 5 = 0
55. x 2 + 4x - 12 = 0
56. x 2 = 6x - 9
57. 2x 2 + 4x = 6
58. x 2 + 2x = -1
59. 3x 2 = 3x + 6
60. x 2 = x + 12
61. 4x 2 + 8x + 4 = 0
Solve using square roots. Check your answer.
62. x 2 = 169
63. x 2 = 121
64. x 2 = 289
65. x 2 = -64
66. x 2 = 81
67. x 2 = -441
68. 4x 2 - 196 = 0
69. 0 = 3x 2 - 48
70. 24x 2 + 96 = 0
71. 10x 2 - 75 = 15
72. 0 = 4x 2 + 144
73. 5x 2 - 105 = 20
Solve. Round to the nearest hundredth.
74. 4x 2 = 160
75. 0 = 3x 2 - 66
76. 250 - 5x 2 = 0
77. 0 = 9x 2 - 72
79. 6x 2 = 78
78. 48 - 2x 2 = 42
Complete the square to form a perfect-square trinomial.
80. x 2 - 8x +
81. x 2 + x +
82. x 2 + 10x +
83. x 2 - 5x +
85. x 2 - 7x +
84. x 2 + 6x +
Solve by completing the square.
86. x 2 + 6x = 91
87. x 2 + 10x = -16
88. x 2 - 4x = 12
89. x 2 - 8x = -12
90. x 2 - 12x = -35
91. -x 2 - 6x = 5
92. -x 2 - 4x + 77 = 0
93. -x 2 = 10x + 9
94. -x 2 + 63 = -2x
Solve using the quadratic formula.
95. x 2 + 3x - 4 = 0
96. x 2 - 2x - 8 = 0
98. x 2 - x - 10 = 0
99. 2x 2 - x - 4 = 0
97. x 2 + 2x - 3 = 0
100. 2x 2 + 3x - 3 = 0
Find the number of real solutions of each equation using the discriminant.
101. x 2 + 4x + 1 = 0
102. 2x 2 - 3x + 2 = 0
103. x 2 - 5x + 2 = 0
104. 2x 2 - 4x + 2 = 0
105. x 2 + 2x - 5 = 0
106. 2x 2 - 2x - 3 = 0
Extra Practice
S21
Use the circle graph for Exercises 5–7.
5. Which candidate received the fewest votes?
äÈ
äx
Óä
6œÌˆ˜}ʜÀÊ-ÌÕ`i˜Ì‡œ`ÞÊ*ÀiÈ`i˜Ì
7. A total of 400 students voted in the election.
How many votes did Velez receive?
10-2
ä{
9i>À
6. Which two candidates received approximately
the same number of votes?
Lesson
Óä
Óä
ä£
4. Estimate the amount by which the population
decreased from 2005 to 2006.
äÎ
3. During which one-year period did the
population increase by the greatest amount?
Óä
£x
£ä
x
ä
Óä
2. Estimate the population in 2005.
*œ«Õ>̈œ˜Êœvʈ`ۈi
äÓ
10-1
Use the line graph for Exercises 1–4.
1. In what year was the population the greatest?
Óä
Lesson
Skills Practice
Óä
Chapter 10
The daily high temperatures in degrees Celsius
during a two-week period in Madison, Wisconsin,
are given at right.
8. Use the data to make a stem-and-leaf plot.
9. Use the data to make a frequency table with intervals.
10. Use the frequency table from Exercise 9 to make a
histogram for the data.
>À˜iÃ
£ä¯
6iiâ
În¯
>VŽÃœ˜
Óx¯
9>˜}
Óǯ
High Temperatures (oC)
22
25
28 33
29
24
19
19
18
25 32
30
32
25
11. Use the data to make a cumulative frequency table.
Lesson
10-3
Find the mean, median, mode, and range of each data set.
12. 42, 45, 48, 45
13. 66, 68, 68, 62, 61, 68, 65, 60
Identify the outlier in each data set, and determine how the outlier affects the
mean, median, mode, and range of the data.
14. 4, 8, 15, 8, 71, 7, 6
15. 36, 7, 50, 40, 38, 48, 40
Use the data to make a box-and-whisker plot.
16. 7, 8, 10, 2, 5, 1, 10, 8, 5, 5
17. 54, 64, 50, 48, 53, 55, 57
Lesson
10-4
18. The graph shows the ages of people who listen to a radio program.
a. Explain why the graph is misleading.
}iÃʜvÊ,>`ˆœÊ*Àœ}À>“ʈÃÌi˜iÀÃ
b. What might someone believe because of the
graph?
c. Who might want to use this graph? Explain.
ÓxÊ̜ÊÎÈ
19. A researcher surveys people at the Elmwood
library about the number of hours they spend
reading each day. Explain why the following
statement is misleading: “People in Elmwood
read for an average of 1.5 hours per day.”
S22
Extra Practice
Îä¯
1˜`iÀÊ£n
£x¯
£nÊ̜ÊÓ{
£x¯
Chapter 10
Lesson
10-5
Skills Practice
20. Identify the sample space and the outcome shown for the spinner at right.
Write impossible, unlikely, as likely as not, likely, or certain to describe
each event.
21. Two people sitting next to each other on a bus have the same birthday.
22. Dylan rolls a number greater than 1 on a standard number cube.
An experiment consists of randomly choosing a fruit snack from
a box. Use the results in the table to find the experimental
probability of each event.
23. choosing a blueberry fruit snack
Cherry
8
Peach
6
24. choosing a cherry fruit snack
Blueberry
6
Outcome
Frequency
25. not choosing a cherry fruit snack
Lesson
10-6
Find the theoretical probability of each outcome.
26. rolling an even number on a number cube
27. tossing two coins and both landing tails up
28. randomly choosing a prime number from a bag that contains ten slips of paper
numbered 1 through 10
29. The probability of choosing a green marble from a bag is __37 . What is the probability of
not choosing a green marble?
30. The odds against winning a game are 8 : 3. What is the probability of winning the game?
Lesson
10-7
Tell whether each set of events is independent or dependent. Explain your answer.
31. You pick a bottle from a basket containing chilled drinks, and then your friend
chooses a bottle.
32. You roll a 6 on a number cube and you toss a coin that lands heads up.
33. A number cube is rolled three times. What is the probability of rolling three numbers
greater than 4?
34. An experiment consists of randomly selecting a marble from a bag, replacing it, and
then selecting another marble. The bag contains 3 blue marbles, 2 orange marbles,
and 5 yellow marbles. What is the probability of selecting a blue marble and then a
yellow marble?
35. Madeleine has 3 nickels and 5 quarters in her pocket. She randomly chooses one
coin and does not replace it. Then she randomly chooses another coin. What is the
probability that she chooses two quarters?
Lesson
10-8
For Exercises 36 and 37, tell whether each situation involves combinations or
permutations. Then give the number of possible outcomes.
36. How many different ways can three photographs be arranged in a row on a wall?
37. How many different smoothies can be made by blending two of the following fruits:
oranges, bananas, strawberries, and peaches?
38. There are 6 entrants in a livestock competition at a county fair. How many different
ways can the first-place, second-place, and third-place ribbons be awarded?
39. An amusement park has 7 roller coasters. How many different ways can Jacinto
choose 4 different roller coasters to ride?
Extra Practice
S23
Chapter 11
Lesson
11-1
Skills Practice
Find the next three terms in each geometric sequence.
1. 1, 5, 25, 125 …
2. 736, 368, 184, 92, …
3. -2, 6, -18, 54, …
1
1
1, _
1 , 1, 3, …
_
_
4. 8, 2, , , …
5. 7, -14, 28, -56, …
6. _
2 8
9 3
7. The first term of a geometric sequence is 2, and the common ratio is 3. What is the
8th term of the sequence?
8. What is the 8th term of the geometric sequence 600, 300, 150, 75, …?
Lesson
11-2
Tell whether each set of ordered pairs satisfies an exponential function. Explain
your answer.
⎧
⎧
1 , 0, 2 , 1, 8 , 2, 32 ⎫⎬
1 , 0, 0 , 1, _
1 , 2, 4 ⎫⎬
9. ⎨ -1, _
10. ⎨ -1, - _
)
( ) ( ) (
( )
( )
2
2
2
⎩
⎭
⎩
⎭
(
(
)
( )(
)
⎧
1 , 2, _
1 ⎫⎬
11. ⎨(-1, 4), (0, 1), 1, _
4
16 ⎭
⎩
⎧
⎫
12. ⎨(0, 0), (1, 3), (2, 12), (3, 27)⎬
⎩
⎭
Graph each exponential function.
x
1 (4)x
13. y = 3(2)
14. y = _
2
x
x
1
1
_
16. y = - (2)
17. y = 5 _
2
2
()
Lesson
11-3
( )
)
15. y = -3 x
18. y = -2(0.25)
x
Write an exponential growth function to model each situation. Then find the value
of the function after the given amount of time.
19. The rent for an apartment is $6600 per year and increasing at a rate of 4% per year;
5 years.
20. A museum has 1200 members and the number of members is increasing at a rate of
2% per year; 8 years.
Write a compound interest function to model each situation. Then find the balance
after the given number of years.
21. $4000 invested at a rate of 4% compounded quarterly; 3 years
22. $5200 invested at a rate of 2.5% compounded annually; 6 years
Write an exponential decay function to model each situation. Then find the value of
the function after the given amount of time.
23. The cost of a stereo system is $800 and is decreasing at a rate of 6% per year; 5 years.
24. The population of a town is 14,000 and is decreasing at a rate of 2% per year; 10 years.
Lesson
11-4
S24
Graph each data set. Which kind of model best describes the data?
⎧
⎫
25. ⎨(0, 3), (1, 0), (2, -1), (3, 0), (4, 3)⎬
⎭
⎧⎩
⎫
26. ⎨(-4, -4), (-3, -3.5), (-2, -3), (-1, -2.5), (0, -2), (1, -1.5)⎬
⎩⎧
⎭
⎫
27. ⎨(0, 4), (1, 2), (2, 1), (3, 0.5), (4, 0.25)⎬
⎩
⎭
Look for a pattern in each data set to determine which kind of model best describes
the data.
⎧
⎫
28. ⎨(-1, -5), (0, -5), (1, -3), (2, 1), (3, 7)⎬
⎧⎩
⎫ ⎭
29. ⎨(0, 0.25), (1, 0.5), (2, 1), (3, 2), (4, 4)⎬
⎩⎧
⎭ ⎫
30. ⎨(-2, 11), (-1, 8), (0, 5), (1, 2), (2, -1)⎬
⎩
⎭
Extra Practice
Chapter 11
Lesson
11-5
Skills Practice
Find the domain of each square-root function.
31. y = √
x+1
32. y = √
x-2+4
x
34. y = √
3x - 6
35. y = 1 + _
3
Graph each square-root function.
37. y = √
x+2
38. y = √
x-3
√
40. y = - √
x
Lesson
Simplify each expression.
11-6
43.
46.
128
√_
2
3
_
√ 48
33. y = √
4+x
36. y = √
4x - 1
39. y = √
3x + 1
41. y = 2 √
x+1
42. y = 3 √
x-2
2
44. √7
+ 24 2
45.
47.
y 2 + 4y + 4
√
(4 - x)2
√
48. √
52 - 42
Simplify. All variables represent nonnegative numbers.
72
49. √
50.
75x 4y 3
√
51.
64
√_
x
53.
16a
_
√
25b
54.
52.
Lesson
11-7
Lesson
11-8
6
4
2
11
_
√
81
18x
_
√
49x
4
3
Add or subtract.
55. 5 √
7 + 3 √7
56. 6 √
2 + √
2
57. 5 √
3 - 2 √3
- 9 √
58. √
5 + 7 √5
5
59. 2 √
y + 4 √
y - 3 √
y
- 3 √
60. 5 √
3 + 4 √2
3
Simplify each expression.
61. √
75 + √
27
62. √
45 - √
20
63. 2 √
12 + √
18
64. 3 √
27x + √
48x
65. 5 √
20y - 2 √
80y
66. √
28a + 2 √
63a - √
175a
67. √
50y - 2 √
18y + 3 √
8y
68. √
12x - √
27x - √
5x
69. 5 √
180s - 6 √
80s
Multiply. Write each product in simplest form.
70. √
5 √
10
71. √
6 √
12
73. (2 √
7)
2
76. 2 √
5 ( √
20 + 3)
79. (3 + √
5 )(8 - √
5)
74. √
6x √
15x
77. √
2x (3 + √
8x )
80. (4 + √
2)
2
72. (3 √
3)
2
)
75. √
3 (2 + √27
78. (4 + √
3 )(1 - √
3)
81. (5 - √
3)
2
Simplify each quotient.
√
5
82. _
√
3
√
5
7
85. _
√
50
Lesson
11-9
2 √
7
83. _
√
5
√
12a
86. _
√
32
Solve each equation. Check your answer.
88. √x
89. √
3x = 9
= 11
√
3
84. _
√
20
√
200x
87. _
√
28
90. √
-2x = 10
91. 5 = √
-4x
92.
94. √
3x + 1 = 4
95. √
2x + 5 = 3
96. √
x-4+1=7
97. √
6 - 3x - 2 = 4
98. √
6 - x - 5 = -3
99. 4 √
x = 20
√
x
+ 5 = 12
93.
√
x
-4=1
Extra Practice
S25
Chapter 12
Lesson
12-1
Skills Practice
Tell whether each relationship is an inverse variation. Explain.
1.
x
y
4
2.
x
y
8
2
8
16
16
32
3.
x
y
6
-1
24
3
4
2
-12
32
6
2
4
-6
64
12
1
8
-3
4. 3xy = 10
5. y - x = 6
6. 6xy = -1
7. Write and graph the inverse variation in which y = 4 when x = 3.
1 when x = 6.
8. Write and graph the inverse variation in which y = _
2
9. Let x 1 = 6, y 1 = 8, and x 2 = 12. Let y vary inversely as x. Find y 2.
10. Let x 1 = -4, y 1 = -2, and y 2 = 16. Let y vary inversely as x. Find x 2.
Lesson
12-2
Identify the excluded values for each rational function.
16
11. y = _
x
3
13. y = - _
x+5
20
14. y = _
x + 20
8
16. y = _
x+5
7
17. y = _
-6
3x - 2
3
18. y = _
+4
2x - 2
1
19. y = _
x+3
1
20. y = _
x-2
1 +4
21. y = _
x
3
22. y = _
x-2
1 +2
23. y = _
x-3
1 -6
24. y = _
x-5
1 +5
25. y = _
x+2
1 +1
26. y = _
x+5
1
12. y = _
x-1
Identify the asymptotes.
2
15. y = _
x-4
Graph each function.
Lesson
Find the excluded value(s) of each rational expression.
12-3
3
27. _
7x
-2
28. _
x2 - x
6
29. __
x 2 + x - 12
p+1
30. __
2
p + 4p - 5
Simplify each rational expression, if possible. Identify excluded values.
4m 2
31. _
12m
7x 5
32. _
28x
4x 2 - 8x
33. _
x-2
2y
34. _
y-1
5x 3 + 20x 2
35. _
x+4
a+1
36. _
a-2
3y 3 + 3y
37. _
y2 + 1
x 3 + 4x
38. _
x2 + 4
Simplify each rational expression, if possible.
S26
b+2
39. __
b 2 + 5b + 6
x-3
40. __
x 2 - 6x + 9
y 2 - 4y - 5
41. __
y 2 - 2y - 3
(m + 2)2
42. __
m 2 - 6m - 16
x2 - 9
43. __
2
x + x - 12
2-m
44. _
3m 2 - 6m
x-4
45. _
12x 2 - 3x 3
6 - 3x
46. __
x 2 - 6x + 8
Extra Practice
Chapter 12
Skills Practice
Lesson
Multiply. Simplify your answer.
12-4
ab
4a 3 · _
47. _
3
b
6a 2
x-2 ·_
2x - 10
50. _
x-5
3
1 (x 2 - 2x - 8)
53. _
2x + 4
4r 3 + 8r _
· 2r
56. _
r3
3r + 6
4b 2 + 4 b 2 - 1
59. _ · _
b-1
8b 2 + 8
6x 3y _
8x 2yz 2
48. _
·
4y 4
3xz 5
x-3 ·_
8
49. _
2
4x - 12
2 3
9b 2
b ·_
_
51. a
6a 3c 12b 5c 2
3x (x 2 + x - 30)
54. _
4x - 20
3x + 6
3x · _
52. _
2x + 4
9
2y
55. _ (y 2 + 10y + 25)
3y + 15
x2 + x
x-3
57. _
·_
2
x - x - 6 6x 2 + 6x
pq + 2q
3pq + 3
60. _ · _
pq + 1 pq 2 + 2q 2
2
2
- 3a - 10 · a
- 2a - 3
__
__
58. a
2
a-5
a -a-6
x 2 + 4x + 3
63. __
÷ (x 2 - 1)
3x 3 + 9x 2
p 2 - 2p
p-1
__
÷
64. __
p 2 + 4p - 5
p 2 + 3p - 10
x2 + 1
1 - 3x
66. _ + _
x-1
x-1
2x
2x 2
+ __
67. __
2
x - 2x - 3 x 2 - 2x - 3
r 2 + 3r + 2
2r + 6
61. _ · _
4r + 4
r 2 - 2r - 8
Divide. Simplify your answer.
3x 2y 3
6y 4
_
62. _
÷
x 2z 5
x 3z 2
Lesson
Add. Simplify your answer.
12-5
5x
3x + _
65. _
4x 3 4x 3
Subtract. Simplify your answer.
5 -_
2
68. _
6y 4 6y 4
Lesson
12-6
5a 2 + 1
15a + 1
m 2 + 2m
m + 12
69. _
-_
70. _
-_
2
2
a -a-6
a -a-6
m2 - 9
m2 - 9
Find the LCM of the given expressions.
71. 8x 5y 8, 6x 4y 9
72. x 2 - 4, x 2 + 7x + 10
73. d 2 - 2d - 3, d 2 + d - 12
Add or subtract. Simplify your answer.
3
5
5 -_
1
75. _
+_
74. _
y 2 4y 2
x2 - x - 6 x + 2
3x - _
x
76. _
x-2 2-x
Divide.
77. (12y 5 - 16y 2 + 4y) ÷ 4y 2 78. (6m 4 - 18m + 3) ÷ 6m 2 79. (16x 4 + 20x 3 - 4x) ÷ -4x 3
2
- 4b - 5
__
80. b
b+1
2x 2 + 9x + 4
81. __
x+4
Divide using long division.
83. (a 2 - 5a - 6) ÷ (a + 1)
84. (2x 2 + 10x + 8) ÷ (x + 4) 85. (3y 2 - 11y + 10) ÷ (y - 2)
a 2 - 13a - 5
__
82. 6
3a + 1
86. (3x 2 - 2x - 7) ÷ (x - 2) 87. (2x 2 + 2x - 9) ÷ (x + 3) 88. (5x 3 + 2x 2 - 4) ÷ (x - 2)
Lesson
12-7
Solve. Check your answer.
5 =_
10
4
4 =_
89. _
90. _
t
x+1 x-1
t+9
8
3
4 =_
1
92. _
93. _
=_
y
a-2 a+1
2y + 4
x
3
3
1
1
2
96. _ + _ = _
95. _ + _ = - _2
x
2 2m
2 2
m
Solve. Identify any extraneous solutions.
x+2
3 =_
x-5
2=_
98. _
99. _
x
x+4
x+4
x2 - 4
8
6
_
91. _
m = m+1
5
6
94. _
=_
4w - 2 5w - 2
3 _
10
97. 1 - _
a = a2
4x - 7 = _
16
100. _
x-4
x-4
Extra Practice
S27
Extra Practice
Chapter 1
Applications Practice
Biology Use the following information for
Exercises 1 and 2.
In general, skin cells in the human body contain
46 chromosomes. (Lesson 1-1)
1. Write an expression for the number of
chromosomes in c skin cells.
2. Find the number of chromosomes in 8, 15, and
50 skin cells.
3. On a winter day in Fairbanks, Alaska, the
temperature dropped from 12 °F to -16 °F.
How many degrees did the temperature
drop? (Lesson 1-2)
4. Geography The elevation of the Dead Sea in
Jordan is -411 meters. The greatest elevation
on Earth is Mt. Everest, at 8850 meters. What is
the difference in elevation between these two
locations? (Lesson 1-2)
5. Jeremy is raising money for his school
by selling magazine subscriptions. Each
subscription costs $16.75. During the first
week, he sells 12 subscriptions. How much
money does he raise? (Lesson 1-3)
6. As a service charge, Nadine’s checking account
is adjusted by -$3 each month. What is the
total amount of the adjustment over the
course of one year? (Lesson 1-3)
7. To go from one figure to the next in the
sequence of figures, each square is split into
four smaller squares. How many squares will
be in Figure 5? (Lesson 1-4)
10. An art museum exhibits a square painting
that has an area of 75 square feet. Find its side
length to the nearest tenth. (Lesson 1-5)
11. Travel The base of the Washington
Monument in Washington, D.C., is a square
with an area of 336 yards. Find the length
of one side of the monument’s base to the
nearest tenth. (Lesson 1-5)
12. The toll to cross a bridge is $2 for cars, $5 for
trucks, and $10 for buses. The total amount
of money collected can be found using the
expression 2C + 5T + 10B. Use the table to
find the total amount of money collected
between 10 A.M. and 11 A.M. (Lesson 1-6)
Bridge Tolls, 10 A.M. to 11 A.M.
Type of Vehicle
Car C
Truck T
Bus B
Number
104
20
3
13. The expression __59 (F - 32) converts a
temperature F in degrees Fahrenheit to a
temperature in degrees Celsius. Convert 77 °F
to degrees Celsius. (Lesson 1-6)
Use the following information for Exercises 14
and 15.
An airplane has 12 rows of seats in first class and
35 rows of seats in coach. Each row has the same
number of seats. (Lesson 1-7)
14. The total number of seats in the plane is
12x + 35x, where x is the number of seats in a
row. Simplify the expression.
ˆ}ÕÀiÊä
S28
ˆ}ÕÀiÊ£
ˆ}ÕÀiÊÓ
15. Find the total number of seats given that the
plane has 6 seats per row.
8. When you fold a sheet of paper in half and
then open it, the crease creates 2 regions.
Folding the paper in half 2 times creates 4
regions. How many regions do you create
when you fold a sheet of paper in half 5
times? (Lesson 1-4)
Use the following information for Exercises 16
and 17.
A sales representative earns $680 per week plus a
$40 commission for each sale. (Lesson 1-8)
9. Dan began his stamp collection with just 5
stamps in the first year. Every year thereafter,
his collection grew 5 times as large as the
year before. How many stamps were in Dan’s
collection after 4 years? (Lesson 1-4)
17. Write ordered pairs for the representative’s
weekly earnings for 5, 8, and 10 sales.
Extra Practice
16. Write a rule for the sales representative’s
weekly earnings.
Chapter 2
Applications Practice
1. Economics In 2004, the average price of an
ounce of gold was $47 more than the average
price in 2003. The 2004 price was $410. Write
and solve an equation to find the average price
of an ounce of gold in 2003. (Lesson 2-1)
2. During a renovation, 36 seats were removed
from a theater. The theater now seats 580
people. Write and solve an equation to find
the number of seats in the theater before the
renovation. (Lesson 2-1)
3. A case of juice drinks contains 12 bottles and
costs $18. Write and solve an equation to find
the cost of each drink. (Lesson 2-2)
4. Astronomy Objects weigh about 3 times
as much on Earth as they do on Mars. A
rock weighs 42 lb on Mars. Write and solve
an equation to find the rock’s weight on
Earth. (Lesson 2-2)
9. Charles is hanging a poster on his wall. He
wants the top of the poster to be 84 inches
from the floor but would be happy for it to be
3 inches higher or lower. Write and solve an
absolute-value equation to find the maximum
and minimum acceptable heights.
(Lesson 2-6)
10. The ratio of students to adults on a school
trip is 9 : 2. There are 6 adults on the trip. How
many students are there? (Lesson 2-7)
11. A cheetah can reach speeds of up to
103 feet per second. Use dimensional analysis
to convert the cheetah’s speed to miles per
hour. Round to the nearest tenth. (Lesson 2-7)
12. Write and solve a proportion to find the height
of the flagpole. (Lesson 2-8)
5. The county fair’s admission fee is $8 and each
ride costs $2.50. Sonia spent a total of $25.50.
How many rides did she go on? (Lesson 2-3)
6. At the beginning of a block party, the
temperature was 84°. During the party, the
temperature dropped 3° every hour. At the end
of the party, the temperature was 66°. How
long was the party? (Lesson 2-3)
7. Consumer Economics A health insurance
policy costs $700 per year, plus $15 for each
visit to the doctor’s office. A different plan
costs $560 per year, but each office visit is $50.
Find the number of office visits for which the
two plans have the same total cost.
(Lesson 2-4)
8. Geometry The formula A = __12 bh gives the
area A of a triangle with base b and height
h. (Lesson 2-5)
a. Solve A =
1
__
bh
2
for h.
b. Find the height of a triangle with an area of
30 square feet and a base of 6 feet.
¶
x°{ÊvÌ
n°£ÊvÌ
ÓÇÊvÌ
13. Paul has 8 jazz CDs. The jazz CDs are 5%
of his collection. How many CDs does Paul
have? (Lesson 2-9)
14. Miguel earns an annual salary of $38,000 plus
a 3.5% commission on sales. His sales for one
year were $90,000. Find his total salary for the
year. (Lesson 2-10)
15. How long would it take $3600 to earn simple
interest of $450 at an annual interest rate of
5%? (Lesson 2-10)
16. A store sells swimsuits at a 30% discount. What
is the final price of a swimsuit that originally
sold for $28? (Lesson 2-11)
17. Mei sells jam at a farmer’s market for $4.20 per
jar. Each jar costs Mei $3 to make. What is the
markup as a percent? (Lesson 2-11)
Extra Practice
S29
Chapter 3
Applications Practice
1. At a food-processing factory, each box of
cereal must weigh at least 15 ounces. Define
a variable and write an inequality for the
acceptable weights of the cereal boxes. Graph
the solutions. (Lesson 3-1)
8. The admission fee at an amusement park is
$12, and each ride costs $3.50. The park also
offers an all-day pass with unlimited rides for
$33. For what numbers of rides is it cheaper to
buy the all-day pass? (Lesson 3-4)
2. In order to qualify for a discounted entry fee at
a museum, a visitor must be less than 13 years
old. Define a variable and write an inequality
for the ages that qualify for the discounted
entry fee. Graph the solutions. (Lesson 3-1)
9. The table shows the cost of Internet access at
two different cafes. For how many hours of
access is the cost at Cyber Station less than the
cost at Web World? (Lesson 3-5)
3. A restaurant can seat no more than 102
customers at one time. There are already
96 customers in the restaurant. Write and
solve an inequality to find out how many
additional customers could be seated in the
restaurant. (Lesson 3-2)
4. Meteorology A hurricane is a tropical
storm with a wind speed of at least 74 mi/h.
A meteorologist is tracking a storm whose
current wind speed is 63 mi/h. Write and solve
an inequality to find out how much greater the
wind speed must be in order for this storm to
be considered a hurricane. (Lesson 3-2)
Hobbies Use the following information for
Exercises 5–7.
When setting up an aquarium, it is recommended
that you have no more than one inch of fish per
gallon of water. For example, in a 30-gallon tank,
the total length of the fish should be at most
30 inches. (Lesson 3-3)
Freshwater Fish
Name
Length (in.)
Red tail catfish
3.5
Blue gourami
1.5
5. Write an inequality to show the possible
numbers of blue gourami you can put in a
10-gallon aquarium.
6. Find the possible numbers of blue gourami
you can put in a 10-gallon aquarium.
7. Find the possible numbers of red tail catfish
you can put in a 20-gallon aquarium.
S30
Extra Practice
Internet Access
Cafe
Cost
Cyber
Station
$12 one-time membership fee
$1.50 per hour
Web
World
No membership fee
$2.25 per hour
10. Larissa is considering two summer jobs. A
job at the mall pays $400 per week plus $15
for every hour of overtime. A job at the movie
theater pays $360 per week plus $20 for every
hour of overtime. How many hours of overtime
would Larissa have to work in order for the
job at the movie theater to pay a higher salary
than the job at the mall? (Lesson 3-5)
11. Health For maximum safety, it is
recommended that food be stored at a
temperature between 34 °F and 40 °F
inclusive. Write a compound inequality
to show the temperatures that are within
the recommended range. Graph the
solutions. (Lesson 3-6)
12. Physics Color is determined by the
wavelength of light. Wavelengths are
measured in nanometers (nm). Our eyes see
the color green when light has a wavelength
between 492 nm and 577 nm inclusive.
Write a compound inequality to show the
wavelengths that produce green light. Graph
the solutions. (Lesson 3-6)
13. Allison ran a mile in 8 minutes. She wants
to run a second mile within 0.75 minute of
her time for the first mile. Write and solve an
absolute-value inequality to find the range of
acceptable times for the second mile.
(Lesson 3-7)
Chapter 4
Applications Practice
1. Donnell drove on the highway at a constant
speed and then slowed down as she
approached her exit. Sketch a graph to
show the speed of Donnell’s car over time.
Tell whether the graph is continuous or
discrete. (Lesson 4-1)
2. Lori is buying mineral water for a party. The
bottles are available in six-packs. Sketch a
graph showing the number of bottles Lori
will have if she buys 1, 2, 3, 4, or 5 six-packs.
Tell whether the graph is continuous or
discrete. (Lesson 4-1)
3. Health To exercise effectively, it is important
to know your maximum heart rate. You can
calculate your maximum heart rate in beats
per minute by subtracting your age from
220. (Lesson 4-2)
a. Express the age x and the maximum heart
rate y as a relation in table form by showing
the maximum heart rate for people who are
20, 30, 35, and 40 years old.
7. The function y = 3.5x describes the number
of miles y that the average turtle can walk in
x hours. Graph the function. Use the graph to
estimate how many miles a turtle can walk in
4.5 hours. (Lesson 4-4)
8. Earth Science The Kangerdlugssuaq glacier
in Greenland is flowing into the sea at the
rate of 1.6 meters per hour. The function
y = 1.6x describes the number of meters y
that flow into the sea in x hours. Graph the
function. Use the graph to estimate the
number of meters that flow into the sea in
8 hours. (Lesson 4-4)
9. The scatter plot shows a relationship between
the number of lemonades sold in a day and
the day’s high temperature. Based on this
relationship, predict the number of lemonades
that will be sold on a day when the high
temperature is 96 °F. (Lesson 4-5)
i“œ˜>`iÊ->iÃ
b. Is this relation a function? Explain.
Season Statistics
Wins
Home Runs
95
185
93
133
80
140
93
167
5. Michael uses 5.5 cups of flour for each loaf
of bread that he bakes. He plans to bake a
maximum of 4 loaves. Write a function to
describe the number of cups of flour used.
Find a reasonable domain and range for the
function. (Lesson 4-3)
6. A gym offers the following special rate. New
members pay a $425 initiation fee and then
pay $90 per year for 1, 2, or 3 years. Write
a function to describe the situation. Find
a reasonable domain and range for the
function. (Lesson 4-3)
Õ«ÃÊ܏`
4. Sports The table shows the number of games
won by four baseball teams and the number
of home runs each team hit. Is this relation a
function? Explain. (Lesson 4-2)
nä
Èä
{ä
Óä
ä
Óä
{ä
Èä
nä
ˆ}…ÊÌi“«iÀ>ÌÕÀiÊ­c®
10. In month 1 the Elmwood Public Library had 85
Spanish books in its collection. Each month,
the librarian plans to order 8 new Spanish
books. How many Spanish books will the
library have in month 15? (Lesson 4-6)
11. Nikki purchases a card that she can use to
ride the bus in her town. Each time she rides
the bus $1.50 is deducted from the value of
the card. After her first ride, there is $43.50
left on the card. How much money will be
left on the card after Nikki has taken 12 bus
rides? (Lesson 4-6)
Extra Practice
S31
Chapter 5
Applications Practice
1. Jennifer is having prints made of her
photographs. Each print costs $1.50. The
function f (x) = 1.50x gives the total cost of
the x prints. Graph this function and give its
domain and range. (Lesson 5-1)
7. Sports Competitive race-walkers move at
a speed of about 9 miles per hour. Write a
direct variation equation for the distance y
that a race-walker will cover in x hours. Then
graph. (Lesson 5-6)
2. The Chang family lives 400 miles from Denver.
They drive to Denver at a constant speed of 50
mi/h. The function f (x) = 400 - 50x gives their
distance in miles from Denver after x hours.
(Lesson 5-2)
8. A bicycle rental costs $10 plus $1.50 per hour.
The cost as a function of the number of hours
is shown in the graph. (Lesson 5-7)
ˆVÞViÊ,i˜Ì>Ê
œÃÌÃ
a. Graph this function and find the intercepts.
œÃÌÊ­f®
Îä
b. What does each intercept represent?
3. History The table shows the number of
nations in the United Nations in different
years. Find the rate of change for each time
interval. During which time interval did the
U.N. grow at the greatest rate? (Lesson 5-3)
Year
Ó
{
È
/ˆ“iÊ­…®
1960
1975
a. Write an equation that represents the cost
as a function of the number of hours.
51
60
99
144
b. Identify the slope and y-intercept and
describe their meaning.
"Ûi˜Ê/i“«iÀ>ÌÕÀi
/i“«iÀ>ÌÕÀiÊ­c®
ä
1950
4. The graph shows the temperature of an oven
at different times. Find the slope of the line.
Then tell what the slope represents.
(Lesson 5-4)
{xä
­£ä]Ê{£ä®
Îxä
c. Find the cost of renting a bike for 6 hours.
9. A hot-air balloon is moving at a constant rate.
Its altitude is a linear function of time, as
shown in the table. Write an equation in
slope-intercept form that represents this
function. Then find the balloon’s altitude
after 25 minutes. (Lesson 5-8)
Balloon’s Altitude
­{ä]Êәä®
Óxä
ä
Óä
{ä
/ˆ“iÊ­“ˆ˜®
5. A straight highway connects the towns of
Dale and Winslow. On a map, the coordinates
of Dale are (5, 16), and the coordinates of
Winslow are (9, 24). A rest area is located on
the highway at the midpoint between the
towns. What are the map coordinates of the
rest area? (Lesson 5-5)
6. On a map, a campground is at (3, 5), and a
fishing area is at (8, 4). Each unit on the map
represents 0.1 mile. To the nearest tenth of
a mile, what is the distance between the
campground and the fishing area? (Lesson 5-5)
Extra Practice
£ä
1945
Number of
Nations
S32
Óä
Time (min)
Altitude (m)
0
250
7
215
12
190
10. Geometry Show that the points A(2, 3),
B(3, 1), C (-1, -1), and D(-2, 1) are the
vertices of a rectangle. (Lesson 5-9)
11. A phone plan for international calls costs
$12.50 per month plus $0.04 per minute. The
monthly cost for x minutes of calls is given by
the function f (x) = 0.04x + 12.50. How will the
graph change if the phone company raises the
monthly fee to $14.50? if the cost per minute is
raised to $0.05? (Lesson 5-10)
Chapter 6
Applications Practice
1. Net Sounds, an online music store, charges $12
per CD plus $3 for shipping and handling. Web
Discs charges $10 per CD plus $9 for shipping
and handling. For how many CDs will the cost
be the same? What will that cost be?
(Lesson 6-1)
2. At Rocco’s Restaurant, a large pizza costs $12
plus $1.25 for each additional topping. At
Pizza Palace, a large pizza costs $15 plus $0.75
for each additional topping. For how many
toppings will the cost be the same? What will
that cost be? (Lesson 6-1)
Use the following information for Exercises 3
and 4.
The coach of a baseball team is deciding between
two companies that manufacture team jerseys.
One company charges a $60 setup fee and $25 per
jersey. The other company charges a $200 setup fee
and $15 per jersey. (Lesson 6-2)
3. For how many jerseys will the cost at the two
companies be the same? What will that cost be?
4. The coach is planning to purchase 20 jerseys.
Which company is the better option? Why?
5. Geometry The length of a rectangle is
5 inches greater than the width. The sum of
the length and width is 41 inches. Find the
length and width of the rectangle. (Lesson 6-2)
6. At a movie theater, tickets cost $9.50 for
adults and $6.50 for children. A group of
7 moviegoers pays a total of $54.50. How many
adults and how many children are in the
group? (Lesson 6-3)
7. Business A grocer is buying large quantities
of fruit to resell at his store. He purchases
apples at $0.50 per pound and pears at $0.75
per pound. The grocer spends a total of $17.25
for 27 pounds of fruit. How many pounds of
each fruit does he buy? (Lesson 6-3)
8. Bricks are available in two sizes. Large bricks
weigh 9 pounds, and small bricks weigh 4.5
pounds. A bricklayer has 14 bricks that weigh a
total of 90 pounds. How many of each type of
brick are there? (Lesson 6-3)
9. Sports The table shows the time it took two
runners to complete the Boston Marathon
in several different years. If the patterns
continue, will Shanna ever complete the
marathon in the same number of minutes as
Maria? Explain. (Lesson 6-4)
Marathon Times (min)
2003
2004
2005
2006
Shanna
190
182
174
166
Maria
175
167
159
151
10. Jordan leaves his house and rides his bike at
10 mi/h. After he goes 4 miles, his brother
Tim leaves the house and rides in the same
direction at 12 mi/h. If their rates stay the
same, will Tim ever catch up to Jordan?
Explain. (Lesson 6-4)
11. Charmaine is buying almonds and cashews for
a reception. She wants to spend no more than
$18. Almonds cost $4 per pound, and cashews
cost $5 per pound. Write a linear inequality to
describe the situation. Graph the solutions.
Then give two combinations of nuts that
Charmaine could buy. (Lesson 6-5)
12. Luis is buying T-shirts to give out at a school
fund-raiser. He must spend less than $100 for
the shirts. Child shirts cost $5 each, and adult
shirts cost $8 each. Write a linear inequality
to describe the situation. Graph the solutions.
Then give two combinations of shirts that Luis
could buy. (Lesson 6-5)
13. Nicholas is buying treats for his dog. Beef
cubes cost $3 per pound, and liver cubes
cost $2 per pound. He wants to buy at least
2 pounds of each type of treat, and he wants
to spend no more than $14. Graph all possible
combinations of the treats that Nicholas could
buy. List two possible combinations.
(Lesson 6-6)
14. Geometry The perimeter of a rectangle is
at most 20 inches. The length and the width
are each at least 3 inches. Graph all possible
combinations of lengths and widths that
result in such a rectangle. List two possible
combinations. (Lesson 6-6)
Extra Practice
S33
Chapter 7
Applications Practice
1. The eye of a bee is about 10 -3 m in diameter.
Simplify this expression. (Lesson 7-1)
2. A typical stroboscopic camera has a shutter
speed of 10 -6 seconds. Simplify this expression.
(Lesson 7-1)
3. Space Exploration During a mission that
took place in August, 2005, the Space Shuttle
Discovery traveled a total distance of
9.3 × 10 6 km. The Space Shuttle’s velocity
was 28,000 km/h. (Lesson 7-2)
a. Write the total distance that the Space
Shuttle traveled in standard form.
b. Write the Space Shuttle’s velocity in
scientific notation.
4. There are approximately 10,000,000 grains in
a pound of salt. Write this number in scientific
notation. (Lesson 7-2)
5. A high-speed centrifuge spins at a speed of
2 × 10 4 rotations per minute. How many
rotations does it make in one hour? Write your
answer in scientific notation. (Lesson 7-3)
6. Astronomy Earth travels approximately
5.8 × 10 8 miles as it makes one orbit of
the Sun. How far does Earth travel in
50 years? (Note: One year is one orbit of
the Sun.) Write your answer in scientific
notation. (Lesson 7-3)
7. Geography In 2005, the population of
Indonesia was 2.4 × 10 8. This was 8 times
the population of Afghanistan. What was the
population of Afghanistan in 2005? Write your
answer in standard form. (Lesson 7-4)
8. The Golden Gate Bridge weighs about
1.8 × 10 9 lb. The Eiffel Tower weighs about
2.25 × 10 7 lb. How many times heavier is the
Golden Gate Bridge than the Eiffel Tower?
Write your answer in standard form.
(Lesson 7-4)
10. A rock is thrown off a 220-foot cliff with an
initial velocity of 50 feet per second. The
height of the rock above the ground is given
by the polynomial -16t 2 - 50t + 220, where t
is the time in seconds after the rock has been
thrown. What is the height of the rock above
the ground after 2 seconds? (Lesson 7-6)
11. The sum of the first n natural numbers is
given by the polynomial __12 n 2 + __12 n. Use this
polynomial to find the sum of the first 9
natural numbers. (Lesson 7-6)
12. Biology The population of insects in a
meadow depends on the temperature. A
biologist models the population of insect A
with the polynomial 0.02x 2 + 0.5x + 8 and the
population of insect B with the polynomial
0.04x 2 - 0.2x + 12, where x represents the
temperature in degrees Fahrenheit.
(Lesson 7-7)
a. Write a polynomial that represents the
total population of both insects.
b. Write a polynomial that represents the
difference of the populations of insect B
and insect A.
13. Geometry The length of the rectangle shown
is 1 inch longer than 3 times the width.
a. Write a polynomial that represents the area
of the rectangle.
b. Find the area of the rectangle when the
width is 4 inches. (Lesson 7-8)
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ÎÝÊÊ£
14. A cabinet maker starts with a square piece
of wood and then cuts a square hole from
its center as shown. Write a polynomial that
represents the area of the remaining piece of
wood. (Lesson 7-9)
9. Carl has 4 identical cubes lined up in a row
and wants to find the total length of the cubes.
He knows that the volume of one cube is
1
_
343 in3. Use the formula s = V 3 to find the
length of one cube. What is the length of the
row of cubes? (Lesson 7-5)
S34
Extra Practice
ÝÊÊÎ
ÝÊÊÈ
Chapter 8
Applications Practice
1. Ms. Andrews’s class has 12 boys and 18 girls.
For a class picture, the students will stand in
rows on a set of steps. Each row must have the
same number of students, and each row will
contain only boys or girls. How many rows will
there be if Ms. Andrews puts the maximum
number of students in each row? (Lesson 8-1)
2. A museum director is planning an exhibit of
Native American baskets. There are 40 baskets
from North America and 32 baskets from
South America. The baskets will be displayed
on shelves so that each shelf has the same
number of baskets. Baskets from North and
South America will not be placed together
on the same shelf. How many shelves will
be needed if each shelf holds the maximum
number of baskets? (Lesson 8-1)
3. The area of a rectangular painting is
(3x 2 + 5x) ft 2. Factor this polynomial to find
possible expressions for the dimensions of the
painting. (Lesson 8-2)
4. Geometry The surface area of a cylinder with
radius r and height h is given by the expression
2πr 2 + 2πrh. Factor this expression.
(Lesson 8-2)
5. The area of a rectangular classroom in square
feet is given by x 2 + 9x + 18. The width of the
classroom is (x + 3) ft. What is the length of
the classroom? (Lesson 8-3)
8. A rectangular poster has an area of
(6x 2 + 19x + 15) in 2. The width of the poster
is (2x + 3) in. What is the length of the
poster? (Lesson 8-4)
9. Physics The height of an object thrown
upward with a velocity of 38 feet per second
from an initial height of 5 feet can be modeled
by the polynomial -16t 2 + 38t + 5, where t
is the time in seconds. Factor this expression.
Then use the factored expression to find the
object’s height after __12 second. (Lesson 8-4)
10. A rectangular pool has an area of
(9x 2 + 30x + 25) ft 2. The dimensions of the
pool are of the form ax + b, where a and b are
whole numbers. Find an expression for the
perimeter of the pool. Then find the perimeter
when x = 5. (Lesson 8-5)
11. Geometry The area of a square is
9x 2 - 24x + 16. Find the length of each side of
the square. Is it possible for x to equal 1 in this
situation? Why or why not? (Lesson 8-5)
Architecture Use the following information for
Exercises 12–14.
An architect is designing a rectangular hotel room.
A balcony that is 5 feet wide runs along the length
of the room, as shown in the figure. (Lesson 8-6)
ÓÝÊvÌ
xÊvÌ
Gardening Use the following information for
Exercises 6 and 7.
A rectangular flower bed has a width of (x + 4) ft.
The bed will be enlarged by increasing the length,
as shown. (Lesson 8-3)
­ÝÊÊ{®ÊvÌ
6. The original flower bed has an area of
(x 2 + 9x + 20) ft 2. What is its length?
12. The area of the room, including the balcony,
is (4x 2 + 12x + 5) ft 2. Tell whether the
polynomial is fully factored. Explain.
13. Find the length and width of the room
(including the balcony).
14. How long is the balcony when x = 9?
7. The enlarged flower bed will have an area of
(x 2 + 12x + 32) ft 2. What will be the new
length of the flower bed?
Extra Practice
S35
Chapter 9
Applications Practice
1. The table shows the height of a ball at
various times after being thrown into the
air. Tell whether the function is quadratic.
Explain. (Lesson 9-1)
Time (s)
0
0.5
1
1.5
2
Height (ft)
4
20
28
28
20
2. The height of the curved roof of a camping
tent can be modeled by f (x) = -0.5x 2 + 3x,
where x is the width in feet. Find the height of
the tent at its tallest point. (Lesson 9-2)
3. Engineering A small bridge passes over
a stream. The height in feet of the bridge’s
curved arch support can be modeled by
f (x) = -0.25x 2 + 2x + 1.5, where the x-axis
represents the level of the water. Find the
greatest height of the arch support.
(Lesson 9-2)
4. Sports The height in meters of a football that
is kicked from the ground is approximated
by f (x) = -5x 2 + 20x, where x is the time in
seconds after the ball is kicked. Find the ball’s
maximum height and the time it takes the ball
to reach this height. Then find how long the
ball is in the air. (Lesson 9-3)
5. Physics Two golf balls are dropped, one from
a height of 400 feet and the other from a height
of 576 feet. (Lesson 9-4)
a. Compare the graphs that show the time it
takes each golf ball to reach the ground.
b. Use the graphs to tell when each golf ball
reaches the ground.
6. A model rocket is launched into the air with
an initial velocity of 144 feet per second. The
quadratic function y = -16x 2 + 144x models
the height of the rocket after x seconds. How
long is the rocket in the air? (Lesson 9-5)
7. A gymnast jumps on a trampoline. The
quadratic function y = -16x 2 + 24x models
her height in feet above the trampoline after
x seconds. How long is the gymnast in the
air? (Lesson 9-5)
S36
Extra Practice
8. A child standing on a rock tosses a ball into the
air. The height of the ball above the ground is
modeled by h = -16t 2 + 28t + 8, where h is
the height in feet and t is the time in seconds.
Find the time it takes the ball to reach the
ground. (Lesson 9-6)
9. A fireworks rocket is launched from
the edge of a rooftop. The height of the
rocket above the ground is modeled by
h = - 16t 2 + 40t + 24, where h is the height in
feet and t is the time in seconds. Find the time
it takes the rocket to hit the ground.
(Lesson 9-6)
10. Geometry The base of the triangle in the
figure is five times the height. The area of
the triangle is 400 in 2. Find the height of the
triangle to the nearest tenth. (Lesson 9-7)
Ý
xÝ
11. The length of a rectangular swimming pool is
8 feet greater than the width. The pool has an
area of 240 ft 2. Find the length and width of
the pool. (Lesson 9-8)
12. Geometry One base of a trapezoid is 4 ft
longer than the other base. The height of the
trapezoid is equal to the shorter base. The
trapezoid’s area is 80 ft 2. Find the height.
Hint: A = __12 h(b 1 + b 2) (Lesson 9-8)
(
)
Ý
Ý
ÝÊÊ{
13. A referee tosses a coin into the air at the start
of a football game to decide which team will
get the ball. The height of the coin above the
ground is modeled by h = -16t 2 + 12t + 4,
where h is the height in feet and t is the time in
seconds after the coin is tossed. Will the coin
reach a height of 8 feet? Use the discriminant
to explain your answer. (Lesson 9-9)
Chapter 10
Applications Practice
Geography Use the following information for
Exercises 1–3.
The bar graph shows the areas of the Great
Lakes. (Lesson 10-1)
7. Use the data to make a box-and-whisker plot.
8. The weekly salaries of five employees at a
restaurant are $450, $500, $460, $980, and
$520. Explain why the following statement
is misleading: “The average salary is
$582.” (Lesson 10-4)
Ài>ÃʜvÊ̅iÊÀi>ÌÊ>ŽiÃ
9. The graph shows the sales figures for three
sales representatives. Explain why the graph
is misleading. What might someone believe
because of the graph? (Lesson 10-4)
>ŽiÊ"˜Ì>Àˆœ
>ŽiʈV…ˆ}>˜
->iÃÊvœÀÊ"V̜LiÀ
˜
Ã
iÀ
φ
À
1. Estimate the difference in the areas between
the lake with the greatest area and the lake
with the least area.
˜`
Îä]äää
“
Óä]äää
Ài>Ê­“ˆÓ®
£ä]äää
ˆ>
ä
£n]äää
£Ç]äää
£È]äää
£x]äää
£{]äää
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->iÃÊ­f®
>ŽiÊÕÀœ˜
7
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->iÃÊ,i«ÀiÃi˜Ì>̈Ûi
2. Estimate the total area of the five lakes.
3. Approximately what percent of the total area is
Lake Superior?
4. The scores of 18 students on a Spanish exam
are given below. Use the data to make a stemand-leaf plot. (Lesson 10-2)
Exam Scores
65
94
92
75
71
83
77
73
91
82
63
79
80
77
99
76
80
88
5. The numbers of customers who visited a hair
salon each day are given below. Use the data to
make a frequency table with intervals.
(Lesson 10-2)
Number of Customers Per Day
32
35
29
44
41
25
35
40
41
32
33
28
33
34
Sports Use the following information for
Exercises 6 and 7.
The numbers of points scored by a college football
team in 11 games are given below. (Lesson 10-3)
10. A manager inspects 120 stereos that were built
at a factory. She finds that 6 are defective. What
is the experimental probability that a stereo
chosen at random will be defective?
(Lesson 10-5)
Travel Use the following information for
Exercises 11–13.
A row of an airplane has 2 window seats, 3 middle
seats, and 4 aisle seats. You are randomly assigned
a seat in the row. (Lesson 10-6)
11. Find the probability that you are assigned a
window seat.
12. Find the odds in favor of being assigned a
window seat.
13. Find the probability that you are not assigned
a middle seat.
14. A class consists of 19 boys and 16 girls. The
teacher selects one student at random to
be the class president and then selects a
different student to be vice president. What
is the probability that both students are
girls? (Lesson 10-7)
10 17 17 14 21 7 10 14 17 17 21
6. Find the mean, median, mode, and range of
the data set.
Extra Practice
S37
Chapter 11
Applications Practice
1. Scientists who are developing a vaccine track
the number of new infections of a disease each
year. The values in the table form a geometric
sequence. To the nearest whole number, how
many new infections will there be in the 6th
year? (Lesson 11-1)
Year
Number of New
Infections
1
12,000
2
9000
3
6750
a. Find the investment’s value after 5 years.
b. Approximately how many years will it take
for the investment to be worth $3100?
3. Chemistry Cesium-137 has a half-life of
30 years. Find the amount left from a 200-gram
sample after 150 years. (Lesson 11-3)
4. The cost of tuition at a dance school is $300
a year and is increasing at a rate of 3% a year.
Write an exponential growth function to model
the situation and find the cost of tuition after
4 years. (Lesson 11-3)
5. Use the data in the table to describe how the
price of the company’s stock is changing. Then
write a function that models the data. Use your
function to predict the price of the company’s
stock after 7 years. (Lesson 11-4)
Stock Prices
Price ($)
0
1
2
3
10.00
11.00
12.20
13.31
6. Use the data in the table to describe the rate
at which Susan reads. Then write a function
that models the data. Use your function to
predict the number of pages Susan will read in
6 hours. (Lesson 11-4)
Total Number of Pages Read
S38
Time (h)
1
2
3
4
Pages
48
96
144
192
Extra Practice
8. Geometry Given the surface area, S, of a
S
sphere, the formula r = ___
can be used to
4π
find the sphere’s radius. What is the radius of
a sphere with a surface area of 100 m 2? Use
3.14 for π. Round your answer to the nearest
hundredth of a meter. (Lesson 11-5)
√
2. Finance For a savings account that earns
5% interest each year, the function
x
f (x) = 2000(1.05) gives the value of a
$2000 investment after x years. (Lesson 11-2)
Year
gives the
7. The function f (x) = √1.44x
approximate distance in miles to the horizon
as observed by a person whose eye level is x
feet above the ground. Jamal stands on a tower
so that his eyes are 180 ft above the ground.
What is the distance to the horizon? Round
your answer to the nearest tenth.
(Lesson 11-5)
9. Cooking A chef has a square baking pan with
sides 8 inches long. She wants to know if an
11-inch fish can fit in the pan. Find the length
of the diagonal of the pan. Give the answer as
a radical expression in simplest form. Then
estimate the length to the nearest tenth of
an inch. Tell whether the fish will fit in the
pan. (Lesson 11-6)
10. Alicia wants to put a fence around the irregular
garden plot shown. Find the perimeter of the
plot. Give your answer as a radical expression
in simplest form. (Lesson 11-7)
ÊȖе
£ÓÊÊ
е“
ÊȖе
ÓÇÊÊ
е“
е
ÊȖÎÊÊ
“
ÊȖе
ÇxÊÊ
е“
11. Physics The velocity of an object in meters
√
2 √
E
per second is given by _____
, where E is kinetic
√
m
energy in Joules and m is mass in kilograms.
What is the velocity of an object that has
40 Joules of kinetic energy and a mass of
10 kilograms? Give the answer as a radical
expression in simplest form. Then estimate
the velocity to the nearest tenth.
(Lesson 11-8)
12. A rectangular window has an area of 40 ft 2.
The window is 8 feet long and its height
is √
x + 2 ft. What is the value of x? What is the
height of the window? (Lesson 11-9)
Chapter 12
Applications Practice
1. The inverse variation xy = 200 relates the
number of words per minute x at which a
person types to the number of minutes y
that it takes to type a 200-word paragraph.
Determine a reasonable domain and range
and then graph this inverse variation. Use the
graph to estimate how many minutes it would
take to type the paragraph at a rate of 60 words
per minute. (Lesson 12-1)
7. A committee consists of five more women than
men. The chairperson randomly chooses one
person to serve as secretary and a different
person to serve as treasurer. Write and simplify
an expression that represents the probability
that both people who are chosen are men.
What is the probability of choosing two men
if there are 6 men on the committee?
(Lesson 12-4)
2. Business The owner of a deli finds that the
number of sandwiches sold in one day varies
inversely as the price of the sandwiches. When
the price is $4.50, the deli sells 60 sandwiches.
How many sandwiches can the owner expect
to sell when the price is $3.60? (Lesson 12-1)
8. Transportation A delivery truck makes a
delivery to a town 150 miles away traveling
r miles per hour. On the return trip, the
delivery truck travels 20% faster. Write and
simplify an expression for the truck’s roundtrip delivery time in terms of r. Then find the
round-trip delivery time if the truck travels 55
mi/h on its way to the delivery. (Lesson 12-5)
3. A gardener has $30 in his budget to buy
packets of seeds. He receives 3 free packets of
seeds with his order. The number of packets
30
y he can buy is y = __
x + 3, where x is the
price per packet. Describe the reasonable
domain and range values. Then graph the
function. (Lesson 12-2)
4. Ashley wants to save $1000 for a trip to Europe.
She puts aside x dollars per month, and her
grandmother contributes $10 per month.
The number of months y it will take to save
1000
$1000 is y = _____
. Describe the reasonable
x + 10
domain and range values. Then graph the
function. (Lesson 12-2)
5. Geometry Find the ratio of the area of a
circle to the circumference of the circle. (Hint:
For a circle, A = πr 2 and C = 2πr). For what
radius is this ratio equal to 1? (Lesson 12-3)
6. Geometry For a cylinder with radius r and
height h, the volume is V = πr 2h, and the
surface area is S = 2πr 2 + 2πrh. What is the
ratio of the volume to the surface area for a
cylinder? What is this ratio when r = h = 1?
(Lesson 12-3)
À
…
9. Recreation Jordan is hiking 2 miles to a vista
point at the top of a hill and then back to his
campsite at the base of the hill. His downhill
rate is 3 times his uphill rate, r. Write and
simplify an expression in terms of r for the
time that the round-trip hike will take. Then
find how long the hike will take if Jordan’s
uphill rate is 2 mi/h. (Lesson 12-5)
10. Geometry The volume of a rectangular prism
is the area of the base times the height. A
rectangular prism has a volume given by
(2x 2 + 7x + 5) cm 3 and a height given by
(x + 1) cm. What is the area of the base of the
rectangular prism? (Lesson 12-6)
11. Tanya can deliver newspapers to all of the
houses on her route in 1 hour. Her brother,
Nick, can deliver newspapers along the
same route in 2 hours. How long will it
take to deliver the newspapers if they work
together? (Lesson 12-7)
12. Agriculture Grains are harvested using a
combine. A farm has two combines—one that
can harvest the wheat field in 9 hours and
another that can harvest the wheat field in
11 hours. How long will it take to harvest the
wheat field using both combines?
(Lesson 12-7)
Extra Practice
S39
Problem Solving Handbook
Draw a Diagram
Problem Solving Strategies
You can draw a diagram to help you visualize what
the words of a problem are describing.
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
EXAMPLE
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
A gardener wants to plant a 2.5-foot-wide border of
flowers around a rectangular herb garden. The herb
garden is 12 feet long and 7.5 feet wide. What is the
area of the border?
1
Understand the Problem
You need to find the area of the garden’s border. You are given the garden’s
dimensions and the width of the border.
2 Make a Plan
Draw and label a diagram of the herb garden with the surrounding border. Find the
dimensions of the outer rectangle. Then find the area of the inner rectangle and
subtract to find the area of the border.
3 Solve
length of outer rectangle: 2.5 ft + 12 ft + 2.5 ft = 17 ft
width of outer rectangle: 2.5 ft + 7.5 ft + 2.5 ft = 12.5 ft
Ó°xÊvÌ
Ó°xÊvÌ
£ÓÊvÌ
Ó°xÊvÌ
Find the area of each rectangle:
area of outer rectangle: 17 ft × 12.5 ft = 212.5
area of inner rectangle: 12 ft × 7.5 ft = 90 ft2
ft2
Ç°xÊvÌ
Subtract:
area of border: 212.5
ft2
- 90
ft2
= 122.5
ft2
4 Look Back
Ó°xÊvÌ
To check your answer, solve the problem in a different way.
Divide the border into four parts and find the area of
each part. Then add the areas.
17 ft × 2.5 ft = 42.5 ft 2
7.5 ft × 2.5 ft = 18.75 ft 2
17 ft × 2.5 ft = 42.5 ft
7.5 ft × 2.5 ft = 18.75 ft
£ÇÊvÌ
Ó°xÊvÌ
Ó°xÊvÌ
2
2
42.5 ft 2 + 42.5 ft 2 + 18.75 ft 2 + 18.75 ft 2 + = 122.5 ft2
PRACTICE
1. A circular fish pond is surrounded by a circular border of stones that is 18 inches wide.
The fish pond is 4 feet in diameter. What is the area of the border? (Use 3.14 for π.)
2. Thirty-two teams are in the first round of a softball tournament. A team is eliminated
as soon as it loses a game. How many games need to be played to determine the
winner? (Hint: Use a tree diagram.)
PS2
Problem Solving Handbook
Ç°xÊvÌ
Make a Model
You can make a model, or representation of the
objects in a problem, to help you solve it.
EXAMPLE
Problem Solving Strategies
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
Mr. Duncan is using blue and white square tiles to create a pattern on his kitchen
wall. The entire design will have 8 rows with 15 tiles in each row. The bottom row
alternates colors starting with blue, and the row above that alternates colors
starting with white. He will continue this alternating pattern so that the same two
colors are never next to each other. How many of each color tile does Mr. Duncan
need to complete the entire design?
1
Understand the Problem
You need to find how many of each color tile are needed. You know the number of
rows and the number of tiles in each row. The colors alternate so that the same two
colors are never next to each other.
2 Make a Plan
Use blocks (preferably blue and white, but any two colors would work) to make a
model of the first two rows. Count how many of each color you use. Then multiply to
find how many of each color would be used in the entire design.
3 Solve
Create the bottom row. Start with a blue block and alternate
colors across the row until you have used 15 blocks.
Create the row above the bottom row. Start with a white block.
You could build all 8 rows
and just count the number of
each color, but each group
of two rows will be the
same, so this way is quicker.
There will be a total of 8 rows: 4 that start with blue and 4 that start with white.
Count how many of each color are used above and multiply each number by 4.
blue: 15 × 4 = 60
white: 15 × 4 = 60
Mr. Duncan needs 60 blue tiles and 60 white tiles.
4 Look Back
The grid is 15 units by 8 units, so there are 15 × 8 = 120 squares in the grid. Add the
number of blue and white tiles to see if the sum is 120: 60 + 60 = 120.
PRACTICE
1. Mr. Duncan decides to tile another area of his kitchen wall. This design will have 12 rows
with 10 tiles in each row. The bottom row will repeat this pattern: blue, white, blue, blue,
white. The row above the bottom row will repeat this pattern: white, green, white, white,
green. He will use these two patterns for each of the remaining rows so that the first colors
of each row always alternate. How many of each color tile will Mr. Duncan need?
Problem Solving Handbook
PS3
Guess and Test
Problem Solving Strategies
The guess and test strategy can be used when you
cannot think of another way to solve the problem.
Begin by making a reasonable guess, and then test it
to see whether your guess is correct. If not, adjust the
guess accordingly and test again. Keep guessing and
testing until you correctly solve the problem.
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
EXAMPLE
The manager of a college computer lab purchased 24 printers at a total cost of
$3120. Some of the printers were laser, and some were ink jet. The laser printers
cost $250 each, and the ink jet printers cost $70 each. How many of each type of
printer did the manager purchase?
1
Understand the Problem
You know the cost of each type of printer and the total number of printers.
You need to find the number of each type of printer purchased.
2 Make a Plan
Make reasonable first guesses for each type of printer. The sum must be 24. Then
multiply each guess by the cost of each printer. Find the total and compare it to
$3120. Adjust the guess as needed and continue until you find the solution.
3 Solve
Use a table to organize your guesses.
Laser
Printers
Ink Jet
Printers
Total
Priners
1st guess
12
12
24
12($250) + 12($70)
$3000 + $840 = $3840
Too high—try fewer laser
printers.
2nd guess
6
18
24
6($250) + 18($70)
$1500 + $1260 = $2760
Too low—try more laser
printers.
3rd guess
8
16
24
8($250) + 16($70)
$2000 + $1120 = $3120
Correct!
Total Cost
The manager purchased 8 laser printers and 16 ink jet printers.
4 Look Back
The total spent is $3120, and the total number of printers is 24. The solution
is correct.
PRACTICE
1. All 350 seats were sold for a concert in the park. Adult tickets cost $15, and
child tickets cost $5. Ticket sales totaled $4350. How many of each type of
ticket were sold?
2. Jane is 3 times as old as Theo. Luke is 5 years older than Theo. Zoe is 8 years
younger than twice Theo’s age, and Cassie is 6 years younger than Theo. The sum
of their ages is 71. How old is each person?
PS4
Problem Solving Handbook
Work Backward
You can work backward to solve a problem when
you know the ending value and are asked to find the
initial value.
EXAMPLE
Problem Solving Strategies
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
Lee Ann is taking a vacation in Paris, France. Her flight arrived in Paris at 9:35 A.M.
on Tuesday. The plane left New York City and flew for 7 hours and 55 minutes to
Nice, France, where there was a layover of 1 hour 12 minutes. From Nice the plane
flew 1 hour and 25 minutes to Paris. Paris time is 6 hours ahead of New York City
time. What time did the plane leave New York City?
1
Understand the Problem
You are asked to find the time that the plane left New York City. You know when the
flight arrived in Paris, the length of the stops that were made along the way, and the
time difference between New York City and Paris.
2 Make a Plan
Work backward from the time the plane arrived in Paris, using inverse operations.
Then apply the time difference between the two cities.
3 Solve
Subtract the length of time it took to fly from Nice to Paris from the time Lee Ann
arrived in Paris.
9:35 A.M. - 1 hour 25 minutes = 8:10 A.M.
Subtract the length of the layover in Nice.
8:10 A.M. - 1 hour 12 minutes = 6:58 A.M.
Subtract the length of the flight from New York to Nice.
6:58 A.M. – 7 hours 55 minutes = 11:03 P.M. Monday
Since Paris time is ahead of New York time, subtract the time difference.
11:03 P.M., Monday - 6 hours = 5:03 P.M. Monday
Lee Ann’s flight left New York City on Monday at 5:03 P.M.
4 Look Back
Work forward to check your answer.
5:03 P.M. Monday + 6 h + 7 h 55 min + 1 h 12 min + 1 h 25 min
= 5:03 P.M. Monday + 16 h 32 min
= 9:35 A.M. Tuesday
This matches the information given in the problem.
PRACTICE
1. A bus arrives in Dallas, Texas, at 10:59 A.M. on Friday. The bus left Atlanta,
Georgia, and took 12 hours and 15 minutes to arrive in Shreveport, Louisiana,
where there was a 45-minute layover. From Shreveport it took 4 hours and 29
minutes to get to Dallas. Dallas time is 1 hour behind Atlanta time. What time did
the bus leave Atlanta?
2. Carolina bought a DVD player that was on sale for 90% of the original price. The
total amount she paid was $135.72, which included a sales tax of $5.22. What was
the original price of the DVD player?
Problem Solving Handbook
PS5
Find a Pattern
Problem Solving Strategies
If a problem involves a sequence of numbers or
figures, it is often necessary to find a pattern to
solve the problem.
EXAMPLE
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
Darian created the following sequence of stars:
How many stars will be in the 6th figure?
1
Understand the Problem
You need to find the number of stars in the 6th figure. You can find the number in
the first four figures by counting.
2 Make a Plan
Count the number of stars in each of the first four figures. Use the information to
find a pattern and determine a general rule.
3 Solve
Look for a pattern between the position of each figure in the sequence and the
number of stars in that figure.
Position
1
2
3
4
Stars
2
6
12
20
The number of stars is the square of the
position number plus the position number.
This rule written algebraically is n 2 + n.
Evaluate the expression for n = 6: n 2 + n
6 2 + 6 = 36 + 6 = 42
There will be 42 stars in the 6th figure.
4 Look Back
Look for another pattern. The number of stars in each position increases by 4, then
by 6, then by 8. That is, the amount of increase always increases by 2. So the number
of stars in the 5th position will be 20 + 10, or 30, and the number of stars in the 6th
position will be 30 + 12, or 42.
PRACTICE
1. The first three figures of a pattern are shown.
How many circles will be in the 10th figure?
PS6
Problem Solving Handbook
2. Lily drew the first four figures of a pattern.
How many squares will be in the 7th figure?
Make a Table
Problem Solving Strategies
You can make a table to solve problems because
the rows and columns can help you arrange
information. Sometimes this also allows you to discover
relationships that might otherwise be hard to notice.
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
EXAMPLE
A scientist begins a culture with 500 bacteria. The number of bacteria triples every
1
30 minutes. How many bacteria are in the culture after 2 __
hours?
2
1
Understand the Problem
You are asked to find the number of bacteria in the culture after 2 __12 hours.
You know the initial number of bacteria, and you know that the population
triples every half hour.
2 Make a Plan
Make a table with rows for time and number of bacteria. Start with the initial
number in the culture. Increase the time in 30-minute increments and triple the
number of bacteria with each increase. Keep extending the table until the time is
2 __12 hours (150 minutes).
3 Solve
Time (min)
Bacteria
0
30
60
90
120
150
500
1500
4500
13,500
40,500
121,500
There are 121,500 bacteria in the culture after 2 __12 hours.
4 Look Back
Check your answer by solving a simpler problem. The number of bacteria in the
culture triples five times (150 min ÷ 30 min = 5). Start with 5 instead of 500 and triple
the number five times.
5 × 3 = 15
15 × 3 = 45
45 × 3 = 135
135 × 3 = 405
405 × 3 = 1215
Multiply by 100 to find the total if you had started with 500; 1215 × 100 = 121,500
PRACTICE
1. A dietician’s report states that a 125-pound woman needs to eat about 1750
Calories a day to maintain her weight. It also states that a 132-pound woman
needs 1848 Calories and a 139-pound woman needs 1946 Calories a day. Based
on these values, how many Calories does a 160-pound woman need to eat each
day to maintain her weight?
2. Simon opened a savings account with an initial deposit of $200. At the end of
each year, the account earns 4% interest. If he does not deposit or withdraw any
additional money, what would his balance be at the end of 6 years?
Problem Solving Handbook
PS7
Solve a Simpler Problem
Sometimes a problem contains numbers that make
it seem difficult to solve. You can solve a simpler
problem by rewriting the numbers so they are easier
to compute.
Problem Solving Strategies
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
EXAMPLE
During a skating competition, Jules skated around the track 35 times. One lap is
0.9 mile. If Jules finished in 1 hour 30 minutes, what was his average speed?
1
Understand the Problem
You are asked to find Jules’s average speed for 35 laps. You know the distance of each
lap and the amount of time it took him to finish the competition.
2 Make a Plan
Solve a simpler problem by using easier numbers to do the computations.
3 Solve
Find the total distance skated.
35(0.9)
There were 35 laps that measured 0.9 mile.
35(1 - 0.1)
Write 0.9 as 1 - 0.1
35(1) - 3.5(0.1)
Use the Distributive Property.
35 - 3.5
31.5
Use the distance formula to find the average speed.
d = rt
1 hour 30 minutes = 1.5 hours
31.5 = r × 1.5
31.5 = r
_
Solve for r.
1.5
315 = r
Multiply the numerator and denominator by 10
_
15
to eliminate the decimals.
1 (315) = r
_
15
1 (300 + 15) = r
_
Write 315 as 300 + 15.
15
1 (300) + _
1 (15) = r
_
Use the Distributive Property.
15
15
20 + 1 = r
21 = r
Jules skated at an average speed of 21 miles per hour.
4 Look Back
Each lap is a little less than 1 mile, so 35 laps is a little less than 35 miles. Round this
distance to 30 miles and use d = rt to find the rate when the time is 1.5 hours:
30 mi = (1.5 h)r r = 20 mi/h. This is close to 21 mi/h.
PRACTICE
1. Diana swam 24 laps in the pool today. One lap is 200 feet. She swam at an
average rate of 4 feet per second. How many minutes did Diana swim?
PS8
Problem Solving Handbook
Use Logical Reasoning
Problem Solving Strategies
Use logical reasoning to solve problems when
you are given several facts and can use common sense
to find a missing fact.
EXAMPLE
1
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
Five players on a baseball team wear the numbers 2, 12, 15, 34, and 42. Their
positions are pitcher, catcher, first base, left field, and center field. The pitcher’s
number is less than the left fielder’s number. The center fielder’s number is greater
than 25, and the left fielder wears an even number. The catcher wears number 34.
What is the pitcher’s number?
1
Understand the Problem
You want to find the jersey number of the pitcher. You know there are five positions
and five jersey numbers. Some information about who wears which number is given.
2 Make a Plan
Organize the information in a table. Start with the fact that the catcher wears
number 34 and use logical reasoning to determine the numbers of the other
positions.
3 Solve
The catcher wears number 34. No other player wears 34, and the catcher wears no
other number. Enter Y’s and N’s in the chart as shown.
The center fielder’s number is greater than 25, so he must wear number 42.
The left fielder cannot wear number 15 (because it is odd), and he cannot have the
least number (the pitcher’s number is less than his). The left fielder must wear 12.
The pitcher’s number is less than 12 (the left fielder’s), so he must wear number 2.
2
12
15
34
42
Pitcher
Y
N
N
N
N
Catcher
N
N
N
Y
N
First Base
N
N
Y
N
N
Left Fielder
N
Y
N
N
N
Center Fielder
N
N
N
N
Y
Y = yes; N = no
Once you enter Y in a cell, enter N in the
remaining cells for the row and the
column that include it.
The pitcher wears number 2.
4 Look Back
Complete the chart if needed. Read the problem while looking at the chart to make
sure there are no contradictions.
PRACTICE
1. Rose, Jill, Gaby, and Chloe bowled the scores 110, 125, 144, and 150. Jill did not
bowl the 110. The person who bowled the 150 is Rose’s sister and Jill’s aunt. Chloe
bowled the 125. What score did Jill bowl?
Problem Solving Handbook
PS9
Use a Venn Diagram
You can use a Venn diagram to display
relationships among sets of numbers. Circles are used
to represent the individual sets.
EXAMPLE
Problem Solving Strategies
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
At a local supermarket, 194 people were given samples of two brands of orange
juice. Their opinions were as follows: 120 people liked brand A, 101 people liked
brand B, and 15 people did not like either brand. How many people liked only
brand A?
1
Understand the Problem
The total number of people was 194, and 15 of them did not like either brand. The
statement “120 people liked brand A” means some of the 120 people liked only brand A
and some liked brand A and brand B. The statement “101 people liked brand B” means
some of the 101 people liked only brand B and some liked brand A and brand B.
2 Make a Plan
Use a Venn diagram to show the relationship among the groups of people.
3 Solve
Draw and label two intersecting circles to show the sets of
people who liked brand A and brand B. Write 15 in the area
labeled “Neither.”
Out of 194 people, 15 liked neither brand. Subtract 15 from
194 to find how many people liked at least one brand:
194 - 15 = 179.
Add the number of people who liked brand A to the number
of people who liked brand B: 120 + 101 = 221. You know there
are only 179 people who liked at least one brand, so subtract
179 from 221: 221 - 179 = 42. This means 42 people were
counted twice, and that 42 people liked both brands. Write 42
in the area labeled both.
Out of 120 people who liked brand A, 42 also liked brand B.
Subtract 42 from 120 to find the number of people who liked
only brand A: 120 - 42 = 78.
So 78 people liked only brand A.
4 Look Back
À>˜`Ê
À>˜`Ê
œÌ…
iˆÌ…iÀ\Ê£x
À>˜`Ê
À>˜`Ê
œÌ…
Çn
iˆÌ…iÀ\Ê£x
Find the number of people who liked brand B only: 101 - 42 = 59. Add all the
numbers in the Venn diagram. The sum of the number who liked only brand A,
the number who liked only brand B, the number who liked both brands, and the
number who liked neither brand should be the total number of people surveyed:
78 + 59 + 42 + 15 = 194.
PRACTICE
In a group of 138 people, 55 own a cat, 27 own a cat and a dog, and 42 own
neither pet.
1. How many people own only a cat?
2. How many people own a dog?
PS10
Problem Solving Handbook
{Ó
x™
Make an Organized List
Problem Solving Strategies
If a problem asks you to find all the possible ways
in which something can happen, you can make an
organized list to keep track of the outcomes.
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
EXAMPLE
A fair coin is tossed 4 times. What is the probability that it lands heads up at least
3 times?
1
Understand the Problem
You need to find the probability that a coin tossed 4 times lands heads up 3 or 4
times.
2 Make a Plan
The formula for probability is:
number of favorable outcomes
probability = ___
total number of outcomes
The total number of outcomes is the number of items in the list. The number of
favorable outcomes is the number of times the coin lands heads up 3 or 4 times.
Make an organized list of the coin tosses to find the total number of outcomes.
3 Solve
Start with heads for all 4 tosses, then heads for the first 3 tosses, then heads for the
first 2 tosses, and then heads for the first toss. Repeat the pattern for tails.
HHHH
HTHH
TTTT
THTT
HHHT
HTHT
TTTH
THTH
HHTH
HTTH
TTHT
THHT
HHTT
HTTT
TTHH
THHH
There are 16 total
outcomes.
There are 5 favorable
outcomes.
5
The probability that the coin lands heads up 3 or 4 times is __
.
16
4 Look Back
Double-check that each combination is listed and that no combination is written
more than once. You can also use the Fundamental Counting Principle to check
the total number of outcomes. For each of the 4 coin tosses, there are 2 possible
outcomes, so the total number of outcomes is 2 × 2 × 2 × 2 = 16.
PRACTICE
1. A beagle, a fox terrier, an Afghan hound, and a golden retriever are competing in
the finals of a dog show. How many ways can the dogs finish in first, second, and
third place?
2. Two number cubes are rolled. What is the probability that the sum of the
numbers rolled is an odd number?
Problem Solving Handbook
PS11
Selected Answers
Chapter 1
81. c divided by d; the quotient of c
and d 83. __52 85. 280
1-3
Check It Out! 1a. -7 1b. 44
1-1
Check It Out! 1a. 4 decreased by
n; n less than 4 1b. the quotient of
t and 5; t divided by 5 1c. the sum
of 9 and q; q added to 9 1d. the
product of 3 and h; 3 times h
2a. 65t 2b. m + 5 2c. 32d 3a. 6
3b. 7 3c. 3 4. a. 63s, b. 756 bottles;
1575 bottles; 3150 bottles
Exercises 1. variable 3. the
quotient of f and 3; f divided
by 3 5. 9 decreased by y; y less
than 9 7. the sum of t and 12; t
increased by 12 9. x decreased by
3; the difference of x and 3
11. w + 4 13. 12 15. 6 17. the
product of 5 and p; 5 groups
of p 19. the sum of 3 and x; 3
increased by x 21. negative 3
times s; the product of negative 3
and s 23. 14 decreased by t; the
difference of 14 and t 25. t + 20
27. 1 29. 2 31a. h - 40, b. 0; 4; 8;
12 33. 2x 35. y + 10 37. 9w; 9 in 2;
72 in 2; 81 in 2; 99 in 2 39. 13; 14; 15;
16 41. 6; 10; 13; 15 43a. 47.84 + m;
b. 58.53 - s 45. x + 7; 19; 21
47. x + 3; 15; 17 49. F 51. 36
53. 1 55. 45° 57. 90° 59. __12 61. 1
63. Multiply the previous term by 3;
729, 2187, 6561.
1-2
Check It Out! 1a. 4 1b. -10
1c. 1.5 2a. -12 2b. -35.8
2c. -16 3a. -8 3b. 4 3c. -2
4. 13,018 ft
Exercises 1. opposite 3. -8.5
9
1
5. 9 __14 7. 1 9. __
11. -4.1 13. 1__
16
10
15. 4 17. -11 __34 19. -30 21. 14
23. -10 25. 13.4 27. 23 ˚F 29. 0.75
31. -12 __25 33. -12 35. 37 37. 0
1
39. __
41. > 43. > 45. <
10
47. 11,331 ft 49. never 51. A
55. F 57. -9 59. 2 61. Subtract 4;
-2, -6, -10 63. 12,660.5 ft
−
−−
65. 0.2 67. 0.36 69. 720˚
SA2
Selected Answers
1
1c. -42 2a. __
2b. - __14 2c. - __12
12
3a. 0 3b. undefined 3c. 0 4. 7.875 mi
Exercises 3. -121 5. 7 7. 2
9. undefined 11. 0 13. about
9
$210,000,000 15. -32 17. __
10
19. -3 21. 0 23. 0 25. -15 °F
27. -4 29. -62 31. 18.75
33. 1 35. -12 37. __32 39. negative
41. negative 43. positive
45. undefined 47. 1 49. __12
51. - __15 53. __98 55. 15 h per
semester 57. < 59. > 61. =
63a. positive b. negative c. The
product of two negative numbers
is positive. The product of that
positive number and a negative
1
number is negative. d. no 65. 75 __
15
1
67. -121 __
69. never 73. C
11
25
75. 16 quarter notes 77. __
49
27
__
79. 5 81. 1 83. - 64 85. Multiply by
-2; -16, 32, -64. 87. The numbers
are alternating positive and
negative multiples of 5; 30,
-35, 40. 89. $85 91. 1510 in2
93. < 95. =
( )
( )
1-4
Check It Out! 1a. 2 2 1b. x 3
27
2a. -125 2b. -36 2c. __
3a. 8 2
64
3
8
3b. (-3) 4. 2 = 256
Exercises 1. the number of times
to use the base as a factor 3. 2 3
5. 49 7. -32 9. 9 2 11. (-4)3
13. 3 4 15. 3 5 = 243 17. 3 3 19. 27
21. -16 23. 7 2 25. (-2) 3 27. 4 3
29. 2 4 = 16 31. < 33. = 35. =
1
37. > 39. 8 41. -64 43. -1 45. __
27
2
2
2
47a. 36 in b. 9 in c. 27 in 49. 6 2
3
51. (-1)4 53. __19
55. between
( )
8000 cm 3 and 15,625 cm 3 57. 2
59. 4 61. 2 63. 4 65a. 100, 1000,
10,000 b. The exponent is the
same as the number of zeros in
the answer. 67. C 69. B 71. 64
73. 65,536 75a. 4 · 4; 4 · 4 · 4
b. 4 · 4 · 4 · 4 · 4 = 4 5 c. 2 + 3 = 5;
the sum of the exponents in 4 2 and
4 3 is the exponent in the product 4 5.
77. 5 79. 5 minus x; x less than 5
1-5
Check It Out! 1a. 2 1b. -5 1c. 3
2a. __23 2b. __12 2c. -__27 3. 3.0 ft
4a. rational number, repeating
decimal 4b. rational number,
terminating decimal, integer
4c. irrational number 4d. natural
number, whole number, integer,
rational number, terminating
decimal
Exercises 3. -15 5. 5 7. -3
9. -4 11. __23 13. __38 15. __14 17. -__15
19. rational number, terminating
decimal, integer 21. irrational
number 23. 11 25. -10 31. 4.1 cm
33. rational number, terminating
decimal, integer, whole number,
natural number 35. irrational
number 37. > 39. < 41. 45;
rational number, terminating
decimal, integer, whole number,
natural number 43. 34.625; rational
number, terminating decimal
45. always 47. always 51. 18
53. A 55. D 57. 0.9 59. -0.1
3
61. 4 63. 65 65a. no 67. __12 69. -__
16
8
71. -3.5 73. -___
75. 64
125
1-6
Check It Out! 1a. 48 1b. 2.6
1c. 2 2a. 15 2b. 3 3a. 1 3b. -3
3c. 21 4. 6.2(9.4 + 8) 5. 400
Exercises 3. 15 5. -9
7. 14 9. 1 11. 14 13. 92 15. 1.5
17. -3 19. -22 21. 12(-2 + 6)
23. 188.4 ft2 25. 19 27. -15 29. 3
31. -5 33. 24 35. 17 37. -9
39. 17 41. -7 43. 0 45. __14 47. 1
49. 6 51. 3 - __25 53. 8 - ⎪3 · 5⎥
55a. 55 b. 498 c. 250 d. 10 e. 30
√7
f. 70 57. 2⎡⎣9 + (-x)⎤⎦ 59. ____
3 · 10
63. 3 · 5 - 6 · 2 = 3 69. H 71. -3
73. 6 77. 20 79. acute 81. 100
83. -11 85. 8 87. __67
1-7
Check It Out! 1a. 21 1b. 560
1c. 28 2a. 9(50) + 9(2) = 468
2b. 12(100) - 12(2) = 1176
2c. 7(30) + 7(4) = 238 3a. 100p
3b. -28.5t 3c. 3m 2 + m 3
4a. 6x - 15 4b. 3a - 16x
Exercises 1. Associative Property of
Addition 3. 24 5. 56 7. 118,000 9. 304
11. 456 13. 763 15. 20x 17. -9r
19. 7.9x 21. 9a - 31 23. 7x - 3x 2
25. 2a + 2 39. -3x - 14
43. 13y - 10 45a. Amy: 98:21;
Julie: 81:12; Mardi: 83:39; Sabine:
63:47 b. Sabine, Julie, Mardi, Amy
47. Commutative Property of
Addition 49. Distributive
Property 51. Distributive Property
53. 6p + 9 57a. equal b. 96π cm2
c. 2(16π) + 96π = 128π cm2
59. J 61. 12x + 116 63. -3b - 7
65a. Commutative Property of
Addition b. Associative Property
of Addition c. Distributive
Property d. Rule for subtraction
67. 36 ft 2 69. 64 71. -__18 73. 3
75. -2
1-8
Check It Out!
1a.
1b.
1c.
I (-2, 6) 8
4
-8
-4
0
-4
n
H (0, 2)
4
m
8
G (2, -3)
-8
2a. none 2b. I 2c. III 2d. II
3. y = 10 + 20x; (1, 30), (2, 50),
(3, 70), (4, 90) 4a. (-4, -6),
(-2, -5), (0, -4), (2, -3), (4, -2);
line 4b. (-3, 30), (-1, 6), (0, 3),
(1, 6), (3, 30); U shape 4c. (0, 2),
(1, 1), (2, 0), (3, 1), (4, 2); V shape
Exercises 7. none 9. none 11. I
13. (-2, 0), (-1, 1), (0, 2), (1, 3),
(2, 4); line 15. (-2, -4), (-1, -2),
(0, 0), (1, -2), (2, -4); V shape
21. none 23. none 25. II
27. y = 500 + 0.10x; (500, 550),
(3000, 800), (5000, 1000),
(7500, 1250) 29. (-2, -4), (-1, -1),
(0, 0), (1, -1), (2, -4); U shape
31. (-2, 7), (-1, 4), (0, 3), (1, 4),
(2, 7); U shape 33. triangle
35. rectangle 37a. c = 2.90f b. f is
input; c is output.
c.
f
c
1
2
3
4
5
6
7
8
2.90
5.80
8.70
11.60
14.50
17.40
20.30
23.20
d. 7 yards 39. y = __12 x + (-3);
(-4,-5), (-2, -4), (0, -3), (2, -2),
(4, -1); line 41a. y = 50 + 1.5x
b. (100, 200); (150, 275); (200, 350);
(250, 425); (300, 500) 43. line
45. line 51. G 53. H 57. (-4, 4)
59. The points make a horizontal
line at y = 6. 61. (-4, 5); 42 square
units 63. cylinder 65. pentagon
67. irrational 69. rational,
terminating decimal, integer
71. x 2 + 3x
Study Guide: Review
1. constant 2. whole numbers
3. coefficient 4. origin 5. 1.99g
6. t + 3 7. 5 8. 5 9. 6 10. 150 ÷ m;
30; 25; 15 11. -14 12. -4.6
13. 4 __12 14. -1 15. -24 16. 14.3
17. 2231 ft 18. 90 19. 0 20. -15.2
21. -8 22. 0 23. undefined 24. 9
15
25. - __23 26. __
27. 3,650,000 steps
7
28. 64 29. -27 30. 81 31. -25
8
16
32. __
33. __
34. 2 4 35. (-10)3
27
25
2
36. (-8) 37. 12 1 38. 729 in3 39. 6
40. 4 41. -7 42. -12 43. __56 44. __13
45. rational number, terminating
decimal, integer, whole number,
natural number 46. rational
number, terminating decimal,
integer, whole number 47. rational
number, terminating decimal,
integer 48. rational number,
terminating decimal 49. irrational
number 50. rational number,
repeating decimal 51. 3.6 ft 52. 23
53. 8 54. 6 55. __12 56. -18 57. 0
58. 62 59. 10 60. 8 61. 10
12
62. 8 + 7(-2) 63. ____
64. 4 √
20 - x
8+3
65. 168 ft 66. 40 67. 270
68. 13(100) + 13(3) = 1339
69. 18(100) - 18(1) = 1782 70. 4x
71. 7y 2 72. 4x + 24 73. 2x 2 + 2
74. -4y + 3y 2 75. 8y - a
76. $9
77–80.
B
y
C
A
x
D
81. I 82. IV 83. I 84. II 85. III
1
86. IV 87. y = p + __
p; ($2, $2.10);
20
($15, $15.75); ($30, $31.50); ($40,
(
)
$42.00) 88. (-4, 4), -1, __14 , (0, 0),
1, __14 , (4, 4); U shape
( )
Chapter 2
2-1
Check It Out! 1a. 8.8 1b. 0 1c. 25
2a. __12 2b. -10 2c. 8 3a. 9.3 3b. 2
3c. 44 4. 35 years old
Exercises 3. 21 5. 16.3 7. __12 9. 0
17
11. 2.3 13. 1.2 15. 32 17. 3.7 19. __
6
4
__
21. 9 23. 17 25. 7 27. 10.5 29. 9
31. 0 33. -17 35. -3100 37. -0.5
39. 0.05 41. 15 43. 1545 45. 30
47. __13 49. a + 500 = 4732; $4232
51. x - 10 = 12; x = 22 53. x + 8 =
16; x = 8 55. 5 + x = 6; x = 1
57. x - 4 = 9; x = 13 59. m + 560 =
1680; $1120 61. 63 + x = 90;
x = 27 63. x + 15 = 90; x = 75
12
65. h - 47 = 28; 75 69. J 71. - __
5
13
__
73. - 12 75. 10 77. 90 79. 9 81. 72
83. 6 ft 85. -80 87. -3
2-2
Check It Out! 1a. 50 1b. -39
1c. 56 2a. 4 2b. -20 2c. 5 3a. - __54
3b. 1 3c. 612 4. 15,000 ft
Exercises 1. 32 3. 14 5. 19 7. 7
9. 5 11. 2.5 13. 14 15. -9 17. __18
19. 16c = 192; $12 21. 24 23. -36
25. -150 27. 55 29. -3 31. 1
33. 13 35. 0.3 37. 2 39. -16 41. -3.5
7
43. -2 45. __
s = 392; $560
10
49. 4s = 84; 21 in. 51. 4s = 16.4;
4.1 cm 53. -3x = 12; x = -4
55. __3x = -8; x = -24 57. 6.25h = 50;
8 h 59. 0.05m = 13.80; 276 min
61. -2 63. 0; 8y = 0; 0 65a. number
of data values c. 185,300 acres
3
67. 7 69. 605 71. __
73. 5.7
16
2
__
75. 3 g = 2; 3 g 77. D 79. B
Selected Answers
SA3
81a. 6c = 4.80 b. c = $0.80 83. 2
85. 9 87. 2 89. -20 91. -132
93. Multiply both sides by a. 95. 12
97. 25 99. 6 years old 101. 6 103. 16
2-3
Check It Out! 1a. 1 1b. 6 1c. 0
55
2a. __
2b. __12 2c. 15 3a. - __56 3b. 5
4
3c. 8 4. $60 5. -42
Exercises 1. 2 3. -18 5. 2 7. 66
9. __54 11. -12 13. 16 15. -3.2 17. 4
19. 15 passes 21. 4 23. -4 25. 4
27. 5 29. -9 31. __14 33. 1 35. 3
28
37. __
39. 3 41. 8 43. 7 45. - __12
5
47. x = 40 49. x = 35
51. 8 - 3n = 2; n = 2
53a. 1963 - 5s = 1863; s = 20 53b. 3
55. 8 57. 4.5 59. -10 61. 10
63. 5k - 70 = 60; 26 in. 65. Stan: 36;
Mark: 37; Wayne: 38 67a. 45,000;
112,500; 225,000; 337,500; 225n
67b. c = 225n 71. H 73. 27 75. 6 __15
77. 14.5 79. -6 81. irrational
83. repeating decimal, rational
85. 8(60) + 8(1) = 488
87. 11(20) + 11(8) = 308
89. 13 91. -18
2-4
Check It Out! 1a. -2 1b. 2
2a. 4 2b. -2 3a. no solution
3b. all real numbers 4. 10 years old
Exercises 3. 1 5. 40 7. - __23
9. 3 11. no solution 13. all real
numbers 15. 6 17. 6 19. 2.85
21. 10 23. 6 25. 14 27. __34
29. -4 31. no solution 33a. 15 weeks
33b. 180 lb 35. x - 30 = 14 - 3x;
x = 11 37. -4 39. 7 41. -3 43. 2
45. 1 47. - __75 49. 4 51. no solution
53. 9 59. F 61. H 63. 2 65. no
solution 67. -20 69. 6, 7, 8 71. $1.68
73. 3y cm 75. -63 77. 4 79. 2
81. -125 83. 15 85. 3
2-5
Check It Out! 1. about 1.46 h
5-b
m
2. i = f + gt 3a. t = ____
3b. V = __
2
D
V
3. w = __
5. m = 4n + 8
Exercises
h
10
7. a = ____
9. I = A - P
b+c
k+5
x-2
____
13. ____
=y
11. x =
y
z
y-b
15. x = 5(a + g) 17. x = ____
m
PV
21. T = M + R
19. T = ___
nR
c - 2a
_____
25. r = 7 - ax
23. b =
2
SA4
Selected Answers
5 - 4y
t-g
27. x = _____
31. a = ______
3
-0.0035
(
)
is equal to the ratio of the
corresponding sides.
35. C 37. D 39. a = __52 c + __34 b
(
)
v -u
41. d = 500 t - __12 43. s = ______
2a
2
2
45. 120 s 47. 12 49. -6 51. 20 53. 12
2-6
Check It Out! 1a. -7, 7 1b. -5.5,
10.5 2a. no solutions 2b. 4
3. |x - 134 = 0.18; minimum
height: 133.82 m; maximum
height: 134.18 m
Exercises 1. – 6, 6 3. -2, 2
5. -__32 , __12 7. no solutions 9. no
solutions 11. 2.8 13. ⎪x - 207⎥ = 2;
mile markers 205 and 209 15. -9,
13 ft 17. -2, 2 19. 18.8, 65.28
14
21. -__
, 4 23. 0 25. 0 27. __23
3
29. ⎪x - 5⎥ = 0.001; 4.999 mm;
5.001 mm 31. ⎪x - 7⎥ = 2; 5, 9
33. ⎪x - 1500⎥ = 75; 1575 bricks;
1425 bricks 35. ⎪x⎥ = 3
37. ⎪x - 2⎥ = 3 39. sometimes
41. always 43a. ⎪t - 24⎥ = 5
43b. 19; 29 43c. yes 43d. The
measurements are correct to within
5 mi/h. 47. C 49. B 51. Division
Property of Equality; Subtraction
Property of Equality; Division
Property of Equality 53. 1__34 ft
55. -8__12 57. -1.4 59. T = S – R
61. w = xz - 3 63. N = 6M – S
Exercises 3. 10 ft 7. 7 in. 11. 480 ft2
13. 4 15. 2.8 ft 17. 4 cm
1.5
4.5
2
___
21. ___
x = 36 ; 12 m 23. k 25. G
27. w = 4; x = 7.5; y = 8
29. 16.6 cm 31. -12 33. -46
35. (-2, 4); (-1, 1); (0, 0); (1, 1);
(2, 4) 37. (-2, -7); (-1, -4); (0, -1);
(1, 2); (2, 5) 39. 32 41. 3.5
2-9
Check It Out! 1a. 12 1b. 16.8
1c. 1.44 2a. 20% 2b. 300% 3a. 75
3b. 320 4. 10 karats
Exercises 3. 21 5. 5.6 7. 80%
9. 12.5% 11. 175 13. 36 15. 48
17. 2.5 19. 25% 21. 50% 23. 40
25. 511.1 27. 100 mg 29. 2% 31. 8%
33. 64% 35. 85% 37. 85% 39. 0.52;
13
90
28
__
41. 90.0; ___
43. 1.12; __
25
100
25
3
3
47. 0.006; ___
49. 0.5 is
45. 0.06; __
50
500
greater than __12 % because __12 % = 0.005.
1
51. 0.001, 1%, __
, 11%, 1.1 53. 0.49,
10
5 __
4
__
,
,
82%,
0.94
55a.
40%
9 5
b. action c. 3% d. 36.9%
57. box 1: 200; 100; 50
box 2: 12; 24; 148; 96
box 3: 25; 50; 100; 200
x
40
59a. __
= ___
; $36 b. $54 61. F
90
100
63. G 65. 17.2% 67. 88.5 71. 120
73. 160 75. 6 in. 77. 3
2-7
Check It Out! 1. 12 2. $7.50/h
3. 20.5 ft/s 4a. -20 4b. 5.75
5. 6 in.
Exercises 1. The ratios are
equivalent. 3. 682 trillion 5. 18,749
lb/cow 7. 0.075 page/min
9. 18 mi/gal 11. __35 13. 39 15. 6.5
h
; 2.94 m 21. 72
17. 23 19. __35 = ___
4.9
23. $403.90/oz 25. 2498.4 km/h
27. 10 29. -1 31. 13 33. 1.2 35. __19
37. 45 39. $84 43. 1.625 45. 3
11
47. - __27 49. __
51. 3 53. 24
3
55. -120 59. A 61. D 63. 40°; 50°
65. 0.0006722 people/m2 67. -27
1
69. - __
71. 10 2 73. -5 75. 8
32
nRT
____
77. V = P
2-8
Check It Out! 1. 2.8 in.
150
45
5.5
3.5
___
___
___
2a. ___
x = 195 ; 650 cm 2b. x = 28 ;
44 ft 3. The ratio of the perimeters
2-10
Check It Out! 1. $462.80 2a. $270
2b. $7650 3a. about $3.30
3b. about $5.60
Exercises 3. $41,775 5. 4 __12 yr
7. about $6.45 9. $462.50
11. $266.75 13. 5 yr 15. about $30
17. $50,400 19. 2 yr 21. $2.89 25. A
27. D 29. 900 31. $47.17 33. $93
14
35. > 37. > 39. 12, 2 41. 2, __
3
43. 50% 45. 80
2-11
Check It Out! 1a. 45% decrease
1b. 20% increase 1c. 43.75% increase
2a. 90 2b. 6 3a. $88 3b. 20%
4a. $15.30 4b. 130%
Exercises 3. 20% decrease
5. 12.5% increase 7. 20% decrease
9. 61.8 11. 8 13. 70% 15. 90%
17. 25% decrease 19. 400% increase
21. 30% increase 23. 15% decrease
25. 20% increase 27. 8 __13 % decrease
29. 252 31. 7.6 33. 15% 35. 650%
37. 50% 41. 18 43. 200% increase
45. 20 47. 60 49. 25% decrease
60
18
51a. 60% b. ___
= __
x ; x = $30
100
53. H 55. G 57. 200 59. 625
61. 64 fl oz 63. $9.43 65. 80°; 170°
67. 60°; 150° 69. -20 71. 57 73. 36
75. -100 77. about $4.20
Study Guide: Review
1. literal equation 2. ratio 3. 36
4. -2 5. -21 6. 18 7. __98 8. __73
10
9. 27 + s = 108; 81 10. 7 11. - __
3
12. -90 13. 13 14. 0 15. -2
16. 17.5 17. -5 18. 40 19. -3
20. - __12 21. 15 22. 18 23. 1
24. 41; 123°; 57° 25. -2 26. -2 27. 1
28. - __23 29. no solution 30. all real
360
numbers 31. 9 32. n = ___
c
2S
33. a = __
n - 34. 3.7 gal
35. x = 15, -27 36. y = 7, 3
37. y = 9, -9 38. x = 17.4, -6.6
39. g = -4, -8 40. x = __57 , - __57
41. |x - 55| = 5; minimum speed:
50 mi/h; maximum speed: 60 mi/h
42. 3 __13 c 43. 1080 m/h 44. 0.85 mi/
min 45. 1.6 46. 54 47. 5 48. -3
49. 2.5 cm 50. 16 ft 51. The ratio of
the areas is the square of the ratio of
the radii. 52. 5.29 53. 3105
54. 66.7% 55. 400% 56. 133.3
57. 240 58. 80% 59. $48,500
60. $9000 61. about $5.60
62. about $2.20 63. 37% increase
64. 33% decrease 65. 91
66. 127.5 67. $3.75; $6.25 68. 37.5%
Chapter 3
3-1
Check It Out! 1. all real numbers
greater than 4
2a.
2
2.5
3
3.5
4
2b.
2c.
3. x < 2.5 4. d = amount employee
can earn per hour; d ≥ 8.25
Exercises 1. A solution of an
inequality makes the inequality
true when substituted for the
variable. 3. all real numbers greater
than -3 5. all real numbers greater
than or equal to 3 11. b > -8 __12
13. d < -7 15. f ≤ 14 17. r < 140
where r is positive 19. all real
numbers less than 2 21. all real
numbers less than or equal to 12
27. v < -11 29. x > -3.3 31. z ≥
9 33. y = years of experience;
y ≥ 5 35. h is less than -5. 37. r is
greater than or equal to -2.
39. p ≤ 17 41. f > 0 43. p = profits;
p < 10,000 45. e = elevation;
e ≤ 5000 51. D 53. C 59. D 61. C
65. < 71. 10 73. 7 75. 3x + 3
77. g = 2b; 16
79. b = 9 81. no solutions
3-2
Check It Out! 1a. s ≤ 9
Ê??
1b. t < 5 __12
Ê??
1c. q < 11
2. 11 + m ≤ 15; m ≤ 4 where m is
nonnegative; Sarah can consume
4 mg or less without exceeding the
RDA. 3. 250 + p > 282; p > 32; Josh
needs to bench press more than
32 additional pounds to break the
school record.
Exercises 1. p > 6 3. x ≤ -15
5. 102 + t ≤ 104; t ≤ 2 where t is
nonnegative 7. a ≥ 5 9. x < 15
11. 1400 + 243 + w ≤ 2000;
w ≤ 357 where w is nonnegative
13. x - 10 > 32; x > 42
15. r - 13 ≤ 15; r ≤ 28 17. q > 51
19. p ≤ 0.8 21. c > -202 23. x ≥ 0
25. 21 + d ≤ 30; d ≤ 9 where d is
nonnegative 27. x < 3; B
29. x ≤ 3; D 31. p ≤ 40,421 where p
is nonnegative 35. a. 411 + 411 =
882 miles b. 822 + m ≤ 1000
c. m ≤ 178, but m cannot be negative.
1
37. F 39. J 41. r ≤ 5 __
10
43. sometimes 45. always
c-2
47. y = 3 - __23 x 49. a = ____
b
51. k = 2s - 11
53. x = 10 55. x ≥ -1
3-3
Check It Out! 1a. k > 6
1b. q ≤ -10
1c. g > 36
2a. x ≥ -10
2b. h > -17
3. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or
12 servings
Exercises 1. b > 9 3. d > 18
5. m ≤ 1.1 7. s > -2 9. x > 5
11. n > -0.4 13. d > -3 15. t > -72
17. 0, 1, 2, 3, 4, 5, or 6 nights 19. j ≤
12 21. d < 7 23. h ≤ __87 25. c ≤
1
-12 27. b ≥ __
29. b ≤ -16
10
31. r < - __32 33. y < 2 35. t > 4
37. z < -11 39. k ≤ -7 41. p ≥
-12 43. x > -3 45. x < 20
47. p ≤ -6 49. b < 2 51. 7x ≥ 21;
x ≥ 3 53. - __45 b ≤ -16; b ≥ 20 57. C
14
4
59. A 67. B 71. g ≤ - __
73. m > __
5
15
75. x = 5 79. 2 3 81. $1.89/gal
83. 35 words/min 85. t < 1
3-4
Check It Out! 1a. x ≤ -6
1b. x < -11
1c. n ≤ -10
2a. m > 10
2b. x > -4
2c. x > 2 __13
Ê??
Selected Answers
SA5
95 + x
≥ 90; 95 + x ≥ 180; x ≥ 85;
3. _____
2
Jim’s score must be at least 85.
Exercises 1. m > 6 3. x ≤ -2
5. x > -16 7. x ≥ -9 9. x > - __12
11. x ≤ 19 13. x > 1 15. sales of
more than $9000 17. x ≤ 1
19. w < -2 21. x < -6 23. f <
-4.5 25. w > 0 27. v > __23 29. x >
-5 31. x < -2 33. a ≥ 11 35. x > 3
37. starting at 29 min 39. x ≤ 2
41. x < 4 43. x < -6 45. r < 8
47. x < 7 49. p ≥ 18 51. __12 x + 9 < 33;
x < 48 53. 4(x + 12) ≤ 16; x ≤ -8
55. B 57. A 59. 24 months or more
61a.
Number
Process
Cost
1
350 + 3
353
2
350 + 3(2)
356
3
350 + 3(3)
359
10
350 + 3(10)
380
n
350 + 3n
350 + 3n
b. c = 350 + 3n c. 350 + 3n ≤ 500;
n ≤ 50; 50 CDs or fewer 65. G 67. 59
69. x > 5 71. x > 0 73. x ≥ 0
75. -3x > 0 77. 7 79. __23 81. -9
83. 25 + 2m = 10 + 2.5m; m = 30
85. a ≥ 6
3-5
Check It Out! 1a. x ≤ -2
1b. t < -1
2. more than 160 flyers
3a. r ≤ 2
3b. x < 3
4a. no solutions 4b. all real numbers
Exercises 1. x < 3 3. x < 2
5. c < -2 7. at least 34 pizzas
9. p < -17 11. x > 3 13. t < 6.8
15. no solutions 17. all real
numbers 19. no solutions
21. y > -2 23. b ≥ -7 25. m > 5
27. x ≥ 2 29. w ≥ 6 31. r ≥ -4
33. no solutions 35. all real numbers
37. all real numbers 39. t < -7
41. x > 3 43. x < 2 45. x > -2
SA6
Selected Answers
47. x ≤ -6 49. 27 s
51a. 400 + 4.50n b. 12n c. 400 +
4.50n < 12n; n > 53 __13 ; 54 CDs or
more 53. 5x - 10 < 6x - 8; x > -2
55. __34 x ≥ x - 5; x ≤ 20
59. x can never be greater than
itself plus 1. 61. D 63. A
67. x < -3 69. w ≥ -1 __67
73. 26 in. 75. y = years; y ≥ 14
Check It Out! 1. 1.0 < c < 3.0
3a. r < 10 OR r > 14
3b. x ≥ 3 OR x < -1
4a. -9 < y < -2
4b. x ≤ -13 OR x ≥ 2
Exercises 1. intersection
3. -5 < x < 5 5. 0 < x < 3
7. x < -8 OR x > 4 9. n < 1 OR n > 4
11. -5 ≤ a ≤ -3 13. c < 1 OR c ≥ 9
15. 16 ≤ k ≤ 50 17. 3 ≤ n ≤ 6
19. 2 < x < 6 21. x < 0 OR x > 3
23. x < -3 OR x > 2
25. q < 0 OR q ≥ 2 27. -2 < s < 1
29a. 225 + 80n gives the cost of the
studio and technicians; the band
will spend between $200 and $550.
b. -0.3125 ≤ n ≤ 4.0625; n cannot
be a negative number c. $155
31. 1 ≤ x ≤ 2 33. -10 ≤ x ≤ 10
35. t < 0 OR t > 100 37. -2 < x < 5
39. a < 0 OR a > 1 41. n < 2 OR n > 5
43. 7 ≤ m ≤ 60 47. D 49. B
51. 0.5 < c < 3 53. s ≤ 6 OR s ≥ 9
55. -1 ≤ x ≤ 3 57. 4x - 5 59. 3a + 3
61. (-2, 3), (-1, 0), (0, -1),
(1, 0), (2, 3); U-shaped 63. m < 2
65. x ≤ -2
3. ⎪p - 125⎥ ≤ 75; 50 ≤ p ≤ 200
Exercises 1. -3 ≤ x ≤ 3
3. -2 < x < 2 5. 4 < x < 6
7. x < -22 OR x > 22 9. x ≤ -4
OR x ≥ 4 11. x ≤ 1 OR x ≥ 5
13. ⎪x - 55⎥ ≤ 25; 30 ≤ x ≤ 80
15. no solutions 17. all real
numbers 19. no solutions
21. 2 < x < 4 23. -3 < x < 3
25. -6 < x < 0 27. x ≤ -10 OR
x ≥ 10 29. x ≤ -10 OR x ≥ 6
31. x < -1 OR x > 2 33. no
solutions 35. all real numbers
37. no solutions 39. always
41. sometimes 43. ⎪x - 2⎥ ≤ 3;
-1 ≤ x ≤ 5 45. ⎪a⎥ ≤ 2 47. ⎪c⎥ ≥ 6__12
49a. 10,010 Hz 49b. ⎪x - 10,010⎥ ≤
9990 51. ⎪n - 23⎥ > 12 53. k ≤ 1;
the inequality is equivalent to
⎪x⎥ < k - 1, and this has no solutions
when the expression on the right side
is less than or equal to 0 (i.e., when
k - 1 ≤ 0 or k ≤ 1). 55. B 57. B
61. 1__12 63. 2__12 65. all real numbers less
than 2 67. all real numbers greater
than or equal to -6 69. 0 < x < 4
71. x < 1 OR x > 4
Study Guide: Review
1. inequality 2. union 3. compound
inequality 4. intersection
5. solution of an inequality
6.
7.
8.
Check It Out! 1a. -3 ≤ x ≤ 3
9.
2b. x ≤ -6 OR x ≥ 1
3-7
4a. all real numbers 4b. no
solutions
2b. -3 ≤ n < 2
2a. x ≤ -2 OR x ≥ 2
2a. 1 < x < 5
3-6
1b. -15 ≤ x ≤ 9
10.
11.
2b. continuous;
12. a < 2 13. k ≥ -3.5
14. q < -10 15. t = temperature;
t ≥ 72 16. s = students; s ≤ 12
where s is a natural number
17. m = minutes; m < 30 where m
is nonnegative 18. t < 7 19. k ≤ 2
20. m > -5 21. x ≥ 4.5 22. w < 9.5
23. a < 5 24. h < 1 25. v < -2
26. 5.5 mi or more 27. $18 or
less 28. a ≤ 5 29. t > -3 30. p > 8
31. x ≤ -25 32. n > 6 33. g < -12
34. k > -7 35. r < -9 36. h < -3
37. g < 2.5 38. 0, 1, 2, 3, 4, 5,
6, 7 39. at least 334 lanyards
40. x < 5 41. t ≥ 6 42. m > -11
43. x < -1 44. h > -3
45. x > 1 __12 46. b ≤ 10 47. y >
3 __12 48. n > -15 49. 0, 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, or 13 50. m <
-1 51. y ≥ -2 52. c < -3
53. q ≤ -4 54. x > 2 55. t < 3
56. no solutions 57. all real numbers
58. p > - __12 59. all real numbers
60. k > 2 61. no solutions
62. 8.75 > m 63. -10 < t < 4
64. -6 < k ≤ 7 65. r > 7 OR r < -2
66. no solutions 67. -2 < p ≤ 5
68. all real numbers 69. 68 ≤ t ≤ 84
70. 102 ≤ n ≤ 183.6 71. -22 ≤ x ≤
22 72. x < -12 OR x > 4 73. -4 ≤
x ≤ 4 74. -18 < x < 0 75. x ≤
-3 OR x ≥ 3 76. -3 < x < 3 77. x
< -13.9 OR x > 13.9 78. -12.5 <
x < 2.1 79. x ≤ 5 OR x ≥ 9 80. x ≤
-4 OR x ≥ 4 81. no solutions
82. -16.8 ≤ x ≤ 5.8 83. |d 72| ≤ 4; 68 ≤ d ≤ 76
Chapter 4
4-1
-1 -1
4IME
3. Possible answer: When the
number of students reaches a
certain point, the number of pizzas
bought increases.
Exercises 1. continuous 3. B
5. C 11. A 13. continuous 19. The
point of intersection represents the
time of day when you will be the
same distance from the base of the
mountain on both the hike up and
the hike down. 23. C 27. C 29. -8
31. __19 33. (-2, 5), (-1, 3), (0, 1),
(1, -1), (2, -3); line 35. (-2, 6),
(-1, 3), (0, 2), (1, 3), (2, 6);
U-shape 37. n - 5 = -2; 3
7ORDSPERMINUTE
7EEKS
0
0
1
-1
2
-4
17. D: {3}; R: 1 ≤ y ≤ 5
19. D: -2 ≤ x ≤ 2; R: 0 ≤ y ≤ 2;
yes 21. yes 23. yes 25. yes
27. no 29a. D: 0 ≤ t ≤ 5; R: 0 ≤ v ≤
750 b. yes c. (2, 300); (3.5, 525)
33. G 35a. {(-3, 5), (-1, 7), (0, 9),
(1, 11), (3, 13)} b. D: {-3, -1, 0, 1,
3}; R: {5, 7, 9, 11, 13} c. yes
37. all real numbers 39. 27 cm
41. x ≥ 19
4-3
Check It Out! 1. y = 3x
4-2
Check It Out! 1. x
y
1
3
2
4
3
5
2a. independent: time; dependent:
cost 2b. independent: pounds;
dependent: cost 3a. independent:
pounds; dependent: cost; f (x) =
1.69x 3b. independent: people;
dependent: cost; f (x) = 6 + 29.99x
4a. 1; -7 4b. -5; 101 5. f (x) =
500x; D: {0, 1, 2, 3}; R: {0, 500, 1000,
1500}
y
x
2a. D: {6, 5, 2, 1}; R: {-4, -1, 0}
2b. D: {1, 4, 8}; R: {1, 4}
3a. D: {-6, -4, 1, 8}; R: {1, 2, 9};
yes; each domain value is paired
with exactly one range value. 3b.
D: {2, 3, 4}; R: {-5, -4, -3}; no; the
domain value 2 is paired with both
-5 and -4.
3.
+EYBOARDING
36, 81} 11. D: {1}; R: {-2, 0, 3, 8};
no 13. D: {-2, -1, 0, 1, 2}; R: {1};
yes
15. x
y
-2 -4
Exercises
Check It Out! 1. C
2a. discrete;
7ATERLEVEL
7ATER4ANK
5.
x
y
x
y
1
1
-7
7
1
2
-3
3
-1
1
5
-5
Exercises 1. dependent
3. y = x - 2 5. independent: size of
bottle; dependent: cost of water
7. independent: hours; dependent:
cost; f (h) = 75h 9. 2; 9 11. -1; -15
13. y = -2x 15. independent: size
of lawn; dependent: cost
17. independent: days late;
dependent: total cost; f(x) = 3.99
+ 0.99x 19. independent: gallons
of gas; dependent: miles; f(x) =
28x 21. 7; 10 23. f(n) = 2n + 5;
D: {1, 2, 3, 4}; R: {$7, $9, $11, $13}
25.
z
1
2
3
4
g(z)
-3 -1
1
3
7. D: {-5, 0, 2, 5}; R: {-20, -8, 0,
7} 9. D: {2, 3, 5, 6, 8}; R: {4, 9, 25,
Selected Answers
SA7
27. f (-6.89) ≈ -16; f (1.01) ≈ 8;
f (4.67) ≈ 20 33. D 35. 3.5
37. 44.1 m 39. y = -3 41. x = 2
43. D: x ≥ 0; R: all real numbers; no
3.
21.
y
4-4
Check It Out! 1a.
5.
y
x
25. y = 5
y
y
29.
y
x
x
9.
y
x
31.
y
x
x
y
x
13.
y
33.
11. y = -1
y
2b.
y
15.
x
x
17.
4IMEH
x
Exercises
1.
19.
y
y
x
x
SA8
Selected Answers
37. x = 1 39. y = -8 41. yes;
yes 43. no; yes 45. no; yes; yes
47. yes; no; yes 55a. v = 10,000 1500h b. 8500 gal
c.
y
y
y
!VERAGE3PEEDOF,AVA&LOW
35.
3. x = 3 4. Possible answer: about
32.5 mi
x
$ISTANCEMI
x
x
2a.
7.
23.
x
1b.
y
x
x
y
y
Time
(h)
Volume
(gal)
0
10,000
1
8,500
2
7,000
3
5,500
4
4,000
59. J 61. J 63. y = 4x + 64
65. 2 3 67. p < -4 69. b ≥ 20
71. -3; 7
41.
Check It Out!
1.
5. continuous
y
x
(EIGHTOFBALL
4-5
4IME
4-6
1b. no 2a. -343 2b. 19.6 3. 750 lb
'AME
2. positive 3a. No correlation;
the temperature in Houston has
nothing to do with the number of
cars sold in Boston. 3b. Positive;
as the number of family members
increases, more food is needed, so
the grocery bill increases too.
3c. Negative; as the number of
times you sharpen your pencil
increases, the length of the pencil
decreases. 4. Graph A; it cannot
be graph B because graph B shows
negative minutes; it cannot be
graph C because graph C shows the
temperature of the pie increasing,
a positive correlation. 5. about
75 rolls
Exercises 3. no 5. positive
7. negative 9. positive 11. A
15. positive 17. positive 19. A
23. positive 25. B
*UANS4RIP
$ISTANCEMI
27a.
4IMEMIN
b. positive 29. C 35. 5(n + 2) =
2n - 8; n = -6 37. no solution
y
Exercises 1. common difference
3. yes; -0.7; -0.7, -1.4, -2.1
5. no 7. -53 9. no 11. yes; -9;
-58, -67, -76 13. 5.9 15. 9500
mi 17. __14 19. -2.2 21. 0.07
23. - __38 , - __1, - __5, - __43 25. -0.2,
2
8
-0.7, -1.2, -1.7 27. -0.3, -0.1,
0.1, 0.3 29. 22 31. 122 33b. $9,
$11, $13, $15; a n = 2n + 7 c. $37
20
d. no 35. -104.5 37. __
39a. a n =
3
6 + 3(n - 1) b. 48 c. $7800 d. a n =
7 + 3(n - 1); $8200
41a.
Time
Interval
Mile
Marker
1
520
2
509
3
498
4
487
5
476
6
465
b. a n = 520 + (n - 1)(-11)
c. number of miles per interval
d. 421 43. F 45. 173 and 182; 20th
and 21st terms 47a. session 16; yes
b. Thursday 49. 20 51. t < -2 OR
t > 2 53. negative
Study Guide: Review
(EIGHT
Check It Out! 1a. yes; __12 ; __54 , __74 , _94_
39.
6. continuous
x
1. domain 2. negative
correlation 3. term
4. continuous
4IME
7. Possible answer: A family buys a
fish tank and some fish. After two
weeks, they buy some more fish.
After two more weeks, they buy
more fish. 8. Possible answer: A
monkey swings from a high branch
to a lower branch. He climbs along
the branch. Then he jumps to a
higher branch and takes a nap.
9.
10.
x
-1
0
2
y
0
1
1
x
-2 -1
2
3
y
-1
3
4
1
11. D: {-4, -2, 0, 2}; R: { -1, 1, 3, 5}
12. D: {-2, -1, 0, 1, 2}; R: {-1, 0}
13. D: {0, 1, 4}; R: {-2, -1, 0, 1, 2}
14. D: -4 ≤ x ≤ 3; R: -3 ≤ y ≤ 5
15. D: {-5, -3, -1, 1}; R: {-3, -2,
-1, 0}; yes 16. D: {-4, -2, 0, 2};
R: {-2, 1}; yes 17. D: {1, 2, 3 ,4};
R: {-1, 0, 1, 2, 3}; no 18. {(1, 5.00),
(2, 6.50), (3, 8.00), (4, 9.50),
(5, 11.00)}; yes 19. yes 20. y is 7
less than x; y = x - 7. 21. y is
9 times x; y = 9x. 22. independent:
number of cakes; dependent: cost;
f (c) = 6c 23. independent: number
of CDs Raul will buy; dependent:
number of CDs Tim will buy; g(n)
= 2n 24. 14 25. -11 26. 6; -1
27.
Distance walked
0OINTSSCORED
&OOTBALL4EAM3CORES
y
x
Time
Selected Answers
SA9
28.
3a.
y
x
notebooks are purchased;
y-intercept: number of notebooks
that can be purchased
if no pens are purchased
3a. yes
y
29.
x
y
x
y
y
3b. yes
3b.
x
y
x
x
y
x
31.
x
y
3c. no
4.
Rental Payment
24
16
8
0
2
4
6
Manicures
D: {0, 1, 2, 3, …}
R: {$10, $13, $16, $19, …}
32.
x
33. Possible answer: $44
34. negative 35. Possible answer:
33 36. yes; -6; -4, -10, -16
37. no 38. no 39. yes; 2.5; 2, 4.5, 7
40. 105 41. -62 42. 20 43. $408
44. -15.5 °C
Chapter 5
Exercises 1. No; it is not in the
form Ax + By = C. 3. yes; yes
5. yes 7. yes 9. yes 11. no 15. yes;
no 17. yes; no 19. yes 23. no
27. yes; yes 29. yes; yes
31. yes; -4x + y = 2; A = -4;
B = 1; C = 2 33. no 35. yes; x = 7;
A = 1; B = 0; C = 7 37. yes; 3x - y
= 1; A = 3; B = -1; C = 1 39. yes;
5x - 2y = -3; A = 5, B = -2,
C = -3 41. no 55. no 57. C
63. not linear 65. -1 67. __19
69. 2 71. 9
domain value is paired with exactly
one range value; yes 1b. Yes;
each domain value is paired with
exactly one range value; yes 1c. No;
each domain value is not paired
with exactly one range value.
2. Yes; a constant change of +2
in x corresponds to a constant
change of -1 in y.
Selected Answers
y-intercept: 3 1b. x-intercept: -10;
y-intercept: 6 1c. x-intercept: 4;
y-intercept: 8
2a.
3CHOOL3TORE0URCHASES
.OTEBOOKS
Check It Out! 1a. Yes; each
Exercises 1. y-intercept
3. x-intercept: 2; y-intercept: -4
5. x-intercept: 2; y-intercept: -1
7. x-intercept: 2; y-intercept: 8
13. x-intercept: -1; y-intercept: 3
15. x-intercept: -4; y-intercept: 2
17. x-intercept: -4; y-intercept: 2
19. x-intercept: 2; y-intercept: 8
21. x-intercept: __18 ; y-intercept: -1
35. A 37. B 41. F 47. x-intercept:
950; y-intercept: -55 49. c > 4
51. m ≥ -6 53. yes
5-3
Check It Out! 1. day 1 to day 6:
-53; day 6 to day 16: -7.5; day 16
to day 22: 0; day 22 to day 30:
-4.375; from day 1 to day 6
2.
"ANK"ALANCE
DAY
DAY
DAY
DAY
5-2
Check It Out! 1a. x-intercept: -2;
5-1
SA10
"ALANCE
Rental payment ($)
30.
0ENS
x-intercept: 30; y-intercept: 20
2b. x-intercept: number of pens
that can be purchased if no
$AY
3. - __25 4a. undefined 4b. 0
5a. undefined 5b. positive
Exercises 1. constant 5. - __34
7. undefined 9. undefined
11. positive 15. 1 17. 0
17
19. positive 23. __
29. C 31. G
18
35. -2 37. D: {3}; R: {4, 2, 0, -2};
no 39. x-intercept: 3; y-intercept: 6
41. x-intercept: __14 ; y-intercept: __12
Exercises 1. direct variation
5-4
Check It Out! 1a. m = 0 1b. m = 3
1c. m = 2 2a. m = __12 2b. m = -3
2c. m = 2 2d. m = - __3 3. m = __1 ;
2
3. yes; -4 5. no 7. 18 9. y = 7x
11. yes; __14 13. yes 15. -16
17. y = 2.50x 19. no 21. y = -3x
2
9. - __59 11. -4 13. undefined
9
13
19. - __
15. - __34 17. - ____
5
5000
23a. Car 1; 20 mi/h b. The speed
and the slope are both equal to the
distance divided by time. c. 20 mi/h
25a. y = 220 - x 27. G 29. - __ab
5-5
(
)
Check It Out! 1. __32 , 0 2. (4, 3)
3. 6.71 4. 17.7 mi
Exercises 3. (1__12 , -4) 5. (3, 3)
7. 5.39 9. 10.82 11. (-2, -3)
13. (17, -23) 15. (2, -8) 17. 8.94
19. 9.22 21. 6.1 mi 25. 13.42
27. 14.32 29a. 7.2 b. (0, -2) c. 23
−− −− −−
31. CD, EF, AB 35. B 37. D
39. 12.5 square units 41. ±8
43. 26 45. x > -2 47. z > -4
49. __14
5-6
4. y = 4x
0ERIMETEROFA3QUARE
y
0ERIMETER
x
3b. y = -3x + 5
y
x
x
The value of k is 2, and the graph
shows that the slope of the line is 2.
29. k = - __29 x
3c. y = -4
y
x
y
x
The value of k is - __29 , and the
graph shows that the slope of the
line is - __29 .
33. y = -6x
4a. y = 18x + 200 4b. slope: 18;
cost per person; y-intercept: 200;
fee 4c. $3800
Exercises
1.
y
x
y
x
The value of k is -6, and the graph
shows that the slope of the line is
-6. 41. C 43. B 47. p = 7 - 4q
4 - 2y
5. y = x - 2 7. y = -3
9. y = -2x - 1 11. y = 3x - 1
13a. y = 18x + 10 b. slope: 18;
Helen’s speed; y-intercept: 10;
distance she has already biked
c. 46 mi
15.
y
x
5-7
Check It Out!
y
49. x = _____
51. y = -2x 53. -4
y
55. __12
The value of k is -3, and the
graph shows that the slope of the
line is -3.
25. y = 2x
Check It Out! 1a. no 1b. yes; - __34
1c. yes; -3 2a. No; possible
y
answer: the value of __x is not the
same for each ordered pair. 2b. Yes;
y
possible answer: the value of __x is
the same for each ordered pair.
2c. No; possible answer: the value
y
of __x is not the same for each
ordered pair. 3. 90
x
2a. y = -12x - __12 2b. y = x
2c. y = 8x - 25
3a. y = __23 x
x
31. __32 - y 33. x = __12 35. x = -3
37. x = 0 39. -11 < x < -5
41. x ≤ -7 OR x ≥ 7 43. x ≤ -8 OR
x ≥ 6 45. yes
y
the height of the plant is increasing
at a rate of 1 cm every 2 days.
4. m = - __23
1
Exercises 1. 1 3. - __12 5. 10 7. ___
540
y
1b.
1a.
3IDELENGTH
y
x
Selected Answers
SA11
17.
y
x
19. y = 5x - 9 21. y = - __12 x + 7
23. y = __1 x + 4 25. y = -2x + 8
2
29. possible 31. impossible
33. C 37. B 39. B 41. y = __13 x - 3
43. -6 45. h = hours; h ≤ 2 where
h is nonnegative 47. n ≤ 8
49. t < -3
3
Exercises 1. parallel 3. y = __
x-1
4
3
2
__
__
and y - 3 = (x - 5) 5. y = x - 4
5-8
Check It Out!
(
1a. y - 1 = 2 x - __12
4
)
1b. y + 4 = 0(x - 3)
2a.
y
x
y
2
x
-2
0
2
-2
3a. y = __13 x + 2 3b. y = 6x - 8
4. x-intercept: -3; y-intercept: 9
5. y = 2.25x + 6; $53.25
Exercises 1. y + 6 = __15 (x - 2)
3. y + 7 = 0(x - 3) 7. y = -__13 x + 7
9. y = -x 11. y = -__1 x + 4
2
13. x-intercept: 3; y-intercept: -3
15. x-intercept: -1; y-intercept: 3
17. y - 5 = __29 (x + 1) 19. y - 8 =
8(x - 1) 23. y = -__27 x + 1 25. y =
11
-6x + 57 27. y = - __
x + 18
2
29. y = 2x - 6 31. x-intercept: 1;
y-intercept: –2 33. x-intercept: -6;
1
y-intercept: 9 35. y = -___
x + 212;
500
200 °F 41. never 43a. y - 11 = 2.5
(x - 2) b. 6 in. c. 16__58 in.
47. y = -8; x = 4 49. A 53a. (0, 12)
and (6, 8) b. y = -__23 x + 12
11
c. 18 min 55. H 57. y = -3x + __
4
59. -6 ≤ x ≤ -1 63. y = -2x + 8
5-9
Check It Out! 1a. y = 2x + 2 and
y = 2x + 1 1b. y = 3x and y - 1 =
−−
3(x + 2) 2. slope of AB = 0; slope
−− __5
−−
of BC = 3 ; slope of CD = 0; slope
SA12
Selected Answers
4.
and y = -6x - 8; y = 3x - 2 and
3y = -x - 11 17. y = - __67 x 19. neither
21. parallel 23. y = __12 x - 5
25. y = 2x + 5 27. y = 3x + 13
29. y = -x + 5 31. y = 4x - 23
33. y = - __34 x 35. y = -x + 1
31
2
11
37. y = __
x - __
39. y = - __15 x - __
5
5
5
1
1
1
__
__
__
41. y = - 2 x - 2 43. y = 2 x + 6
45. y = x - 3 47. y = -4 51a. y =
50x b. y = 50x + 30 53. H 57. - __15
59. 94 + t > 112; t > 18 63. y = __23 x - 5
65. y = - __12 x - __12 67. y = 3
y
x
gx
fx
reflection across y-axis and
translation 2 units up 5. The graph
will be rotated about (0, 175) and
become less steep; the graph will
be translated 5 units up.
Exercises 1. translation
3.
y
fx
3
and y = - __32 x + 2; y = -1 and x = 3
9. x = 7 and x = -9; y = - __56 x + 8
and y = - __56 x - 4 11. y = -3x + 2
and 3x + y = 27; y = __12 x - 1 and
-x + 2y = 17 13. y = 6x and y = - __16 x;
y = __1 x and y = -6x 15. x - 6y = 15
6
2b.
−−
−−
−−−
of AD = __53 ; AB is parallel to CD
because they have the same slope.
−−
−−
AD is parallel to BC because
they have the same slope. Since
opposite sides are parallel, ABCD is
a parallelogram. 3. y = -4 and x = 3;
y - 6 = 5(x + 4) and y = - __15 x + 2
−−
−−
4. slope of PQ = 2; slope of QR = -1;
−−
−−
1
slope of PR = - __2 ; PQ is
−−
perpendicular to PR because the
product of their slopes is -1. Since
PQR contains a right angle, PQR is
a right triangle. 5a. y = __45 x + 3
5b. y = - __15 x + 2
x
gx
translation 4 units down
7.
y
fx
gx
x
rotation about (0, 0) (less steep)
9.
y
gx
fx
x
5-10
Check It Out!
1.
y
fx
rotation about (0, -2) (steeper)
y
13.
x
x
gx
fx
g(x) = - __13 x - 6
x
17.
gx y
fx
rotation about (0, -1) (less steep)
n
3.
[ (m)
4
x
2
m
0
gx
translation 6 units down
2.
y
2
\(m)
g(x) = - __23 x + 2
rotation about (0, 0) (steeper) and
translation 1 unit up
23.
y fx
gx
x
different changes in x. 8. 5x +
y = 1; A = 5; B = 1; C = 1
9. x + 6y = -2; A = 1; B = 6; C =
-2 10. 7x - 4y = 0; A = 7; B = -4;
C = 0 11. y = 9; A = 0; B = 1; C = 9
12.
#UPCAKE3ALES
x
y
45.
!MOUNTEARNED
rotation about (0, 2) (less steep)
27.
y
f x
gx
x
rotation about (0, 0) (steeper) and
translation 5 units down
31. rotation about (0, 0) (steeper)
y
fx
x
46. y = __13 x + 5 47. y = 4x - 9
48.
$ISTANCEFT
y
x
FT
S
FT
S
FT
S
4IMES
(
50. y = 2x + 1 51. y = -5x -26
52. y = 2x + 2 53. y = 2x + 8
54. y = - __13 x and y = - __13 x - 6
55. y - 2 = -4 (x - 1) and
y = -4x - 2 56. y - 1 = -5(x - 6)
and y = __15 x + 2 57. y - 2 = 3(x + 1)
and y = - __13 x 58. y = 2x - 3
59.
y
x
60.
y fx
gx
x
reflection across y-axis
61.
fx
gx
fx
translation 4 units up
)
-ALEKAS"ABYSITTING%ARNINGS
y
x
44.
units 36. 15.65 units 37. 2.2 mi
38. yes; -6 39. yes; 1 40. no
41. yes; -__12 42. -12
43. y = 8x
-ONEYEARNED
1. translation; rotation; reflection
2. y-intercept 3. slope; y-intercept
4. No; a constant change of +2 in
x corresponds to different changes
in y. 5. Yes; a constant change
of +1 in x corresponds to a
constant change of +2 in y.
6. Yes; a constant change of +1 in x
corresponds to a constant change
of -2 in y. 7. No; a constant
change of -1 in y corresponds to
x
gx
20. 5 21. - __43 22. -3 23. - __12
24. 3 25. 7 26. 4 27. -5 28. -1
29. 1 30. 2 31. undefined 32. 0
33. (15, 15) 34. -__12 , -9 35. 7.81
Study Guide: Review
FT
S
g(x) = __16 x - 4 39. translation 9
units down 41. rotation about (0, 0)
(steeper) 43. rotation about (0, 0)
(steeper) 45a. $300 b. 20%
c. Commission changes to 25%.
Base pay changes to $400. 49. D
53. 15x 55. positive 57. negative
59. y = - __35 x and y + 1 = - __35 (x - 2)
61. x = 4 and y = -3; 2y + x = 6 and
y = 2x + 3
y
49.
g x
37.
x
#UPCAKESSOLD
D: whole numbers;
R: nonnegative multiples of 0.5
13. x-intercept: 2; y-intercept: -4
14. x-intercept: 5; y-intercept: 6
15. x-intercept: 3; y-intercept: -9
16. x-intercept: -__12 ; y-intercept: 1
17. x-intercept: -18; y-intercept: 3
18. x-intercept: __13 ; y-intercept: -__14
19.
2ATE
They have different slopes and the
same y-intercept.
y
4IMEH
reflection across y-axis
62. translation 2 units up;
rotation about (0, 3) (steeper)
y
x
Selected Answers
SA13
Chapter 6
6-1
2a. (-2, 3) 2b. (3, -2) 3. 5 movies;
$25
Exercises 1. an ordered pair that
satisfies both equations 3. yes
5. (2, 1) 7. (-4, 7) 9. no 11. yes
13. (3, 3) 15. (3, -1)
⎧y = 2x
⎩ y = 16 + 0.50x
17a. ⎨
#ARNATION3ALES
b.
#OST
Add the two equations:
-7x + y = -2
+7x - y = _
2
_
0+ 0 = 0
6-3
Check It Out! 1a. yes 1b. no
&LORISTSPRICE
3CHOOLBANDSPRICE
#ARNATIONS
It represents how many carnations
need to be sold to break even.
c. No, because the solution is not
a whole number of carnations; 11
carnations. 19. (-2.4, -9.3)
21. (0.3, -0.3) 23. 45 white; 120
pink 25. 8 yr 29. C 31. month 11;
400 33. 42 35. 2.2 37. numbers
less than 5 39. numbers greater
than 6 41. c ≤ -9
6-2
Check It Out! 1. (-2, 4) 2. (4, 1)
0 =
3a. (2, 0) 3b. (3, 4) 4. 9 lilies;
4 tulips
Exercises 1. (-4, 1) 3. (-2, -4)
5. (-6, 30) 7. (3, 2) 9. (4, -3)
11. (-1, -2) 13. (1, 5) 15. 6, - __12
17. (-1, 2) 19. (-1, 2)
(
)
⎧ - w = 2
; length: 11 units;
21. ⎨
⎩ 2 + 2w = 40 width: 9 units
(
)
(
1b. (0, 2) 1c. (3, -10) 2. (-1, 6)
3. 10 months; $860; the first option;
the first option is cheaper for the
first 9 months; the second option is
cheaper after 10 months.
Exercises 1. (9, 35) 3. (3, 8)
5. (-3, -9) 7a. 3 months; $136
b. Green Lawn 9. (-4, 2)
11. (-1, 2) 13. (1, 5)
15. (3, -2) 17. 6 months; $360; the
second option 19. (2, -2)
21. (8, 6) 23. (-9, -14.8)
25. 12 nickels; 8 dimes
⎧x + y = 1000
27. ⎨
; $200 at 5%;
⎩ 0.05x + 0.06y = 58 $800 at 6%
29. x = 60°; y = 30°
35. Possible estimate: (1.75, -2.5);
(1.8, -2.4) 37. F 39. r = 5; s = -2;
t = 4 41. a = 9; b = 5; c = 0
45. x-intercept: 2; y-intercept: -6
Selected Answers
0✔
9. inconsistent; no solutions
)
46 __
15 __
25. (3, 3) 27. __
, 8 29. __
,9
7 7
7 7
⎧3A + 2B = 16
31a. ⎨
b. A = 4; B = 2
⎩ 2A + 3B = 14
c. Buying the first package will save
$8; buying the second package will
save $7. 33. A 35a. s = number
of student tickets; n = number of
nonstudent tickets;
⎧s + n = 358
⎨
⎩ 1.50s + 3.25n = 752.25
b. s = 235; n = 123; 235 student
tickets, 123 nonstudent tickets
37. x = 4; y = -1; z = 10
⎧x + y = 5
39. ⎨
; x = 1; y = 4;
⎩ 3(10x + y) = 42
the number is 14.
41. y = 3x 43. yes; __12 45. no 47. (4, 9)
6-4
Check It Out! 1a. (-2, 1)
SA14
47. x-intercept: 8; y-intercept: 10
49. yes
Check It Out! 1. Possible answer:
Substitute -2x + 5 for y in the
second equation: 2x + (-2x +
5) = 1; 5 = 1 ✘ 2. Possible answer:
Substitute x - 3 for y in the
second equation: x - (x - 3) 3 = 0; 3 - 3 = 0; 0 = 0 ✔
3a. consistent, dependent;
infinitely many solutions
3b. consistent, independent;
one solution 3c. inconsistent; no
solution 4. Yes; the graphs of the
two equations have different
slopes so they intersect.
11. yes 13. Possible answer:
Substitute -x - 1 for y in the first
equation: x + (-x - 1) = 3; -1 =
3 ✘ 15. Possible answer: Compare
slopes and intercepts. -6 + y =
2x → y = 2x - 6; y = 2x - 36; the
lines have the same slope and
different y-intercepts. Therefore
the lines are parallel. 17. Possible
answer: Substitute x - 2 for y in
the second equation: x - (x 2) - 2 = 0; 2 - 2 = 0; 0 = 0 ✔
19. Possible answer: Compare
slopes and intercepts. -9x – 3y =
-18 → y = -3x + 6; 3x + y = 6 →
y = -3x + 6; the lines have
the same slope and the same
y-intercepts. Therefore the graphs
are one line. 21. consistent,
independent; one solution 23. Yes;
the graphs of the two equations
have different slopes, so they
intersect. 27. They will always
have the same number; both
started with 2 and add 4 every
year. 29. The graph will be 2
parallel lines. 31. A 33. D 35. p =
q; p ≠ q 37. 11 km 39. no
41. d = -1__12 ; -6, -7__12 , -9
43. (-2, -4)
6-5
Check It Out! 1a. no 1b. yes
2a.
x
2b.
y
x
Exercises 1. consistent 3. Possible
answer: Substitute -3x + 2 for y in
the first equation: 3x + (-3x + 2) =
6; 2 = 6 ✘ 5. Possible answer:
Substitute -x + 3 for y in the
second equation: x + (-x + 3) 3 = 0; 0 = 0 ✔ 7. Possible answer:
y
2c.
y
x
3a. 2.5b + 2g ≤ 6
hot dogs) 21. y ≤ - __15 x + 3
3b.
23.
/LIVE#OMBINATIONS
y
x
'REENOLIVES
Possible answer: solutions: (3, 3),
(4, 4); not solutions: (-3, 1),
(-1, -4)
y
2b.
x
25.
x
Possible answer: solutions: (0, 0),
(3, -2); not solutions: (4, 4), (1, -6)
y
3a.
29.
y
no solutions
y
3b.
x
x
y
31.
y
x
7.
x
y
x
Exercises 3. yes
5.
y
"LACKOLIVES
3c. Possible answer: (1 lb black, 1 lb
green), (0.5 lb black, 2 lb green)
4a. y < -x 4b. y ≥ -2x - 3
all points between and on the
parallel lines
33.
y
3c.
y
x
9a. r + p ≤ 16
b.
Punch Combinations
x
35.
12
y
x
8
4
4.
4
8
12 16
Orange juice (c)
c. Possible answer: (2 c orange,
2 c pineapple), (4 c orange, 10 c
pineapple) 11. y ≥ x + 5 13. yes
37. 7a + 4s ≥ 280 41. A 43. B 45. C
47.
y
19a. 3x + 2y ≤ 30
b.
(OTDOGSLB
&OOD#OMBINATIONS
c. Possible answer: (3 lb hamburger,
2 lb hot dogs), (5 lb hamburger, 6 lb
2
2
4
6
8
Pepper jack cheese (lb)
49. y ≥ __12 x + 3 51. yes 53. yes
Possible answer: (3 lb pepper jack,
2 lb cheddar), (2.5 lb pepper jack,
4 lb cheddar)
55. y = __34 x + __74 57. y = 3x + 1
Exercises 1. all 3. yes
59. y = -__12 x + __12 61. (-2, 15)
63. (2, 5) 65. (12, 3)
5.
y
6-6
Check It Out! 1a. yes 1b. no
2a.
x
y
x
(AMBURGERMEATLB
4
0
x
6
8
x
y
Cheese Combinations
0
15.
same as solutions of
y > -2x + 3
Cheddar cheese (lb)
Pineapple juice (c)
16
Possible answer: solutions: (3, 3),
(4, 3); not solutions: (0, 0), (2, 1)
Selected Answers
SA15
7.
y
23.
y
x
49.
x
Possible answer: solutions: (0, 4),
(1, 4); not solutions: (2, -1), (3, 1)
9.
25.
51. 25 cm 2 53. 12.5 cm 2 55. no
57. yes
y
x
x
All points are solutions.
no solutions
27.
y
y
x
x
y
x
24
(6, 13)
16
(10, 10)
(OURSATPHARMACY
Possible answer:
(0 h at pharmacy, 9 h babysitting),
(8.5 h at pharmacy, 10 h
babysitting)
31.
8
0
19.
y
x
8
16 24
Lemonade (c)
Possible answer: (6 lemonade,
13 cupcakes), (10 lemonade, 10
cupcakes) 17. yes
33.
x
x
y
Possible answer: solutions: (-2, 0),
(-3, 1); not solutions: (0, 0), (1, 4)
)
17. (-5, 2) 18. (6, 6) 19. 10 h;
$1350; Motor Works 20. (-1, 3)
21. (5, -3) 22. (11, 1) 23. (0, 3)
24. (-2, 8) 25. (3, -5) 26. (4, -6)
27. (2, 2) 28. no solution
29. infinitely many solutions
30. (-2, -4) 31. infinitely many
solutions 32. infinitely many
solutions 33. (-1, -3) 34. no
35. consistent, independent; one
solution 36. inconsistent; no
solution 37. consistent, dependent;
infinitely many solutions
38. inconsistent; no solution
39. consistent, independent;
one solution 40. consistent,
dependent; infinitely many
solutions 41. inconsistent; no
solution 42. no 43. yes 44. yes
45. no
46.
y
x
21.
7. yes 8. yes 9. no 10. (-1, -1)
11. (3, 4) 12. 8 h; $10 13. (-9, -6)
y
y
same solutions as y > 2x - 1
Sales Goals
15.
Cupcakes
,INDAS7ORK(OURS
(OURSBABYSITTING
13.
29.
1. independent system 2. system
of linear equations 3. solution of a
system of linear inequalities
4. inconsistent system
5. independent system 6. no
(
same solutions as y > 2
all points between the parallel lines
and on the solid line
Study Guide: Review
14. __12 , -2 15. (-1, 6) 16. (4, -5)
no solutions
y
x
11.
y
⎧y > x + 1
35. ⎨
⎩y<x+3
⎧y < 2
37. ⎨
⎩ x ≥ -2
39. Student B 45. G
47. about 12 square units
47.
y
x
y
48.
x
x
Possible answer: solutions: (-1, 3),
(0, 5); not solutions: (0, 0), (1, 4)
SA16
Selected Answers
49.
58.
y
x
Exercises 3. 0.00001 5. 100,000,000
y
50.
y
x
Possible answer: solutions: (8, -8),
(9, 0); not solutions: (0, 0), (0, -4)
59.
y
x
51.
y
y
60.
x
x
52. x = slices of pizza; y = bottles of
soda; 2x + y ≥ 450
"OTTLESOFLEMONADE
&UNDRAISING.EEDS
Check It Out! 1a. 7 12 1b. 3 × 5 10
m
1
1c. ___
1d. __
2. 6.696 × 10 8 mi
4
7
5
n
3LICESOFPIZZA
Possible answer: (200, 50),
(150, 150) 53. no 54. yes
55.
1
1
mi 7. y 32 9. __
, or __
11. x 7 y 13
4
81
3
1
___
125
1
1
2c. - __
2d.
2b. __
16
32
1
m 2a. _____
10,000
1
1
- __
3a. __
32
64
2
1
4b. ___
4c. g 4h 6
3b. 2 4a. ___
3
3
m
7r
1
1
Exercises 1. ________
m 3. 1 5. __
10,000,000
27
1
1
1
9. 1 11. __
13. ___
7. - ___
512
16
256
3
1
17. __
19. x 10d 3 21. __4
15. - __
4
32
p7
f
1
1
__
__
23. __
q 25. 1 27. 81 29. - 36 31. 1
2g 10
b
5
1
45. __
47. - __
49. ____
43. __
4
3
7
2
3
Possible answer: solutions: (-6, 6),
(-10, 0); not solutions: (0, 0), (4, -4)
y
51. s 5t 12 53. 1 55.
x
1
__
q2
h
57. _____
2
3
6m k
a
1
61. - __16 63. 3 65. 3 67. __
59. __
2
16
2
y
x
Possible answer: solutions: (0, 0),
(-5, 0); not solutions: (8, 0), (3, -3)
y
3
b
1
_____
3x 8y 12
79. never 81. sometimes
83. sometimes 87. 81 89. 1 91. -3
93. -1 95. D 97. A 103. -2
105. 4 107. 28 111. y = __13 x + 5
113. y = -4x + 9
7-2
Check It Out! 1a. 0.01 1b. 100,000
x
k
w
5
3
__
69. ___
y 71. - 6 73. 2a b 75.
g6
k
1
1
39. 1 41. ___
33. - __13 35. 4 37. ___
256
144
57.
y
Exercises 1. 2 5 3. n 8 5. 7.5 × 10 8
y
x
Possible answer: solutions: (-6, 2),
(-8, 1); not solutions: (0, 0), (4, 1)
x
3a. 3 20 3b. 1 3c. a 18 4a. 64p 3
1
4b. 25t 4 4c. __
4
Chapter 7
Check It Out! 1.
56.
7-3
7-1
7. 10 -6 9. 650,300,000 11. 0.092
13. 5.85 × 10 -3, 2.5 × 10 -1,
8.5 × 10 -1, 3.6 × 10 8, 8.5 × 10 8
15. 0.000000001
17. 100,000,000,000,000 19. 10 6
21. 92,000 23. 0.00042
25. 10,000,000,000,000 27. 1.23 ×
10 -3, 1.32 × 10 -3, 3.12 × 10 -3,
2.13 × 10 -1, 2.13 × 10 1, 3.12 × 10 2
29. 2.7 × 10 7 31. 2.35 × 10 5
33. 6 × 10 -7 35. 4.12 × 10 -2
37. yes 39. no; 2.5 × 10 2 41. yes
43. yes 47. 10 -3 51. F 55. Let
m = number of minutes; m ≥ 45
3
1
59. (-2, 1) 61. __
63. ___
16
125
1c. 10,000,000,000 2a. 10 8 2b. 10 -4
2c. 10 -1 3a. 85,340,000 3b. 0.00163
4a. 1.43 × 10 5 km 4b. 13,000 m/s
5. 2 × 10 -12, 4 × 10 -3, 5.2 × 10 -3,
3 × 10 14, 4.5 × 10 14, 4.5 × 10 30
13. 36k 2 15. -8x 15 17. b 10 19. 6 8
x
a 12
1
21. __
23. 2 9, or 512 25. __
27. ___
5
3
2
y
b
x
29. 27x 3 31. p 28q 14 33. -256x 12
35. 6 37. 3 39. 8 41. 2x 3 43. 2m 10n 6
45. 108x 13 47. 125x 6 49. 3a 6
a7
51. 10 3, or 1000 57. __
59. 15m 12n 9
b5
2 7
7
61. 9s t 63. t 67. yes 69. 17k 2
71. 6x 4 73. 15a 2b 3 75a. 6 × 10 -7 m
b. 3 × 10 8 m/s c. Associative
and Commutative Properties of
2
2
1
Multiplication 77. (6ab) 79. _____
2kmn
81. H 83. F 85. 3 2x 87. x + 1
2
89. x 3y + 3z 91. x x 93. x = 4
95. x = 4 97. 1.728 × 10 -6 99. 15
101. no 103. 7,800,000 105. 98.3
(
)
7-4
n
n
1
1b. __
1c. ___
Check It Out! 1a. ___
m5
m5
y3
3
3
3
2. 1.1 × 10 -2 3. $12,800
1d. __
16
64
a b
a
9
2
, or __
4b. _____
4c. __
5a. __
,
4a. __
4
10 15
3
3
81
5 20
6
or
3
729
___
64
3
c d
b c
t
5b. ____
5c. __
4
4
8 12
16a
b
3
4
s
Exercises 1. 25 3. 3 5. 7 × 10 2 7. 1
16
2b
4
1
9. __
11. ____
13. __
15. ___
17. 27
6 4
2
25
9
2
a b
3a
19. x 5 21. 5 × 10 -7 23. 7 × 10 -3
y3
a
29. __6
25. 2 × 10 27 kg 27. ___
6
12
b
31.
39.
x
y 25
196
3x 5
___
33. ___
35. 2d 2 37. ___
10
2
4
x
9x
c4
25
1
__
41. __
43. ___
45.
-1
4
2
100
a
p
Selected Answers
SA17
47. 2000: 3 × 10; 1995: 2.84 × 10;
1990: 2.65 × 10 1 51. 3 53. 3; 4
55. B 57. A 59. 3 61. m; (-n);
m; -n; Definition of negative
exponent; a n 63. 1 65. 12 67 x = - __12
69. 1 71. -125x 12
7-5
Check It Out! 1a. 3 1b. 15
y
2a. 8 2b. 1 2c. 81 3. 1944
4a. xy 3 4b. xy
7-7
Exercises 1. 5 3. 4 5. 3 7. 6 9. 5
11. 10 13. 4 15. 32 17. 125 19. 256
21. 0 23. x 2y 25. x 3y 3 27. a 2 29. 1
31. 10 33. 8 35. 2 37. 2 39. 14
41. 8 43. 8 45. 64 47. 1000
49. 243 51. 2g 53. 2m 55. 3x 2
57. ab 4 59. a 8b 61. 1 63. 0 65. 625
8
67. 3 69. __23 71. __14 73. __49 75. ___
343
2
__
16
1
79. ___
81. 1.86 in. 83. n 3
77. __
27
625
will be less than n because __23 < 1.
3
__
n 2 will be greater than n because
3
__
> 1. 85a. 10 in. 85b. The distance
2
doubles (20 in.). 87. B 89. C 91. a
93. x 3 95. 3 97. 36π cm 2; both
volume and surface area are
described by 36π (although the units
are different). 99. -1 101. n < 3
103. y ≤ -2 105. D: {-2, -1, -0, 1};
R: {0, 1, 2, 3}; yes; each domain
value is paired with exactly one
range value. 107. D: 1 ≤ x ≤ 4;
R: 2 ≤ y ≤ 4; yes; each domain
value is paired with exactly one
range value.
7-6
Check It Out! 1a. 3 1b. 1 1c. 3
2a. 1 2b. 5 3a. x 5 + 9x 3 - 4x 2 +
16; 1 3b. -3y 8 + 18y 5 + 14y; -3
4a. cubic polynomial 4b. constant
monomial 4c. 8th degree trinomial
5. 1606 ft
Exercises 1. d 3. a 5. 3 7. 0 9. 8
11. 3 13. 4 15. -8a 9 + 9a 8; -8
17. 3x 2 + 2x - 1; 3 19. 5c 4 + 5c 3 +
3c 2 - 4; 5 21. linear binomial
23. quartic polynomial 25. quartic
trinomial 27. 4 29. 6 31. 7 33. 1
35. 4 37. 2 39. 3 41. 4.9t 3 - 4t 2 +
t + 2.5; 4.9 43. x 10 + x 7 - x 5 +
x 3 - x; 1 45. 5x 3 + 3x 2 + 5x - 4; 5
47. -d 3 + 3d 2 + 4d + 5; -1
49. -x 5 - x 3 + 4x 2 + 1; -1
51. linear monomial 53. quadratic
SA18
Selected Answers
trinomial 55. quartic trinomial
57. quadratic monomial 59. always
61. never 63a. 58.5 in3 b. 66 in3
c. 0 d. yes 65. -48; 0; 3270 75. A is
incorrect 77. J 79a. 58 cm; 65 cm
b. 50.310 cm 81. 90 - m
83. inconsistent; no solutions
85. consistent and independent;
p8
x2
one solution 87. __
89. __
5
16
Check It Out! 1a. 5s 2 + s
1b. 20z 4 - 6 1c. x 8 + 6y 8 1d. b 3c 2
2. 12a 3 + 15a 2 - 16a 3. -2x 2 - x
4. -0.05x 2 + 46x - 3200
Exercises 1. -3a 2 + 9a 3. 0.26r 4 +
0.32r 3 5. 3b 3c 7. 23n 3 + 3n + 15
9. 9x 2 - x - 6 11. 4c 4 + 8c + 6
13. -3r + 11 15. 8a 2 + 5a + 9
17. 12n 2 + 6n - 3m 19. d 5 + 1
21. 5x 23. 2x 3 - 5 25. 10t 2 + t
27. x 5 + x 4 29. -2t 3 + 8t 2
31. -6m 3 + 2m 2 + 5m + 3
33. 4w 2 + 6w + 4 35. t - 5
37. 2n -2 39. 6x 2 - x - 1
41. -u 3 + 3u 2 + 3u + 6 43. x = __32 ,
or 1.5 45. B is incorrect. 47. 3x + 6
49. 6x + 14 51. 2x 2 + x - 5 55. G
57. 3x 2 - 2 69. b 11 71. 9z 12
7-8
Check It Out! 1a. 18x 5 1b. 10r 2t 4
1c. 4x 5y 5z 7 2a. 8x 2 + 2x + 6
2b. 15a 3b + 3ab 2 2c. 5r 3s 2 - 15r 2s 3
3a. a 2 - a - 12 3b. x 2 - 6x + 9
3c. 2a 2 + 7ab 2 - 4b 4 4a. x 3 - x 2 6x + 18 4b. 3x 3 - 4x 2 + 11x + 10
5a. x 2 - 4x 5b. 12 m 2
Exercises 1. 14x 6 3. 3r 5s 5t 5
5. 21x 7y 3 7. 4x 2 + 8x + 4
9. 6a 5b 2 + 2a 4b 3
11. 10x 3y 4 - 5x 2y 2 13. x 2 - x - 2
15. x 2 - 4x + 4 17. 4a 4 - 2ab 12a 3b 2 + 6b 3 19. x 3 + 3x 2 - 7x +
15 21. -6x 4 + 12x 3 + 4x 2 - 18x +
20 23. x 3 - 4x 2 - 4x - 5
25a. 2x 2 - 3x b. 20 in 2 27. -12r 5s 5
29. 10a 4 31. -6a 5b 6 33. -12a 7b 7c 8
35. 9s 2 + 54s 37. 27x 3 - 12x 2
39. 10s 3t 3 - 15s 2t 5 41. -10x 3 +
15x 2 + 5x 43. -14x 6y 3 + 7x 5y 4
45. x 2 + 8x + 16 47. 5x 2 + 13x - 6
49. 10x 2 - x - 2 51. 7x 2 - 52x - 32
53. x 3 - x 2 - x + 10
3
55. -10x 4 + 2x + 20x 2 - 19x + 3
57. 8x 5 - 12x 3 - 2x 4 + 17x 2 - 21
59. x 3 - 3x - 2 61. -x 3 + 3x 2 - 3x
+ 1 63. 16x 2 - 48x + 36 65a. 3; 2;
10x 5 + 5x 3; 5 b. 2; 2; x 4 - x 3 + 2x 2
- 2x; 4 c. 1; 3; x 4 - 5x 3 + 6x 2 + x 3; 4 d. m + n 67. 12x 2 + 12x + 3
69a. 2x 2 b. 800 m 2 71. 2x 2 - 7x 30 73. 8x 2 - 16xy + 6y 2 75. 6x 2 9x - 6 77. x 3 + 3x 2 79. 2x 3 7x 2 - 10x + 24 81. 8p 3 - 36p 2q +
54pq 2 - 27q 3 87. C 89. D
91. -x 2 - 6 93. a. x 2 - 1 b. 8x +
16 95. x 3 + 3x 2 + 2x 97. a = 2
99. 3.61 101. 9.49
7-9
Check It Out! 1a. x 2 + 12x + 36
1b. 25a2 + 10ab + b2 1c. 1 + 2c 3 + c6
2a. x 2 - 14x + 49 2b. 9b 2 - 12bc +
4c 2 2c. a 4 - 8a 2 + 16 3a. x 2 - 64
3b. 9 - 4y 4 3c. 81 - r 2 4. 25
Exercises 3. 4 + 4x + x 2 5. 4x 2 +
24x + 36 7. 4a 2 + 28ab + 49b 2
9. x 2 - 4x + 4 11. 64 - 16x + x 2
13. 49a 2 - 28ab + 4b 2 15. x 2 - 36
17. 4x 4 - 9 19. 4x 2 - 25y 2
21. x 2 + 6x + 9 23. x 4 + 2x 2y 2 + y 4
25. 4 + 12x + 9x 2 27. s 4 - 14s 2 + 49
29. a 2 - 16a + 64 31. 9x 2 - 24x +
16 33. a 2 - 100 35. 49x 2 - 9
37. 25a 4 - 81 39. π x 2 + 8π x + 16π
41. x 2 + 2xy + y 2 43. x 4 - 16
45. x 4 - 8x 2 + 16 47. 1 + 2x + x 2
49. x 6 - 2a 3x 3 + a 6 51. 36a 2 25b 2 53. 4; 4 55. 25; 25 57. 9; 9
59. -5; -5 61. 840 65. 1, 4, 9, 16,
25, 36, 49, 64, 81, 100 67. B
69. D 71. x 3 + 4x 2 - 16x - 64
73. b = ± 2 √c 75. 13 cm
Study Guide: Review
1. cubic 2. standard form of a
polynomial 3. monomial
4. trinomial 5. scientific notation
1
1
1
6. __
in. 7. 1 8. 1 9. ___
10. _____
,
32
125
10,000
27
1
1
12. ___
13. __
or 0.0001 11. __
4
16
256
1
1
15. b 16. -____
17. 2b 6c 4
14. ___
2 4
2
m
2x y
3a
s
19. ___
20. 10,000,000
18. ___
2
2
2
3
4c
qr
21. 0.00001 22. 10 2 23. 10 -11
24. 325,000 25. 1800 26. 0.17
27. 0.000299 28. 5.8 × 10 -7,
6.3 × 10 -3, 2.2 × 10 2, 1.2 × 10 4
29. $38,500,000,000 30. 5 9
31. 2 3 · 3 4 32. b 10 33. r 5 34. x 12
1
1
1
35. 1 36. __
, or __18 37. __
, or ___
4
3
625
2
5
1
38. ____
39. g 12h 8 40. x 4y 2
6
16b
44. __1
41. -x y 42. x y 43. j k
5
45. m 8n 30 46. 8 × 10 11 47. 9 × 10 7
48. 1 × 10 10 49. 2.8 × 10 15
50. 6 × 10 1 51. 1.8 × 10 -8
7
52. 3.55 × 10 7 53. 64 54. m 5 55. __
32
3 4
1
56. 6b 57. t v 58. 16 59. 5 ×10
60. 2.5 × 10 7 61. 9 62. 7 63. 16
64. 8 65. z 2 66. 5x 2 67. x 4y 3
68. m 2n 4 69. 0 70. 3 71. 6 72. 1
73. 3n 2 + 2n - 4; 3 74. -a 6 - a 4 +
3a 3 + 2a; -1 75. linear binomial
76. quintic monomial 77. quartic
trinomial 78. constant monomial
79. -4t + 3 80. -6x 6 - x 5 81. 3h 3 3h 2 + 5 82. 2m 2 - 5m - 1 83. p 2 +
5p + 8 84. -7z 2 - z + 10 85. 3g 2 +
2g + 4 86. -x 2 + 4x + 8 87. 8r 2
88. 6a 6b 89. 18x 3y 2 90. 3s 6t 14
91. 2x 2 - 8x + 12 92. -3a 2b 2 +
6a 3b 2 - 15a 2b 93. a 2 - 3a - 18
94. b 2 - 6b - 27 95. x 2 - 12x + 20
96. t 2 - 1 97. 8q 2 + 34q + 30
98. 20g 2 - 37g + 8 99. p 2 - 8p + 16
100. x 2 + 24x + 144 101. m 2 +
12m + 36 102. 9c 2 + 42c + 49
103. 4r 2 - 4r + 1 104. 9a 2 - 6ab +
b 2 105. 4n 2 - 20n + 25 106. h 2 26h + 169 107. x 2 - 1 108. z 2 - 225
109. c 4 - d 2 110. 9k 4 - 49
4 2
6 15
6 9
Chapter 8
8-1
Check It Out! 1a. 2 3 · 5 1b. 3 · 11
1c. 7 2 1d. 19 2a. 4 2b. 5 3a. 9g 2
3b. 1 3c. 1 4. 7
Exercises 3. 3 2 · 2 2 5. 3 3 · 2 7. 7
(prime) 9. 3 · 5 2 11. 7 13. x 2
15. 1 17. 2 · 3 2 19. 2 2 · 3
21. 17 23. 7 2 25. 9 27. 10 29. 9s
31. 5 33. 4x 2 35. 2n 39. 15
rows 41. 8 and 20; 4 43. 63 and
105; 21 45. 54 and 72; 18 47. 36; 2;
9; 3 49. 105; 5; 7 51. 2; 2; 27; 3
53. 24; 2; 6; 3 55. 2; 2; 10, 5 57. D
61. 9y 63. 2p 2r 65. 4a 2b 3 69. 3.12 h
71. ≈ 0.10 mi/yr 73. 40 75. 3x 2 +
14x - 3
1d. 2x 2(4x 2 + 2x - 1)
2. 2x cm; (x + 2) cm
3a. (4s - 5)(s + 6) 3b. (7x + 1)(2x + 3)
3c. cannot be factored 3d. (5x - 2)2
4a. (2b 2 + 3)(3b + 4)
4b. (4r + 1)(r 2 + 6)
5a. (5x 2 - 4)(3 - 2x)
5b. (8 + x)(y - 1)
Exercises 1. 5a(3 - a) 3. 7(-5x + 6)
5. 2h(6h 3 + 4h - 3) 7. m(9m + 1)
9. 3(12f + 6f 2 + 1) 11. 16t(-t + 20)
13. (2b + 5)(b + 3)
15. (x 2 + 2)(x + 4)
17. (2b 2 + 5)(2b - 3)
19. (7r 2 + 6)(r - 5)
21. 2(r - 2)(r - 3)
23. (7q - 2)(2q - 3)
25. (2m 2 - 3)(m - 3) 27. 9y(y + 5)
29. x 2(-14x 2 + 5)
31. -d 2(4d 2 - d + 3)
33. 7c(3c + 2) 35. -5g 2(g + 3)
37. cannot be factored
39. (6y + 1)(y - 7)
41. (-3 + 4b)(b + 2)
43. (2a 2 + 3)(a - 4)
45. (6x 2 + 1)(x + 3)
47. (n 2 + 5)(n - 2)
49. (2m 2 - 3)(m - 1)
51. (2f 2 - 5)(3f - 4)
53. (b 2 - 2)(b + 4) 55. 3v
57. 2k 59. 2; binomial; x(x + 5)
61. 3; trinomial; a 2(a 2 + a + 1)
63a. 100x 3; 200x 2; 400x
b. 100x 3 + 200x 2 + 400x + 800
c. 100(x 2 + 4)(x + 2); $1603.12
67. The sum of opposite binomials
is 0. 69a. Commutative Property
of Addition b. Associative Property
of Addition c. Distributive
Property d. Distributive Property
71. D 73. C 75. -9ab(8ab + 5)
77. (a + c)(b + d)
79. (x 2 + 3)(x - 4) 81. 11 83. 5
8-3
Check It Out! 1a. (x + 4)(x + 6)
1b. (x + 4)(x + 3) 2a. (x + 6)(x + 2)
2b. (x - 6)(x + 1) 2c. (x + 6)(x + 7)
2d. (x - 8)(x - 5) 3a. (x + 5)(x - 3)
3b. (x - 4)(x - 2) 3c. (x - 10)(x + 2)
4.
n
n 2 - 7n + 10
0
0 2 - 7 (0) + 10 = 10
Check It Out! 1a. b(5 + 9b 2)
1
1 2 - 7(1) + 10 = 4
1b. cannot be factored
1c. -y 2(18y + 7)
2
2 2 - 7(2) + 10 = 0
3
3 2 - 7(3) + 10 = -2
4
4 2 - 7(4) + 10 = -2
8-2
n
0
1
2
3
4
(n - 5)(n - 2)
(0 - 5)(0 - 2) = 10
(1 - 5)(1 - 2) = 4
(2 - 5)(2 - 2) = 0
(3 - 5)(3 - 2) = -2
(4 - 5)(4 - 2) = -2
Exercises 1. (x + 4)(x + 9)
3. (x + 4)(x + 10) 5. (x + 2)(x + 8)
7. (x - 1)(x - 6) 9. (x - 3)(x - 8)
11. (x + 9)(x - 3) 13. (x - 2)(x + 1)
15. (x - 9)(x + 5) 17. (x + 3)(x + 10)
19. (x + 4)(x + 12)
21. (x + 2)(x + 14) 23. (x - 1)(x - 5)
25. (x - 4)(x - 8) 27. (x + 7)(x - 3)
29. (x - 13)(x + 1) 31. (x - 7)(x + 5)
33. C 35. D 37. They are inverse
operations. 39. (x - 2)(x - 9)
41. (x + 1)(x + 9) 43. (x + 6)(x + 7)
45. (x + 2)(x + 9) 47. (x - 3)(x + 8)
49. (x - 5)(x + 9) 51. approximately
1.5 55. x 2 + 6x + 8; (x + 4)(x + 2)
57. Positive; - , - ; Both negative
59. Negative; + , - ; Positive;
Negative 61a. d = t 2 b. d = 4t
c. t(t - 4) 63. true 65. false 67. 4
69. 4 71a. (x + 10) ft b. = (x + 14) ft;
w = (x + 6) ft c. A = (x 2 + 20x + 84) ft 2
73. D 75. C 77. (x 2 + 9)(x 2 + 9)
79. (d 2 + 21)(d 2 + 1)
81. (de - 5)(de + 4) 83. 16; 11; 29
85a. (x + 7) ft b. (4x + 26) ft c. $92.00
d. $36.96 e. $128.96 87. x 5 89. t 12
91. (x + 2)(x 2 + 5)
93. (p - 2)(2p 3 + 7)
8-4
Check It Out! 1a. (3x + 1)(2x + 3)
1b. (3x + 4)(x - 2)
2a. (2x + 5)(3x + 1)
2b. (3x - 4)(3x - 1)
2c. (3x + 4)(x + 3)
3a. (3x - 1)(2x + 3)
3b. (4n + 3)(n - 1)
4a. -1(2x + 3)(3x + 4)
4b. -1(3x + 2)(x + 5)
Exercises 1. (2x + 5)(x + 2)
3. (5x - 3)(x + 2) 5. (3x + 4)(x - 6)
7. (x + 2)(5x + 1)
9. (4x - 5)(x - 1)
11. (5x + 4)(x + 1)
13. (2a - 1)(2a + 5)
15. (2x - 3)(x + 2)
17. (10x + 1)(x - 1)
19. (2x + 3)(4 - x)
Selected Answers
SA19
21. -1(5x + 3)(x - 2)
23. -1(2x - 1)(2x + 5)
25. (3x + 2)(3x + 1)
27. (n + 2)(3n + 2)
29. (4c - 5)(c - 3)
31. (2x + 5)(4x + 1)
33. (5x - 6)(x + 3)
35. (10n - 7)(n - 1)
37. (7x + 1)(x + 2)
39. (3x - 4)(x - 5)
41. (x - 7)(4x - 3)
43. (4y - 1)(3y + 5)
45. (2x - 1)(2x + 3)
47. (3x + 5)(x - 3)
49. -1(2x - 3)(2x + 5)
51. -1(3x - 2)(x + 1)
53. 2x 2 - 5x + 2; (x - 2)(2x - 1)
55. (9n + 8)(n + 1)
57. (2x - 1)(2x - 5)
59. (3x + 8)(x + 2)
61. (3x + 4)(2x - 3)
63. (2x - 3)(2x - 3)
65. (2x + 3)(3x + 2)
69a. -16t 2 + 20t + 6
b. -2(4t + 1)(2t - 3) c. 10 ft 71. D
73. B 77. B 79. A
81. (2x + 1)(2x + 1)
83. (9x + 1)(9x + 1)
85. (5x + 2)(5x + 2) 87. -7; -5; 5; 7
89. -6; 6 95. (x + 1)(x - 9)
8-5
Check It Out! 1a. yes; (x + 2)2
1b. yes; (x - 7) 2 1c. no; -6x ≠
2(3x)(2) 2. 4(3x + 1) m; 40 m
3a. yes; (1 - 2x)(1 + 2x)
3b. yes; (p 4 + 7q 3)(p 4 - 7q 3)
3c. No; 4y 5 is not a perfect square.
Exercises 1. yes; (x - 2)2 3. yes;
(3x - 2)2 5. yes; (x - 3) 2
7. 4(x + 12); 88 yd 9. yes; (s + 4)
(s - 4) 11. yes; (2x 2 + 3y)(2x 2 - 3y)
13. yes; x 3 + 3x 3 - 3 15. No; the
last term must be positive. 17. no;
10x ≠ 2(5x)(2) 19. yes; (4x - 5)2
21. yes; (1 + 2x)(1 - 2x) 23. No;
4x and 9y are not perfect squares.
25. yes; (9 - 10x 2)(9 + 10x 2)
27. 49 29. 4y 2
31. (10x + 9y)(10x - 9y); difference
of 2 squares
33. (2r 3 + 5s 3) (2r 3 - 5s 3);
difference of 2 squares
35. (x 7 + 12)(x 7 - 12); difference of
2 squares 39. c = 32 41a. 5z - 4
b. 20z - 16 c. 11; 44; 121
43a. 0; 0; 100; 100; 0 b. 16; 16; 36;
SA20
Selected Answers
36; -24 c. 25; 25; 25; 25; -25
d. 36; 36; 16; 16; -24 e. 100; 100; 0;
0; 0 45. a - b; a + b 47. C 49. 1
51a. a = 2; b = v + 2
b. [2 + (v + 2)][2 - (v + 2)] =
(v + 4)(-v) = -v 2 - 4v 53. a = 3y;
b = y; (3y - 4)(9y 2 + 12y + 16)
55. D: {5, 4, 3, 2}; R: {2, 1, 0, -1};
yes 57. D: {2}; R: {-8, -2, 4, 10}; no
59. 6a 3 + 14a 2 - 10a 61. t 2 - 8t + 16
63. 8
8-6
Check It Out! 1a. yes 1b. no;
4(x + 1) 2 2a. 4x(x + 2) 2
2b. 2y(x - y)(x + y)
3a. (3x + 4)(x + 1)
3b. 2p 3(p + 6)(p - 1)
3c. 3q 4(3q + 4)(q + 2) 3d. 2(x 4 + 9)
Exercises 1. yes 3. yes 5. no;
4(2p 2 + 1)(2p 2 - 1)
7. 3x 3(x + 2)(x - 2) 9. 2p(2q + 1) 2
11. mn(n 2 + m)(n 2 - m)
13. 3x 2(2x - 3)(x + 1)
15. (p 3 + 1)(p 2 + 3)
17. unfactorable 19. no;
2xy(y 2 - 4y + 5)
21. no; 3n 2(n + 5)(n - 5) 23. yes
25. -4x(x - 3)2 27. 5(d - 3)(d - 9)
29. 2x(7x + 5y)(7x - 5y)
31. unfactorable
33. (p 2 + 4)(p + 2)(p - 2)
35. (k 2 + 3)(2k + 3) 37. x 2 + 12x +
36 = (x + 6)2 39. s 2 - 16s + 28 =
(s - 2)(s - 14) 41. b 2 - 49 =
(b + 7)(b - 7) 45. (3x - 1)(x + 7)
47. (3x + y - 3)(3x - y - 7)
53. 8 55. C 57. C 59a. V =
8p⎡⎣π(3p + 1)2⎤⎦ b. r = (3p + 1) cm
c. h = 8 cm; V = 128π cm 3
61. h 2(h 4 + 1)(h 2 + 1)
63. x n + 3(x 2 + x + 1) 65. -2n
23
67. 12.3r 69. __
= __3x ; 34.5 cm
2
71. (2x - 1)(2x + 3)
Study Guide: Review
1. prime factorization 2. greatest
common factor 3. 2 2 · 3 4. 2 2 · 5
5. 2 5 6. prime 7. 2 3 · 5 8. 2 6
9. 2 · 3 · 11 10. 2 · 3 · 19 11. 5
12. 12 13. 1 14. 27 15. 4 16. 3
17. 2x 18. 9b 2 19. 25r 20. 6 boxes;
13 rows 21. 5x(1 - 3x 2)
22. 16(-b + 2) 23. -7(2v + 3)
24. 4(a 2 - 3a - 2)
25. 5g(g 2 - 3)(g 2 + 1)
26. 10(4p 2 - p + 3)
27. (6x + 5) ft by x ft
28. (2x + 9)(x - 4)
29. (t - 6)(3t + 5)
30. (5 - 3n)(6 - n)
31. (b + 2)(b + 4)
32. (x 2 + 7)(x - 3)
33. (n 2 + 1)(n - 4)
34. (2b + 5)(3b - 4)
35. (2h 2 - 7)(h + 7)
36. (3t + 1)(t + 6)
37. (5m - 1)(2m + 3)
38. (4p - 3)(2p 2 + 1)
39. -1(r - 5)(r - 2)
40. (b 2 - 5)(b - 3)
41. (t + 4)(-t 2 + 6)
42. -1(3h - 1)(h - 4)
2
43. -1(d - 1) 44. (2 - b)(5b - 6)
45. (t + 1)(5 - t)
46. (2b 2 + 5)(4 - b)
47. -1(3r - 1)(r - 1)
48. left rectangle: 2x 2 + 3x; right
rectangle: 8x + 12; combined:
2x 2 + 8x + 3x + 12; (2x + 3)(x + 4)
49. (x + 1)(x + 5) 50. (x + 2)(x + 4)
51. (x + 3)(x + 5) 52. (x - 6)(x - 2)
53. (x + 5) 2 54. (x - 2)(x - 11)
55. (x + 4)(x + 20) 56. (x - 6)(x - 20)
57. (x + 12)(x - 7) 58. (x + 3)(x - 8)
59. (x + 4)(x - 7) 60. (x - 1)(x + 5)
61. (x + 3)(x - 2) 62. (x + 5)(x - 4)
63. (x - 8)(x + 6) 64. (x - 9)(x + 4)
65. (x - 12)(x + 6)
66. (x - 10)(x + 7)
67. (x + 20)(x - 6)
68. (x + 7)(x - 1) 69. (y + 3) m
70. (2x + 1)(x + 5)
71. (3x + 7)(x + 1)
72. (2x - 1)(x - 1)
73. (3x + 2)(x + 2)
74. (5x + 3)(x + 5)
75. (2x - 3)(3x - 5)
76. (4x + 5)(x + 2)
77. (3x + 4)(x + 2)
78. (7x - 2)(x - 5)
79. (3x + 2)(3x + 4)
80. (2x + 1)(x - 1)
81. (3x + 1)(x - 4)
82. (2x - 1)(x - 5)
83. (7x + 2)(x - 3)
84. (5x + 1)(x - 2)
85. -1(2x - 1)(3x + 2)
86. (6x + 5)(x - 1)
87. (3x - 2)(2x + 7)
88. -1(2x + 1)(2x - 5)
89. -1(2x - 3)(5x + 2)
90. 12x 2 - 11x - 5; (4x - 5)(3x + 1)
2
91. yes; (x + 6)2 92. no; 5x ≠ 2(x)(5)
93. no; -2x ≠ 2(2x)(1)
94. yes; (3x + 2)2 95. no; 8x ≠
2(4x)(2) 96. yes; (x + 7)2 97. yes;
(10x - 9)(10x + 9) 98. No; 2 is
not a perfect square. 99. No; 5
and 10 are not perfect squares.
100. yes; (-12 + x 3)(-12 - x 3)
101. no; terms must be subtracted
102. yes; (10p - 5q)(10p + 5q)
103. (x - 5)(x + 5); difference of 2
squares 104. (x + 10) 2; perfectsquare trinomial 105. (j - k2)(j + k2);
difference of 2 squares
106. (3x - 7)2; perfect-square
trinomial 107. (9x + 8) 2;
perfect-square trinomial
108. (4b 2 - 11c 3)(4b 2 + 11c 3);
difference of 2 squares
109. no; 2(2x + 3)(x + 1)
110. yes
111. no; (b 2 + 9)(b - 3)(b + 3)
112. yes 113. 4(x - 4)(x + 4)
114. 3b 3(b - 4)(b + 2)
115. a 2b 3(a - b)(a + b)
116. t 4 (t 8 + 1)(t 4 + 1)(t 2 + 1)
(t + 1)(t - 1) 117. 5(x + 3)(x + 1)
118. 2x 2(x - 5)(x + 5)
119. 2(s + 4)(t + 4)
120. 5m(5m + 2)(m - 4)
121. 4x(4x 2 + 1)(2x - 3)
122. 6s 2t (s + t 2)
123. 2(m + 3)(m - 3)(5m + 2)
Chapter 9
4a. vertex: (-2, 5); maximum: 5
4b. vertex: (3, -1); minimum: -1
5a. D: all real numbers; R: y ≥ 4
5b. D: all real numbers; R: y ≤ 3
1b.
x
Exercises 1. minimum 3. yes
5. no
7.
Exercises 1.
9.
3.
x
2b.
y
x
11. upward; a > 0 13. upward;
a > 0 15. downward; a < 0
17. (-3, -4); minimum: -4 19. D:
all real numbers; R: y ≤ 4 21. D: all
real numbers; R: y ≥ -4 23. yes
25. yes 31. upward; a > 0
33. vertex: (0, -5); minimum: -5
35. D: all real numbers; R: y ≤ 0
37. D: all real numbers; R: y ≥ -2
39. never 41. always
43. sometimes 45. no 47. yes
49. yes 53. quadratic 55. quadratic
57. neither 59. linear 61b. t ≥ 0
c. 16 ft d. 2 s 65. C 67. yes
71. (-2)4 73. 42 __34 mi
5.
7. x = 2 9. x = -2 11. x = - __34
13. (1, 8) 15. (-2, -2) 17. (-2, -9)
19. no zeros 21. -8, -2
23. x = 6 25. x = - __12 27. x = 5
29. (-3.5, -12.25) 31. (-2, 5)
33. (1, 4.5) 35. x = 0. 37. 0 39. 2
41. B 43. 2 45. 25 ft; 100 ft
47. y = -2x + 3 49. y = -4x + 2
51. yes
9-3
1a.
y
y
x
7. maximum height: 144 ft at 3 s;
time in the air: 6 s
y
9.
x
11.
9-2
x
3a. Because a < 0, the parabola
opens downward. 3b. Because
a > 0, the parabola opens upward.
x
Check It Out!
y
Exercises 3. -1 5. no zeros
x
differences are constant. 1b. Yes;
the function can be written in the
form y = ax 2 + bx + c.
y
y
x
x
1b. 3 2a. x = -3 2b. x = 1
3. x = - __14 4. (2, -14) 5. 7 ft
y
9-1
Check It Out! 1a. Yes; the second
2. maximum height: 16 ft at 0.5 s;
time it takes to reach the pool: 1.5 s
y
y
x
Check It Out! 1a. no zeros
2a.
y
13.
y
x
15. x = 4; (4, -16) 17. x = 0; (0, 4)
15
19. x = - __12 ; - __12 , - __
21. vertex:
4
(0, 0); axis of symmetry: x = 0
23. vertex: (3, -5); axis of symmetry:
x = 3 25. vertex: (0, -4); axis of
symmetry: x = 0 27b. D: 0 ≤x ≤
3.16; R: 0 ≤ y ≤ 50 c. about 3.16 s
31. 12 cm/s 35a. h(t) = -16t 2 +
45t + 50 b. approximately
(1.4, 81.6) 37. A 39. D 41. -1
43. x-intercept: 3; y-intercept: 6
(
)
Selected Answers
SA21
45. no x-intercept; y-intercept: 3
47. (3, -1) 49. (-1, -16) 51. (-1, 4)
9-4
Check It Out! 1a. f (x), g(x)
1b. g(x), f (x), h (x) 2a. same
width, same axis of symmetry,
opens upward, translated 4 units
down 2b. narrower, same axis
of symmetry, opens upward,
translated 9 units up 2c. wider,
same axis of symmetry, opens
upward, translated 2 units up
3a. The graph of the ball that is
dropped from a height of 100 ft is
a vertical translation of the graph
of the ball that is dropped from
a height of 16 ft. The y-intercept
of the graph of the ball that is
dropped from 100 ft is 84 units
higher. 3b. The ball that is dropped
from 16 ft reaches the ground in
1 s. The ball that is dropped from
100 ft reaches the ground in 2.5 s.
Exercises 1. f (x), g (x) 3. h (x),
g (x), f (x) 5. same width, same
axis of symmetry, opens upward,
translated 6 units up 7. wider,
same axis of symmetry, opens
upward, same vertex
9a. h 1(t) = -16t 2 + 16, h 2(t) =
-16t 2 + 256; the graph of h 2 is a
vertical translation of the graph
of h 1. The y-intercept of h 2 is 240
units higher. b. The baseball that
is dropped from 256 ft reaches the
ground in 4 s. The baseball that
is dropped from 16 ft reaches the
ground in 1 s. 11. f (x), g(x) 13. f(x),
h (x), g (x) 19. always 21. never
25. f (x) = 3x 2 - 6 29. B 31. A
39. D 41. 0 43a. f (x) = x 2 - 7
b. f (x) = -x 2 + 2 c. f (x) = __12 x 2 + 1
45. no correlation
9-5
Check It Out! 1a. x = -4 1b. no
real solutions 1c. x = -2 or
x = 2 2. 2 s
Exercises 1. zeros, x-intercepts
3. -4, 4 5. 6 7. -2, 4 9. -1
11. no real solutions 13. no real
solutions 15. x = -4 or x = 4
17. x = __12 or x = 5 19. x = -3
21. no real solutions 23. no real
solutions 25. always 27. always
SA22
Selected Answers
29. never 31a. 4 s b. 10 ft 33. 1.4 s
35. -1, 1 41. C 45. no real solutions
47. x ≈ -1.6 or x ≈ 0.86 49. y - 4 =
-3(x + 2) 51. 27 53. x 11 55. a 3b 3
27b 2
57. ____
6
8a
9-6
Check It Out! 1a. 0, -4 1b. -4, 3
2a. 3 2b. 1, -5 2c. - __53 2d. __13 , 1
3. 1.5 s
Exercises 1. -2, 8 3. -7, -9
5. -11, 0 7. -6, 2 9. 2, 3
11. -8, -2 13. 4 15. 6 17. -2, -__32
19. 1 s 21. -4, -7 23. 0, 9
25. - __12 , __13 27. -2, 4 29. -__13 , 1
31. -2 33. 1 35. 1 37. 1 39. B
41. 6 m 43. 6 s 45. no 47a. 3 s
b. 64 ft c. yes 49. F 51. -5, -1
53. __12 , -3 55. -2, 5 57. -1, 0
59. x 2 - x - 12 61. x 2 - 4x - 12 =
48; x = 10 63. -8 65. 2 67. (-9,
-16) 69. ±7 71. 3, 5
23. -13, -2 25. -6, 8 27. -2, 3
-1 ± √
5
-15 ± √
105
31. _________
33. 4 in.
29. _______
2
2
7 37. -3, __12 39. -10, 2
35. 1 ± √
49
41. 81 43. __
45. 9
4
47a. (10 + 2x)(24 + 2x) = 640
b. 3 ft 51. -6 ± √
26 53. -6 55. no
real solutions 57. no real solutions
61a. -16t 2 + 64t + 32 = 0 b. 4
c. 4.4 s 63. B 65. B 67. - __32 , __23
√
7
√
7
, - __23 + ___
71. 0, - __ab
69. - __23 - ___
3
3
2
2
77. x - 8x + 16 79. t - 8t + 16
81. 64b 4 - 4 83. ±1 85. ±4 87. ±15
89. ±1.55 91. ±5.10 93. ±1.48
9-9
Check It Out! 1a. 2, - __13 1b. 2, -__15
8 - √
56
8 + √
56
≈ 0.13, ______
≈ 3.87
2. ______
4
4
3a. 0 3b. 1 3c. 2 4. No; for the
equation 45 = -16t 2 + 20t + 0,
the discriminant is negative, so the
weight will not ring the bell.
5a. -2, -5 5b. -2, 7
-4 - √
184
9-7
Check It Out! 1a. ±11
1b. 0 1c. no real solutions
2a. no real solutions 2b. 9, 1
3a. ±9.49 3b. ±5.66 3c. no real
solutions 4. 45 ft
Exercises 1. ±15 3. no real
solutions 5. no real solutions 7. ±5
9. no real solutions 11. 11, -5
13. ±5.20 15. ±4.47 17. ±13
19. no real solutions 21. no real
13
solutions 23. ± __29 25. ± __58 27. ± __
7
29. ±4.69 31. ±10.20 33. ±7.07
2d
__
35. 6.1 s 37. t = √
39. a = -6
a
and b = -3 or a = 6 and b = 3
41. about 4.2 ft by 8.4 ft 43. A
45. sometimes 47a. a > 0
b. a = 0 c. a < 0 49. no; x =
1
± √__
, irrational 51. yes; x = ± __12 ,
2
8
rational 55. H 57. ± __14 59. ± __
11
61. 13 63. 2, 4 65. -2, 7
9-8
-4 + √
184
5c. ________
≈ -4.39, ________
≈
4
4
2.39
Exercises 1. no 3. __12 , 3 5. -4, -10
-6 - √
24
1 __
7. - __
, 3 9. ________
≈ -5.45,
2
2 2
-6 - √
24
________
≈ -0.55
2
-1 - √
61
-1 + √
61
≈ 1.14, ________
≈
11. ________
6
6
√
-1 - 41
≈ -1.85,
-1.47 13. ________
4
-1 + √
41
________
≈ 1.35 15. 1 17. 0
4
19. 2 21. 0 23. yes 25. -3
27. - __12 29. -3, __32 31. 1, 9
√
4 - 12
≈ 0.27,
33. ______
2
-7 - √
17
4 + √
12
______
≈ 3.73 35. ________
≈ 2.78,
4
2
-7 + √
17
________
≈ -0.72 37. 2
4
39. no 41. -5, 3 43. no real
solutions 45. 2 solutions; -2, __14
47. 1 solution; __23 49. 2 x-intercepts;
7
__
, -3 51. 1 x-intercept; 5 57. A
2
59a. 1 b. -1 c. -1 d. -1 63. -10, 4
65. (r 3 + t)(s 2 + 5)
67. (n 4 - 2)(n - 6) 69. f (x), g (x)
25
Check It Out! 1a. 36 1b. __
1c. 16
4
Study Guide: Review
Exercises 3. 4 5. -5, -1 7. -6, 5
1. vertex 2. minimum; maximum
3. zero of a function 4. discriminant
5. completing the square 6. Yes; it
is in standard form. 7. No; a = 0
8. Yes; it is in standard form. 9. No;
a quadratic function does not have
a power of x greater than 2.
2a. -9, -1 2b. 4 ± √
21 3a. - __13 , 2
3b. no real solutions 4. 16.4 ft by
24.4 ft
-5 ± 3 √5
13. no real
9. 1, 9 11. ________
2
solutions 15. 4 ± √
10 17. 7.2 m;
11.2 m 19. 1 21. -2, 12
10.
25.
y
Chapter 10
y
10-1
x
11.
x
x
y
x
y
27.
x
13.
y
x
14. upward 15. downward
16. (-2, -4); minimum: -4
17. -5 and 2 18. -1 and 2
19. x = 6; (6, 4) 20. x = -1;
(-1, -18)
21.
y
x
22.
x
y
y
x
24.
y
-4
-4
-8
x
/THER
3LEEPING
3PORTS
(OMEWORK
Check It Out! 1a. bread
1b. cheese and mayonnaise
2. 2001, 2002, and 2005; about
13,000 3. about 18 °F 4. Prices
increased from January through
July or August, and then prices
decreased through November.
5. 31.25%
6.
6ERAS$AY
7ATER&OUNTAIN
(EIGHTM
12.
26.
y
23.
4IMES
In 2 seconds, the water reaches its
maximum height of 20 meters. It
takes a total of 4 seconds for the
water to reach the ground. 28. g(x),
f (x) 29. The graphs have the same
width. 30. h (x), f (x), g (x) 31. same
width, same axis of symmetry,
opens upward, vertex translated
5 units up 32. narrower, same
axis of symmetry, opens upward,
vertex translated 1 unit down
33. narrower, same axis of
symmetry, opens upward, vertex
translated 3 units up 34. x = -3
or x = -1 35. x = -3 36. no real
solutions 37. x = 1 or x = 5
38. x = 4 39. x = 1 or x = -1
40. no real solutions 41. x = -5 or
x = -1 42. x = -7 or x = -2
43. x = -3 or x = 5 44. x = -1 or
x = 2 45. x = -5 46. x = 4.5
47. x 2 + 2x = 48; 6 ft 48. x = -8
or x = 8 49. x = -12 or x = 12
50. no real solutions 51. x = 0
52. x = -5 or x = 5 53. x = - __52 or
x = __52 54. 4 ft 55. x = -8 or x = 6
56. x = -7 or x = 3 57. x = 1 or
x = 5 58. x = 5 ± √
5 59. 16 ft
by 12 ft 60. x = -1 or x = 6
61. x = - __12 or x = 5 62. x = 1
6 ± √
8
64. 1 65. 0 66. 2 67. 2
63. x = ______
2
3CHOOL
%ATING
A circle graph shows parts of a
whole.
Exercises 1. one part of a whole
3. 82 animals 5. $15 7. Prices at
stadium A are greater than prices at
stadium B. 9. between weeks 4 and
5 11. 18% 13. purple 15. blue and
green 17. 225,000 19. Friday
21. 3.5 times 23. games 3, 4, and 5
25. Stock Y changed the most
between April and July of 2004.
27. 8 __13 % 31. double line 33. circle
35a. Greece; about 40% b. United
States; about 15% 37. D 41. 19
girls 43. D: {-3, -1, 0, 1, 3}; R: {0,
1, 3}; yes 47. quadratic binomial
10-2
Check It Out!
1.
4EMPERATUREª#
3TEM ,EAVES
Key: 1]9 means 19
2.
Interval
Frequency
4–6
5
7–9
4
10–12
4
13–15
2
-12
Selected Answers
SA23
3.
11a.
6ACATION
.UMBER
Interval Frequency
Cumulative
Frequency
36–38
4
4
39–41
6
10
42–44
5
15
45–47
1
16
b. 10
n
4a.
n n n
,ENGTHDAYS
15a.
Interval
Frequency
160–169.9
2
170–179.9
4
Interval
Frequency
Cumulative
Frequency
180–189.9
3
28–31
2
2
190–199.9
1
32–35
7
9
200–209.9
2
36–39
5
14
210–219.9
1
40–43
3
17
4b. 9
Exercises 1. stem-and-leaf plot
3. !USTIN 3TEM .EW9ORK
"REATHING)NTERVALS
5.
&REQUENCY
n
n n
n
4IMEMIN
3UMMER 3TEM 7INTER
Key: ]2]1 means 21
Ê Ê 7]2] means 27
SA24
Check It Out! 1. mean: 14 lb;
median: 14 lb; modes: 12 lb and
16 lb; range: 4 lb 2. 3; the outlier
decreases the mean by 3.7 and
increases the range by 18. It has no
effect on the median and mode.
3a. mode: 7 3b. Median: 81; the
median is greater than either the
mean or the mode.
9.
10-3
4.
7.
19. G 21. 8; 8; 41; 66 23. -2.3
25. 0.5 in. 27. books
Interval
Frequency
2.0–2.4
2
2.5–2.9
7
3.0–3.4
5
3.5–3.9
3
Selected Answers
5a. The data set for 2000; the
distance between the points for the
least and greatest values is less for
2000 than for 2007. 5b. about $40
million
Exercises 3. mean: 31.5; median:
33.5; mode: 44; range: 32 5. mean:
78.25; median: 78; mode: 78; range:
15 7. 13; the outlier decreases the
mean by 11.15 and the median by
4. It increases the range by 51 and
has no effect on the mode.
9. Median: 83; the median is
greater than the mean, and there
is no mode. 13. Simon; about 3000
points 15. mean: 2.5; median: 2.5;
modes: 2 and 3; range: 3 17. mean:
60; median: 60; mode: 60; range: 5
19. 23; the outlier increases the
mean by 3, the median by 2.5, and
the range by 15. It has no effect
on the mode. 21. Mean: 153; the
mean is greater than the median,
and there is no mode. 25. Sneaks
R Us; the middle half of the data
doesn’t vary as much at Sneaks R
Us as at Jump N Run. 27. mean:
5.5; median: 5.5; mode: none;
range: 9 29. mean: 3.5; median:
3.4; mode: none; range: 5.3
31. mean: 24.4; median: 25; modes:
23 and 25; range: 3 33. mean: 15__16 ;
median: 12__12 ; mode: none; range:
35 37. sometimes 39. always
41. Median; the mean is affected
by the outlier of 1218, and there
is no mode. 43. Median or mode;
the store wants their prices to
seem low, and the median and
mode are both $2.80 less than the
mean. 49. Mean: $32,000; median:
$25,000; median; the outlier
of $78,000 increases the mean
significantly. 51. 96 53. increase
the mean; decrease the mean
55. G 57. The mean decreases by
6.6 lb. 61. 32; typing speed is 32
words per minute. 63. 2 65. 5
67. length: 5 yd; width: 3 yd
10-4
Check It Out! 1. Possible
answer: company D; the fertilizer
from company D appears to be
more effective than the other
fertilizers. 2. Possible answer: taxi
drivers; the drivers could justify
charging higher rates by using
this graph, which seems to show
that gas prices have increased
dramatically. 3. Possible answer:
Smith; Smith might want to show
that he or she got many more
votes than Atkins or Napier. 4. The
sample size is much too small.
Exercises 3a. The vertical scale
does not start at 0, and the
categories on the horizontal scale
are not at equal time intervals.
b. Possible answer: Tourism is
decreasing rapidly. 5. The sample
size is too small. 7a. The vertical
scale does not start at 0.
b. Possible answer: Single men
pay significantly more than single
women.
10-5
Check It Out! 1. sample space:
{1, 2, 3, 4, 5, 6}; outcome shown: 3
7
13
2. certain 3a. __
3b. __
4a. 99.8%
20
20
4b. 34,930
Exercises 3. sample space: {blue,
red, yellow, green}; outcome shown:
red 5. impossible 7. unlikely
9. __35 11a. 30% b. 54 13. sample
space: {blue, red, yellow}; outcome
shown: blue 15. as likely as not
6
17. likely 19. __
21a. 5% b. 21
25
27. about 1%; about 57 29. B 31. B
33. as likely as not 35. unlikely
37a. 7 b. 8 39. $14 41. reflected
across the x-axis 43. mean: 6;
median: 5; mode: 5
10-8
9.
Check It Out! 1. 15,600
2a. permutation; 6
2b. combination; 6 3. 362,880
4. 792
Exercises 1. combination 3. 8
5. combination; 6 7. 20,160 9. 35
11. 15,504 13. 441,000
15. combination; 10 17. 120
19. 133,784,560 21. nP r
1
25a. _________
b. about 85,810 h
308,915,776
27. J 29. 260,130 31. 168 33a. 28
b. 28 37. 5x + 5 39. independent:
minutes; dependent: volume
of water in tub; f (x) = 15x
41. independent: hours; dependent:
total fee; f (x) = 300 + 80x
1. outcome 2. interquartile range
3. independent events 4. 2003
5. 14 more boys
6. 3TEM ,EAVES
Check It Out! 1a. 50% 1b. 33 __13 %
1
2. 0.4 3. __
25
Key: 1]2 means 12
7. #OMEDY
#AMP
9
23. __15 25. __45
17. __59 19. 70% 21. __
10
1
__
27. 4:1 31. 2 35. D 37. B
10-7
Check It Out! 1a. Independent;
Exercises 1. dependent
3. independent 5. independent
1
7. __18 9. __16 11. __
13. __27
16
1
15. dependent 17. __18 19. __
12
27
1
b. __
25. dependent
21. __29 23a. __
64
64
3
b. __16
27. independent 29a. __
20
9
1
2
___
__
__
c. 100 d. 15 31. 15 35. D 37. A
39. 72.9% 41. 80% 43. 48%
5
45. 24 47. wider 49. __35 51. __
26
$AYSAND
$AYS
13
1
45. __
41. (4x - 3)(x - 1) 43. __
10
20
the result of rolling the number
cube the first time does not affect
the result of the second roll.
1b. Dependent; choosing the first
student leaves fewer students to
choose from the second time. 2. __14
8
3. __
87
n
n n n
#APACITYGAL
10. mean: 14; median: 12; mode:
12; range: 28 11. Median; the mean
is higher than 4 of the 5 prices; the
mode is the lowest price.
12.
Study Guide: Review
10-6
Exercises 1. complement 3. 25%
9
5. __12 7. __23 9. __
11. 1:12 13. __14 15. 50%
10
'AS4ANK#APACITIES
&REQUENCY
9a. The sectors of the graph do not
add to 100%.
b. Possible answer: Nearly half
of the state’s spending was for
welfare. 15. B 19. w ≤ 1500
21. b ≤ 20 23. t ≥ -4
Chapter 11
Gas Tank Capacities
Capacity
Key: ]12]8 means 128
Ê Ê 4]10] means 104
8.
13. The scale on the vertical axis is
too large. This makes the slopes of
the segments less steep.
14. Someone might believe that the
price has been relatively stable
when in fact it has doubled.
15. 99.5% 16. 24,875 17. 250 18. __12
19. __14 20. __58 21. independent
22. independent 23. dependent
2048
256
24. ____
25. 0 26. ____
27. 60
9555
2401
28. permutation; 604,800
29. combination; 220
30. combination; 1365
Tally
Frequency
10−14
IIII I
6
15−19
IIII IIII
20−24
III
3
25−29
III
3
10
11-1
Check It Out! 1a. 80, -160, 320
1b. 216, 162, 121.5 2. 7.8125
3. $1342.18
Exercises 3. 25, 12.5, 6.25
5. 1,000,000,000 7. 4 9. 162, 243,
364.5 11. 2058; 14,406; 100,842
5 ___
5
13. __
, 5 , ___
15. 0.0000000001, or
32 128 512
1
1 × 10 -10 17. 80; 160 19. __13 21. __17 ; __
49
1
23. 6; -48 25. 4913 27. yes; __3 29. no
31. no 33a. 1.28 cm b. 40.96 cm
35. -2, -8, -32, -128
3
37. 2, 4, 8, 16 39. 12, 3, __34 , __
16
43a. $3993; $4392.30 b. 1.1
c. $2727.27 45. J 47. x 4, x 5, x 6
49. 1, y, y 2 51. -400 53. the 7th
term 55. b > 10 57. c < - __13
61. f (x) = x 2 + 4
Selected Answers
SA25
11-2
Check It Out! 1. 3.375 in. 2a. no
2b. yes
3a.
y
x
3b.
y
x
4a.
y
x
Exercises 1. exponential growth
t
3. y = 300(1.08) ; 441 5. A =
4200(1.007)4t; $4965.43 7. y =
t
10(0.84) ; 4.98 mg 9. 5.5 g 11. y =
t
1600(1.03) ; 2150 13. A = 30(1.078)t;
47 members 15. A = 7000(1.0075)4t;
t
$9438.44 17. A = 12,000(1.026) ;
t
$17,635.66 19. y = 58(0.9) ; $24.97
21. growth; 61% 23. decay; 33 __13 %
25. growth; 10% 27. growth; 25%
t
29. y = 58,000,000(1.001) ;
t
58,174,174 31. y = 8200(0.98) ;
t
$7118.63 33. y = 970(1.012) ; 1030
35. B 37. 18 yr 39. A; B 45. D 47. D
49. about 20 yr 51. 100 min, or 1 h
40 min 53. $225,344 55. 16 ft
11-4
Check It Out!
1a.
y
4b.
y
y
1b.
5a.
y
x
x
Exercises 1. no 3. no 17. about
2023 19. 289 ft 21. yes 23. no
35. y = 4.8(2)x 41. -0.125
43a. $2000 b. 8% c. $2938.66
45. C 45. C 47. D 49. 3 51. The
value of a is the y-intercept.
53. 25 55. 9x 2
quadratic
2. quadratic 3. The oven
temperature decreases by 50 °F
every 10 min; y = -5x + 375; 75 °F
3. linear 5. exponential 7. Grapes
cost $1.79/lb; y = 1.79x ; $10.74
11. linear 13. exponential
15. = 6k ; linear 17. linear
x
19. y = 0.2(4) 21. linear 27. C
145
29. C 33. ___
g
35. 5, -5 37. __94 , - __94
11-5
Check It Out! 1a. 40 ft/s
11-3
Check It Out! 1. y = 1200(1.08)t;
$1904.25 2a. A = 1200(1.00875)4t;
$1379.49 2b. A = 4000(1.0025)12t;
t
$5083.47 3. y = 48,000(0.97) ; 38,783
3a.
SA26
Selected Answers
27. x ≥ - __32 37. 3.61 units
39. Mercury: 4214 m/s; Venus:
10,361 m/s; Earth: 11,200 m/s;
Mars: 5016 m/s 45. A 47. C
49. x ≤ -5 OR x ≥ 5
51. x ≤ -4 OR x ≥ __32
53. D: x ≤ 3; R: y ≤ 4 57a. 2 and 4
b. 3, 1 59. y = -__12 x + 2
61. 9x 2 - 6x + 1
63. a 2 - 2ab 2c + b 4c 2 65. 9r 2 - 4s 2
67. A = 42,000(1.0125)4t; $48,751.69
y
x
Check It Out! 1a. 8 1b. 7 1c. 13
1d. ⎪3 - x⎥ 2a. 8 √
2 2b. xy √
x
y3
6
3b 3a. __23 3b. __
3c. __
2c. 4a √
2
2
z 2 √z
x
p3
__
ft;
5. 60 √2
q5
17
6
2 9. 4x 2y √
2y 11. ____
13. ___
7. 18 √
5
7
√x
6 √3
√
√
√
16 2
5x 2
17. ____
19. _____
21. _____
15. ___
7
3
9
13
41 mi; 32 mi 25. 20 27. 9
23. 5 √
29. ⎪x + 1⎥ 31. ⎪x - 3⎥ 33. 20 √
10
3 √5
√
x √x
y
14
5 37. ____
39. ____
41. ____
35. 8rs √
3
2
3
8s √
3
47. 15x √
45. -20 √3
7
43. ____
7
x 3 √x
√
√
49. x 51. ____
53.
36
;
6
55.
50 ;
3
59. √
2 57. √3
20 ; 2 √
5 67. C
5 √
69. C 71. x √
x + 1 73a. ⎪x⎥ b. x 2
c. ⎪x 3⎥ d. x 4 e. ⎪x 5⎥ f. x n; ⎪x n⎥
75. no 77. exponential
11-7
Check It Out! 1a. - √
7 1b. 3 √3
1b. 30.98 ft/s 2a. x ≥ __12 2b. x ≥ __53
4a. 1.5625 mg 4b. 0.78125 g
Exercises 1. There is no variable
under the square-root sign.
3. x ≥ -6 5. x ≥ 0 7. x ≥ -3
15. 49.96 mi/h 17. x ≥ 0 19. x ≤ __32
21. x ≥ - __53 23. x ≥ 40 25. x ≥ 9
√
Exercises 1. exponential
6. after about 13 yr
Exercises 1. 3x - 6 3. 7 5. 6 √
5
x
y
84.9 ft
5b.
x
2 √
5
4b. ____
4c.
4a. ____
7
5y
y
11-6
exponential
x
x
3b.
+ 8 √
1c. 8 √n
5s 2a. 5 √
6
1d. √2s
2b. 12 √
3 - 3 √
2 2c. 5 √
3y
3. 10 √
b in.
+ 5 √
Exercises 3. 10 √
5 5. 3 √7
2
+ 6 √5a
9. 13 √3 11. - √5x
7. 5 √6a
- 4 √3t
15. 6 √3 17. -3 √
13. 8 √2t
11
√
√
19. -4 √n
21.
7
7
23.
12
2
25. 3 √
7x - 12 √
3x 27. 3 √
5j
31. 12 √7 33. 0 35. 7 √3
29. 2 √3m
37. 7 √
2 41. 2 √
3 + 5 √
5+5
√
43. 8 √
7x - 70x 45. 35 √
5k
47. 5 √
3 + 5 √
5 51. 9 53. 18
in.; 8 √
55. 36x 2 57. 16 √3
3 in.;
24 √
3 in. 59. B 61. A 63. √
x (x + 2)
65. 0 67. (x + 2) √
x-1
1
69. 3 √
x + 1 - x √
x + 2 73. __
12
75. x ≥ -3
11-8
Check It Out! 1a. 5 √
2 1b. 63
- 3 √6
2a. 4 √3
1c. 2m √7
2b. 5 √
2 + 4 √
15 2c. 7 √
k - 5 √
7k
3a. 21 + 5 √
2d. 150 - 20 √5
3
3b. 83 + 18 √
2 3c. 11 - 6 √2
√
√
65
21a
3 4a. ____
4b. _____
3d. 17 - √
5
6
8 √
35
4c. _____
7
Exercises 1. √
6 3. 125 5. 3 √
30a
9. √
7. 2 √
6 + √42
35 - √
21
11. 5 √
3y + 4 √
5y 13. 12 + 7 √
2
17. 81 - 30 √
15. -5 - 2 √3
2
33. no solution 35. 4 37. 2 39. no
solution 41. 48 43. -25 45. 71
47. -8 49. 36 51. -16 53. 8 55. 9
57. 2 59. -5 61. 5 63. 1 65. 1
67. x = 144; 12 in. 69. √
x - 3 = 4; 49
71. x = √
x + 6 ; 3 73. 3 in. by 1 in.
75a. 54.88 joules b. 0 joules
77. 1690 ft 79. x = 25; y = 16
81. sometimes 87. A 89. C
91. A 93. 1, 2 95. 2 97. 0
101. 51.2 mi 103. 10,000
Study Guide: Review
1. square-root function
2. exponential decay 3. common
ratio 4. exponential function
5. 81, 243, 729 6. 48, -96, 192
7. 5, 2.5, 1.25 8. -256, -1024, -4096
9. 7,812,500 10. 19,131,876 11. yes
12. no
13.
π √6
3x + x 69. ____
s ≈ 1.9 s
67. 3 + 2 √
4
71. 269.5 ft 2 75. B 77. D
+ 4 √5
81. -5 - 2 √
6
79. -4 √3
85. 2 √
83. 2 - √3
6 + 2 √5
87. translation 4 units down
89. (x - 3)(x + 10)
91. (x + 4)(x - 4)
93. 2(x 2 + 3)(x 2 - 3)
7x
95. 6 √
10 97. ___
2
8y
38.
y
x
39.
y
x
40.
y
x
x
41.
y
14.
y
x
42.
Check It Out! 1a. 36 1b. 3 1c. __13
y
y
x
43.
y
x
18
33. x ≥ __34
31. x ≥ __72 32. x ≥ - __
5
34. x ≥ 1
35.
x
t
15. y = 9(1.15) ; 24
t
16. y = 24,500(0.96) ; 3182
17. quadratic 18. linear
19. exponential 20. exponential
21. quadratic 22. linear
23. y = 1.5x; 15 h 24. 4.74 cm
25. x ≥ 0 26. x ≥ -4 27. x ≥ 0
28. x ≥ -2 29. x ≥ __43 30. x ≥ -3
11-9
2a. 9 2b. 18 2c. 3 3a. 121 3b. 64
11
3c. 100 4a. 2 4b. __
5a. no solution
2
5b. no solution 5c. 4 6. 8; 3 cm
7 √
2x
51. 2 √y 53. 180 in 2
49. _____
2
55. (6 √
10 - 2 √
5 ) cm 2 57. √
30
61. 3 √
59. -5 - 2 √3
2
63. 134 √
3 + 96 65. x - 2 √
xy + y
√
5x
25. ____
27. 3 √
10 29. 8
5x
x
y
y
√
√
2 √7
26
33
21. ____
23. ____
19. ____
7
2
18
7 33. 4 √
5 - 5 √
2
31. 6d √
35. 2 √
3 - 2 √
5 37. 3 √f + 12 √
3f
39. 75 + 19 √
15 41. 10 - √
2
√
5 √
6
3x
43. 67 + 16 √
3 45. ____
47. ____
x
2
37.
44.
y
x
x
36.
45. 11 46. n 2 47. x + 3 48. 5
49. 6d 50. y 3 √
x 51. 2 √
3
√5
t
2 √
55. __2
52. 4b 2ab 53. ___ 54. __
y
Exercises 1. No; it does not contain
a variable under the radical sign.
3. -8 5. -144 7. 27 9. 50 11. -2
13. 9 15. 64 17. 16 19. 16 21. __49
23. 100 25. 5 27. 13 29. 6 31. 2
2
10
3
4p √
2
t √t
2b 3 √2
56. _____
57. ____
58. _____
s
7
5
2
x
4
61. 3 √
7 60. 3 √3
2 + 2 √
3
59. 9 √
62. √
5t 63. 2 √
2 64. 2 √
3 + 2 √
5
Selected Answers
SA27
67. √
65. -2 √
5x 66. 10 √6
14
√
68. 3 √
2 69. 6 7x 70. 150
4 √
5
71. 4 √
2 - 4 72. 71 + 16 √
7 73. ____
5
2
17. y = __
x
12-1
Check It Out! 1a. No; the product
xy is not constant. 1b. Yes; the
product xy is constant. 1c. No; the
equation cannot be written in the
form y = __kx . 2. y = __5x
y
x
x
2
x
200
0
100 200 300
Members
41. C 43. C
47. approximately 888.9 watts
49. D: {-4, -2, 0, 2, 4}; R: {1, 3, 5};
yes 51. -1, 7 53. 2 √
10 cm
3
x
25.
2a. x = 5; y = 0 2b. x = -4; y = 5
2c. x = -77; y = -15
3a.
27a. D: x > 0; R: natural numbers > 5
b.
x
3b.
0ARTS
0RESSUREATM
y
4a. D: x > 0; R: natural
numbers > 10
x
-8
9. 4 11. 16 teeth
13. yes 15. no
12-3
#OPIES
Exercises 1. excluded value 3. -3
5. 4 7. x = -5; y = 0 9. x = -9;
y = -10
Selected Answers
!VERAGEPRICEOFPARTS
29. 7 31. -__12 37. x = -1; y = 0
39. x = 2; y = 5 41. B 43. C 51. D:
x > 2 53. D: x > - __15 55. I and III; II
and IV 59. J 61a. yes b. D: all real
numbers c. R: 0 < y ≤ 1 d. no
3
63. y = ____
+ 3 65. -2, 3
x+2
67. 0.46875 cm 69. no
4b.
8
0RICE
0
y
x
-6
7. y = ___
x
x
y
y
Exercises 3. no 5. yes
SA28
15. 0 17. 0 19. x = 4; y = 0
21. x = 3; y = 4
23.
y
100
4. 3 5. 80.625 lb
-8
y
Check It Out! 1a. 0 1b. 1 1c. -4
3. D: x > 0; R: y > 0; 2.5 mm
x
8
12-2
6OLUMEOFGASMM
13.
300
0
-10
19. 2 21. 12 yd 23. direct; 8
25. neither 27. inverse; 12
10
29. inverse; 15 31. d = __
n ; inverse
2000
33. neither 35. y = ____
x ; D: natural
numbers; R: y > 0
Contribution ($)
Chapter 12
0
y
2
-4
2
√3
√
√6
10n
3a √
2
75. ___
76. _____
77. ___
74. _____
2
3
2n
2
3 79. 64 80. 8 81. 3 82. 25
78. - √
83. -81 84. 100 85. 3 86. no
solution 87. x = 4 88. x = 6
19
89. x = 7 90. x = __
91. x = 12
2
92. x = 3 93. x = 4 94. x = 5
11.
y
Check It Out! 1a. -5 1b. 0, -5
m
1c. -3, -4 2a. __
; m ≠ 0 2b. 6p
3
3n
b-5
1
; n ≠ 2 3a. ____
3b. ____
2c. ____
r+5
n-2
b+5
3
3
1
4b. - ____
4c. _____
4a. - ____
4+x
x+1
x + 11
5. The barrel cactus with a radius
of 3 inches has less of a chance to
survive. Its surface-area-to-volume
ratio is greater than for a cactus
with a radius of 6 inches.
2
Exercises 3. 0, 8 5. __a2 ; a ≠ 0 7. ____
;
y+3
h
1
____
____
; h ≠ -2 11.
y ≠ -3 9.
h+2
j-5
c+2
2
15. ____
17. - ____
13. ____
c-4
8+n
j-3
b+1
2
b1 + b2
1
; x ≠ ±4
65. - ____
x+4
p-7
z-1
2
35. ____
37. ____
33. ____
p-5
x-4
z+1
12-6
p+6
3w + 7
1
43. ____
45. __13 47. _____
39. - ____
12
3
b+7
5+x
53a. __6
49. 1 51. - ____
s
b. 3 c. 1 57. F 59. sometimes
a-3
65. ±14
61. sometimes 63. ____
a+5
67. -2, 0 73. -6
5x 2y 4
Check It Out! 1a. - __94 1b. ____
6
p 2 - p - 20
n+4
3m - 15
______
______
_________
3b. 3
2. m - 6 3a. 2
n + 2n
p + 16p
3x - 15
2w 6
x
______
____
________
4a.
4b.
4c.
x5
v 2x 3
x 2 + 5x + 6
5. approximately 0.23
m 2 - 10m
7. 3y -6 9. ________
11. a 3 + 10a 2 +
2
a + 6b
2r + 28
1
_____
__
15. 17. ______
19. b
25a 13.
9
37. b
b. ___
236
5
1b. x 2 + __13 - __
2a. k + 5 2b. b - 7
2x
2c. s + 6 3a. 2y + 1 3b. a - 2
13
20
4a. 3m - 5 + _____
4b. y + 6 + ____
m+3
y-3
r-4
10y + 20
- 4 25. _______
3y + 15
7r
3n 2 - 3n
29. _______
31. 1
n+8
x2
___________
2
3p 8q 2
_____
4(4x 2 + 8x - 1)
1
1
39. ___
41. ___
2m
16x
x
1
51. __13 53. __
c. 4 47. H 49. ______
2
2
3x + 9x
z
1
57. m ≤ 9
55. _____
2a + 2
11. x + 1 13. c + 3 15. x - 2
-1
-1
17. a + 2 + ____
19. n + 4 + ____
n+4
a+2
-2
21. 4n - 5 + _____
2n + 1
3a. 15f 2h 2
4d - 3
3b. (x - 6)(x + 2)(x + 5) 4a. _____
2
a+8
5
5. __
h or 12.5 min
4b. ____
a-2
24
3d
1
2
Exercises 1. __2y 3. ____
5. ____
x-4
a+1
7. 6x 3y 2z 9. (y + 4)(y - 4)(y + 9)
x+3
260
1
__
13a. ___
11. ____
r b. 6 2 h 15. a - 1
x+2
9. y = __1x
y
35
23. -2x 2 + 6x - 15 + ____
x+3
x
-10
27. 4k 2 - 4k + 2 + ____
k+1
2
29. 3t + 4 - __2t 31. -4p + 1 + __
3
33. 4t + 3 35. x - 3 37. 3a - 1
14
39. 3x + 4 + ____
x-2
-2
41. 3x + 1 + _____
2x - 1
-216
43. 2t 2- 6t + 25 + _____
3t + 9
10. -15 11. $13,200 12. -4; x = -4
and y = 0 13. -1; x = -1 and y = 3
14. -3; x = -3 and y = -4 15. __74 ;
x = __74 and y = 5
16.
2y
x
√
15
65. 3 m 67. ____
15
73. 2k 2 + 5k + 2
2a. -4 2b. -4 2c. 1, 3 3. 22 __29 min
4a. 5; 7 is extraneous.
4b. 1 and 5; no extraneous
solutions 4c. 4; 0 is extraneous.
y
1
61. 3x - __
+ __
x 63. x + 2
2y
12-7
p
Check It Out! 1a. 2 1b. 1 1c. - __76
Check It Out! 1a. 2 1b. 3y
y
x
12-5
2b.
8. y = - __4x
x
5 - 5 √
2 71. 4(x + 1)
69. 6 √
2a.
1. rational expression 2. rational
function 3. rational equation
4. inverse variation 5. discontinuous
function 6. Yes; the product xy is
constant. 7. No; the product xy is
not constant.
14
Exercises 1. 2x - __12 3. 7b - __
+ __b8
3
3
__
5. 2x + 4 + 7. 2x - 3 9. 2y + 5
3t 61. x ≠ 3; x = 3 and y = 0
59. 8 √
63. x ≠ 0; x = 0 and y = 3 65. x ≠ 0;
x = 0 and y = 0
4b + 12
________
b 2 + 3b - 4
Study Guide: Review
-7
5a. x 2 - 2x - 4 + ____
x-2
3
45. -20 47. 2x - 5 + ____
x+1
51. 0.5m + 1 57. C 59. B
43. 1 45a. 64 cm b. 80 cm
3
____
a-2
31. 5 33. __32 35. -4, 3 37. 6 h
39. 2; 3 is extraneous. 41. No
solution; 4 is extraneous.
240
43. ___
; t - 2; 40 mi/h
t
1
1
45a. __
= __
+ __1y b. 40 cm
15
24
c. It will increase to 72 cm. 49. F
53. Eddie: 6 h; Luke 3 h; Ryan: 4 h
1
55. y = -2x and y = __
x + 4 are
2
perpendicular. 59. 5
3
25. m + 1 + _____
m-1
6h
2x - 4
Exercises 1. ___
3. _____
5. __a6
3
5jk 2
2a
Check It Out! 1a. -2p + 1 - __p3
21. __12 h, or 30 min 23. - __43 ; 1 is
extraneous. 25. -2 27. 0 29. __45
-7
5b. 2p 2 - 2p + 6 + ____
p+1
12-4
1
35a.
33. -___
3
(x + 4)(x + 2)
2
simplified; m ≠ 4 31. __8t ; t ≠ 0
27. 4m 2 - 4m
2b
8x + 20
43. A
41. __________
x - 4y
25. 0 27. - __12 , 4 29. already
1
21. ______
23.
3x - 15
4(m - 2)
3
x-5
1
b. 14 h 35. ____
37. ___
39. ____
2
7+c
3
)(
(
)
az + by +cx
_________
55.
; x ≠ 0, y ≠ 0, and
xyz
z ≠ 0 61. __1 , 4 63. 2; t ≠ ±2
b2
will be the same: ______
.
b2
27. ______
3 (y - 3 )
19
-m 2 - 6m
700
________
29. ___
31.
33a. ___
2
r
21z
; x ≠ y and x ≠ -y
53. __________
x+y x-y
b
x+2
y+2
(y + 4)
47. 4x 2; 8x 2; 8x 3 49. A 51. D
5
2
21. - _____
23a. ______
b. They
10 + q
b +b
1
17. m 19. 3a + 1 21. 36a(3a + 1)
23. 10xy 3z 25. (y + 5)(y + 2)
17.
y
x
Exercises 1. rational equation
11
3. -24 5. - __83 7. __32 9. __35 11. - __
5
15
4
__
__
13. 19 15. 3, - 3 17. -2, 3 19. -1, __32
Selected Answers
SA29
18.
x
x-2
π
x
20. D: x > 0; R: y > 0
2b 2 + 8b
n 2 - n - 42
y
x 2 + 2x - 3
12n 3
1
________
40. ____
41. __
42. ____
m 43. 4x 2 - 16
3
b-3
(b + 8)(b + 7)
44. ____________ 45. 10a 2b 2
2(b + 4)(2b + 7)
b2 + 8
46. 10x (x - 3) 47. _____
2
2b
8p - 2
3x 2 + 2x - 4
48. _________
49. _________
2
2
x -2
,ENGTHCM
p - 4p + 2
n -1
10m
h 2 + 5h - 1
40
54. __
55. 2n2 - 3n - 5
53. _________
3r
h-5
5
57. x + 2 58. 3n + 1
56. x - __2x + __
2
x
SA30
3
4x 2 - 12x
15b 2
- 3c 3
36. ____
37. ____
35. _______
3
2
4d 2
b+2
n 2 + 3n + 2
_______
________
39.
38.
7m + 2
5b - 1
-10
51. _____
52. ______
50. _____
2
2
7-b
34
18
12
70. - __34 71. __
72. - __
17 + ____
7
x+2
11
-2
; x ≠ ±3
2 30. ____
x+3
x+3
3
____
31. x - 1 ; x ≠ -5 and x ≠ 1 32. ____
;
x-5
2
2b + 2b
4
__
_______
x ≠ -6 and x ≠ 5 33. 34.
y
18
69. -4x 2 + 10x 8 + ____
b-2
1
; k ≠ 0 and
r ≠ 0 28. _____
2k - 3
3
1
__
____
; x ≠ -6 and x ≠
k ≠ 29.
2
19.
36
67. 2n + 7 + ____
68. 3b 2 + 6b +
n-5
21. 0 22. 7 23. 0, 1 24. -1, 5
1
25. 5, -5 26. 4, 7 27. __
;
3r
y
7IDTHCM
Selected Answers
59. h + 12 60. 3x + 2 61. m - 6
62. 3m + 4 63. x + 2 64. x + 6
-3
65. p - 2 66. 2x - 1 + ____
x+2
73. - __76 ; 0 is extraneous. 74. - __23 ; 1
is extraneous. 75. - __13 , 1
1
79. -2; 4 is
76. -3 77. ±1 78. - __
12
extraneous. 80. 4, 5 81. -12, 1
82. -19 83. 0; 2 is extraneous.
Glossary/Glosario
KEYWORD: MA7 Glossary
A
ENGLISH
absolute value (p. 14) The
absolute value of x is the distance
from zero to x on a number line,
denoted ⎪x⎥.
⎧x
if x ≥ 0
⎩ -x if x < 0
SPANISH
valor absoluto El valor absoluto
de x es la distancia de cero a x
en una recta numérica, y se
expresa ⎪x⎥.
⎧x
si x ≥ 0
⎩ -x si x < 0
EXAMPLES
⎪3⎥ = 3
⎪-3⎥ = 3
⎪x⎥ = ⎨
⎪x⎥ = ⎨
absolute-value equation (p. 112)
An equation that contains
absolute-value expressions.
ecuación de valor absoluto Ecuación
que contiene expresiones de valor
absoluto.
⎪x + 4⎥ = 7
absolute-value function (p. 378)
A function whose rule contains
absolute-value expressions.
función de valor absoluto Función
cuya regla contiene expresiones de
valor absoluto.
y = ⎪x + 4⎥
absolute-value inequality (p. 212)
An inequality that contains
absolute-value expressions.
desigualdad de valor absoluto
Desigualdad que contiene
expresiones de valor absoluto.
⎪x + 4⎥ > 7
acute angle An angle that
measures greater than 0° and less
than 90°.
ángulo agudo Ángulo que mide más
de 0° y menos de 90°.
acute triangle A triangle with
three acute angles.
triángulo acutángulo Triángulo con
tres ángulos agudos.
Addition Property of Equality
(p. 79) For real numbers a, b, and
c, if a = b, then a + c = b + c.
Propiedad de igualdad de la
suma Dados los números reales a, b
y c, si a = b, entonces a + c = b + c.
x-6= 8
+6 +6
−−−− −−−
x
= 14
Addition Property of Inequality
(p. 176) For real numbers
a, b, and c, if a < b, then
a + c < b + c. Also holds true
for >, ≤, ≥, and ≠.
Propiedad de desigualdad de la
suma Dados los números reales a, b
y c, si a < b, entonces a + c < b + c.
Es válido también para >, ≤, ≥ y ≠.
x-6< 8
+6 +6
−−−− −−−
x
< 14
additive inverse (p. 15) The
opposite of a number. Two
numbers are additive inverses if
their sum is zero.
inverso aditivo El opuesto de un
número. Dos números son inversos
aditivos si su suma es cero.
The additive inverse of
5 is -5.
algebraic expression (p. 6) An
expression that contains at least
one variable.
expresión algebraica Expresión que
contiene por lo menos una variable.
algebraic order of operations See
order of operations.
orden algebraico de las operaciones
Ver orden de las operaciones.
The additive inverse of
-5 is 5.
2x + 3y
4x
Glossary/Glosario
G1
ENGLISH
SPANISH
AND (p. 204) A logical operator
representing the intersection of
two sets.
Y Operador lógico que representa
la intersección de dos conjuntos.
angle A figure formed by two rays
with a common endpoint.
ángulo Figura formada por dos
rayos con un extremo común.
EXAMPLES
A = {2, 3, 4, 5} B = {1, 3, 5, 7}
The set of values that are in A
AND B is A B = {3, 5}.
area The number of
nonoverlapping unit squares of a
given size that will exactly cover
the interior of a plane figure.
área Cantidad de cuadrados
unitarios de un determinado
tamaño no superpuestos que
cubren exactamente el interior
de una figura plana.
arithmetic sequence (p. 276)
A sequence whose successive
terms differ by the same nonzero
number d, called the common
difference.
sucesión aritmética Sucesión
cuyos términos sucesivos difieren
en el mismo número distinto de
cero d, denominado diferencia
común.
Associative Property of Addition
(p. 46) For all numbers a, b, and c,
(a + b) + c = a + (b + c).
Propiedad asociativa de la suma
Dados tres números cualesquiera
a, b y c, (a + b) + c = a + (b + c).
Associative Property of
Multiplication (p. 46) For all
numbers a, b, and c, (a · b) · c =
a · (b · c).
Propiedad asociativa de la
multiplicación Dados tres números
cualesquiera a, b y c, (a · b) · c =
a · (b · c).
asymptote (p. 878) A line that a
graph gets closer to as the value
of a variable becomes extremely
large or small.
asíntota Línea recta a la cual se
aproxima una gráfica a medida
que el valor de una variable
se hace sumamente grande o
pequeño.
x
Ó
The area is 10 square units.
4,
7,
10,
+3+3 +3 +3
d=3
(5 + 3) + 7 = 5 + (3 + 7)
(5 · 3) · 7 = 5 · (3 · 7)
Þ
{
asymptote
Ý
ä
{
average See mean.
promedio Ver media.
axis of a coordinate plane (p. 54)
One of two perpendicular number
lines, called the x-axis and the
y-axis, used to define the location
of a point in a coordinate plane.
eje de un plano cartesiano
Una de las dos rectas numéricas
perpendiculares, denominadas eje
x y eje y, utilizadas para definir la
ubicación de un punto en un
plano cartesiano.
axis of symmetry (p. 378, p. 620)
A line that divides a plane figure
or a graph into two congruent
reflected halves.
eje de simetría Línea que divide
una figura plana o una gráfica en
dos mitades reflejadas congruentes.
13, 16, …
{
y-axis
0
x-axis
݈ÃʜvÊÃޓ“iÌÀÞ
Þ
{ ÞÊNÝN
Ó
Ý
{
Ó
ä
Ó
{
G2
Glossary/Glosario
Ó
{
B
ENGLISH
back-to-back stem-and-leaf
plot (p. 709) A graph used to
organize and compare two sets of
data so that the frequencies can
be compared. See also stem-andleaf plot.
SPANISH
diagrama doble de tallo y hojas
Gráfica utilizada para organizar
y comparar dos conjuntos de
datos para poder comparar las
frecuencias. Ver también
diagrama de tallo y hojas.
EXAMPLES
Data set A: 9, 12, 14, 16, 23, 27
Data set B: 6, 8, 10, 13, 15, 16, 21
Set A
Set B
9 0 68
642 1 0356
73 2 1
Key: ⎪2⎥ 1 means 21
7 ⎪2⎥ means 27
gráfica de barras Gráfica con
barras horizontales o verticales
para mostrar datos.
-՘ˆ}…̽ÃÊ/À>ÛiÊ/ˆ“i
̜Ê*>˜iÌÃ
{nää
xäää
ÓÈää
À˜
À
ÌÕ
->
Ìi
«ˆ
>À
Ã
ÇÈä
Õ
>
ÀÌ
…
{äää
Îäää
Óäää
£äää xää
/ˆ“iʭî
bar graph (p. 700) A graph that
uses vertical or horizontal bars to
display data.
*>˜iÌ
3 4 = 3 · 3 · 3 · 3 = 81
3 is the base.
base of a power (p. 26) The
number in a power that is used as
a factor.
base de una potencia Número de
una potencia que se utiliza como
factor.
base of an exponential
function (p. 796) The value of b in
a function of the form f (x) = ab x,
where a and b are real numbers
with a ≠ 0, b > 0, and b ≠ 1.
base de una función exponencial
Valor de b en una función del tipo
f(x) = ab x, donde a y b son
números reales con a ≠ 0,
b > 0 y b ≠ 1.
biased sample (p. 733) A sample
that does not fairly represent the
population.
muestra no representativa
Muestra que no representa
adecuadamente una
población.
To find out about the exercise
habits of average Americans, a
fitness magazine surveyed its
readers about how often they
exercise. The population is all
Americans and the sample is
readers of the fitness magazine.
This sample will likely be biased
because readers of fitness
magazines may exercise more
often than other people do.
binomial (p. 497) A polynomial
with two terms.
binomio Polinomio con dos
términos.
x+y
2a 2 + 3
4m 3n 2 + 6mn 4
boundary line (p. 428) A line that
divides a coordinate plane into
two half-planes.
línea de límite Línea que divide
un plano cartesiano en dos
semiplanos.
In the function f (x) = 5(2) ,
the base is 2.
x
Î
Þ
œÕ˜`>ÀÞʏˆ˜i
Ý
Î
ä
Î
Î
Glossary/Glosario
G3
ENGLISH
box-and-whisker plot (p. 718) A
method of showing how data are
distributed by using the median,
quartiles, and minimum and
maximum values; also called a
box plot.
SPANISH
gráfica de mediana y rango Método
para mostrar la distribución de datos
utilizando la mediana, los cuartiles
y los valores mínimo y máximo;
también llamado gráfica de caja.
EXAMPLES
&IRSTQUARTILE
-INIMUM
ä
Ó
{
4HIRDQUARTILE
-EDIAN
È
n
£ä
£Ó
-AXIMUM
£{
C
Cartesian coordinate system
See coordinate plane.
sistema de coordenadas
cartesianas Ver plano cartesiano.
center of a circle The point
inside a circle that is the same
distance from every point on
the circle.
centro de un círculo Punto dentro
de un círculo que se encuentra a la
misma distancia de todos los
puntos del círculo.
central angle of a circle An angle
whose vertex is the center of a
circle.
ángulo central de un círculo Ángulo
cuyo vértice es el centro de un
círculo.
circle The set of points in a plane
that are a fixed distance from a
given point called the center of the
circle.
círculo Conjunto de puntos en
un plano que se encuentran a
una distancia fija de un punto
determinado denominado
centro del círculo.
circle graph (p. 702) A way to
display data by using a circle
divided into non-overlapping
sectors.
gráfica circular Forma de mostrar
datos mediante un círculo dividido
en sectores no superpuestos.
,iÈ`i˜ÌÃʜvÊiÃ>]Ê<
Èx³
£Î¯
{xqÈ{
Óǯ
1˜`iÀÊ
£n
£™¯
££¯
Îä¯
£nqÓ{
Óxq{{
circumference The distance
around a circle.
circunferencia Distancia alrededor
de un círculo.
ˆÀVՓviÀi˜Vi
G4
coefficient (p. 48) A number that
is multiplied by a variable.
coeficiente Número que se
multiplica por una variable.
In the expression 2x + 3y, 2 is
the coefficient of x and 3 is the
coefficient of y.
combination (p. 761) A selection
of a group of objects in which
order is not important. The
number of combinations of r
objects chosen from a group of n
objects is denoted nCr.
combinación Selección de un
grupo de objetos en la cual el
orden no es importante. El número
de combinaciones de r objetos
elegidos de un grupo de n objetos
se expresa así: nCr.
For objects A, B, C, and D,
there are 6 different
combinations of 2 objects.
AB, AC, AD, BC, BD, CD
commission (p. 139) Money
paid to a person or company for
making a sale, usually a percent of
the sale amount.
comisión Dinero que se paga a una
persona o empresa por realizar una
venta; generalmente se trata de un
porcentaje del total de la venta.
Glossary/Glosario
ENGLISH
SPANISH
EXAMPLES
common difference (p. 276) In an
arithmetic sequence, the nonzero
constant difference of any term
and the previous term.
diferencia común En una sucesión
aritmética, diferencia constante
distinta de cero entre cualquier
término y el término anterior.
In the arithmetic sequence 3,
5, 7, 9, 11, …, the common
difference is 2.
common factor (p. 545) A factor
that is common to all terms of
an expression or to two or more
expressions.
factor común Factor que es común
a todos los términos de una
expresión o a dos o más
expresiones.
Expression: 4x 2 + 16x 3 - 8x
Common factor: 4x
Expressions: 12 and 18
Common factors: 2, 3, and 6
common ratio (p. 790) In a
geometric sequence, the constant
ratio of any term and the previous
term.
razón común En una sucesión
geométrica, la razón constante
entre cualquier término y el
término anterior.
Commutative Property of
Addition (p. 46) For any two
numbers a and b, a + b = b + a.
Propiedad conmutativa de la
suma Dados dos números
cualesquiera a y b, a + b = b + a.
Commutative Property of
Multiplication (p. 46) For any two
numbers a and b, a · b = b · a.
Propiedad conmutativa de la
multiplicación Dados dos números
cualesquiera a y b, a · b = b · a.
complement of an event (p. 745)
The set of all outcomes that are
not the event.
complemento de un suceso
Todos los resultados que no
están en el suceso.
complementary angles Two angles
whose measures have a sum
of 90°.
ángulos complementarios Dos
ángulos cuyas medidas suman 90°.
In the geometric sequence
32, 16, 8, 4, 2, . . ., the
1
common ratio is __
.
2
3+4=4+3=7
3 · 4 = 4 · 3 = 12
In the experiment of
rolling a number cube, the
complement of rolling a 3 is
rolling a 1, 2, 4, 5, or 6.
ÎÇÂ
xÎÂ
completing the square (p. 663) A
process used to form a perfectsquare trinomial. To complete
()
2
the square of x 2 + bx, add __b2 .
completar el cuadrado Proceso
utilizado para formar un trinomio
cuadrado perfecto. Para completar
el cuadrado de x 2 + bx, hay que
()
2
sumar __b2 .
complex fraction (p. 904) A
fraction that contains one or more
fractions in the numerator, the
denominator, or both.
fracción compleja Fracción que
contiene una o más fracciones en el
numerador, en el denominador, o en
ambos.
composite figure (p. 83) A plane
figure made up of triangles,
rectangles, trapezoids, circles,
and other simple shapes, or a
three-dimensional figure made
up of prisms, cones, pyramids,
cylinders, and other simple threedimensional figures.
figura compuesta Figura plana
compuesta por triángulos,
rectángulos, trapecios, círculos
y otras figuras simples, o figura
tridimensional compuesta por
prismas, conos, pirámides, cilindros
y otras figuras tridimensionales
simples.
x 2 + 6x +
( ) = 9.
6
Add _
2
2
x 2 + 6x + 9
1
_
2
_
2
1+_
3
Glossary/Glosario
G5
ENGLISH
SPANISH
compound event (p. 761) An
event made up of two or more
simple events.
suceso compuesto Suceso formado
por dos o más sucesos simples.
compound inequality (p. 204) Two
inequalities that are combined
into one statement by the word
and or or.
desigualdad compuesta Dos
desigualdades unidas en un
enunciado por la palabra y u o.
EXAMPLES
In the experiment of tossing
a coin and rolling a number
cube, the event of the coin
landing heads and the
number cube landing on 3.
x ≥ 2 AND x < 7 (also
written 2 ≤ x < 7)
ä
Ó
{
È
n
x < 2 OR x > 6
ä
compound interest (p. 860)
Interest earned or paid on both
the principal and previously
earned interest. The formula
for compound interest is
A = P(1 +
)
r nt
__
n
, where A is the
final amount, P is the principal, r
is the interest rate expressed as a
decimal, n is the number of times
interest is compounded, and t is
the time.
G6
interés compuesto Intereses
ganados o pagados sobre el capital
y los intereses ya devengados. La
fórmula de interés compuesto es
r
A = P(1 + __
n ) , donde A es la
nt
cantidad final, P es el capital, r
es la tasa de interés expresada
como un decimal, n es la cantidad
de veces que se capitaliza el
interés y t es el tiempo.
compound statement (p. 203) Two
statements that are connected by
the word and or or.
enunciado compuesto Dos
enunciados unidos por la palabra
y u o.
cone (p. 894) A three-dimensional
figure with a circular base and a
curved surface that connects
the base to a point called the
vertex.
cono Figura tridimensional con
una base circular y una superficie
lateral curva que conecta la base
con un punto denominado vértice.
congruent Having the same
size and shape, denoted
by .
congruente Que tiene el mismo
tamaño y la misma forma, expresado
por .
conjugate of an irrational
number (p. 845) The conjugate of
is
a number in the form a + √b
√
a - b.
conjugado de un número irracional
El conjugado de un número en la
.
forma a + √
b es a - √b
consistent system (p. 420)
A system of equations or
inequalities that has at least one
solution.
sistema consistente Sistema de
ecuaciones o desigualdades que
tiene por lo menos una solución.
constant (p. 6) A value that does
not change.
constante Valor que no cambia.
Glossary/Glosario
Ó
{
È
n
If $100 is put into an
account with an interest
rate of 5% compounded
monthly, then after 2 years,
the account will have
(
100 1 +
0.05 12·2
____
) = $110.49.
12
The sky is blue and the grass is
green.
I will drive to school or I will
take the bus.
+
,
*
−− −−
PQ RS
-
The conjugate of 1 + √
2 is
1 - √
2.
⎧x + y = 6
⎨
⎩x - y = 4
solution: (5, 1)
3, 0, π
SPANISH
EXAMPLES
constant of variation (p. 336) The
constant k in direct and inverse
variation equations.
constante de variación La constante
k en ecuaciones de variación directa
e inversa.
continuous graph (p. 235) A graph
made up of connected lines or
curves.
gráfica continua Gráfica compuesta
por líneas rectas o curvas
conectadas.
y = 5x
constant of variation
˜}iˆµÕi½ÃÊi>ÀÌÊ,>Ìi
Þ
i>ÀÌÊÀ>Ìi
ENGLISH
Ý
/ˆ“i
convenience sample (p. 733) A
sample based on members of
the population that are readily
available.
muestra de conveniencia Una
muestra basada en miembros de
la población que están fácilmente
disponibles.
conversion factor (p. 121) The
ratio of two equal quantities, each
measured in different units.
factor de conversión Razón entre
dos cantidades iguales, cada una
medida en unidades diferentes.
coordinate (p. 54) A number used
to identify the location of a point.
On a number line, one coordinate
is used. On a coordinate plane,
two coordinates are used, called
the x-coordinate and the
y-coordinate.
coordenada Número utilizado
para identificar la ubicación
de un punto. En una recta
numérica se utiliza una
coordenada. En un plano
cartesiano se utilizan dos
coordenadas, denominadas
coordenada x y coordenada y.
A reporter surveys people he
personally knows.
12 inches
_
1 foot
A
{ Î Ó £
ä
£
Ó
Î
{
x
È
The coordinate of A is 2.
{
Þ
Ó
Ý
{
ä
Ó
Ó
{
The coordinates of B are
(-2, 3).
coordinate plane (p. 54) A plane
that is divided into four regions
by a horizontal line called the
x-axis and a vertical line called
the y-axis.
plano cartesiano Plano dividido
en cuatro regiones por una línea
horizontal denominada eje x y una
línea vertical denominada eje y.
correlation (p. 266) A measure of
the strength and direction of the
relationship between two
variables or data sets.
correlación Medida de la
fuerza y dirección de la
relación entre dos variables
o conjuntos de datos.
އ>݈Ã
ä
݇>݈Ã
*œÃˆÌˆÛiÊVœÀÀi>̈œ˜
Þ
œÊVœÀÀi>̈œ˜
Þ
Ý
i}>̈ÛiÊVœÀÀi>̈œ˜
Þ
Ý
Ý
corresponding angles of
polygons (p. 127) Angles in the
same relative position in polygons
with an equal number of angles.
ángulos correspondientes de los
polígonos Ángulos que se ubican
en la misma posición relativa en
polígonos que tienen el mismo
número de ángulos.
∠A and ∠D are corresponding angles.
Glossary/Glosario
G7
ENGLISH
corresponding sides of
polygons (p. 127) Sides in the
same relative position in polygons
with an equal number of sides.
cosine (p. 928) In a right triangle,
the cosine of angle A is the ratio
of the length of the leg adjacent
to angle A to the length of the
hypotenuse.
SPANISH
lados correspondientes de los
polígonos Lados que se ubican
en la misma posición relativa en
polígonos que tienen el mismo
número de lados.
EXAMPLES
−−
−−
AB and DE are corresponding sides.
coseno En un triángulo rectángulo,
el coseno del ángulo A es la
razón entre la longitud del cateto
adyacente al ángulo A y la longitud
de la hipotenusa.
…Þ«œÌi˜ÕÃi
>`>Vi˜Ì
adjacent
cos A = _________
hypotenuse
_1 = _3
cross products (p. 121) In the
c
statement __ab = __
, bc and ad are the
d
cross products.
productos cruzados En el
c
enunciado __ab = __
, bc y ad son
d
productos cruzados.
Cross Product Property (p. 121)
For any real numbers a, b, c, and
c
d, where b ≠ 0 and d ≠ 0, if __ab = __
,
d
then ad = bc.
Propiedad de productos cruzados
Dados los números reales a, b, c
c
y d, donde b ≠ 0 y d ≠ 0, si __ab = __
,
d
entonces ad = bc.
cube A prism with six square
faces.
cubo Prisma con seis caras
cuadradas.
cube in numeration (p. 26) The
third power of a number.
cubo en numeración Tercera
potencia de un número.
cube root (p. 32) A number,
3
written as √
x
, whose cube is x.
raíz cúbica Número, expresado
3
como √
x
, cuyo cubo es x.
cubic equation (p. 681) An
equation that can be written
in the form a x 3 + bx 2 + cx +
d = 0, where a, b, c, and d are real
numbers and a ≠ 0.
ecuación cúbica Ecuación que se
puede expresar como a x 3 + bx 2 +
c x + d = 0, donde a, b, c, y d son
números reales y a ≠ 0.
cubic function (p. 680) A function
that can be written in the form
f (x) = a x 3 + bx 2 + c x + d, where
a, b, c, and d are real numbers and
a ≠ 0.
función cúbica Función que
se puede expresar como
f (x) = a x 3 + b x 2 + c x + d, donde
a, b, c, y d son números reales
y a ≠ 0.
cubic polynomial (p. 497) A
polynomial of degree 3.
polinomio cúbico Polinomio de
grado 3.
x3 + 4 x2 - 6x + 2
cumulative frequency (p. 711) The
frequency of all data values that
are less than or equal to a given
value.
frecuencia acumulativa
Frecuencia de todos los valores
de los datos que son menores
que o iguales a un valor dado.
For the data set 2, 2, 3, 5, 5, 6,
7, 7, 8, 8, 8, 9, the cumulative
frequency table is shown below.
2
6
Cross products: 2 · 3 = 6
and 1 · 6 = 6
_
10 , then 4x = 60,
If 4 = _
x
6
so x = 15.
8 is the cube of 2.
3
√
64 = 4, because 4 3 = 64;
4 is the cube root of 64.
4 x 3 + x 2 - 3x - 1 = 0
f ( x) = x 3 + 2 x 2 - 6 x + 8
Data
2
3
5
6
7
8
9
G8
Glossary/Glosario
Frequency
2
1
2
1
2
3
1
Cumulative
Frequency
2
3
5
6
8
11
12
ENGLISH
cylinder (p. 894) A threedimensional figure with two
parallel congruent circular
bases and a curved surface
that connects the bases.
SPANISH
EXAMPLES
cilindro Figura tridimensional con
dos bases circulares congruentes
paralelas y una superficie lateral
curva que conecta las bases.
D
data Information gathered
from a survey or experiment.
datos Información reunida en una
encuesta o experimento.
degree measure of an angle
A unit of angle measure; one
1
degree is ___
of a circle.
360
medida en grados de un ángulo
Unidad de medida de los ángulos;
1
un grado es ___
de un círculo.
360
degree of a monomial (p. 496)
The sum of the exponents of the
variables in the monomial.
grado de un monomio Suma de
los exponentes de las variables del
monomio.
degree of a polynomial (p. 496)
The degree of the term of the
polynomial with the greatest
degree.
grado de un polinomio Grado
del término del polinomio con el
grado máximo.
dependent events (p. 750)
Events for which the occurrence
or nonoccurrence of one event
affects the probability of the other
event.
sucesos dependientes Dos sucesos
son dependientes si el hecho de
que uno de ellos ocurra o no afecta
la probabilidad del otro suceso.
dependent system (p. 421) A
system of equations that has
infinitely many solutions.
sistema dependiente Sistema de
ecuaciones que tiene infinitamente
muchas soluciones.
dependent variable (p. 250) The
output of a function; a variable
whose value depends on the value
of the input, or independent
variable.
variable dependiente Salida de una
función; variable cuyo valor depende
del valor de la entrada, o variable
independiente.
diameter A segment that has
endpoints on the circle and that
passes through the center of the
circle; also the length of that
segment.
diámetro Segmento que
atraviesa el centro de un círculo
y cuyos extremos están sobre
la circunferencia; longitud de
dicho segmento.
difference of two cubes (p. 584) A
polynomial of the form a 3 - b 3,
which may be written as the
product (a - b)(a 2 + ab + b 2).
diferencia de dos cubos Polinomio
del tipo a 3 - b 3, que se puede
expresar como el producto
(a - b)(a 2 + ab + b 2).
x 3 - 8 = (x - 2)(x 2 + 2x + 4)
difference of two squares (p. 523)
A polynomial of the form a 2 - b 2,
which may be written as the
product (a + b)(a - b).
diferencia de dos cuadrados
Polinomio del tipo a 2 - b 2, que se
puede expresar como el producto
(a + b)(a - b).
x 2 - 4 = (x + 2)(x - 2)
4x 2y 5z 3 Degree: 2 + 5 + 3 = 10
5 = 5x 0 Degree: 0
3x 2y 2 + 4xy 5 - 12x 3y 2
Degree 4 Degree 6 Degree 5
Degree 6
From a bag containing 3 red
marbles and 2 blue marbles,
draw a red marble, and then
draw a blue marble without
replacing the first marble.
⎧x + y = 2
⎨
⎩ 2x + 2y = 4
For y = 2x + 1, y is the
dependent variable.
input: x output: y
Glossary/Glosario
G9
ENGLISH
SPANISH
EXAMPLES
dimensional analysis (p. 121) A
process that uses rates to convert
measurements from one unit to
another.
análisis dimensional Un proceso que
utiliza tasas para convertir medidas
de unidad a otra.
direct variation (p. 336) A
linear relationship between two
variables, x and y, that can be
written in the form y = kx, where
k is a nonzero constant.
variación directa Relación lineal
entre dos variables, x e y, que puede
expresarse en la forma y = kx, donde
k es una constante distinta de cero.
1 qt
_
= 6 qt
12 pt ·
2 pt
{
Þ
Ó
Ý
{ Ó
Ó
{
{
y = 2x
discontinuous function (p. 878) A
function whose graph has one or
more jumps, breaks, or holes.
función discontinua Función cuya
gráfica tiene uno o más saltos,
interrupciones u hoyos.
Þ
{
Ý
ä
{
descuento Cantidad por la que se
reduce un precio original.
discrete graph (p. 235) A graph
made up of unconnected points.
gráfica discreta Gráfica compuesta
de puntos no conectados.
Theme Park Attendance
People
discount (p. 145) An amount by
which an original price is reduced.
{
Years
discriminant (p. 672) The
discriminant of the quadratic
equation ax 2 + bx + c = 0 is
b 2 - 4ac.
discriminante El discriminante
de la ecuación cuadrática
ax 2 + bx + c = 0 es b 2 - 4ac.
Distance Formula (p. 331) In a
coordinate plane, the distance
from (x 1, y 1) to (x 2, y 2) is
Fórmula de distancia En un plano
cartesiano, la distancia desde (x 1, y 1)
hasta (x 2, y 2) es
d = √
(x 2 - x 1)2 + (y 2 - y 1) 2 .
d = √
(x 2 - x 1)2 + (y 2 - y 1) 2 .
The discriminant of
2x 2 - 5x - 3 = 0 is
(-5) 2 - 4(2)(-3) or 49.
Þ
­Ó]Êx®
{
­£]Ê£®
{
Ó
Ó
Ý
ä
Ó
{
The distance from (2, 5) to (-1, 1) is
(-1 - 2)2 + (1 - 5)2
d = √
=
(-3) 2 + (-4) 2
√
= √
9 + 16 = √
25 = 5.
Distributive Property (p. 47) For
all real numbers a, b, and c,
a(b + c) = ab + ac, and
(b + c)a = ba + ca.
Propiedad distributiva Dados los
números reales a, b y c,
a(b + c) = ab + ac, y
(b + c)a = ba + ca.
Division Property of Equality
(p. 86) For real numbers a, b,
and c, where c ≠ 0, if a = b,
then __ac = __bc .
Propiedad de igualdad de la
división Dados los números reales
a, b y c, donde c ≠ 0, si a = b,
entonces __ac = __bc .
G10
Glossary/Glosario
3(4 + 5) = 3 · 4 + 3 · 5
(4 + 5)3 = 4 · 3 + 5 · 3
4x = 12
4x = _
12
_
4
4
x=3
ENGLISH
SPANISH
Division Property of Inequality
(p. 182, p. 183) If both sides of
an inequality are divided by the
same positive quantity, the new
inequality will have the same
solution set. If both sides of an
inequality are divided by the
same negative quantity, the new
inequality will have the same
solution set if the inequality
symbol is reversed.
Propiedad de desigualdad de la
división Cuando ambos lados de una
desigualdad se dividen entre el mismo
número positivo, la nueva desigualdad
tiene el mismo conjunto solución.
Cuando ambos lados de una
desigualdad se dividen entra el
mismo número negativo, la nueva
desigualdad tiene el mismo conjunto
solución si se invierte el símbolo de
desigualdad.
domain (p. 240) The set of all
first coordinates (or x-values) of a
relation or function.
dominio Conjunto de todos los
valores de la primera coordenada
(o valores de x ) de una función o
relación.
EXAMPLES
4x ≥ 12
12
4x ≥ _
_
4
4
x≥3
-4x ≥ 12
-4x ≤ 12
-4
-4
x ≤ -3
_ _
The domain of the function
{(-5, 3), (-3, -2), (-1, -1),
(1, 0)} is {-5, -3, -1, 1}.
E
element Each member in a set or
matrix. See also entry.
elemento Cada miembro en un
conjunto o matriz. Ver también
entrada.
elimination method (p. 411) A
method used to solve systems of
equations in which one variable
is eliminated by adding or
subtracting two equations of the
system.
eliminación Método utilizado para
resolver sistemas de ecuaciones
por el cual se elimina una variable
sumando o restando dos ecuaciones
del sistema.
empty set (p. 102) A set with no
elements.
conjunto vacío Conjunto sin
elementos.
The solution set of ⎪x⎥ < 0 is
the empty set, { }, or .
entry (p. 770) Each value in a
matrix; also called an element.
entrada Cada valor de una matriz,
también denominado elemento.
3 is the entry in the first row
and second column of
⎡2 3⎤
A=⎢
, denoted a 12.
⎣0 1⎦
equally likely outcomes (p. 744)
Outcomes are equally likely if
they have the same probability of
occurring. If an experiment has n
equally likely outcomes, then the
1
probability of each outcome is __
n.
resultados igualmente probables Los
resultados son igualmente probables
si tienen la misma probabilidad de
ocurrir. Si un experimento tiene n
resultados igualmente probables,
entonces la probabilidad de cada
1
resultado es __
n.
If a fair coin is tossed, then
P(heads) = P(tails) = 1 .
2
So the outcome “heads”
and the outcome “tails”
are equally likely.
equation (p. 77) A mathematical
statement that two expressions
are equivalent.
ecuación Enunciado matemático
que indica que dos expresiones son
equivalentes.
x+4=7
2+3=6-1
(x - 1)2 + (y + 2)2 = 4
equilateral triangle A triangle
with three congruent sides.
triángulo equilátero Triángulo con
tres lados congruentes.
equivalent ratios (p. 120) Ratios
that name the same comparison.
razones equivalentes Razones que
expresan la misma comparación.
_
_1 and _2 are equivalent ratios.
2
4
Glossary/Glosario
G11
ENGLISH
SPANISH
EXAMPLES
evaluate (p. 7) To find the value
of an algebraic expression by
substituting a number for each
variable and simplifying by using
the order of operations.
evaluar Calcular el valor de una
expresión algebraica sustituyendo
cada variable por un número y
simplificando mediante el orden de
las operaciones.
Evaluate 2x + 7 for x = 3.
2x + 7
2(3) + 7
6+7
13
event (p. 737) An outcome or set
of outcomes of an experiment.
suceso Resultado o conjunto de
resultados en un experimento.
In the experiment of rolling
a number cube, the event
“an odd number” consists of
the outcomes 1, 3, and 5.
excluded values (p. 878) Values
of x for which a function or
expression is not defined.
valores excluidos Valores de x para
los cuales no está definida una
función o expresión.
The excluded values of
(x + 2)
__
(x - 1)(x + 4)
are x = 1 and x = -4,
which would make the
denominator equal to 0.
experiment (p. 737) An operation,
process, or activity in which
outcomes can be used to estimate
probability.
experimento Una operación,
proceso o actividad en la que se
usan los resultados para estimar
una probabilidad.
experimental probability (p. 738)
The ratio of the number of times
an event occurs to the number of
trials, or times, that an activity is
performed.
probabilidad experimental
Razón entre la cantidad de
veces que ocurre un suceso
y la cantidad de pruebas,
o veces, que se realiza una
actividad.
exponent (p. 26) The number that
indicates how many times the
base in a power is used as a factor.
exponente Número que indica la
cantidad de veces que la base de
una potencia se utiliza como factor.
exponential decay (p. 807) An
exponential function of the form
f (x) = ab x in which 0 < b < 1.
If r is the rate of decay, then the
function can be written
t
y = a (1 - r) , where a is the initial
amount and t is the time.
decremento exponencial Función
exponencial del tipo f (x) = ab x
en la cual 0 < b < 1. Si r es la tasa
decremental, entonces la función
se puede expresar como
t
y = a (1 - r) , donde a es la
cantidad inicial y t es el tiempo.
Tossing a coin 10 times and
noting the number of heads
Kendra attempted 27 free
throws and made 16 of them.
The experimental probability
that she will make her next free
throw is P(free throw) =
16 ≈ 0.59.
number made
__
=_
27
number attempted
3 4 = 3 · 3 · 3 · 3 = 81
4 is the exponent.
()
1
f (x) = 3 _
2
x
Þ
Ý
exponential expression
An algebraic expression in which
the variable is in an exponent
with a fixed number as the base.
expresión exponencial Expresión
algebraica en la que la variable está
en un exponente y que tiene un
número fijo como base.
exponential function (p. 796) A
function of the form f (x) = ab x,
where a and b are real numbers
with a ≠ 0, b > 0, and b ≠ 1.
función exponencial Función del
tipo f (x) = ab x, donde a y b son
números reales con a ≠ 0, b > 0
y b ≠ 1.
2 x+1
f (x) = 3 · 4 x
Þ
Ý
G12
Glossary/Glosario
ENGLISH
SPANISH
EXAMPLES
exponential growth (p. 805) An
exponential function of the form
f (x) = a b x in which b > 1. If r
is the rate of growth, then the
function can be written
t
y = a(1 + r) , where a is the initial
amount and t is the time.
crecimiento exponencial Función
exponencial del tipo f (x) = ab x
en la que b > 1. Si r es la tasa de
crecimiento, entonces la función se
t
puede expresar como y = a(1 + r) ,
donde a es la cantidad inicial y t es el
tiempo.
expression (p. 6) A mathematical
phrase that contains operations,
numbers, and/or variables.
expresión Frase matemática que
contiene operaciones, números y/o
variables.
extraneous solution (p. 848) A
solution of a derived equation
that is not a solution of the
original equation.
solución extraña Solución de una
ecuación derivada que no es una
solución de la ecuación original.
To solve √
x = -2, square
both sides; x = 4.
4 = -2 is false; so 4
Check √
is an extraneous solution.
factor (p. 544) A number or
expression that is multiplied by
another number or expression to
get a product. See also factoring.
factor Número o expresión que
se multiplica por otro número o
expresión para obtener un
producto. Ver también factoreo.
12 = 3 · 4
3 and 4 are factors of 12.
factorial (p. 762) If n is a positive
integer, then n factorial, written n!,
is n · (n - 1) · (n - 2) · … · 2 · 1.
The factorial of 0 is defined to
be 1.
factorial Si n es un entero positivo,
entonces el factorial de n, expresado
como n!, es n · (n - 1) · (n - 2) · …
· 2 · 1. Por definición, el factorial de
0 será 1.
factoring (p. 544) The process
of writing a number or algebraic
expression as a product.
factorización Proceso por el que
se expresa un número o expresión
algebraica como un producto.
fair (p. 744) When all outcomes
of an experiment are equally
likely.
justo Cuando todos los resultados When tossing a fair coin, heads
de un experimento son igualmente and tails are equally likely.
Each has a probability of __12 .
probables.
family of functions (p. 369) A set
of functions whose graphs have
basic characteristics in common.
Functions in the same family are
transformations of their parent
function.
familia de funciones Conjunto de
funciones cuyas gráficas tienen
características básicas en común.
Las funciones de la misma familia
son transformaciones de su
función madre.
f (x) = 2 x
Þ
Ó
Ý
Ó ä
Ó
6x + 1
F
x 2 - 1 = (x - 1)(x + 1)
(x - 1) and (x + 1) are
factors of x 2 - 1.
7! = 7 · 6 · 5 · 4 · 3 · 2 · 1
= 5040
x 2 - 4x - 21 = (x - 7)(x + 3)
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Glossary/Glosario
G13
ENGLISH
first differences (p. 610) The
differences between y-values of
a function for evenly spaced
x-values.
SPANISH
EXAMPLES
primeras diferencias Diferencias
entre los valores de y de una
función para valores de x
espaciados uniformemente.
Constant change in x-values
+1
+1
+1
+1
x
0
1
2
3
4
y = x2
0
1
4
9
16
+1
+3
+5
+7
First differences
first quartile (p. 718) The median
of the lower half of a data set,
denoted Q 1. Also called lower
quartile.
primer cuartil Mediana de la mitad
inferior de un conjunto de datos,
expresada como Q 1. También se
llama cuartil inferior.
FOIL (p. 513) A mnemonic
(memory) device for a method of
multiplying two binomials:
Multiply the First terms.
Multiply the Outer terms.
Multiply the Inner terms.
Multiply the Last terms.
FOIL Regla mnemotécnica para
recordar el método de multiplicación
de dos binomios:
Multiplicar los términos Primeros
F
L
(First).
Multiplicar los términos Externos (x + 2)(x - 3) = x 2 - 3x + 2x - 6
= x2 - x - 6
(Outer).
I
Multiplicar los términos Internos
O
(Inner).
Multiplicar los términos Últimos
(Last).
formula (p. 107) A literal equation
that states a rule for a relationship
among quantities.
fórmula Ecuación literal que
establece una regla para una relación
entre cantidades.
fractional exponent See rational
exponent.
exponente fraccionario Ver
exponente racional.
frequency (p. 710) The number
of times the value appears in the
data set.
frecuencia Cantidad de veces que
aparece el valor en un conjunto
de datos.
frequency table (p. 710) A table
that lists the number of times, or
frequency, that each data value
occurs.
tabla de frecuencia Tabla
que enumera la cantidad
de veces que ocurre cada
valor de datos, o la
frecuencia.
function (p. 241) A relation in
which every domain value is
paired with exactly one range
value.
función Relación en la que a
cada valor de dominio corresponde
exactamente un valor de rango.
Lower half
18, 23, 28,
First quartile
A = πr 2
In the data set 5, 6, 6, 7, 8,
9, the data value 6 has a
frequency of 2.
Data set: 1, 1, 2, 2, 3, 4, 5, 5, 5, 6, 6, 6, 6
Frequency table:
Data
1
2
3
4
5
6
Frequency
2
2
1
1
3
4
È
x
Ó
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G14
Glossary/Glosario
Upper half
29, 36, 42
{
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ENGLISH
SPANISH
EXAMPLES
function notation (p. 250) If x is
the independent variable and y is
the dependent variable, then the
function notation for y is f (x),
read “f of x,” where f names the
function.
notación de función Si x es la
variable independiente e y es la
variable dependiente, entonces
la notación de función para y es
f (x), que se lee “f de x,” donde f
nombra la función.
function rule (p. 250) An algebraic
expression that defines a
function.
regla de función Expresión
algebraica que define una
función.
Fundamental Counting
Principle (p. 760) If one event
has m possible outcomes and
a second event has n possible
outcomes after the first event has
occurred, then there are mn total
possible outcomes for the two
events.
Principio fundamental de conteo
Si un suceso tiene m resultados
posibles y otro suceso tiene n
resultados posibles después de
ocurrido el primer suceso, entonces
hay mn resultados posibles en total
para los dos sucesos.
equation: y = 2x
function notation: f(x) = 2x
f (x) = 2x 2 + 3x - 7
function rule
If there are 4 colors of shirts,
3 colors of pants, and 2 colors
of shoes, then there are
4 · 3 · 2 = 24 possible outfits.
G
geometric sequence (p. 790) A
sequence in which the ratio of
successive terms is a constant r,
called the common ratio, where
r ≠ 0 and r ≠ 1.
sucesión geométrica Sucesión en
la que la razón de los términos
sucesivos es una constante r,
denominada razón común,
donde r ≠ 0 y r ≠ 1.
graph of a function (p. 256) The
set of points in a coordinate plane
with coordinates (x, y), where x is
in the domain of the function
f and y = f(x).
gráfica de una función Conjunto
de los puntos de un plano
cartesiano con coordenadas
(x, y), donde x está en el
dominio de la función f e
y = f (x).
1,
2,
4,
8, 16, …
·2 ·2 ·2 ·2
{
r=2
Þ
Þ
Ó
Ý
{
Ó
ä
Ó
Ý
n
{
ä
{
Ó
{
{
n
{
n
Þ
graph of a system of linear
inequalities (p. 435) The region
in a coordinate plane consisting
of points whose coordinates are
solutions to all of the inequalities
in the system.
gráfica de un sistema de
desigualdades lineales Región de
un plano cartesiano que consta
de puntos cuyas coordenadas
son soluciones de todas las
desigualdades del sistema.
graph of an inequality in one
variable (p. 171) The set of
points on a number line that are
solutions of the inequality.
gráfica de una desigualdad en una
variable Conjunto de los puntos de
una recta numérica que representan
soluciones de la desigualdad.
graph of an inequality in two
variables (p. 428) The set of
points in a coordinate plane
whose coordinates (x, y) are
solutions of the inequality.
gráfica de una desigualdad en dos
variables Conjunto de los puntos
de un plano cartesiano cuyas
coordenadas (x, y) son soluciones
de la desigualdad.
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y≤x+1
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Glossary/Glosario
G15
ENGLISH
SPANISH
EXAMPLES
graph of an ordered pair (p. 54)
For the ordered pair (x, y), the
point in a coordinate plane that
is a horizontal distance of x units
from the origin and a vertical
distance of y units from the
origin.
gráfica de un par ordenado Dado
el par ordenado (x, y), punto en un
plano cartesiano que está a una
distancia horizontal de x unidades
desde el origen y a una distancia
vertical de y unidades desde el
origen.
greatest common factor
(monomials) (GCF) (p. 575) The
product of the greatest integer and
the greatest power of each variable
that divide evenly into each
monomial.
máximo común divisor (monomios)
(MCD) Producto del entero mayor y
la potencia mayor de cada variable
que divide exactamente cada
monomio.
greatest common factor (numbers)
(GCF) The largest common factor
of two or more given numbers.
máximo común divisor (números)
(MCD) El mayor de los factores
The GCF of 27 and 45 is 9.
comunes compartidos por dos o más
números dados.
grouping symbols (p. 40) Symbols
such as parentheses ( ), brackets
[ ], and braces { } that separate
part of an expression. A fraction
bar, absolute-value symbols, and
radical symbols may also be used
as grouping symbols.
símbolos de agrupación Símbolos
tales como paréntesis ( ), corchetes
[ ] y llaves { } que separan parte
de una expresión. La barra de
fracciones, los símbolos de valor
absoluto y los símbolos de radical
también se pueden utilizar como
símbolos de agrupación.
{
Þ
Ó
Ý
{
Ó
ä
Ó
{
Ó
{
-
S(2, -4)
The GCF of 4x 3y and 6x 2y is
2x 2y.
⎧
⎫
6 + ⎨3 - ⎡⎣(4 - 3) + 2⎤⎦ + 1⎬ - 5
⎧⎩
⎫
⎩
⎭
6 + ⎨3 - ⎡⎣1 + 2⎤⎦ + 1⎬ - 5
6 + {3 - 3 + 1} - 5
6+1-5
2
⎭
H
half-life (p. 807) The half-life of a
substance is the time it takes for
one-half of the substance to decay
into another substance.
vida media La vida media de
una sustancia es el tiempo que
tarda la mitad de la sustancia en
desintegrarse y transformarse en
otra sustancia.
half-plane (p. 428) The part of the
coordinate plane on one side of a
line, which may include the line.
semiplano La parte del plano
cartesiano de un lado de una línea,
que puede incluir la línea.
Carbon-14 has a half-life of
5730 years, so 5 g of an initial
amount of 10 g will remain
after 5730 years.
Î
Ý
Î
ä
Î
Heron’s Formula (p. 834) A
triangle with side lengths a, b, and
c has area
A = √
s(s - a)(s - b)(s - c) ,
where s is one-half the perimeter,
or s = __12 (a + b + c).
G16
Glossary/Glosario
fórmula de Herón Un triángulo con
longitudes de lado a, b y c tiene un
área A = √
s(s - a)(s - b)(s - c) ,
donde s es la mitad del perímetro
ó s = __12 (a + b + c).
Þ
Î
histogram (p. 710) A bar graph
used to display data grouped in
intervals.
SPANISH
histograma Gráfica de barras
utilizada para mostrar datos
agrupados en intervalos de clases.
EXAMPLES
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ENGLISH
{ä
Îä
Óä
£ä
ä
n
n n n
->>ÀÞÊÀ>˜}iʭ̅œÕÃ>˜`Êf®
horizontal line (p. 316) A line
described by the equation
y = b, where b is the y-intercept.
línea horizontal Línea descrita
por la ecuación y = b, donde b
es la intersección con el eje y.
y=4
Þ
Ó
Ý
{ Ó ä
hypotenuse The side opposite the
right angle in a right triangle.
hipotenusa Lado opuesto al ángulo
recto de un triángulo rectángulo.
Ó
{
…Þ«œÌi˜ÕÃi
I
identity (p. 101) An equation
that is true for all values of the
variables.
identidad Ecuación verdadera para
todos los valores de las variables.
inclusive events (p. 758) Events
that have one or more outcomes
in common.
sucesos inclusivos Sucesos que
tienen uno o más resultados en
común.
inconsistent system (p. 420)
A system of equations or
inequalities that has no solution.
sistema inconsistente Sistema de
ecuaciones o desigualdades que
no tiene solución.
independent events (p. 750)
Events for which the occurrence
or nonoccurrence of one event
does not affect the probability of
the other event.
sucesos independientes Dos sucesos
son independientes si el hecho de
que se produzca o no uno de ellos
no afecta la probabilidad del otro
suceso.
independent system (p. 421)
A system of equations that has
exactly one solution.
sistema independiente Sistema
de ecuaciones que tiene sólo una
solución.
independent variable (p. 250)
The input of a function; a variable
whose value determines the
value of the output, or dependent
variable.
variable independiente Entrada de
una función; variable cuyo valor
determina el valor de la salida, o
variable dependiente.
3=3
2(x - 1) = 2x - 2
In the experiment of rolling a
number cube, rolling an even
number and rolling a number
less than 3 are inclusive events
because both contain the
outcome 2.
⎧x + y = 0
⎨
⎩x + y = 1
From a bag containing 3 red
marbles and 2 blue marbles,
draw a red marble, replace
it, and then draw a blue
marble.
⎧x + y = 7
⎨
⎩x - y = 1
Solution: (4, 3)
For y = 2x + 1, x is the
independent variable.
Glossary/Glosario
G17
ENGLISH
SPANISH
n
√
x,
EXAMPLES
n
√
x,
index (p. 488) In the radical
which represents the nth root of
x, n is the index. In the radical √
x,
the index is understood to be 2.
índice En el radical
que
representa la enésima raíz de x, n
es el índice. En el radical √x
, se da
por sentado que el índice es 2.
indirect measurement (p. 128) A
method of measurement that uses
formulas, similar figures, and/or
proportions.
medición indirecta Método
de medición en el que se usan
fórmulas, figuras semejantes
y/o proporciones.
inequality (p. 170) A statement
that compares two expressions by
using one of the following signs:
<, >, ≤, ≥, or ≠.
desigualdad Enunciado que
compara dos expresiones utilizando
uno de los siguientes signos:
<, >, ≤, ≥, o ≠.
input (p. 55) A value that is
substituted for the independent
variable in a relation or function.
entrada Valor que sustituye a la
variable independiente en una
relación o función.
input-output table A table that
displays input values of a function
or expression together with the
corresponding outputs.
tabla de entrada y salida Tabla
que muestra los valores de entrada
de una función o expresión junto
con las correspondientes salidas.
integer (p. 34) A member of the
set of whole numbers and their
opposites.
entero Miembro del conjunto de
números cabales y sus opuestos.
intercept See x-intercept and
y-intercept.
intersección Ver intersección con el
eje x e intersección con el eje y.
interest (p. 139) The amount of
money charged for borrowing
money or the amount of money
earned when saving or investing
money. See also compound
interest, simple interest.
interés Cantidad de dinero que se
cobra por prestar dinero o cantidad
de dinero que se gana cuando se
ahorra o invierte dinero. Ver también
interés compuesto, interés simple.
interquartile range (IQR)
(p. 718) The difference of the third
(upper) and first (lower) quartiles
in a data set, representing the
middle half of the data.
rango entre cuartiles Diferencia
entre el tercer cuartil (superior) y
el primer cuartil (inferior) de un
conjunto de datos, que representa
la mitad central de los datos.
intersection (p. 205) The
intersection of two sets is the
set of all elements that are
common to both sets, denoted
by .
intersección de conjuntos La
A = {1, 2, 3, 4}
intersección de dos conjuntos es el
B = {1, 3, 5, 7, 9}
conjunto de todos los elementos
A B = {1, 3}
que son comunes a ambos
conjuntos, expresado por .
inverse operations Operations
that undo each other.
operaciones inversas Operaciones
que se anulan entre sí.
G18
Glossary/Glosario
3
The radical √
8 has an
index of 3.
x≥2
{ Î Ó £
ä
£
Ó
Î
{
x
È
For the function f (x) = x + 5,
the input 3 produces an output
of 8.
Input
x
1
2
3
Output
y
4
7
10 13
4
…, -3, -2, -1, 0, 1, 2, 3, …
Lower half
Upper half
18, 23, 28, 29, 36, 42
First quartile Third quartile
Interquartile range:
36 - 23 = 13
Addition and subtraction of
the same quantity are inverse
operations:
5 + 3 = 8, 8 - 3 = 5
Multiplication and division by
the same quantity are inverse
operations:
2 · 3 = 6, 6 ÷ 3 = 2
ENGLISH
SPANISH
EXAMPLES
inverse variation (p. 871) A
relationship between two
variables, x and y, that can be
written in the form y = __kx , where k
is a nonzero constant and x ≠ 0.
variación inversa Relación entre
dos variables, x e y, que puede
expresarse en la forma y = __kx , donde
k es una constante distinta
de cero y x ≠ 0.
8
y=_
x
irrational number (p. 34) A real
number that cannot be expressed
as the ratio of two integers.
número irracional Número real que
no se puede expresar como una
razón de enteros.
√
2 , π, e
isolate the variable (p. 77) To
isolate a variable in an equation,
use inverse operations on both
sides until the variable appears by
itself on one side of the equation
and does not appear on the other
side.
despejar la variable Para despejar
la variable de una ecuación, utiliza
operaciones inversas en ambos lados
hasta que la variable aparezca sola
en uno de los lados de la ecuación y
no aparezca en el otro lado.
isosceles triangle A triangle with
at least two congruent sides.
triángulo isósceles Triángulo
que tiene al menos dos lados
congruentes.
10 = 6 - 2x
-6 -6
−− −−−−−−
4=
-2x
4 = -2x
-2
-2
-2 = x
_ _
L
leading coefficient (p. 497) The
coefficient of the first term of a
polynomial in standard form.
coeficiente principal Coeficiente del
primer término de un polinomio en
forma estándar.
least common denominator
(LCD) (p. 907) The least common
multiple of the denominators of
two or more given fractions or
rational expressions.
mínimo común denominador
(MCD) Mínimo común múltiplo de
los denominadores de dos o más
fracciones dadas o expresionnes
racionales.
least common multiple
(monomials) (LCM)
(p. 906) The product of the
smallest positive number and the
lowest power of each variable that
divide evenly into each monomial.
mínimo común múltiplo (monomios)
(MCM) El producto del número
positivo más pequeño y la menor
The LCM of 6 x 2 and 4 x is 12 x 2.
potencia de cada variable que divide
exactamente cada monomio.
least common multiple (numbers)
(LCM) The smallest whole number,
other than zero, that is a multiple
of two or more given numbers.
mínimo común múltiplo (números)
(MCM) El menor de los números
cabales, distinto de cero, que es
múltiplo de dos o más números
dados.
leg of a right triangle
One of the two sides of a right
triangle that form the right angle.
cateto de un triángulo
rectángulo Uno de los dos lados de
un triángulo rectángulo que forman
el ángulo recto.
3x 2 + 7x - 2
Leading coefficient: 3
5 is 12.
3 and _
The LCD of _
4
6
The LCM of 10 and 18 is 90.
i}
i}
Glossary/Glosario
G19
ENGLISH
SPANISH
EXAMPLES
like radicals (p. 835) Radical
terms having the same radicand
and index.
radicales semejantes Términos
radicales que tienen el mismo
radicando e índice.
2x and √
2x
3 √
like terms (p. 47) Terms with the
same variables raised to the same
exponents.
términos semejantes Términos con
las mismas variables elevadas a los
mismos exponentes.
3a 3b 2 and 7a 3b 2
line A straight path that has
no thickness and extends forever.
línea Un trazo recto que no tiene
grosor y se extiende infinitamente.
line graph (p. 701) A graph that
uses line segments to show how
data changes.
gráfica lineal Gráfica que se
vale de segmentos de recta para
mostrar cambios en los datos.
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linear equation in one variable
An equation that can be written in
the form ax = b where
a and b are constants and a ≠ 0.
ecuación lineal en una variable
Ecuación que puede expresarse en
la forma ax = b donde a y b son
constantes y a ≠ 0.
x+1=7
linear equation in two variables
(p. 302) An equation that can be
written in the form Ax + By = C
where A, B, and C are constants
and A and B are not both 0.
ecuación lineal en dos variables
Ecuación que puede expresarse en
la forma Ax + By = C donde A, B y
C son constantes y A y B no son
ambas 0.
2x + 3y = 6
linear function (p. 300) A function
that can be written in the form
y = mx + b, where x is the
independent variable and m and
b are real numbers. Its graph is a
line.
función lineal Función que puede
expresarse en la forma y = mx + b,
donde x es la variable independiente
y m y b son números reales. Su
gráfica es una línea.
y=x-1
{
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linear inequality in one variable
An inequality that can be written
in one of the following forms:
ax < b, ax > b, ax ≤ b, ax ≥ b,
or ax ≠ b, where a and b are
constants and a ≠ 0.
desigualdad lineal en una variable
Desigualdad que puede expresarse
de una de las siguientes formas:
ax < b, ax > b, ax ≤ b, ax ≥ b o
ax ≠ b, donde a y b son constantes
y a ≠ 0.
3x - 5 ≤ 2(x + 4)
linear inequality in two
variables (p. 428) An inequality
that can be written in one of the
following forms: Ax + By < C,
Ax + By > C, Ax + By ≤ C,
Ax + By ≥ C, or Ax + By ≠ C,
where A, B, and C are constants
and A and B are not both 0.
desigualdad lineal en dos variables
Desigualdad que puede expresarse
de una de las siguientes formas:
Ax + By < C, Ax + By > C, Ax + By
≤ C, Ax + By ≥ C o Ax + By ≠ C,
donde A, B y C son constantes y A y
B no son ambas 0.
2x + 3y > 6
G20
Glossary/Glosario
È
ENGLISH
SPANISH
EXAMPLES
d = rt
1h b + b
A=_
( 1 2)
2
literal equation (p. 108) An
equation that contains two or
more variables.
ecuación literal Ecuación que
contiene dos o más variables.
lower quartile See first quartile.
cuartil inferior Ver primer cuartil.
M
mapping diagram (p. 240)
A diagram that shows the
relationship of elements in the
domain to elements in the range
of a relation or function.
diagrama de correspondencia
Diagrama que muestra la relación
entre los elementos del dominio
y los elementos del rango de una
función.
markup (p. 145) The amount by
which a wholesale cost is increased.
margen de ganancia Cantidad que
se agrega a un costo mayorista.
matrix (p. 770) A rectangular
array of numbers.
matriz Arreglo rectangular de
números.
>««ˆ˜}ʈ>}À>“
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Ó
⎢-2
⎣ 7
máximo de una función Valor
de y del punto más alto en la
gráfica de la función.
mean (p. 716) The sum of all the
values in a data set divided by the
number of data values. Also called
the average.
media Suma de todos los valores
de un conjunto de datos dividida
entre el número de valores de
datos. También llamada promedio.
measure of an angle Angles are
measured in degrees. A degree
1
is ___
of a complete circle.
360
medida de un ángulo Los ángulos
se miden en grados. Un grado es
1
___
de un círculo completo.
360
measure of central tendency
(p. 716) A measure that describes
the center of a data set.
medida de tendencia dominante
Medida que describe el centro de
un conjunto de datos.
median (p. 716) For an ordered
data set with an odd number of
values, the median is the middle
value. For an ordered data set
with an even number of values,
the median is the average of the
two middle values.
mediana Dado un conjunto de
datos ordenado con un número
impar de valores, la mediana es el
valor medio. Dado un conjunto
de datos con un número par de
valores, la mediana es el promedio
de los dos valores medios.
midpoint (p. 330) The point
that divides a segment into two
congruent segments.
punto medio Punto que divide
un segmento en dos segmentos
congruentes.
⎡ 1
maximum of a function (p. 380,
p. 612) The y-value of the highest
point on the graph of the function.
,>˜}i
0
3⎤
2 -5
-6
3⎦
­ä]ÊÓ®
The maximum of the function is 2.
Data set: 4, 6, 7, 8, 10
4 + 6 + 7 + 8 + 10
Mean: __
5
35 = 7
=_
5
ÓÈ°nÂ
mean, median, or mode
8,
9,
9,
12, 15
Median: 9
4,
6,
7,
10, 10, 12
7 + 10
Median: _ = 8.5
2
−−
Point B is the midpoint of AC.
Glossary/Glosario
G21
ENGLISH
minimum of a function (p. 380,
p. 612) The y-value of the lowest
point on the graph of the function.
SPANISH
EXAMPLES
mínimo de una función Valor de
y del punto más bajo en la gráfica
de la función.
­ä]ÊÓ®
The minimum of the function
is -2.
mode (p. 716) The value or values
that occur most frequently in a
data set; if all values occur with
the same frequency, the data set is
said to have no mode.
moda El valor o los valores que se
presentan con mayor frecuencia
en un conjunto de datos. Si todos
los valores se presentan con la
misma frecuencia, se dice que el
conjunto de datos no tiene moda.
monomial (p. 496) A number or a
product of numbers and variables
with whole-number exponents, or
a polynomial with one term.
monomio Número o producto de
números y variables con exponentes
de números cabales, o polinomio
con un término.
Multiplication Property of
Equality (p. 86) If a, b, and c are
real numbers and a = b, then
ac = bc.
Propiedad de igualdad de la
multiplicación Si a, b y c son
números reales y a = b,
entonces ac = bc.
Multiplication Property of
Inequality (p. 182, p. 183) If
both sides of an inequality are
multiplied by the same positive
quantity, the new inequality will
have the same solution set.
If both sides of an inequality are
multiplied by the same negative
quantity, the new inequality will
have the same solution set if the
inequality symbol is reversed.
Propiedad de desigualdad de la
multiplicación Si ambos lados de
una desigualdad se multiplican
por el mismo número positivo,
la nueva desigualdad tendrá el
mismo conjunto solución.
Si ambos lados de una desigualdad
se multiplican por el mismo número
negativo, la nueva desigualdad
tendrá el mismo conjunto solución si
se invierte el símbolo de desigualdad.
multiplicative inverse (p. 21) The
reciprocal of the number.
inverso multiplicativo Recíproco de
un número.
mutually exclusive events (p. 758)
Two events are mutually exclusive
if they cannot both occur in the
same trial of an experiment.
sucesos mutuamente excluyentes
Dos sucesos son mutuamente
excluyentes si ambos no pueden
ocurrir en la misma prueba de un
experimento.
Data set: 3, 6, 8, 8, 10 Mode: 8
Data set: 2, 5, 5, 7, 7 Modes: 5
and 7
Data set: 2, 3, 6, 9, 11 No mode
3x 2 y 4
1x =7
_
3
1 x = (3)(7)
(3) _
3
x = 21
( )
1x >7
_
3
1
_
(3) x > (3)(7)
3
x > 21
( )
-x ≤ 2
(-1)(-x) ≥ (-1)(2)
x ≥ -2
The multiplicative inverse
of 5 is __15 .
In the experiment of rolling
a number cube, rolling a 3
and rolling an even number
are mutually exclusive
events.
N
natural number (p. 34) A counting
number.
número natural Número que se
utiliza para contar.
negative correlation (p. 267)
Two data sets have a negative
correlation if one set of data
values increases as the other set
decreases.
correlación negativa Dos conjuntos
de datos tienen una correlación
negativa si un conjunto de valores de
datos aumenta a medida que el otro
conjunto disminuye.
G22
Glossary/Glosario
1, 2, 3, 4, 5, 6, …
Þ
Ý
ENGLISH
SPANISH
negative exponent (p. 460) For
any nonzero real number x and
1
any integer n, x -n = __
.
xn
exponente negativo Para cualquier
número real distinto de cero x y
1
cualquier entero n, x -n = __
.
xn
negative number A number that is
less than zero. Negative numbers
lie to the left of zero on a number
line.
número negativo Número menor
que cero. Los números negativos
se ubican a la izquierda del cero
en una recta numérica.
negative square root (p. 32) The
opposite of the principal square
root of a number a, written as - √a
.
raíz cuadrada negativa Opuesto de la
raíz cuadrada principal de un número
a, que se expresa como - √a
.
net (p. 894) A diagram of the
faces of a three-dimensional
figure arranged in such a way that
the diagram can be folded to form
the three-dimensional figure.
plantilla Diagrama de las caras de
una figura tridimensional que se
puede plegar para formar la figura
tridimensional.
no correlation (p. 267) Two data
sets have no correlation if there
is no relationship between the
sets of values.
sin correlación Dos conjuntos
de datos no tienen correlación si
no existe una relación entre los
conjuntos de valores.
n th root (p. 488) The nth root
n
of1 a number a, written as √
a
or
__
n
a , is a number that is equal to a
when it is raised to the nth power.
enésima raíz La enésima raíz de un
1
__
n
número a, que se escribe √
a o a n,
es un número igual a a cuando se
eleva a la enésima potencia.
number line (p. 14) A line used to
represent the real numbers.
recta numérica Línea utilizada para
representar los números reales.
numerical expression (p. 6) An
expression that contains only
numbers and/or operations.
expresión numérica Expresión que
contiene únicamente números y
operaciones.
EXAMPLES
1 ; 3 -2 = _
1
x -2 = _
x2
32
-2 is a negative number.
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Î
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The negative square root
of 9 is - √
9 = -3.
10 m
10 m
6m
6m
Þ
Ý
5
√
32 = 2, because 2 5 = 32.
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£
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Î
{
x
È
2 · 3 + (4 - 6)
O
obtuse angle An angle that
measures greater than 90°
and less than 180°.
ángulo obtuso Ángulo que mide más
de 90° y menos de 180°.
obtuse triangle A triangle with
one obtuse angle.
triángulo obtusángulo Triángulo con
un ángulo obtuso.
odds (p. 746) A comparison
of favorable and unfavorable
outcomes. The odds in favor
of an event are the ratio of the
number of favorable outcomes
to the number of unfavorable
outcomes. The odds against an
event are the ratio of the number
of unfavorable outcomes to the
number of favorable outcomes.
probabilidades a favor y en contra
Comparación de los resultados
favorables y desfavorables. Las
probabilidades a favor de un suceso
son la razón entre la cantidad de
resultados favorables y la cantidad
de resultados desfavorables. Las
probabilidades en contra de
un suceso son la razón entre
la cantidad de resultados
desfavorables y la cantidad
de resultados favorables.
The odds in favor of rolling a 3
on a number cube are 1 : 5.
The odds against rolling a 3 on
a number cube are 5 : 1.
Glossary/Glosario
G23
ENGLISH
SPANISH
EXAMPLES
opposite (p. 15) The opposite of
a number a, denoted -a, is the
number that is the same distance
from zero as a, on the opposite
side of the number line. The sum
of opposites is 0.
opuesto El opuesto de un número
a, expresado -a, es el número que
se encuentra a la misma distancia
de cero que a, del lado opuesto de
la recta numérica. La suma de los
opuestos es 0.
opposite reciprocal (p. 364) The
opposite of the reciprocal of a
number. The opposite reciprocal
of any nonzero number a is - __a1 .
recíproco opuesto Opuesto
del recíproco de un número.
El recíproco opuesto de a
es - __a1 .
OR (p. 204) A logical operator
representing the union of two
sets.
O Operador lógico que
representa la unión de dos
conjuntos.
order of operations (p. 40)
A process for evaluating
expressions:
First, perform operations in
parentheses or other grouping
symbols.
Second, simplify powers and
roots.
Third, perform all multiplication
and division from left to right.
Fourth, perform all addition and
subtraction from left to right.
orden de las operaciones Regla
para evaluar las expresiones:
Primero, realizar las operaciones
entre paréntesis u otros símbolos
de agrupación.
Segundo, simplificar las potencias y
las raíces.
Tercero, realizar todas las
multiplicaciones y divisiones de
izquierda a derecha.
Cuarto, realizar todas las sumas y
restas de izquierda a derecha.
ordered pair (p. 54) A pair of
numbers (x, y) that can be used
to locate a point on a coordinate
plane. The first number x indicates
the distance to the left or right of
the origin, and the second number
y indicates the distance above or
below the origin.
par ordenado Par de números (x, y)
que se pueden utilizar para ubicar
un punto en un plano cartesiano. El
primer número, x, indica la distancia
a la izquierda o derecha del origen
y el segundo número, y, indica la
distancia hacia arriba o hacia abajo
del origen.
origin (p. 54) The intersection of
the x- and y-axes in a coordinate
plane. The coordinates of the
origin are (0, 0).
origen Intersección de los ejes
x e y en un plano cartesiano. Las
coordenadas de origen son (0, 0).
outcome (p. 737) A possible result
of a probability experiment.
resultado Resultado posible de un
experimento de probabilidad.
outlier (p. 716) A data value that
is far removed from the rest of the
data.
valor extremo Valor de
datos que está muy alejado
del resto de los datos.
output (p. 55) The result of
substituting a value for a variable
in a function.
salida Resultado de la sustitución
de una variable por un valor en
una función.
G24
Glossary/Glosario
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È x { Î Ó £
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Î
{
x
È
5 and -5 are opposites.
3.
2 is - _
The opposite reciprocal of _
3
2
A = {2, 3, 4, 5} B = {1, 3, 5, 7}
The set of values that are in
A OR B is A B = {1, 2, 3, 4, 5, 7}.
2 + 3 2 - (7 + 5) ÷ 4 · 3
2 + 3 2 - 12 ÷ 4 · 3 Add inside
parentheses.
2 + 9 - 12 ÷ 4 · 3 Simplify the
power.
2+9-3·3
Divide.
2+9-9
Multiply.
11 - 9
Add.
2
Subtract.
{
Þ
Ó
Ý
{
Ó
ä
Ó
{
The ordered pair
(-2, 3) can be used
to locate B.
œÀˆ}ˆ˜
ä
In the experiment of rolling
a number cube, the possible
outcomes are 1, 2, 3, 4, 5, and 6.
-OSTOFDATA -EAN
/UTLIER
For the function f (x) = x 2 + 1,
the input 3 produces an output
of 10.
ENGLISH
SPANISH
EXAMPLES
parabola (p. 611) The shape of the
graph of a quadratic function.
parábola Forma de la gráfica de una
función cuadrática.
parallel lines (p. 361) Lines
in the same plane that do not
intersect.
líneas paralelas Líneas en el mismo
plano que no se cruzan.
parallelogram A quadrilateral
with two pairs of parallel
sides.
paralelogramo Cuadrilátero con dos
pares de lados paralelos.
parent function (p. 369) The
simplest function with the
defining characteristics of the
family. Functions in the same
family are transformations of their
parent function.
función madre La función más
básica que tiene las características
distintivas de una familia. Las
funciones de la misma familia son
transformaciones de su función
madre.
Pascal’s triangle (p. 590) A
triangular arrangement of
numbers in which every row
starts and ends with 1 and each
other number is the sum of the
two numbers above it.
triángulo de Pascal Arreglo
triangular de números en el cual
cada fila comienza y termina con
1 y los demás números son la suma
de los dos valores que están arriba
de cada uno.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
percent (p. 133) A ratio that
compares a number to 100.
porcentaje Razón que compara un
número con 100.
17 = 17%
_
100
percent change (p. 144) An
increase or decrease given as a
percent of the original amount.
See also percent decrease, percent
increase.
porcentaje de cambio Incremento
o disminución dada como un
porcentaje de la cantidad original. Ver
también porcentaje de disminución,
porcentaje de incremento.
percent decrease (p. 144) A
decrease given as a percent of the
original amount.
porcentaje de disminución
Disminución dada como un
porcentaje de la cantidad original.
If an item that costs $8.00 is
marked down to $6.00, the
amount of the decrease is
$2.00, so the percent decrease is
2.00
____
= 0.25 = 25%.
8.00
percent increase (p. 144) An
increase given as a percent of the
original amount.
porcentaje de incremento
Incremento dado como un
porcentaje de la cantidad
original.
If an item’s wholesale cost of
$8.00 is marked up to $12.00,
the amount of the increase is
$4.00, so the percent increase
4.00
is ____
= 0.5 = 50%.
8.00
perfect square (p. 32) A number
whose positive square root is a
whole number.
cuadrado perfecto Número cuya
raíz cuadrada positiva es un número
cabal.
P
À
Ã
f (x) = x 2 is the parent
function for g (x) = x 2 + 4
and h (x) = (5x + 2)2 - 3.
36 is a perfect square
36 = 6.
because √
Glossary/Glosario
G25
ENGLISH
SPANISH
EXAMPLES
perfect-square trinomial (p. 521)
A trinomial whose factored form
is the square of a binomial. A
perfect-square trinomial has the
2
form a 2 - 2ab + b 2 = (a - b) or
2
a 2 + 2ab + b 2 = (a + b) .
trinomio cuadrado perfecto Trinomio
cuya forma factorizada es el
cuadrado de un binomio. Un
trinomio cuadrado perfecto tiene
2
la forma a 2 - 2ab + b 2 = (a - b)
2
o a 2 + 2ab + b 2 = (a + b) .
x 2 + 6x + 9 is a perfectsquare trinomial, because
x 2 + 6x + 9 = (x + 3) 2.
perimeter (p. 52) The sum of the
side lengths of a closed plane
figure.
perímetro Suma de las longitudes
de los lados de una figura plana
cerrada.
18 ft
6 ft
Perimeter = 18 + 6 + 18 + 6 = 48 ft
permutation (p. 761) An
arrangement of a group of objects
in which order is important.
permutación Arreglo de un grupo
de objetos en el cual el orden es
importante.
perpendicular Intersecting to form
90° angles.
perpendicular Que se cruza para
formar ángulos de 90°.
perpendicular lines (p. 363) Lines
that intersect at 90° angles.
líneas perpendiculares Líneas que se
cruzan en ángulos de 90°.
plane A flat surface that has no
thickness and extends forever.
plano Una superficie plana que
no tiene grosor y se extiende
infinitamente.
point A location that has
no size.
punto Ubicación exacta que no
tiene ningún tamaño.
point-slope form (p. 351) The
point-slope form of a linear
equation is y - y 1 = m(x - x 1),
where m is the slope and (x 1, y 1)
is a point on the line.
forma de punto y pendiente La
forma de punto y pendiente de una
ecuación lineal es y - y 1 = m(x - x 1),
donde m es la pendiente y (x 1, y 1) es
un punto en la línea.
polygon (p. 52) A closed plane
figure formed by three or more
segments such that each segment
intersects exactly two other
segments only at their endpoints
and no two segments with a
common endpoint are collinear.
polígono Figura plana cerrada
formada por tres o más segmentos
tal que cada segmento se cruza
únicamente con otros dos
segmentos sólo en sus extremos y
ningún segmento con un extremo
común a otro es colineal con éste.
polynomial (p. 496) A monomial
or a sum or difference of
monomials.
polinomio Monomio o suma o
diferencia de monomios.
G26
Glossary/Glosario
For objects A, B, C, and
D, there are 12 different
permutations of 2 objects.
AB, AC, AD, BC, BD, CD
BA, CA, DA, CB, DB, DC
˜
“
˜
“
*
point P
y - 3 = 2(x - 3)
2x 2 + 3xy - 7y 2
ENGLISH
SPANISH
EXAMPLES
x+1
x + 2 x 2 + 3x + 5
-(x 2 + 2x)
−−−−−−−
x+5
-(x + 2)
−−−−−−
3
x 2+ 3x + 5
3
__
=x+1+_
x+2
x +2
polynomial long division (p. 914)
A method of dividing one
polynomial by another.
división larga polinomial
Método por el que se divide
un polinomio entre otro.
population (p. 732) The entire
group of objects or individuals
considered for a survey.
población Grupo completo de
objetos o individuos que se desea
estudiar.
positive correlation (p. 267)
Two data sets have a positive
correlation if both sets of data
values increase.
correlación positiva Dos conjuntos
de datos tienen correlación positiva
si los valores de ambos conjuntos de
datos aumentan.
positive number A number greater
than zero.
número positivo Número mayor
que cero.
positive square root (p. 32) The
positive square root of a number,
indicated by the radical sign.
raíz cuadrada positiva Raíz cuadrada
positiva de un número, expresada
por el signo de radical.
power (p. 26) An expression
written with a base and an
exponent or the value of such an
expression.
potencia Expresión escrita con una
base y un exponente o el valor de
dicha expresión.
Power of a Power Property
(p. 476) If a is any nonzero real
number and m and n are integers,
n
then (a m) = a mn.
Propiedad de la potencia de una
potencia Dado un número real
a distinto de cero y los números
n
enteros m y n, entonces (a m) = a mn.
(6 7)4 = 6 7·4
Power of a Product Property
(p. 477) If a and b are any nonzero
real numbers and n is any integer,
n
then (ab) = a nb n.
Propiedad de la potencia de un
producto Dados los números reales
a y b distintos de cero y un número
n
entero n, entonces (ab) = a nb n.
(2 · 4)3 = 2 3 · 4 3
Power of a Quotient Property
(p. 483, p. 484) If a and b are any
nonzero real numbers and n is an
Propiedad de la potencia de un
cociente Dados los números reales a
y b distintos de cero y un número
integer, then
a n
__
(b)
n
a
= __
.
bn
entero n, entonces
a n
__
(b)
n
a
= __
.
bn
prediction (p. 739) An estimate or
guess about something that has
not yet happened.
predicción Estimación o suposición
sobre algo que todavía no ha
sucedido.
prime factorization (p. 544) A
representation of a number or
a polynomial as a product of
primes.
factorización prima Representación
de un número o de un polinomio
como producto de números primos.
In a survey about the
study habits of high school
students, the population is
all high school students.
Þ
Ý
2 is a positive number.
{ Î Ó £
ä
£
Ó
Î
{
The positive square root of
36 = 6.
36 is √
2 3 = 8, so 8 is the third
power of 2.
= 6 28
= 8 · 64
= 512
(_35 ) = _53 · _53 · _53 · _53
4
·3·3·3
= 3__
5·5·5·5
34
=_
54
The prime factorization of
60 is 2 · 2 · 3 · 5.
Glossary/Glosario
G27
ENGLISH
SPANISH
EXAMPLES
prime number (p. 544) A whole
number greater than 1 that has
exactly two positive factors, itself
and 1.
número primo Número cabal mayor 5 is prime because its only
que 1 que es divisible únicamente
positive factors are 5 and 1.
entre sí mismo y entre 1.
principal (p. 139) An amount of
money borrowed or invested.
capital Cantidad de dinero que se
pide prestado o se invierte.
prism (p. 894) A polyhedron
formed by two parallel congruent
polygonal bases connected by
faces that are parallelograms.
prisma Poliedro formado por dos
bases poligonales congruentes y
paralelas conectadas por caras
laterales que son paralelogramos.
probability (p. 737) A number
from 0 to 1 (or 0% to 100%) that
is the measure of how likely an
event is to occur.
probabilidad Número entre 0 y 1
(o entre 0% y 100%) que describe
cuán probable es que ocurra un
suceso.
Product of Powers Property
(p. 474) If a is any nonzero real
number and m and n are integers,
then a m · a n = a m+n.
Propiedad del producto de potencias
Dado un número real a distinto de
cero y los números enteros m y n,
entonces a m · a n = a m+n.
Product Property of Square
Roots (p. 830) For a ≥ 0 and
ab = √
a · √
b.
b ≥ 0, √
Propiedad del producto de raíces
cuadradas Dados a ≥ 0 y
b ≥ 0, √
ab = √
a · √
b.
proportion (p. 120) A statement
c
that two ratios are equal; __ab = __
.
d
proporción Ecuación que establece
c
.
que dos razones son iguales; __ab = __
d
pyramid (p. 894) A polyhedron
formed by a polygonal base and
triangular lateral faces that meet at
a common vertex.
pirámide Poliedro formado por
una base poligonal y caras laterales
triangulares que se encuentran en
un vértice común.
Pythagorean Theorem If a right
triangle has legs of lengths a and b
and a hypotenuse of length c, then
a 2 + b 2 = c 2.
Teorema de Pitágoras Dado un
triángulo rectángulo con catetos de
longitudes a y b y una hipotenusa de
longitud c, entonces a 2 + b 2 = c 2.
A bag contains 3 red marbles
and 4 blue marbles. The
probability of randomly
choosing a red marble is __37 .
6 7 · 6 4 = 6 7+4
= 6 11
√
9 · 25 = √
9 · √
25
= 3 · 5 = 15
4
2 =_
_
3
6
£ÎÊV“
xÊV“
£ÓÊV“
5 2 + 12 2 = 13 2
25 + 144 = 169
Pythagorean triple (p. 539) A set
of three positive integers a, b, and
c such that a 2 + b 2 = c 2.
Tripleta de Pitágoras Conjunto de
tres enteros positivos a, b y c tal que
a 2 + b 2 = c 2.
The numbers 3, 4, and 5
form a Pythagorean triple
because 3 2 + 4 2 = 5 2.
Q
quadrant (p. 54) One of the four
regions into which the x- and
y-axes divide the coordinate
plane.
cuadrante Una de las cuatro
regiones en las que los ejes x e y
dividen el plano cartesiano.
+Õ>`À>˜ÌÊ
+Õ>`À>˜ÌÊ
ä
+Õ>`À>˜ÌÊ +Õ>`À>˜ÌÊ6
G28
Glossary/Glosario
ENGLISH
SPANISH
quadratic equation (p. 642) An
equation that can be written in
the form ax 2 + bx + c = 0, where
a, b, and c are real numbers and
a ≠ 0.
ecuación cuadrática Ecuación
que se puede expresar como
ax 2 + bx + c = 0, donde a, b y c
son números reales y a ≠ 0.
Quadratic Formula (p. 670)
fórmula cuadrática La fórmula
-b ± √
b 2 - 4ac
-b ± √
b 2 - 4ac
EXAMPLES
x 2 + 3x - 4 = 0
x2 - 9 = 0
The solutions of 2x 2 - 5x - 3 = 0
are given by
(-5)2 - 4(2)(-3)
-(-5) ± √
x = ___
2(2)
√
5
±
25
+
24
5±7
= __ = _
4
4
1
x = 3 or x = - _
2
The formula x = ____________
,
2a
which gives solutions, or roots, of
equations in the form
ax 2 + bx + c = 0, where a ≠ 0.
x = ____________
, que da
2a
soluciones, o raíces, para
las ecuaciones del tipo
ax 2 + bx + c = 0, donde
a ≠ 0.
quadratic function (p. 610) A
function that can be written in the
form f (x) = ax 2 + bx + c, where
a, b, and c are real numbers and
a ≠ 0.
función cuadrática Función
que se puede expresar como
f (x) = ax 2 + bx + c, donde a,
b y c son números reales
y a ≠ 0.
quadratic polynomial (p. 497) A
polynomial of degree 2.
polinomio cuadrático Polinomio de
grado 2.
quartile The median of the upper
or lower half of a data set. See also
first quartile, third quartile.
cuartil La mediana de la mitad
superior o inferior de un conjunto
de datos. Ver también primer
cuartil, tercer cuartil.
Quotient of Powers Property
(p. 481) If a is a nonzero real
number and m and n are integers,
am
then ___
= a m-n.
an
Propiedad del cociente de
potencias Dado un número real
a distinto de cero y los números
am
enteros m y n, entonces ___
= a m-n.
an
Quotient Property of Square
Roots (p. 830) For a ≥ 0 and
Propiedad del cociente de raíces
cuadradas Dados a ≥ 0 y
√a
a
= ___
.
b > 0, __
b
√b
√a
a
= ___
.
b > 0, __
b
f (x) = x 2 - 6x + 8
x 2 - 6x + 8
&IRSTQUARTILE
-INIMUM
ä
Ó
{
4HIRDQUARTILE
-EDIAN
È
n
£ä
-AXIMUM
£Ó
£{
6 7 = 6 7-4 = 6 3
_
64
√
9
9 =_
3
_
=_
25
√
25
5
√b
R
radical equation (p. 846) An
equation that contains a variable
within a radical.
ecuación radical Ecuación que
contiene una variable dentro de un
radical.
radical expression (p. 829) An
expression that contains a radical
sign.
expresión radical Expresión que
contiene un signo de radical.
radical symbol (p. 32) The
symbol √ used to denote a
root. The symbol is used alone
to indicate a square root or with
n
an index, √, to indicate the nth
root.
símbolo de radical Símbolo √
que se utiliza para expresar una
raíz. Puede utilizarse solo para
indicar una raíz cuadrada, o con un
n
índice, √, para indicar la enésima
raíz.
√
x+3+4=7
√
x+3+4
√
36 = 6
3
√
27 = 3
Glossary/Glosario
G29
ENGLISH
SPANISH
EXAMPLES
radicand (p. 829) The expression
under a radical sign.
radicando Número o expresión
debajo del signo de radical.
radius A segment whose
endpoints are the center of a
circle and a point on the circle;
the distance from the center of a
circle to any point on the circle.
radio Segmento cuyos extremos son
el centro de un círculo y un punto de
la circunferencia; distancia desde el
centro de un círculo hasta cualquier
punto de la circunferencia.
random sample (p. 727) A sample
selected from a population so that
each member of the population
has an equal chance of being
selected.
muestra aleatoria Muestra
seleccionada de una población
tal que cada miembro de ésta
tenga igual probabilidad de ser
seleccionada.
Mr. Hansen chose a random
sample of the class by writing
each student’s name on a slip
of paper, mixing up the slips,
and drawing five slips without
looking.
range of a data set (p. 716) The
difference of the greatest and least
values in the data set.
rango de un conjunto de datos
La diferencia del mayor y menor
valor en un conjunto de datos.
The data set {3, 3, 5, 7, 8, 10,
11, 11, 12} has a range of
12 - 3 = 9.
range of a function or relation
(p. 240) The set of all second
coordinates (or y-values) of a
function or relation.
rango de una función o relación
Conjunto de todos los valores de
la segunda coordenada (o valores
de y) de una función o relación.
The range of the function
{(-5, 3), (-3, -2), (-1, -1),
(1, 0)} is {-2, -1, 0, 3}.
rate (p. 120) A ratio that
compares two quantities
measured in different units.
tasa Razón que compara dos
cantidades medidas en diferentes
unidades.
rate of change (p. 314) A ratio
that compares the amount of
change in a dependent variable
to the amount of change in an
independent variable.
tasa de cambio Razón que compara
la cantidad de cambio de la variable
dependiente con la cantidad
de cambio de la variable
independiente.
Expression: √
x+3
Radicand: x + 3
,>`ˆÕÃ
55 miles = 55 mi/h
_
1 hour
The cost of mailing a letter
increased from 22 cents in
1985 to 25 cents in 1988.
During this period, the rate of
change was
change in cost
25 - 22
___________
= _________
= __3
change in year
1988 - 1985
3
= 1 cent per year.
ratio (p. 120) A comparison of
two quantities by division.
razón Comparación de dos
cantidades mediante una división.
rational equation (p. 920) An
equation that contains one or
more rational expressions.
ecuación racional Ecuación que
contiene una o más expresiones
racionales.
rational exponent (p. 489) An
exponent that can be expressed
m
as __
n such that if m and n are
exponente racional Exponente que
m
se puede expresar como __
n tal que si
m y n son números enteros,
m
_
m
n
n
m
) .
integers, then b n = √b
= ( √b
rational expression (p. 886)
An algebraic expression whose
numerator and denominator
are polynomials and whose
denominator has a degree ≥ 1.
G30
Glossary/Glosario
m
_
1 or 1 : 2
_
2
x+2
__
=6
2
x + 3x - 1
1
_
6
64 6 = √
64
m
n
n
m
entonces b n = √b
= ( √
b) .
expresión racional Expresión
algebraica cuyo numerador y
denominador son polinomios y cuyo
denominador tiene un grado ≥ 1.
x+2
__
x 2 + 3x - 1
ENGLISH
SPANISH
EXAMPLES
rational function (p. 878) A
function whose rule can be
written as a rational expression.
función racional Función cuya
regla se puede expresar como una
expresión racional.
x+2
f(x) = __
2
x + 3x - 1
rational number (p. 34) A number
that can be written in the form __ab ,
where a and b are integers and
b ≠ 0.
número racional Número que se
puede expresar como __ab , donde a y b
son números enteros y b ≠ 0.
−
2, 0
3, 1.75, 0.3, - _
3
rationalizing the denominator
(p. 842) A method of rewriting a
fraction by multiplying by another
fraction that is equivalent to 1
in order to remove radical terms
from the denominator.
racionalizar el denominador Método
que consiste en escribir nuevamente
una fracción multiplicándola por
otra fracción equivalente a 1 a fin de
eliminar los términos radicales del
denominador.
√
√
2
2
1 ·_
_
=_
2
√2
√
2
ray A part of a line that starts at
an endpoint and extends forever
in one direction.
rayo Parte de una recta que
comienza en un extremo y se
extiende infinitamente en una
dirección.
real number (p. 34) A rational or
irrational number. Every point on
the number line represents a real
number.
número real Número racional o
irracional. Cada punto de la recta
numérica representa un número
real.
,i>Ê ՓLiÀÃ
,>̈œ˜>Ê ՓLiÀÃÊ­ύ®
ÚÚÚ
ÊÓÇÊÊÊ
Ü
ä°ÊÎÊ
˜Ìi}iÀÃÊ­ϖ®
{
Î
>ÌÕÀ>Ê ՓLiÀÃÊ­ϊ®
£
i
Ó
recíproco Dado el número real
a ≠ 0, el recíproco de a es __a1 . El
producto de los recíprocos es 1.
rectangle A quadrilateral with
four right angles.
rectángulo Cuadrilátero con cuatro
ángulos rectos.
rectangular prism (p. 894) A prism
whose bases are rectangles.
prisma rectangular Prisma cuyas
bases son rectángulos.
rectangular pyramid (p. 894)
A pyramid whose base is a
rectangle.
pirámide rectangular Pirámide cuya
base es un rectángulo.
reflection (p. 371) A
transformation that reflects, or
“flips,” a graph or figure across a
line, called the line of reflection.
reflexión Transformación en la
que una gráfica o figura se refleja
o se invierte sobre una línea,
denominada la línea de reflexión.
regular polygon A polygon that is
both equilateral and equiangular.
polígono regular Polígono equilátero
de ángulos iguales.
ÊÊȖ££Ê
е
еÊ
е
ÊȖÓÊ
Ê
ä
Î
ÚÚÊxÊÊÊ
{°x
reciprocal (p. 21) For a real
number a ≠ 0, the reciprocal of a
is __a1 . The product of reciprocals
is 1.
е
ÊȖ£ÇÊ
еÊ
Ó
7…œiÊ Õ“LiÀÃÊ­ϓ®
£
ÀÀ>̈œ˜>Ê ՓLiÀÃ
ÊÊÊ
ÊÚÚÚ
棊
££
û
™
Number
Reciprocal
2
1
__
1
1
-1
-1
0
No reciprocal
2
Ī
Ī
Ī
Glossary/Glosario
G31
ENGLISH
SPANISH
relation (p. 240) A set of ordered
pairs.
relación Conjunto de pares
ordenados.
repeating decimal (p. 34) A
rational number in decimal
form that has a nonzero block
of one or more digits that repeat
continuously.
decimal periódico Número racional
en forma decimal que tiene un
bloque de uno o más dígitos que se
repite continuamente.
replacement set (p. 8) A set of
numbers that can be substituted
for a variable.
conjunto de reemplazo Conjunto de
números que pueden sustituir una
variable.
rhombus A quadrilateral with four
congruent sides.
rombo Cuadrilátero con cuatro
lados congruentes.
right angle An angle that
measures 90°.
ángulo recto Ángulo que mide 90°.
rise (p. 315) The difference in the
y-values of two points on a line.
distancia vertical Diferencia entre
los valores de y de dos puntos de
una línea.
rotation (p. 370) A transformation
that rotates or turns a figure
about a point called the center of
rotation.
rotación Transformación que rota
o gira una figura sobre un punto
llamado centro de rotación.
run (p. 315) The difference in the
x-values of two points on a line.
distancia horizontal Diferencia
entre los valores de x de dos
puntos de una línea.
EXAMPLES
⎧
⎫
⎨(0, 5), (0, 4), (2, 3), (4, 0)⎬
⎩
⎭
− − −−
−
1.3, 0.6, 2.14, 6.773
For the points (3, -1) and
(6, 5), the rise is 5 - (-1) = 6.
Ī
Ī
Ī
Ī
For the points (3, -1) and
(6, 5), the run is 6 - 3 = 3.
S
sales tax (p. 140) A percent of the
cost of an item that is charged by
governments to raise money.
impuesto sobre la venta Porcentaje
del costo de un artículo que cobran
los gobiernos para recaudar dinero.
sample (p. 732) A part of the
population.
muestra Una parte de la población.
In a survey about the
study habits of high school
students, a sample is a
survey of 100 students.
sample space (p. 737) The set
of all possible outcomes of a
probability experiment.
espacio muestral Conjunto de
todos los resultados posibles de un
experimento de probabilidad.
In the experiment of rolling
a number cube, the sample
space is {1, 2, 3, 4, 5, 6}.
scalar (p. 771) A number that is
multiplied by a matrix.
escalar Número que se multiplica
por una matriz.
3
[ ] [ ]
3 -6
1 -2
=
2 3
6 9
scalar
scale (p. 122) The ratio between
two corresponding measurements.
G32
Glossary/Glosario
escala Razón entre dos medidas
correspondientes.
1 cm : 5 mi
ENGLISH
scale drawing (p. 122) A drawing
that uses a scale to represent an
object as smaller or larger than
the actual object.
SPANISH
dibujo a escala Dibujo que utiliza
una escala para representar un
objeto como más pequeño o más
grande que el objeto original.
EXAMPLES
A blueprint is an example of
a scale drawing.
scale factor (p. 129) The
multiplier used on each
dimension to change one
figure into a similar figure.
factor de escala El multiplicador
utilizado en cada dimensión para
transformar una figura en una
figura semejante.
6 in.
4 in.
2 in.
3 in.
3 = 1.5
Scale factor: _
2
scale model (p. 122) A threedimensional model that uses a
scale to represent an object as
smaller or larger than the actual
object.
modelo a escala Modelo
tridimensional que utiliza una escala
para representar un objeto como
más pequeño o más grande que el
objeto real.
scalene triangle A triangle with no
congruent sides.
triángulo escaleno Triángulo sin
lados congruentes.
scatter plot (p. 266) A graph with
points plotted to show a possible
relationship between two sets of
data.
diagrama de dispersión Gráfica con
puntos que se usa para demostrar
una relación posible entre dos
conjuntos de datos.
Þ
n
È
{
Ó
Ý
ä
scientific notation (p. 467) A
method of writing very large or
very small numbers, by using
powers of 10, in the form m × 10 n,
where 1 ≤ m < 10 and n is an
integer.
notación científica Método que
consiste en escribir números muy
grandes o muy pequeños utilizando
potencias de 10 del tipo m × 10 n,
donde 1 ≤ m < 10 y n es un número
entero.
second differences (p. 610)
Differences between first
differences of a function.
segundas diferencias Diferencias
entre las primeras diferencias de
una función.
Ó
{
È
12,560,000,000,000 =
1.256 × 10 13
0.0000075 = 7.5 × 10 -6
Constant change in x-values
+1 +1 +1 +1
x
0
1
2
3
4
y = x2
0
1
4
9
16
First differences
+1 +3 +5 +7
Second differences
sequence (p. 276) A list of numbers
that often form a pattern.
sucesión Lista de números que
generalmente forman un patrón.
set-builder notation (p. 170) A
notation for a set that uses a rule
to describe the properties of the
elements of the set.
notación de conjuntos Notación
para un conjunto que se vale de una
regla para describir las propiedades
de los elementos del conjunto.
n
+2 +2 +2
1, 2, 4, 8, 16, …
{x | x > 3} is read “The set of
all x such that x is greater
than 3.”
Glossary/Glosario
G33
ENGLISH
SPANISH
similar (p. 127) Two figures are
similar if they have the same
shape but not necessarily the
same size.
semejantes Dos figuras con
la misma forma pero no
necesariamente del mismo
tamaño.
similarity statement (p. 127) A
statement that indicates that
two polygons are similar by
listing the vertices in the order of
correspondence.
enunciado de semejanza Enunciado
que indica que dos polígonos son
semejantes enumerando los vértices
en orden de correspondencia.
EXAMPLES
È
x
x°{
{
£Ó
£ä
£ä°n
n
quadrilateral ABCD ∼
quadrilateral EFGH
simple event (p. 761) An event
consisting of only one outcome.
suceso simple Suceso que tiene
sólo un resultado.
simple interest (p. 139) A fixed
percent of the principal. For
principal P, interest rate r, and
time t in years, the simple interest
is I = Prt.
interés simple Porcentaje fijo del
capital. Dado el capital P, la tasa de
interés r y el tiempo t expresado en
años, el interés simple es I = Prt.
simplest form of a square root
expression (p. 829) A square root
expression is in simplest form
if it meets the following
criteria:
1. No perfect squares are in the
radicand.
2. No fractions are in the
radicand.
3. No square roots appear in the
denominator of a fraction.
forma simplificada de una expresión
de raíz cuadrada Una expresión
de raíz cuadrada está en forma
simplificada si reúne los siguientes
requisitos:
1. No hay cuadrados perfectos en
el radicando.
2. No hay fracciones en el
radicando.
3. No aparecen raíces cuadradas en
el denominador de una fracción.
See also rationalizing the
denominator.
Ver también racionalizar el
denominador.
simplest form of a rational
expression (p. 887) A rational
expression is in simplest form if
the numerator and denominator
have no common factors.
forma simplificada de una expresión
racional Una expresión racional
está en forma simplificada cuando
el numerador y el denominador no
tienen factores comunes.
In the experiment of rolling
a number cube, the event
consisting of the outcome 3
is a simple event.
If $100 is put into an account
with a simple interest rate of
5%, then after 2 years, the
account will have earned
I = 100 · 0.05 · 2 = $10 in
interest.
Not Simplest
Form
Simplest
Form
√
180
6 √
5
√
216a 2b 2
6ab √
6
√
7
_
√
2
√
14
_
2
x - 1)(x + 1)
x 2 - 1 = (__
_
2
x +x-2
(x - 1)(x + 2)
x
+1
=_
x+2
Simplest form
G34
Glossary/Glosario
ENGLISH
SPANISH
simplest form of an exponential
expression (p. 474) An
exponential expression is in
simplest form if it meets the
following criteria:
1. There are no negative
exponents.
2. The same base does not appear
more than once in a product or
quotient.
3. No powers, products, or
quotients are raised to powers.
4. Numerical coefficients in
a quotient do not have any
common factor other than 1.
forma simplificada de una expresión
exponencial Una expresión
exponencial está en forma
simplificada si reúne los siguientes
requisitos:
1. No hay exponentes negativos.
2. La misma base no aparece más
de una vez en un producto o
cociente.
3. No se elevan a potencias
productos, cocientes ni potencias.
4. Los coeficientes numéricos en un
cociente no tienen ningún factor
común que no sea 1.
simplify (p. 40) To perform all
indicated operations.
simplificar Realizar todas las
operaciones indicadas.
simulation (p. 736) A model of
an experiment, often one that
would be too difficult or timeconsuming to actually perform.
simulación Modelo de un
experimento; generalmente se
recurre a la simulación cuando
realizar dicho experimento sería
demasiado difícil o llevaría mucho
tiempo.
sine (p. 928) In a right triangle,
the ratio of the length of the leg
opposite ∠A to the length of the
hypotenuse.
seno En un triángulo rectángulo,
razón entre la longitud del cateto
opuesto a ∠A y la longitud de la
hipotenusa.
EXAMPLES
Not Simplest
Form
Simplest
Form
78 · 74
7 12
(x 2)-4 · x 5
1
_
x3
a 5b 9
_
(ab)4
ab 5
13 - 20 + 8
-7 + 8
1
…Þ«œÌi˜ÕÃi
œ««œÃˆÌi
opposite
sin A = __
hypotenuse
slope (p. 315) A measure of the
steepness of a line. If (x 1, y 1) and
(x 2, y 2) are any two points on the
line, the slope of the line, known
as m, is represented by the
y2 - y1
equation m = _____
x2 - x1 .
pendiente Medida de la inclinación
de una línea. Dados dos puntos
(x 1, y 1) y (x 2, y 2) en una línea, la
pendiente de la línea, denominada
m, se representa con la ecuación
y2 - y1
m = _____
x2 - x1 .
{
Þ
Ó
­Ó]ÊÓ®
­Ó]Ê£®
{
ä
Ó
Ý
{
Ó
{
_
y - y1
3
-1 - 2 = _
m = x2 - x = _
4
-2 - 2
1
2
slope-intercept form (p. 345) The
slope-intercept form of a linear
equation is y = mx + b, where
m is the slope and b is the
y-intercept.
forma de pendiente-intersección
La forma de pendiente-intersección
de una ecuación lineal es
y = mx + b, donde m es la pendiente
y b es la intersección con el eje y.
y = -2x + 4
The slope is -2.
The y-intercept is 4.
solution of a linear equation in
two variables An ordered pair
or ordered pairs that make the
equation true.
solución de una ecuación lineal en
dos variables Un par ordenado o
pares ordenados que hacen que la
ecuación sea verdadera.
(4, 2) is a solution of
x + y = 6.
Glossary/Glosario
G35
ENGLISH
SPANISH
EXAMPLES
solution of a linear inequality in
two variables (p. 428) An ordered
pair or ordered pairs that make
the inequality true.
solución de una desigualdad lineal
en dos variables Un par ordenado
o pares ordenados que hacen que
la desigualdad sea verdadera.
(3, 1) is a solution of
x + y < 6.
solution of a system of linear
equations (p. 397) Any ordered
pair that satisfies all the equations
in a system.
solución de un sistema de
ecuaciones lineales Cualquier
par ordenado que resuelva todas
las ecuaciones de un sistema.
⎧x + y = -1
⎨
⎩ -x + y = -3
solution of a system of linear
inequalities (p. 435) Any
ordered pair that satisfies all the
inequalities in a system.
solución de un sistema de
desigualdades lineales Cualquier
par ordenado que resuelva todas
las desigualdades de un sistema.
Solution: (1, -2)
⎧y ≤ x + 1
⎨
⎩ y < -x + 4
Þ
­Ó]£®ÊˆÃʈ˜Ê̅iÊ
œÛiÀ>««ˆ˜}Ê
Å>`i`ÊÀi}ˆœ˜Ã]
ÜʈÌʈÃÊ>Ê܏Ṏœ˜°
Ó
ä
Ó
Ó
Ó
solution of an equation in one
variable (p. 77) A value or values
that make the equation true.
solución de una ecuación en una
variable Valor o valores que hacen
que la ecuación sea verdadera.
Equation: x + 2 = 6
Solution: x = 4
solution of an inequality in one
variable (p. 170) A value or values
that make the inequality true.
solución de una desigualdad en una
variable Valor o valores que hacen
que la desigualdad sea verdadera.
Inequality: x + 2 < 6
Solution: x < 4
solution set (p. 77) The set of
values that make a statement
true.
conjunto solución Conjunto de
valores que hacen verdadero un
enunciado.
Inequality: x + 3 ≥ 5
Solution set: {x | x ≥ 2}
{ Î Ó £
square A quadrilateral with four
congruent sides and four right
angles.
cuadrado Cuadrilátero con cuatro
lados congruentes y cuatro ángulos
rectos.
square in numeration (p. 26) The
second power of a number.
cuadrado en numeración La segunda
potencia de un número.
square root (p. 32) A number that
is multiplied by itself to form a
product is called a square root of
that product.
raíz cuadrada El número que se
multiplica por sí mismo para formar
un producto se denomina la raíz
cuadrada de ese producto.
square-root function (p. 822) A
function whose rule contains a
variable under a square-root sign.
función de raíz cuadrada Función
cuya regla contiene una variable
bajo un signo de raíz cuadrada.
standard form of a linear equation
(p. 302) Ax + By = C, where A, B,
and C are real numbers and A and
B are not both 0.
forma estándar de una ecuación
lineal Ax + By = C, donde A, B y C
son números reales y A y B no son
ambos cero.
G36
Glossary/Glosario
ä
£
Ó
Î
{
x
16 is the square of 4.
√
16 = 4, because
4 2 = 4 · 4 = 16.
-5
y = √3x
2x + 3y = 6
È
Ý
ENGLISH
SPANISH
EXAMPLES
standard form of a polynomial
(p. 497) A polynomial in one
variable is written in standard
form when the terms are in order
from greatest degree to least
degree.
forma estándar de un polinomio Un
polinomio de una variable se
expresa en forma estándar cuando
los términos se ordenan de mayor a
menor grado.
standard form of a quadratic
equation (p. 642) ax 2 + bx + c = 0,
where a, b, and c are real numbers
and a ≠ 0.
forma estándar de una ecuación
cuadrática ax 2 + bx + c = 0, donde
a, b y c son números reales y a ≠ 0.
stem-and-leaf plot (p. 709) A
graph used to organize and
display data by dividing each data
value into two parts, a stem and
a leaf.
diagrama de tallo y hojas Gráfica
utilizada para organizar y mostrar
datos dividiendo cada valor de datos
en dos partes, un tallo y una hoja.
stratified random sample (p. 732)
A sample in which a population is
divided into distinct groups and
members are selected at random
from each group.
muestra aleatoria estratificada
Muestra en la que la población está
dividida en grupos diferenciados
y los miembros de cada grupo se
seleccionan al azar.
substitution method (p. 404) A
method used to solve systems of
equations by solving an equation
for one variable and substituting
the resulting expression into the
other equation(s).
sustitución Método utilizado para
resolver sistemas de ecuaciones
resolviendo una ecuación para una
variable y sustituyendo la expresión
resultante en las demás ecuaciones.
Subtraction Property of Equality
(p. 79) If a, b, and c are real
numbers and a = b, then
a - c = b - c.
Propiedad de igualdad de la resta Si
a, b y c son números reales y a = b,
entonces a - c = b - c.
x+6= 8
-6 -6
−−−− −−
x
= 2
Subtraction Property of
Inequality (p. 176) For real
numbers a, b, and c, if a < b, then
a - c < b - c. Also holds true for
>, ≤, ≥, and ≠.
Propiedad de desigualdad de la resta
Dados los números reales a, b y c,
si a < b, entonces a - c < b - c.
Es válido también para >, ≤, ≥ y ≠.
x+6< 8
-6 -6
−−−− −−
x
< 2
supplementary angles Two angles
whose measures have a sum of
180°.
ángulos suplementarios Dos ángulos
cuyas medidas suman 180°.
surface area (p. 520) The total
area of all faces and curved
surfaces of a three-dimensional
figure.
área total Área total de todas las
caras y superficies curvas de una
figura tridimensional.
4x 5 - 2 x 4 + x 2 - x + 1
2x 2 + 3x - 1 = 0
-Ìi“
Î
{
x
i>ÛiÃ
ÓÊÎÊ{Ê{ÊÇʙ
äÊ£ÊxÊÇÊÇÊÇÊn
£ÊÓÊÓÊÎ
iÞ\ÊÎ]Óʓi>˜ÃÊΰÓ
Ms. Carter chose a
stratified random sample
of her school’s student
population by randomly
selecting 30 students
from each grade level.
Îäc
£xäc
£ÓÊV“
ÈÊV“
nÊV“
Surface area
= 2(8)(12) + 2(8)(6) + 2(12)(6)
= 432 cm 2
Glossary/Glosario
G37
ENGLISH
SPANISH
EXAMPLES
system of linear equations
(p. 397) A system of equations
in which all of the equations are
linear.
sistema de ecuaciones lineales
Sistema de ecuaciones en el que
todas las ecuaciones son lineales.
⎧2x + 3y = -1
⎨
⎩ x - 3y = 4
system of linear inequalities
(p. 435) A system of inequalities
in which all of the inequalities are
linear.
sistema de desigualdades lineales
Sistema de desigualdades en el que
todas las desigualdades son lineales.
⎧2x + 3y > -1
⎨
⎩ x - 3y ≤ 4
systematic random sample
(p. 732) A sample based on
selecting one member of the
population at random and then
selecting other members by using
a pattern.
muestra sistemática Muestra en
la que se elige a un miembro de la
población al azar y luego se
elige a otros miembros mediante
un patrón.
Mr. Martin chose a systematic
random sample of customers
visiting a store by selecting one
customer at random and then
selecting every tenth customer
after that.
T
tangent (p. 928) In a right
triangle, the ratio of the length of
the leg opposite ∠A to the length
of the leg adjacent to ∠A.
tangente En un triángulo
rectángulo, razón entre la longitud
del cateto opuesto a ∠A y la longitud
del cateto adyacente a ∠A.
œ««œÃˆÌi
>`>Vi˜Ì
tan A =
opposite
_
adjacent
3x 2 + 6x - 8
term of an expression (p. 47) The
parts of the expression that are
added or subtracted.
término de una expresión Parte de
una expresión que debe sumarse o
restarse.
term of a sequence (p. 276)
An element or number in the
sequence.
término de una sucesión Elemento o
número de una sucesión.
5 is the third term in the
sequence 1, 3, 5, 7, …
terminating decimal (p. 34) A
decimal that ends, or terminates.
decimal finito Decimal con un
número determinados de posiciones
decimales.
1.5, 2.75, 4.0
theoretical probability (p. 744)
The ratio of the number of equally
likely outcomes in an event to
the total number of possible
outcomes.
probabilidad teórica Razón entre el
número de resultados igualmente
probables de un suceso y el número
total de resultados posibles.
In the experiment of rolling a
number cube, the theoretical
probability of rolling an odd
number is __36 = __12 .
third quartile (p. 718) The median
of the upper half of a data set.
Also called upper quartile.
tercer cuartil La mediana de la
mitad superior de un conjunto de
datos. También se llama cuartil
superior.
Lower half
18, 23, 28,
tip (p. 140) An amount of money
added to a bill for service; usually
a percent of the bill.
propina Cantidad que se agrega a una
factura por servicios; generalmente,
un porcentaje de la factura.
transformation (p. 369) A change
in the position, size, or shape of a
figure or graph.
transformación Cambio en la
posición, tamaño o forma de una
figura o gráfica.
Term Term Term
B
B
Preimage
A
C
ABC
G38
Glossary/Glosario
Upper half
29, 36, 42
Third quartile
Image
A
ABC
C
ENGLISH
SPANISH
translation (p. 369) A
transformation that shifts or
slides every point of a figure or
graph the same distance in the
same direction.
traslación Transformación en la
que todos los puntos de una figura
o gráfica se mueven la misma
distancia en la misma dirección.
trapezoid A quadrilateral with
exactly one pair of parallel sides.
trapecio Cuadrilátero con sólo un
par de lados paralelos.
tree diagram (p. 760) A branching
diagram that shows all possible
combinations or outcomes of an
experiment.
EXAMPLES
Ī
Ī
Ī
diagrama de árbol Diagrama
con ramificaciones que muestra
todas las combinaciones o
resultados posibles de un
experimento.
Ī
(
4
The tree diagram shows the
possible outcomes when
tossing a coin and rolling a
number cube.
línea de tendencia Línea en
un diagrama de dispersión que
sirve para mostrar la correlación
entre conjuntos de datos más
claramente.
՘`‡À>ˆÃiÀ
£Óää
œ˜iÞÊÀ>ˆÃi`Ê­f®
trend line (p. 269) A line on a
scatter plot that helps show the
correlation between data sets
more clearly.
£äää
nää
Èää
{ää
Óää
ä
xä £ää £xä Óää
,œÃÊ܏`Ê
In the experiment of rolling
a number cube, each roll is
one trial.
trial (p. 737) Each repetition or
observation of an experiment.
prueba Una sola repetición u
observación de un experimento.
triangle A three-sided polygon.
triángulo Polígono de tres lados.
triangular prism (p. 894) A prism
whose bases are triangles.
prisma triangular Prisma cuyas
bases son triángulos.
triangular pyramid (p. 894) A
pyramid whose base is a
triangle.
pirámide triangular Pirámide cuya
base es un triángulo.
trigonometric ratio (p. 928) Ratio
of the lengths of two sides of a
right triangle.
razón trigonométrica Razón entre
dos lados de un triángulo rectángulo.
"ASES
V
L
>
a , cos A = _
b , tan A = _
a
sin A = _
c
c
b
trinomial (p. 497) A polynomial
with three terms.
trinomio Polinomio con tres
términos.
4x 2 + 3xy - 5y 2
Glossary/Glosario
G39
ENGLISH
SPANISH
EXAMPLES
union (p. 206) The union of two
sets is the set of all elements that
are in either set, denoted by .
unión La unión de dos conjuntos es
el conjunto de todos los elementos
que se encuentran en ambos
conjuntos, expresado por .
A = {1, 2, 3, 4}
B = {1, 3, 5, 7, 9}
A B = {1, 2, 3, 4, 5, 7, 9}
unit rate (p. 120) A rate in which
the second quantity in the
comparison is one unit.
tasa unitaria Tasa en la que la
segunda cantidad de la comparación
es una unidad.
30 mi = 30 mi/h
_
1h
unlike radicals (p. 835) Radicals
with a different quantity under
the radical.
radicales distintos Radicales con
cantidades diferentes debajo del
signo de radical.
unlike terms Terms with different
variables or the same variables
raised to different powers.
términos distintos Términos con
variables diferentes o las mismas
variables elevadas a potencias
diferentes.
upper quartile See third quartile.
cuartil superior Ver tercer cuartil.
U
2 √
2 and 2 √3
4xy 2 and 6x 2y
V
value of a function (p. 251)
The result of replacing the
independent variable with a
number and simplifying.
valor de una función Resultado
de reemplazar la variable
independiente por un número y
luego simplificar.
The value of the function
f (x) = x + 1 for x = 3 is 4.
value of a variable (p. 7) A
number used to replace a variable
to make an equation true.
valor de una variable Número
utilizado para reemplazar una
variable y hacer que una
ecuación sea verdadera.
In the equation x + 1 = 4,
the value of x is 3.
value of an expression (p. 7) The
result of replacing the variables in
an expression with numbers and
simplifying.
valor de una expresión Resultado
de reemplazar las variables de una
expresión por un número y luego
simplificar.
The value of the expression
x + 1 for x = 3 is 4.
variable (p. 6) A symbol used
to represent a quantity that can
change.
variable Símbolo utilizado para
representar una cantidad que
puede cambiar.
In the expression 2x + 3, x is
the variable.
Venn diagram A diagram used to
show relationships between sets.
diagrama de Venn Diagrama
utilizado para mostrar la
relación entre conjuntos.
Brand A
Brand B
Both
Neither: 15
vertex of a parabola (p. 612) The
highest or lowest point on the
parabola.
vértice de una parábola Punto más
alto o más bajo de una parábola.
­ä]ÊÓ®
The vertex is (0, -2).
G40
Glossary/Glosario
ENGLISH
vertex of an absolute-value
graph (p. 378) The point on the
axis of symmetry of the graph.
SPANISH
EXAMPLES
vértice de una gráfica de valor
absoluto Punto en el eje de simetría
de la gráfica.
{
Þ
ÞÊNÝN
Ó
6iÀÌiÝ
{
vertical angles The nonadjacent
angles formed by two intersecting
lines.
ángulos opuestos por el vértice
Ángulos no adyacentes formados
por dos líneas que se cruzan.
Ý
ä
Ó
Ó
{
£
Î {
Ó
∠1 and ∠3 are vertical angles.
∠2 and ∠4 are vertical angles.
vertical line (p. 316) A line whose
equation is x = a, where a is the
x-intercept.
línea vertical Línea cuya ecuación
es x = a, donde a es la intersección
con el eje x.
Þ
x
{
Î
Ó
£
ÝÊÊÓ
x {ÎÓ £
£
vertical-line test (p. 247) A test
used to determine whether a
relation is a function. If any
vertical line crosses the graph of
a relation more than once, the
relation is not a function.
prueba de la línea vertical Prueba
utilizada para determinar si una
relación es una función. Si una
línea vertical corta la gráfica de una
relación más de una vez, la relación
no es una función.
volume (p. 520) The number of
nonoverlapping unit cubes of a
given size that will exactly fill the
interior of a three-dimensional
figure.
volumen Cantidad de cubos
unitarios no superpuestos de un
determinado tamaño que llenan
exactamente el interior de una
figura tridimensional.
voluntary response sample
(p. 733) A sample in which
members choose to be in the
sample.
muestra de respuesta
voluntaria Una muestra en la que
los miembros eligen participar.
£ Ó Î { x
x
Ý
Þ
x
x
Ý
x
Function
Not a function
{ÊvÌ
ÎÊvÌ
£ÓÊvÌ
Volume = (3)(4)(12) = 144 ft 3
A store provides survey
cards for customers who
wish to fill them out.
W
whole number (p. 34) A member
of the set of natural numbers and
zero.
número cabal Miembro del conjunto
de los números naturales y cero.
0, 1, 2, 3, 4, 5, …
X
x-axis (p. 54) The horizontal axis
in a coordinate plane.
eje x Eje horizontal en un plano
cartesiano.
x-axis
0
Glossary/Glosario
G41
ENGLISH
SPANISH
EXAMPLES
x-coordinate (p. 54) The first
number in an ordered pair, which
indicates the horizontal distance
of a point from the origin on the
coordinate plane.
coordenada x Primer número de un
par ordenado, que indica la distancia
horizontal de un punto desde el
origen en un plano cartesiano.
4
x-intercept (p. 307) The
x-coordinate(s) of the point(s)
where a graph intersects the
x-axis.
intersección con el eje x
Coordenada(s) x de uno o más
puntos donde una gráfica corta el
eje x.
y
2
x
-4
0
-2
x-coordinate
2
4
-2
P
(-2, -3) -4
4
-2
y
(2, 0) x
4
0
-2
The x-intercept is 2.
Y
y-axis
y-axis (p. 54) The vertical axis in a
coordinate plane.
eje y Eje vertical en un plano
cartesiano.
0
y-coordinate (p. 54) The second
number in an ordered pair, which
indicates the vertical distance
of a point from the origin on the
coordinate plane.
coordenada y Segundo número
de un par ordenado, que indica la
distancia vertical de un punto desde
el origen en un plano cartesiano.
y-intercept (p. 307) The
y-coordinate(s) of the point(s)
where a graph intersects the
y-axis.
intersección con el eje y
Coordenada(s) y de uno o más
puntos donde una gráfica corta el
eje y.
y
4
2
x
-4
-2
0
2
4
y-coordinate
P -2
(-2, -3) -4
4
y
(0, 2)
-2
0
2
x
-2
The y-intercept is 2.
Z
zero exponent (p. 460) For any
nonzero real number x, x 0 = 1.
exponente cero Dado un número
real distinto de cero x, x 0 = 1.
zero of a function (p. 619) For
the function f, any number x such
that f (x) = 0.
cero de una función Dada la función
f, todo número x tal que f (x) = 0.
50 = 1
Ó
£
x { ÎÓ £
£
­Î]Êä®
Ó
£ Ó Î { x
­£]Êä®
Î
{
x
The zeros are -3 and 1.
Zero Product Property (p. 650) For
real numbers p and q, if pq = 0,
then p = 0 or q = 0.
G42
Glossary/Glosario
Propiedad del producto cero Dados
los números reales p y q, si pq = 0,
entonces p = 0 o q = 0.
If (x - 1)(x + 2) = 0,
then x - 1 = 0 or x + 2 = 0,
so x = 1 or x = -2.
Index
A
Aaron, Hank, 42
Absolute value
definition of, 14
equations, 112–114
functions, 378–381, AT5
inequalities, 212–215
Addition
of matrices, 770
of polynomials, 502–503, 504–506
properties of, 46
of radical expressions, 835–837
of rational expressions,
905–908
of real numbers, 14–17
solving equations by, 77–79
solving inequalities by, 176–179
Addition Property of Equality,
79, 86
Addition Property of Inequality,
176
Additive identity, 15
Additive inverse, 15
Advertising, 721
Agriculture, 677
Air Force Academy, 439
Air Force One, 254
Albers, Josef, 30
Algebraic expressions, 6, 7, 8, 10,
40–42, 72, 248
Algebra Lab, see also Technology Lab
Compound Events, 758–759
Explore Changes in Population,
150–151
Explore Constant Changes, 322–323
Explore Properties of Exponents,
472–473
Explore the Axis of Symmetry, 618
Model Completing the Square, 662
Model Equations with Variables on
Both Sides, 99
Model Factoring, 550, 558–559
Model Growth and Decay, 804
Model Inverse Variation, 870
Model One-Step Equations, 76
Model Polynomial Addition and
Subtraction, 502–503
Model Polynomial Division, 912
Model Polynomial Multiplication,
510–511
Model Systems of Linear Equations,
403
Model Variable Relationships, 248
Simulations, 736
Truth Tables and Compound
Statements, 203
Vertical-Line Test, 247
Algebra tiles, 76, 99, 502–503,
510–511, 550, 558–559, 912
All of the Above, 862–863
Altitude sickness, 356
Amusement parks, 833
Anglerfish, 81
Angles
central, 703
corresponding, 127
exterior, AT21
remote interior, AT21
Animals, 186
Animals Link, 186
Annulus, 557
Answers, choosing combinations of,
936–937
Any Question Type
read the problem for understanding,
450–451
spatial reasoning, 780–781
translate words to math, 600–601
use a diagram, 536–537
Applications
Advertising, 721
Agriculture, 677
Amusement Parks, 833
Animals, 186
Aquatics, 644
Archaeology, 124, 810
Archery, 624
Architecture, 622, 834
Art, 30, 565
Astronomy, 10, 333, 340, 468, 469,
475, 479, 486, 722, 826
Athletics, 281, 423, 646
Automobiles, 713
Aviation, 86, 334, 409, 928
Basketball, 757
Biology, 16, 29, 89, 97, 105, 121, 122,
124, 137, 208, 216, 310, 311, 463,
464, 469, 479, 489, 491, 646, 712,
889, 891, AT4
Business, 18, 58, 98, 139, 142, 193,
197, 347, 349, 372, 416, 422, 433,
437, 438, 506, 721, 722, 817, 883,
AT12
Camping, 185
Career, 658
Carpentry, 534
Chemistry, 123, 138, 204, 209, 416,
470, 486, 713
City Planning, 582
Communication, 115, 193, 470, 877
Construction, 109, 115, 320, 566,
890, 929
Consumer Application, 102, 184, 244,
314, 347
Consumer Economics, 88, 96, 103,
179, 271, 321, 356, 407, 408, 414,
415, 430, 482, 817
Contests, 801
Data Collection, 238, 320, 637
Decorating, 125
Design, 31
Diving, 24, 929
Earth Science, 88
Ecology, 271
Economics, 17, 88, 96, 103, 179, 271,
321, 407, 408, 414, 415, 482, 817
Education, 24, 186, 200
Electricity, 624, 630, 875
Employment, 148
Engineering, 114, 179, 624, 630, 875
Entertainment, 23, 30, 37, 111, 125,
137, 194, 201, 216, 280, 304, 373,
402, 424, 469, 526, 660, 714, 756,
903, AT8, AT12
Environment, 340
Environmental Science, 124, 310, 328
Farming, 439
Finance, 80, 88, 109, 124, 139, 143,
409, 555, 806, 819, 884
Fitness, 79, 349, 401, 838, 910
Fund-raising, 269
Games, 653
Gardening, 33, 124, 192, 573
Gemology, 881
Geography, 19, 332, 478, 486
Geology, 80, 81, 424, 826
Geometry, 10, 30, 36, 43, 44, 45, 50,
57, 81, 88, 96, 104, 109, 126, 185,
194, 199, 201, 244, 306, 334, 362,
363, 365, 366, 367, 409, 416, 433,
478, 479, 487, 491, 492, 493, 499,
500, 507, 508, 509, 517, 518, 519,
525, 529, 534, 549, 556, 557, 565,
572, 573, 583, 591, 646, 654, 655,
659, 660, 661, 666, 667, 669, 749,
757, 825, 836, 837, 838, 839, 843,
844, 845, 849, 850, 851, 891, 892,
918, 919, AT4, AT21
Health, 179, 180, 244, 326, 469
Hiking, 773
History, 97, 356, 731, 765
Hobbies, 130, 374, 433, 667, 675,
AT17
Home Economics, 876
Landscaping, 401
Law Enforcement, 825
Logic, 731, AT21
Manufacturing, 115, 124, 461
Marine Biology, 616
Math History, 255, 416, 526, 590, 834
Measurement, 128, 305, 355, 470,
582, 794
Mechanics, 875, 877
Medicine, 463, 501
Index
IN1
Meteorology, 10, 17, 110, 208, 209,
356, 851
Military, 439
Money, 408
Music, 208, 216, 218, 516, 548, 873
Navigation, 929
Number Sense, 548, 549
Number Theory, 280, 654, 660
Nutrition, 87, 88, 136, 148, 215, 244
Oceanography, 121, 260
Personal Finance, 311, 551, 812
Pet Care, 819, AT8
Photography, 507, 517
Physical Science, 306, 479, 793, AT16
Physics, 498, 555, 573, 589, 636, 638,
654, 660, 673, 800, 844, 851, 874
Population, 126
Probability, 901, 902
Problem-Solving, 28, 94, 177–178,
259, 354, 399, 523–524, 579–580,
627–628, 665–666, 753, 815–816,
921–922
Quality Control, 739
Real Estate, 131
Recreation, 22, 87, 116, 200, 238, 244,
280, 321, 333, 334–335, 408, 439,
832, 908
Recycling, 8
Remodeling, 565
Safety, 214
School, 81, 199, 374, 400, 433, 755
Science, 124, 356, 478, 552, 808
Shipping, 279
Solar Energy, 918
Space Shuttle, 115
Sports, 42, 44, 50, 104, 107, 110, 121,
125, 172, 178, 180, 209, 238, 310,
335, 348, 381, 424, 485, 518, 529,
616, 645, 652, 654, 676, 712, 718,
721, 740, 741, 792, 793, 800, 831
Statistics, 82, 88, 469, 799
Technology, 29, 142, 471, 534, 546,
766, 801
Temperature, 116, 215, 217
Transportation, 96, 124, 179, 209, 254,
261, 271, 304, 305, 341, 500, 851
Travel, 18, 36, 90, 104, 185, 278, 279,
308, 319, 334, 630, 875, 909, 910,
924, AT8
Wages, 305, 339
Waterfalls, 645
Weather, 23, 712, 713, 721
Winter Sports, 876
Applied Sciences major, 402
Approximating solutions, 91
Aquatics, 644
Archaeology, 124, 810
Archery, 624
Architecture, 622, 834
Area
of composite figures, 83
in the coordinate plane, 313
of a rectangle, 83
of a square, 83
IN2
Index
surface, 493, 520
of a trapezoid, 335, 654
of a triangle, 83
Are You Ready?, 3, 73, 167, 231, 297,
393, 457, 541, 607, 695, 787, 867
Arguments, writing convincing, 395
Arithmetic sequences, 276–278
Art, 30, 565
Art Link, 565
Assessment
Chapter Test, 66, 158, 224, 288, 386,
448, 534, 598, 688, 778, 860, 934
College Entrance Exam Practice
ACT, 159, 289, 449, 599
SAT, 67, 387, 535
SAT Mathematics Subject Tests, 689,
779, 861, 935
SAT Student-Produced Responses,
225
Cumulative Assessment, 70–71,
162–163, 228–229, 292–293,
390–391, 452–453, 538–539,
602–603, 692–693, 782–783,
864–865, 938–939
Multi-Step Test Prep, 38, 60, 118, 152,
188, 218, 264, 282, 342, 376, 426,
442, 494, 528, 576, 592, 640, 678,
734, 768, 820, 854, 896, 926
Multi-Step Test Prep questions are also
found in every exercise set. Some
examples: 10, 18, 24, 30, 36
Ready to Go On?, 39, 61, 153, 189,
219, 265, 283, 343, 377, 427, 443,
495, 529, 577, 593, 641, 679, 735,
769, 821, 855, 897, 927
Standardized Test Prep, 70–71,
162–163, 228–229, 292–293,
390–391, 452–453, 538–539,
602–603, 692–693, 782–783,
864–865, 938–939
Study Guide: Preview, 4, 74, 168, 232,
298, 394, 458, 542, 608, 696, 788,
868
Study Guide: Review, 62–65, 154–157,
220–223, 284–287, 382–385,
444–447, 530–533, 594–597,
684–687, 774–777, 856–859,
930–933
Test Prep
Test Prep questions are found
in every exercise set. Some
examples: 11, 19, 25, 31, 37
Test Tackler
Any Question Type
Read the Problem for
Understanding, 450–451
Spatial Reasoning, 780–781
Translate Words to Math, 600–601
Use a Diagram, 536–537
Extended Response
Explain Your Reasoning, 690–691
Understand the Scores, 290–291
Gridded Response
Fill in Answer Grids Correctly, 68–69
Multiple Choice
Choose Combinations of Answers,
936–937
Eliminate Answer Choices, 160–161
None of the Above or All of the
Above, 862–863
Recognize Distracters, 388–389
Short Response
Understand Short Response Scores,
226–227
Associative Properties of Addition
and Multiplication, 46
Astronomy, 10, 333, 340, 468, 469,
475, 479, 486, 722, 826
Astronomy Link, 10, 340
Asymptotes
definition of, 878
graphing rational functions using, 880
identifying, 879
Athletics, 281, 423, 646
Atoms, 470
Automobiles, 713
Automobiles Link, 713
Average, 716–719
Aviation, 86, 334, 409, 928
Axes, 54
Axis of symmetry
in absolute-value functions, 378
in a parabola, 618, 620
definition of, 618
exploring, 618
finding
by using the formula, 621
by using zeros, 620
B
Back-to-back stem-and-leaf plot,
709
Bald eagles, 891
Bamboo, 311
Bar graphs, 698–699, 700
Bases of numbers, 26
Basketball, 757
Biased samples, 733
Binomial(s)
cubic, 532
definition of, 497
division of polynomials by, 914–915
opposite, 554, 888
special products of, 521–525
Biology, 16, 29, 89, 97, 105, 121, 122,
124, 137, 208, 216, 310, 311, 463,
464, 469, 479, 489, 491, 646, 712,
889, 891, AT4
Biology Link, 105, 311, 464, 646, 891
Biology major, 106
Biostatistics major, 767
Blood loss, 464
Blood volume, 121
Biology Link, 105, 311, 464, 646, 891
Biology major, 106
Biostatistics major, 767
Blood loss, 464
Blood volume, 121
Boiling point, 356
Box-and-whisker plot, 718–719, 725
Boyle’s law, 874
Braces, 40
Brackets, 40
Business, 18, 58, 98, 139, 142, 193, 197,
347, 349, 372, 416, 422, 433, 437,
438, 506, 721, 722, 817, 883, AT12
C
Calculator, see Graphing calculator
Camping, 185
Career Path
Applied Sciences major, 402
Biology major, 106
Biostatistics major, 767
Culinary Arts major, 202
Data Mining major, 358
Environmental Sciences major, 567
Carpentry, 534
Cartesian plane, 58
Caution!, 27, 48, 86, 139, 184, 301,
308, 314, 315, 406, 421, 437, 461,
477, 552, 571, 586, 610, 613, 621,
636, 744, 792, 799, 831, 887, 906
Central angles of circles, 703
Central tendency, measure of, 716
Changes
constant, 322–323
percent, 144
rate of
constant and variable, AT14–AT15
decrease, 803
definition of, 314, AT14
identifying linear and nonlinear
functions from, AT15–AT16
increase, 803
slope and, 314–317
Changing dimensions, 53, 129
Chapter Test, 66, 158, 224, 288, 386,
448, 534, 598, 688, 778, 860, 934,
see also Assessment
Charts, reading and interpreting,
697
Cheetahs, 105
Chemistry, 123, 138, 204, 209, 416,
470, 486, 713
Chemistry Link, 209, 470
Choosing
combinations of answers, 936–937
factoring methods, 586–588
models
graphing data for, 813–814
using patterns for, 814
Circle graphs, 698–699, 702
City Planning, 582
Coefficients
definition of, 48
leading, of polynomials, 497
opposite, 411
Coincident lines, 421
College Entrance Exam Practice,
see also Assessment
ACT, 159, 289, 449, 599
SAT, 67, 387, 535
SAT Mathematics Subject Tests, 689,
779, 861, 935
SAT Student-Produced Responses, 225
Combinations
of answers, choosing, 936–937
definition of, 761
and permutations, 760–763
Combining like radicals, 835
Combining like terms, 48
Commission, 139
Common difference, 276–277
Common ratio, 790, 792
Communicating math
choose, 567, 675
compare, 11, 147, 172, 184, 201, 238,
357, 425, 485, 492, 516, 668, 669,
811, 824, 850, 919
construct, 730
create, 244, 245, 305, 714
define, 547
describe, 19, 31, 50, 56, 79, 82, 89,
135, 137, 148, 192, 195, 207, 215,
217, 238, 242, 245, 261, 272, 305,
320, 327, 328, 333, 347, 349, 364,
372, 374, 375, 424, 433, 506, 546,
556, 582, 623, 625, 636, 638, 644,
653, 659, 667, 675, 705, 707, 721,
808, 818, 826, 844, 875, 884, 918,
AT4, AT6, AT10, AT11, AT12
determine, 25, 36, 423, 425, 792, 876,
AT4, AT12, AT17
explain, 8, 17, 19, 22, 24, 30, 37, 42,
43, 44, 45, 50, 57, 80, 82, 88, 95,
104, 105, 109, 110, 114, 116, 123,
125, 131, 135, 137, 141, 142, 174,
175, 179, 180, 185, 186, 199, 201,
202, 209, 210, 217, 236, 242, 244,
246, 252, 254, 260, 261, 271, 272,
279, 280, 306, 321, 328, 334, 340,
341, 366, 367, 372, 399, 400, 402,
409, 415, 416, 417, 424, 425, 431,
434, 439, 440, 464, 469, 470, 471,
477, 479, 486, 493, 498, 500, 501,
508, 518, 526, 527, 547, 548, 554,
556, 557, 563, 565, 573, 574, 583,
590, 591, 629, 630, 631, 637, 638,
646, 653, 659, 661, 668, 676, 677,
706, 707, 708, 711, 713, 714, 719,
728, 729, 730, 739, 748, 749, 755,
756, 763, 765, 766, 794, 795, 808,
809, 818, 825, 826, 833, 838, 842,
852, 876, 884, 892, 901, 903, 908,
910, 918, 923, 925, AT4
express, 28
find, 106, 549, 566, 567, 574, 637, 918
give (an) example(s), 29, 56, 125, 187,
236, 246, 254, 500, 519, 527, 588,
638, 708, 719, 727, 739, 754, 827,
837, 838, 892
identify, 35, 349, 381, 506, 809, 884,
891
list, 439, 876
make, 713, 729, 853
name, 129, 175, 252, 309, 433, 704,
874
Reading and Writing Math, 5, 75, 169,
233, 299, 395, 459, 543, 609, 697,
789, 869, see also Reading Math;
Reading Strategies; Study Strategies;
Writing Math; Writing Strategies
show, 80, 81, 82, 88, 89, 97, 104, 124,
126, 142, 148, 179, 180, 187, 367,
492, 556, 566, 573, 590, 616, 766,
832
tell, 49, 87, 90, 103, 116, 216, 238,
329, 381, 431, 468, 616, 625, 636,
666, 741, 747, 800, 837, 852, 882
write, 8, 9, 98, 105, 136, 312, 401,
518, 573, 584, 591, 617, 794, 809,
833, 885, 889, 911
Write About It
Write About It questions are found
in every exercise set. Some
examples: 9, 19, 24, 29, 31
Communication, 115, 193, 470, 877
Commutative Properties of
Addition and Multiplication, 46
Compatible numbers, 46
Complement of an event, 745
Completing the square, 663–666,
674
Complex fractions, 904
Composite figures, areas of, 83
Compound events, 758–759, 761
Compound inequalities, 204–207,
212
Compound interest, 806
Compound statements, 203
Conditional statements, AT21
Cones, 520, 894
Congruent segments, 330
Conjecture, making a, 322, 323, 368,
632, 648, 662, 828, 870, 893, AT18
Conjugates, rationalizing
denominators using, 845
Connecting Algebra
to Data Analysis, 275, 360, 698–699,
732–733
to Geometry, 52–53, 83, 211, 313,
520, 803, 894, 895
to Number Theory, 418–419, 585
Index
IN3
Consistent systems, 420–421
Constant
definition of, 6
in trinomial factoring, 560–562
of variation, 336, 871
Constant changes, exploring,
322–323
Construction, 109, 115, 320, 566, 890,
929
Consumer Application, 102, 184, 244,
314, 347
Consumer Economics, 88, 96, 103,
179, 271, 321, 356, 407, 408, 414,
415, 430, 482, 817
Contact lenses, 180
Contests, 801
Continuous graphs, 235
Contrapositives, AT21
Convenience sample, 733
Conversion factors, 121, 609
Converting between probabilities
and odds, 746
Convincing arguments/
explanations, writing, 395
Coordinate plane
area in, 313
distance in, 331–332
locating points in, 54
Correlation, 266
Corresponding angles, 127
Corresponding sides, 127
Cosine, 928
Counterexamples, AT18–AT19
Crash test dummies, 500
Create a table to evaluate
expressions, 12–13
Critical Thinking
Critical Thinking questions are found in
every exercise set. Some examples:
11, 18, 23, 24, 30
Cross products
in proportions, 121
solving rational equations by using,
920
Cross Products Property, 121
Cube roots, 32
Cubes, difference of, 584
Cubic binomials, 532
Cubic equations, 681–683
Cubic functions, 680–683
Cubic polynomials, 497
Culinary Arts major, 202
Cumulative Assessment,
70–71,162–163, 228–229, 292–293,
390–391, 452–453, 538–539,
602–603, 692–693, 782–783,
864–865, 938–939
Cumulative frequency, 711
Cylinders
properties of, 894
surface area of, 520
volume of, 520, 661
IN4
Index
D
Data
displaying, 700–704
distributions, 716–719
graphing, to choose a model, 813–814
organizing, 700–704
Data Analysis, Connecting Algebra
to, 275, 360, 698–699, 732–733
Data Collection applications, 238,
320, 637
Data Mining major, 358
Death Valley National Park, 18
Decay, exponential, 805–808
Decorating, 125
Deductive reasoning, AT19
Degrees
of monomials, 496
of polynomials, 496
of power functions, AT3
Denominators
like, 905, 906
rationalizing, 842, 845
unlike, 905, 906
Density Property of Real Numbers,
37
Dependent events, 750–754
Dependent systems, 421
Dependent variables, 250, 251, 252,
253, 254
Descartes, Rene, 58
Design, 31
Devon Island, 10
Diagrams
ladder, 544
mapping, 240
reading and interpreting, 697
tree, 760
using, 536–537
Difference(s)
of cubes, 584
first, 610
second, 610
of two squares, 523, 580
Dimensional analysis, 121
Dimensions
changing, 53, 129
of a matrix, 770
Direct variation, 336–339
Discontinuous functions, 878
Discount, 145
Discrete graphs, 235
Discriminant, 672
Quadratic Formula and, 670–675
Displaying data, 700–704
Distance Formula, 331–332
Distracters, recognizing, 388–389
Distributions, data, 716–719
Distributive Property, 47, 551
Diving, 24, 929
Diving Link, 24
Division
long, 914–915
of polynomials, 912, 913–917
of radical expressions, 840–842
of rational expressions, 898–901
of real numbers, 20–22
of signed numbers, 20
solving equations by, 84–87
solving inequalities by, 182–184
by zero, 21–22
Division properties of exponents,
481–485
Division Property of Equality, 86
Division Property of Inequality,
182, 183
Domain(s), 240, 241, 242, 243, 244,
245, 246, 252, 253, 254, 255, 256,
257, 259, 260, 263, 264, 265, 839, 877
of linear functions, 303
of quadratic functions, 613
reasonable, 252, 253, 254, 255, 259,
265, 287, 288, 303, 308, 616, 873,
876, 881, 883, 884
of square-root functions, 823–824
Double-bar graphs, 701
Double-line graphs, 702
Dow Jones Industrial Average
(DJIA), 17
Draw a diagram, PS2
Drum Corps International, 548
E
Eagles, 891
Earned run average (ERA), 110
Earth Science, 88
Ecology, 271
Ecology Link, 271
Economics, 17, 88, 96, 103, 179, 271,
321, 407, 408, 414, 415, 430, 482, 817
Education, 24, 186, 200
Electricity, 624, 630, 875
Electricity Link, 844
Elimination, solving systems of
linear equations by, 411–415
Ellipsis, 276
Employment, 148, see also Career Path
Empty set, 102
End behavior, AT4
Engineering, 114, 179, 624, 630, 875
Engineering Link, 624
Entertainment, 23, 30, 37, 111, 125,
137, 194, 201, 216, 280, 304, 373,
402, 424, 469, 526, 660, 714, 756,
903, AT8, AT12
Entry of a matrix, 770
Environment, 340
Environmental Science, 124, 310,
328, 567
Environmental Sciences major, 567
Equality
Power Property of, 846
properties of, 79, 86
Equally likely, 744
Equations
absolute value, 112–114
cubic, 681–683
definition of, 77
finding slope from, 326
linear
definition of, 302
point-slope form of, 351–355
slope-intercept form of, 344–347,
350
solving, by using a spreadsheet, 396
standard form of, 302
systems of, 420–423
literal, 108
model
one-step, 76
with variables on both sides, 99
point-slope form, 351–355
quadratic
definition of, 642
discriminant of, 672
related function of, 642
roots of, 648–649
solving
by completing the square, 664,
674
by factoring, 650–653, 674
by graphing, 642–644, 674
by using square roots, 656–659,
674
by using the Quadratic Formula,
670–675
standard form of, 642
radical, 846–850
rational, 920–923, 926
slope-intercept form, 344–347, 350
solutions of, 77
solving
by addition, 77–79
by division, 84–87
by elimination, 411–415
by graphing, 91, AT9
by multiplication, 84–87
multi-step, 92–95
by subtraction, 77–79
two-step, 92–95
by using cross products, 920
with variables on both sides,
100–103
standard form, 302, 642
systems of
classification of, 421–422
identifying solutions of, 397
with infinitely many solutions, 421
modeling, 403
with no solution, 420–421
solving
by elimination, 411–415
by graphing, 397–399
by substitution, 404–407
for trend lines, 360
Equivalent ratios, 120
Equivalents, common, 133
Error Analysis, 18, 58, 80, 88, 125, 142,
174, 186, 201, 245, 254, 261, 272,
328, 357, 416, 425, 434, 439, 464,
479, 492, 501, 508, 526, 557, 573,
583, 590, 630, 654, 660, 668, 713,
730, 748, 810, 818, 838, 852, 884,
903, 910, 918
Escape velocity, 826
Estimating
cube roots, 33
with percents, 140
solutions using the Quadratic Formula,
671
square roots, 33
Estimation, 30, 50, 110, 136, 148, 180,
187, 254, 262, 272, 311, 320, 334,
366, 410, 440, 479, 519, 565, 590,
616, 660, 721, 741, 811, 826, 852,
876, 918
Evaluating
exponential functions, 796–797
expressions, 7, 12–13
factored and polynomial forms, 563
functions, 251
Events
compound, 758–759, 761
definition of, 737
dependent, 750–754
inclusive, 758
independent, 750–754
mutually exclusive, 758
simple, 761
Exam preparation, 869
Excluded values, 868, 878
Experimental probability, 737–739
Experiments, 737
Explanations
convincing, 395
for your reasoning in extended
responses, 690–691
Exploring
axis of symmetry of a parabola, 618
constant changes, 322–323
properties of exponents, 472–473
roots, zeros, and x-intercepts, 648–649
Exponent(s)
definition of, 26
division properties of, 481–485
integer, 460–462
and powers of ten, 466–467
in scientific notation, 467–468
using patterns to investigate, 460
multiplication properties of, 474–477
negative, 460, 466
powers and, 26–28
properties of, 472–473
rational, 488–490
reading, 27
zero, 460
Exponential decay, 805–808
Exponential expressions,
simplifying, 474
Exponential functions, 796–799
definition of, 796
evaluating, 796–797
general form of, 815
graphs of, 797–799, AT9
Exponential growth, 805–808
Exponential models, 813–816
Expressions
algebraic, 6, 7, 8, 10, 40–42, 72, 248
create a table to evaluate, 12–13
exponential, simplifying, 474
numerical, 6
radical, see Radical expressions
rational, see Rational expressions
simplifying, 46–49
square-root, 829, 841
variables and, 6–8
Extended Response, 71, 163, 229,
246, 290–291, 293, 306, 391, 450,
451, 453, 509, 539, 549, 603, 677,
690–691, 693, 783, 865, 911, 939
Explain Your Reasoning, 690–691
Understand the Scores, 290–291
Extension
Absolute-Value Functions, 378–381
Cubic Functions and Equations,
680–683
Matrices, 770–773
Trigonometric Ratios, 928–929
Exterior angles, AT21
Extra Practice, S4–S39
Extraneous solutions, 848–849, 922
F
Factor(s)
common, 545
definition of, 544
greatest common, 545–546
prime factorization, 544
Factorial, 762
Factoring polynomials
choosing a method, 586–588
common binomial factors in, 553
factored form, 551
factoring ax 2 + bx + c, 568–571
factoring perfect-square trinomials,
578–579, 585
factoring the difference of two
squares, 580–581, 585
factoring x 2 + bx + c, 560–563
by graphing, 575
by greatest common factor,
551–554, 587
by grouping, 553–554, 588
by guess and check, 560, 568
knowing when an expression is fully
factored, 586
Index
IN5
modeling, 550, 558–559
by multiple methods, 587–588
with opposite binomials, 554
recognizing special products, 585
in simplifying rational expressions,
886–888
solving quadratic equations by,
650–653, 674
steps in, 586
trinomial constant as product of
binomial constants, 560–562
unfactorable polynomials, 587
Factorization, prime, 544
Factor tree, 544
Fair experiments, 744
Families of functions
definition of, 369
of linear functions, 368, 882
of quadratic functions, 632, 882
of rational functions, 882
of square-root functions, 882
Farming, 439
Fibonacci sequence, 280
Figures
composite, 83
reading and interpreting, 697
similar, 127
solid, 894
Final exam preparation, 869
Finance, 80, 88, 109, 124, 139, 143,
409, 555, 806, 819, 884
Find a pattern, PS6
Finding a term of a geometric
sequence, 790–792
Finding a term of an arithmetic
sequence, 276–278
First coordinates, 240
First differences, 610
First quartile, 718
Fitness, 79, 349, 401, 838, 910
Flying fish, AT4
FOIL method, 513, 560, 841
Formula(s), 107
area
of composite figures, 83
in the coordinate plane, 313
of a rectangle, 83
of a square, 83
surface, 493, 520
of a trapezoid, 335, 654
of a triangle, 83
axis of symmetry of a parabola, 621
combinations, 763
compound interest, 806
distance, 331–332
distance from a light source, 492
experimental probability, 738
exponential decay, 807
exponential growth, 805
half-life, 807
Heron’s, 834
midpoint, 330–331
period of a pendulum, 660
IN6
Index
permutations, 762
probability of dependent events, 753
probability of independent events, 751
Quadratic, 670
recursive, AT13
remembering, 789
simple interest, 139
slope, 324
solving for a variable, 108
speed of light, 494
surface area of a sphere, 493
term of a geometric sequence, 790
term of an arithmetic sequence,
276–277
theoretical probability, 744
volume of a cylinder, 661
Foundation plan, 895
Fraction(s)
complex, 904
as exponents, 488–490
finding roots of, 33
negative, 352
Frequency, 709–711
cumulative, 711
Frequency table, 710
Function(s)
absolute-value, 378–381, AT5
cubic, 680–683
definition of, 241
degree of, 496, AT3
discontinuous, 878
end behavior of, AT4
evaluating, 251
exponential, 796–799, 815, AT9
families of, 369, 632, 882
general forms of, 815
graphing, 256–260
greatest-integer, AT7
identifying, AT10–AT12
introduction to, 54–56
linear
definition of, 300, AT15
family of, 368, 882
general form of, 815
graphing, 302
identifying, 300–303
reflections of, 371
rotations of, 370
transformations of, 369–372
vertical translations of, 369
parent
definition of, 369
linear, 369, 882
quadratic, 632, 633, 882
rational, 882
square-root, 882
piecewise, AT5–AT7
power, AT2–AT4
quadratic
characteristics of, 619–623
comparing graphs of, 635
definition of, 610
domain of 613
in families of functions, 632, 882
finding zeros of, 619
general form of, 815
graphing, 626–629, AT9
using a table of values, 611
identifying, 610–613
range of, 613
transformations of, 633–636
radical, 828
rational, 878–882, 893
relations and, 240–242
square-root, 822–824, 882
step, AT6
writing, 249–252
zeros of, 619
Function notation, 250
Function rules, 250, 263
Function table, 55–56, 263
Fundamental Counting Principle,
760
Fund-raising, 269
G
Galileo Galilei, 255
Gallium, 209
Games, 653
Gardening, 33, 124, 192, 573
GCF (greatest common factor),
545-546
factoring by, 551–554
Gemology, 881
General forms of functions, 815
Geodes, 424
Geography, 19, 332, 478, 486
Geology, 80, 81, 424, 826
Geology Link, 81, 424, 826
Geometric models
of powers, 26
of special products, 521, 523
Geometric probability, 910
Geometric sequences, 790–792
Geometry, see also Applications
angles
central, 703
corresponding, 127
exterior, AT21
remote interior, AT21
annulus, 557
area
of composite figures, 83
in the coordinate plane, 313
of a rectangle, 83
of a square, 83
surface, 493, 520
of a trapezoid, 335, 654
of a triangle, 83
changing dimensions, 53, 129
circle, diameter of, 334
cones
properties of, 894
surface area of, 520
volume of, 520
Connecting Algebra to, 52–53, 83,
211, 313, 520, 894
coordinate plane, 54
area in the, 313
distance in the, 331–332
corresponding sides, 127
cosine, 928
cylinders, 520, 894
dimensions, changing, 53, 129
foundation plan, 895
geometric models
of powers, 26
of special products, 521, 523
geometric probability, 910
half-plane, 428
Heron’s formula, 834
indirect measurement, 128
nets, 894
perimeter, 52–53
planes
Cartesian, 58
coordinate, 54
polygons, 52–53
prisms, 520, 894
pyramids, 520, 894
Pythagorean Theorem, 331, 661, 831
rectangles, area of, 83
reflections, 371
rotations, 370
scale, 122
scale drawing, 122
scale factor, 129
scale model, 122
sectors, 702
similar figures, 127
sine, 928
slopes of lines, 361–364
surface-area-to-volume ratio, 889
tangent, 928
transformations
of absolute-value functions,
379–381
of linear functions, 369–372
of quadratic functions, 633–636
translations, 369, 823
Triangle Inequality, 211
trigonometric ratios, 928–929
volume, 520
Get Organized, see Graphic organizers
go.hrw.com, see Online Resources
Graph(s), see also Graphing
bar, 698–699, 700
circle, 698–699, 702
comparing, of quadratic functions, 635
connecting to function rules and
tables, 263
continuous, 235
discrete, 235
double-bar, 701
double-line, 702
finding slope from, 325
finding zeros of quadratic functions
from, 619
identifying linear functions by, 300
line, 701–704
misleading, 726–727
reading and interpreting, 697
turning points on, 680
Graphic Organizers
Graphic Organizers are found in every
lesson. Some examples: 8, 17, 22,
28, 35
Graphics, reading and
interpreting, 697
Graphing, see also Graph(s)
absolute-value functions, 379–381
data to choose a model, 813–814
exponential functions, 799, AT9
to factor polynomials, 575
functions, 256–260
greatest-integer functions, AT7
inequalities, 170–172
linear functions, 302
linear inequalities, 429
midpoints and endpoints, 330–331
piecewise functions, AT5–AT7
point-slope form, 352–353
power functions, AT2–AT3
quadratic functions, 611, 626–629,
AT9
radical functions, 828
rational functions, 880, 893
relationships, 234–236
slope-intercept form, 344–347, 350
solving equations by, 91
solving quadratic equations by,
642–644, 674
solving systems of linear equations by,
397–399
square-root functions, translations
of, 823
step functions, AT6
Graphing Calculator, 12, 16, 91,
263, 274, 368, 370, 371, 398, 401,
441, 575, 617, 625, 632, 634, 635,
636, 637, 639, 643, 644, 645, 646,
647, 648, 649, 664, 668, 671, 674,
724–725, 743, 763, 799, 801, 811,
825, 826, 853, 885, 918, AT9
Greatest common factor (GCF),
545–546
factoring by, 551–554
Greatest-integer function, AT7
Gridded Response, 45, 68–69, 71, 98,
105, 132, 149, 163, 195, 229, 255,
293, 310, 329, 341, 350, 367, 391,
401, 451, 453, 465, 539, 584, 601,
603, 625, 639, 693, 783, 827, 865,
877, 892, 939
Fill in Answer Grids Correctly, 68–69
Griffith-Joyner, Florence, 125
Grouping, factoring by, 553–554, 588
Grouping symbols, 40, 41
Growth
exponential, 805–808
modeling, 804
Guess and test, PS4
H
Half-life, 807
Half-plane, 428
Hamm, Paul, 44
Handball team, 518
Health, 179, 180, 244, 326, 469
Health Link, 180
Helpful Hint, 8, 16, 21, 40, 41, 46, 47,
92, 94, 100, 112, 121, 128, 129, 134,
140, 144, 145, 176, 196, 198, 204,
207, 212, 235, 242, 250, 256, 257,
266, 309, 346, 352, 363, 364, 397,
398, 404, 405, 412, 413, 422, 429,
481, 484, 488, 490, 514, 515, 545,
554, 562, 579, 587, 611, 619, 627,
635, 643, 651, 652, 657, 664, 671,
673, 710, 738, 762, 763, 791, 796,
805, 807, 815, 822, 824, 830, 835,
840, 842, 849, 872, 914, 915, 922,
928, AT10
Heron of Alexandria, 834
Heron’s formula, 834
Hiking, 773
Histograms, 710
frequency and, 709–711
History, 97, 356, 731, 765
History Link, 97, 731, 765
Hobbies, 130, 374, 433, 667, 675, AT17
Hobbies Link, 374
Home Economics, 876
Homework Help Online
Homework Help Online is available for
every lesson. Refer to the go.hrw.com
box at the beginning of each exercise
set. Some examples: 9, 17, 23, 29, 35
Horizontal lines, 363, 879
Hot-air balloons, 20, 22
Hot Tip!, 67, 69, 71, 159, 161, 163, 225,
227, 229, 289, 291, 293, 387, 389,
391, 449, 451, 453, 535, 537, 539,
599, 601, 603, 689, 691, 693, 779,
781, 783, 861, 863, 865, 935, 937, 939
Hurricanes, 851
I
Identifying asymptotes, 879
Identities
equations as, 101
Identity
additive, 15
multiplicative, 21
Inclusive events, 758
Inconsistent systems, 420–421
Independent events, 750–754
Index
IN7
Independent systems, 421
Independent variables, 250–254
Index, in roots, 32, 488
Indirect measurement, 128
Inductive reasoning, AT18–AT19
Inequalities
absolute-value, 212–215
compound, 204–207, 212
definition of, 170
graphing, 170–172, 429
linear
definition of, 428
graphing, 429
solutions of, 428
solving, 428–431
systems of, 435–437, 441
properties of, 176, 182, 183
solutions of, 170
solving
by addition, 176–179
compound, 204–207
by division, 182–184
by multiplication, 182–184
multi-step, 190–192
by subtraction, 176–179
two-step, 190–192
with variables on both sides,
196–199
triangle, 211
writing, 170–172
Input, 55, 249
Input-output table, 55–56
Integer exponents, 460–462
and powers of ten, 466–467
in scientific notation, 467–468
using patterns to investigate, 460
Integers, 33
Intercepts, 307–309, 353, 626–628
Interest
compound, 806
simple, 139
Interpreting
graphics, 697
scatter plots and trend lines, 274, 360
Interquartile range (IQR), 718
Intersection, 205
Inverse operations, 77, 84, 92, 100,
107, 176
Inverse Property of Addition, 15
Inverse Property of Multiplication,
21
Inverses
additive, 15
multiplicative, 21
squaring and square roots, 32
Inverse variation, 871–874, 896
IQR (interquartile range), 718
Irrational numbers, 34
Ishtar Gate, 526
Isolating variables, 77
IN8
Index
K
Kangaroos, 646
Key words, 234
King, Martin Luther, Jr., 97
Kites, 200
Know-It Note
Know-It Notes are found throughout
this book. Some examples: 15, 20,
21, 40, 46
Koopa (turtle), 142
L
Ladder diagram, 544
Landscaping, 401
Landscaping Link, 401
Law Enforcement, 825
LCD (least common denominator),
907, 920–921
LCM (least common multiple),
906–907
Leading coefficients of
polynomials, 497
Leaning Tower of Pisa, 630
Least common denominator (LCD),
907, 920–921
Least common multiple (LCM),
906–907
Leonardo da Vinci, 624
Light, speed of, 494
Light-year, 469
Like denominators, 905, 906
Likely, equally, 744
Like radicals, 835
Like terms, 47
Lincoln, Abraham, 97
Line(s)
coincident, 421
horizontal, 363, 879
median-fit, 275
parallel, 361–364
perpendicular, 361–364
slope of a, 315–317
trend
finding equations for, 360
interpreting, 274, 360
scatter plots and, 266–269, 360
vertical, 361, 879
Linear equations
definition of, 302
point-slope form of, 351–355
slope-intercept form of, 344–347, 350
solving, by using a spreadsheet, 396
standard form of, 302
systems of, 420–423
Linear functions
definition of, 300, AT15
family of, 368, 882
general form of, 815
graphing, 302
identifying, 300–303
reflections of, 371
rotations of, 370
transformations of, 369–372
vertical translations of, 369
Linear inequalities
definition of, 428
graphing, 429
solutions of, 428
solving, 428–431
systems of, 435–437, 441
Linear models, 360, 813–816
Linear systems, 420–423
Line graphs, 701–704
Link
Animals, 186
Art, 565
Astronomy, 10, 340
Automobiles, 713
Biology, 105, 311, 464, 646, 891
Chemistry, 209, 470
Diving, 24
Ecology, 271
Electricity, 844
Engineering, 624
Geology, 81, 424, 826
Health, 180
History, 97, 731, 765
Hobbies, 374
Landscaping, 401
Math History, 58, 416, 526, 590, 834
Meteorology, 851
Military, 439
Music, 548
Number Theory, 280
Recreation, 200
School, 755
Science, 356
Solar Energy, 918
Sports, 44, 125, 238, 518
Statistics, 88
Technology, 142, 801
Transportation, 254, 500
Travel, 319, 630, 910
Winter Sports, 876
Lists of ordered pairs
identifying exponential functions by,
797, 814
identifying linear functions by, 301,
814
identifying quadratic functions by, 610,
814
Literal equations, 108
Logic, 731, AT21
Long division, polynomial 914–915
Lookout Mountain Incline Railway,
319
Lower quartile, see First quartile
Luminosity, 492
M
Magnification, 926
Make a Conjecture, 322, 323, 632,
648, 662, 828, 870, 893, AT18
Make a model, PS3
Make an organized list, PS11
Make a table, PS7
Manufacturing, 115, 124, 461
Mapping diagrams, 240
Maps, 122
Marine Biology, 616
Markup, 145
Mars lander, 340
Math History, 255, 416, 526, 590, 834
Math History Link, 58, 416, 526, 590,
834
Math symbols, 233
Matrices, 770–773
Maximum values
of absolute-value functions, 380
of parabolas, 612
of power functions, AT3
Mean, 716, 717
Measurement
applications, 128, 305, 355, 470, 582,
794
indirect, 128
Measures of central tendency,
716–719
Mechanics, 875, 877
Median, 716, 717
Median-fit line, 275
Medicine, 463, 501
Mental Math, 46, 47, 140, 161, 585
Meteorology, 10, 17, 110, 208, 209,
356, 851
Meteorology Link, 851
Middleton Place Gardens, 401
Midpoint formula, 330–331
Military, 439
Military Link, 439
Minimum values
of absolute-value functions, 380
of parabolas, 612
of power functions, AT3
Misleading graphs and statistics,
726–727
Mode, 716, 717
Model(s)
choosing, 813–814
exponential, 813–816
geometric
of powers, 26
of special products, 521, 523
linear, 360, 813–816
quadratic, 813–816
rectangle, for multiplying polynomials,
514
Modeling
addition and subtraction of real
numbers, 14
completing the square, 662
equations with variables on both
sides, 99
factoring, 550, 558–559
growth and decay, 804
inverse variation, 870
one-step equations, 76
polynomial addition and subtraction,
502–503
polynomial division, 912
polynomial multiplication, 510–511
systems of linear equations, 403
variable relationships, 248
Money, 408
Monomials, 496
Moore’s law, 801
Multiple Choice, 70–71, 160–161,
162–163, 292–293, 390–391, 451,
452–453, 536, 537, 538–539, 601,
602–603, 692–693, 781, 782–783,
862–863, 864–865, 936–937,
938–939
Choose Combinations of Answers,
936–937
Eliminate Answer Choices, 160–161
None of the Above or All of the Above,
862–863
Recognize Distracters, 388–389
Multiple representations, 15, 21, 26,
27, 46, 47, 76, 79, 86, 99, 101, 171,
176, 182, 183, 198, 204, 205, 206,
240, 248, 263, 299, 324, 344, 361,
363, 403, 460, 466, 474, 476, 477,
481, 483, 484, 502, 503, 510, 511,
521, 523, 550, 558, 559, 561, 578,
580, 612, 620, 621, 633, 648, 649,
650, 656, 662, 663, 762, 763, 822,
830, 846, 871, 879, 913
Multiplication
of polynomials, 512–516
by powers of ten, 467
properties of, 46
of radical expressions, 840–842
of rational expressions, 898–901
of real numbers, 20–22
scalar, of matrices, 771
of signed numbers, 20
solving equations by, 84–87
solving inequalities by, 182–184
of square-root expressions containing
two terms, 841
by zero, 21–22
Multiplication properties of
exponents, 474–477
Multiplication Property of
Equality, 86
Multiplication Property of
Inequality, 182, 183
Multiplicative identity, 21
Multiplicative inverse, 21
Multi-Step, 11, 57, 59, 115, 132, 137,
143, 148, 149, 216, 243, 244, 279,
334, 357, 400, 409, 415, 424, 432,
439, 465, 480, 494, 518, 526, 548,
549, 583, 617, 629, 637, 654, 660,
667, 668, 676, 794, 801, 811, 817,
826, 827, 833, 839, 852, 876, 883,
910, 918, 924
Multi-step equations, solving, 92–95
Multi-step inequalities, solving,
190–192
Multi-Step Test Prep, 38, 60, 118, 152,
188, 218, 264, 282, 342, 376, 426,
442, 494, 528, 576, 592, 640, 678,
734, 768, 820, 854, 896, 926
Multi-Step Test Prep questions are also
found in every exercise set. Some
examples: 10, 18, 24, 30, 36
Music, 208, 216, 218, 516, 548, 873
Music Link, 548
Mutually exclusive events, 758
N
Natural numbers, 33
Navigation, 929
Negative correlation, 267
Negative exponents, 460
Negative Power of a Quotient
Property, 484
Negative slope, 316
Nets, 894
Nightingale, Florence, 731
No correlation, 267
None of the Above, 862–863
Nonlinear equations, graphing to
solve, AT9
Nonlinear functions, AT15–AT16
Notation, scientific, 467–468
Null set, 102
Index
IN9
Numbers
compatible, 46
irrational, 34
natural, 33
prime, 544
random, 743
rational, 33
real, see Real numbers
signed, 20
whole, 33
Number Sense, 548, 549
Number Theory, 280, 654, 660
Connecting Algebra to, 418–419, 585
Number Theory Link, 280
Numerical expressions, 6
Nutrition, 87, 88, 136, 148, 215, 244
O
Oceanography, 121, 260
Ocelots, 271
Odds, 746
Okeechobee, Lake, 332
Online Resources
Career Resources Online, 106, 202,
402, 567, 767
Chapter Project Online, 2, 72, 166,
230, 296, 392, 456, 540, 606, 694,
786, 866
Homework Help Online
Homework Help Online is available
for every lesson. Refer to the
go.hrw.com box at the beginning
of each exercise set. Some
examples: 9, 17, 23, 29, 35
Lab Resources Online, 12, 76, 99, 150,
263, 274, 396, 403, 502, 575, 632,
648
Parent Resources Online
Parent Resources Online are
available for every lesson. Refer
to the go.hrw.com box at the
beginning of each exercise set.
Some examples: 9, 17, 23, 29, 35
State Test Practice Online, 70, 162,
228, 292, 390, 452, 538, 602, 692,
782, 864, 938
Operations
inverse, 77, 84, 92, 100, 107, 176
order of, 40–42
Opposite binomials, 554, 888
factoring with, 554
Opposite coefficients, 411
Opposites, 15
Orangutans, 186
Ordered pairs
definition of, 54
identifying exponential functions by
using, 610, 814
IN10
Index
identifying linear functions by using,
301, 814
identifying quadratic functions by
using, 797, 814
showing relations by, 240
Order of operations, 40–42
Organizing data, 700–704
Origin, 54
Outcome, 737
Outliers, 716–717, 725
Output, 55, 249
P
Paella, 336
Parabola(s)
axis of symmetry of a, 620–621
definition of, 611
exploring, 618
identifying the direction of a, 612
vertex of a, 612, 621
vertical translations of a, 635
width of a, 633
Parallel lines, slopes of, 361–364
Parent functions
definition of, 369
linear, 369, 882
quadratic, 632, 633, 882
rational, 882
square-root, 882
Parentheses, 40
Parent Resources Online
Parent Resources Online are available
for every lesson. Refer to the
go.hrw.com box at the beginning of
each exercise set. Some examples:
9, 17, 23, 29, 35
Pascal, Blaise, 590
Pascal’s Triangle, 590
Patterns
in choosing a model, 814
in finding properties of exponents,
472–473
identifying, AT10–AT11
in investigating integer exponents, 460
in investigating powers of ten, 466
recursive, AT10–AT11
Pearl, Nancy, 374
Pendulum, period of a, 660
Percent(s), 133–135
applications of, 139–141
Percent change, 144–146
Percent proportion, 133
Perfect squares, 32, 33
Perfect-square trinomials, 521, 578
Perimeter, 52–53
Period, of a pendulum, 844
Permutations, 760–763
Perpendicular lines, slopes of,
361–364
Personal Finance, 311, 551, 812
Pet Care, 819, AT8
pH, 486
Photography, 507, 517
Physical Science, 306, 479, 793, AT16
Physics, 498, 555, 573, 589, 636, 638,
654, 660, 673, 800, 844, 851, 874
Piecewise functions, AT5–AT7
Pimlico Race Course, 238
Plane(s)
Cartesian, 58
coordinate, 54
Point-slope form of linear
equations, 351–355
Polygons, 52–53
Polynomial(s), see also Factoring
polynomials
addition of, 504–506
cubic, 497, 532
degrees of, 496–497
division of, 913–917
leading coefficients of, 497
long division, 914–915
multiplication of, 512–516
quadratic, 497
standard form of, 497, 916
subtraction of, 502–506
unfactorable, 587
Population, 126
Population density, 486
Positive correlation, 267
Positive Power of a Quotient
Property, 483
Positive slope, 316
Power(s)
definition of, 26
exponents and, 26–28
geometric models of, 26
Negative, of a Quotient Property, 484
Positive, of a Quotient Property, 483
of a Power Property, 476
of a Product Property, 477
of ten, 466–467
Power functions, AT2–AT4
Power of a Power Property, 476
Power of a Product Property, 477
Power Property of Equality, 846
Powers of ten, 466–467
Powers Property
Power of, 476
Product of, 474–475
Quotient of, 481, 898
Prediction, 739
Preparing for your final exam, 869
Prime factorization, 544
Prime numbers, 544
Principal, 139
Principal square root, 32
Prisms, 520, 894
Probability
applications, 901, 902
converting between odds and, 746
definition of, 737
of dependent events, 753
experimental, 737–739
geometric, 910
of inclusive events, 758
of independent events, 751
of mutually exclusive events, 758
theoretical, 744–747
Problem-Solving Applications, 28,
94, 177–178, 259, 354, 399, 523–524,
579–580, 627–628, 665–666, 753,
815–816, 921–922
Problem-Solving Handbook,
PS2–PS11
Draw a diagram, PS2
Find a pattern, PS6
Guess and test, PS4
Make a model, PS3
Make an organized list, PS11
Make a table, PS7
Solve a simpler problem, PS8
Use logical reasoning, PS9
Use a Venn diagram, PS10
Work backward, PS5
Product of Powers Property,
474–475
Product Property
Power of a, 476
of Square Roots, 830
Zero, 886
Product Rule for Inverse Variation,
873
Properties
of addition, 46
of equality, 79, 86
of inequality, 176, 182, 183
of multiplication, 46
of zero, 21
Proportions
applications of, 127–129
cross products in, 121
definition of, 120
percent, 133
rates, ratios and, 120–123
Pyramids, 520, 894
Pythagorean Theorem, 331, 661, 831
Q
Qin Jiushao, 416
Quadrants, 54
Quadratic equations
definition of, 642
discriminant of, 672
related function of, 642
roots of, 648–649
solving
by completing the square, 664, 674
by factoring, 650–653, 674
by graphing, 642–644, 674
by using square roots, 656–659, 674
by using the Quadratic Formula,
670–675
standard form of, 642
Quadratic Formula
discriminant and, 670–675
in estimating solutions, 671–672
solving quadratic equations by,
670–675
Quadratic functions
characteristics of, 619–623
comparing graphs of, 635
definition of, 610
domain of, 613
in families of functions, 632, 882
finding zeros of, 619
general form of, 815
graphing, 626–629, AT9
using a table of values, 611
identifying, 610–613
range of, 613
transformations of, 633–636
Quadratic models, 813–816
Quadratic parent functions, 632,
633, 882
Quadratic polynomials, 497
Quality Control, 739
Quotient of Powers Property,
481, 898
Quotient Property
Negative Power of a, 484
Positive Power of a, 483
of Square Roots, 830
R
Radical equations, 846–850
Radical expressions, 829–832
addition of, 835–837
definition of, 829
division of, 840–842
multiplication of, 840–842
Product Property of, 830–831
Quotient Property of, 830–831
square-root expressions, 829–831, 841
subtraction of, 835–837
Radical functions, graphing, 828
Radicals, like, 835
Radical symbol, 32, 488, 490
Radicand, 829
Rainbows, 494
Random numbers, 743
Random samples, 727, 732
Range, 82, 240, 241, 242, 243, 244, 245,
246, 252, 253, 254, 255, 260, 264,
265, 877
of linear functions, 303
of quadratic functions, 613
reasonable, 252, 253, 254, 255, 259,
265, 287, 288, 303, 308, 372, 617,
873, 876, 881, 883
Range (of a data set), 716, 717
Rate of change
constant and variable, AT14–AT15
decrease, 803
definition of, 314, AT14
identifying linear and nonlinear
functions from, AT15–AT16
increase, 803
slope and, 314–317
Rates, 120–123
Ratio(s)
equivalent, 120
rates, proportions and, 120–123
surface-area-to-volume, 889
trigonometric, 928–929
Rational equations, 920–923, 926
Rational exponents, 488–490
Rational expressions
addition of, 905–908
definition of, 886
division of, 898–901
multiplication of, 898–901
simplifying, 886–889
subtraction of, 905–908
Rational functions, 878–882
definition of, 878
excluded values in, 878
family of, 882
graphing, 880–881, 893
identifying asymptotes, 878–880
Rationalizing denominators,
841, 845
Rational numbers, 33
Reading
graphics, 697
the problem, 459
Reading and Writing Math, 5, 75,
169, 233, 299, 395, 459, 543, 609,
697, 789, 869, see also Reading
Strategies; Study Strategies; Writing
Strategies
Reading Math, 34, 54, 120, 122, 127,
172, 251, 276, 324, 422, 460, 467,
468, 581, 656, 702, 718, 745, 746,
806, 874
Reading Strategies
Read a Lesson for Understanding, 543
Read and Interpret Graphics, 697
Read and Interpret Math Symbols, 233
Read and Understand the Problem,
459
Use Your Book for Success, 5
Index
IN11
Ready to Go On?, 39, 61, 119, 153,
189, 219, 265, 283, 343, 377, 427,
443, 495, 529, 577, 593, 641, 679,
735, 769, 821, 855, 897, 927, see also
Assessment
Real Estate, 131
Real numbers
addition of, 14–17
classification of, 33
definition of, 14
Density Property of, 37
division of, 20–22
integers, 33, 34
irrational, 34
multiplication of, 20–22
natural numbers, 33, 34
rational, 33, 34
square roots and, 32–35
subtraction of, 14–17
whole numbers, 33, 34
Real-World Connections, 164–165,
294–295, 454–455, 604–605,
784–785, 940–941
Reasonable answer, 79, 80, 81, 82,
88, 89, 95, 97, 104, 121, 123, 124,
126, 128, 130, 134, 137, 142, 144,
145, 148, 149, 161, 172, 178, 184,
235, 259, 399, 400, 430, 437, 636,
643, 666, 674, 796, 805, 807, 822
Reasonable domain, 252, 253, 254,
255, 259, 265, 287, 288, 303, 308,
372, 617, 873, 876, 881, 883
Reasonableness, 79, 80, 81, 82, 88, 89,
95, 97, 104, 123, 124, 126, 128, 130,
134, 137, 142, 144, 145, 148, 149,
161, 254, 255, 259, 265, 269, 643, 822
Reasonable range, 252, 253, 254, 255,
259, 265, 287, 288, 303, 308, 372,
617, 873, 876, 881, 883
Reasoning
deductive, AT19
explaining your, in extended responses,
690–691
inductive, AT18–AT19
spatial, 780–781
Reciprocals, 21
Recognizing distracters, 388–389
Recreation, 22, 87, 116, 200, 238, 244,
280, 321, 333, 334–335, 408, 439,
832, 908
Recreation Link, 200
Rectangle model for multiplying
polynomials, 514
Rectangles, area of, 83
Recursive formula, AT13
Recursive patterns, AT10–AT11
Recycling, 8
Reflections, 371
Relations, functions and, 240–242
Relationships
graphing, 234–236
variable, model, 248
IN12
Index
Remember!, 42, 108, 113, 190, 205,
214, 302, 303, 362, 369, 412, 435,
460, 475, 488, 496, 497, 504, 512,
524, 544, 560, 569, 627, 633, 658,
665, 670, 682, 700, 761, 814, 836,
841, 871, 873, 881, 886, 889, 898,
899, 901, 907, 916
Remembering formulas, 789
Remodeling, 565
Remote interior angles, AT21
Repeating decimals, 33
Replacement set, 8
Representations
multiple, 15, 21, 26, 27, 46, 47, 76,
79, 86, 99, 101, 171, 176, 182, 183,
198, 204, 205, 206, 240 248, 263,
299, 324, 344, 361, 363, 403, 460,
466, 474, 476, 477, 481, 483, 484,
502, 503, 510, 511, 521, 523, 550,
558, 559, 561, 578, 580, 612, 620,
621, 633, 648, 649, 650, 656, 662,
663, 762, 763, 822, 830, 846, 871,
879, 913
of solid figures, 894
Rise, 315
Root(s), see also Radical expressions
cube, 32
exploring, 648–649
of fractions, 33
principal square, 32
of quadratic equations, 648–649
square, 32
symbol for, 32
Rotations, 370
Run, 315
S
Safety, 214
Sales tax, 140
Samples, random, 727, 732
Sample space, 737
Sandia Peak Tramway, 308
Scalar, 771
Scale, 122
Scale drawing, 122
Scale factor, 129
Scale model, 122
Scatter plots
interpreting, 274
trend lines and, 266–269, 360
School, 81, 199, 374, 400, 433, 755
School Link, 755
Science, 124, 356, 478, 552, 808
Science Link, 356
Scientific notation, 467–468
Seabiscuit, 238
Sea horses, 121
Second coordinates, 240
Second differences, 610
Sectors, 702
Selected Answers, SA2–SA30
Sequences
arithmetic, 276–278
definition of, 276
geometric, 790–792
recursive, AT10–AT11
Set-builder notation, 170
Sheppard, Alfred, 124
Shipping, 279
Short Response, 25, 71, 82, 90, 143,
163, 187, 202, 226–227, 229, 239,
273, 293, 335, 350, 391, 417, 440,
451, 453, 465, 501, 536, 537, 539,
566, 591, 600, 601, 603, 617, 631,
669, 693, 708, 723, 742, 766, 781,
783, 811, 865, 885, 904, 939
Understand Short Response Scores,
226–227
Signed numbers, multiplication and
division of, 20
Silicon chips, 801
Similar figures, 127
Simple events, 761
Simple interest, 139
Simple random sample, 732
Simplest form of a square-root
expression, 829
Simplifying
exponential expressions, 474
expressions, 46–49
rational expressions, 886–889
Simulations, 736
Sine, 928
Slope(s)
comparing, 317
defined, 315
finding, 325–326
formula, 324–327
negative, 316
of parallel lines, 361–364
of perpendicular lines, 361–364
positive, 316
rate of change and, 314–317
of trend lines, 360
undefined, 316
zero, 316
Slope formula, 324–327
Slope-intercept form of linear
equations, 344–347, 350
Snowshoes, 876
Solar cars, 713
Solar Energy, 918
Solar Energy Link, 918
Solar-powered aircraft, 918
Solid figures, representing, 894
Solutions
of absolute-value equations, 112–114
of absolute-value inequalities,
212–215
approximating, 91
of equations, 77
estimating using the Quadratic
Formula, 671–672
extraneous, 848–849, 922
of inequalities, 170
of linear equations by using a
spreadsheet, 396
of linear inequalities, 428
of rational equations, 920–921
of systems of linear equations, 397
of systems of linear inequalities, 435,
441
Solution set, 77, 102
Solve a simpler problem, PS8
Space Shuttle, 115
Spatial reasoning, 780–781
Special products
of binomials, 521–525
factoring, 578–581
geometric models of, 521, 523
Special systems, solving, 420–423
Speed of light, 494
Speed squares, 534
Spheres
surface area, 493
volume, 493, AT4
Sports, 42, 44, 50, 104, 107, 110, 121,
125, 172, 178, 180, 209, 238, 310,
335, 348, 381, 424, 485, 518, 529,
616, 645, 652, 654, 676, 712, 718,
721, 740, 741, 792, 793, 800, 831
Sports Link, 44, 125, 238, 518
Spreadsheet, 13, 150–151, 396
Square(s)
difference of two, 523, 580
perfect, 32
Square, area of a, 83
Square, completing the, 663–666,
674
Square root(s)
definition of, 32
estimating, 33
of fractions, 33
irrational, 34
principal, 32
Product Property of, 830
Quotient Property of, 830
real numbers and, 32–35
solving quadratic equations by using,
656–659, 674
Square-root expressions, see also
Radical expressions
multiplication of, 841
simplest form of, 829
Square-root functions, 822–824
domain of, 823
family of, 882
graphs of translations of, 823
Square-Root Property, 656
Standard form
of linear equations, 302
of polynomials, 497, 916
of quadratic equations, 642
Standardized Test Prep, 70–71,
162–163, 228–229, 292–293,
390–391, 452–453, 538–539,
602–603, 692–693, 782–783,
864–865, 938–939, see also
Assessment
Statistics
applications, 82, 88, 469, 799
measures of central tendency,
716–719
misleading, 726–727
Statistics Link, 88
Stem-and-leaf plot, 709, 715
Step functions, AT6
Stonehenge II, 124
Stratified random sample, 732
Student to Student, 47, 76, 171, 242,
308, 406, 476, 571, 674, 751, 816, 888
Study Guide: Preview, 4, 74, 168, 232,
298, 394, 458, 542, 608, 696, 788,
868, see also Assessment
Study Guide: Review, 62–65,
154–157, 220–223, 284–287,
382–385, 444–447, 530–533,
594–597, 684–687, 774–777,
856–859, 930–933, see also
Assessment
Study Strategies
Learn Vocabulary, 609
Prepare for Your Final Exam, 869
Remember Formulas, 789
Use Multiple Representations, 299
Use Your Notes Effectively, 169
Use Your Own Words, 75
Substitution, solving systems of
linear equations by, 404–407
Subtraction
of matrices, 771
of polynomials, 502–503, 504–506
of radical expressions, 835–837
of rational expressions, 905–908
of real numbers, 14–17
solving equations by, 77–79
solving inequalities by, 176–179
Subtraction Property of
Equality, 79, 86
Subtraction Property of
Inequality, 176
Surface area, 493, 520
Surface-area-to-volume ratio, 889
Sydney Harbour Bridge, 114
Symmetric Property of
Equality, 187
Symmetry, axis of
in absolute-value functions, 378
in a parabola, 618, 620
definition of, 618
exploring, 618
finding
by using the formula, 621
by using zeros, 620
System of linear inequalities,
435–437
System(s) of linear equations
classification of, 421–422
identifying solutions of, 397
with infinitely many solutions, 421
modeling, 403
with no solution, 420–421
solving
by elimination, 411–415
by graphing, 397–399
by substitution, 404–407
Systematic random sample, 732
T
Tables
connecting to function rules and
graphs, 263
evaluate expressions using, 12–13
finding slope from, 325
frequency, 710
identifying linear functions by, 301
of values in graphing quadratic
functions, 611
Tangent, 928
Technology, 29, 142, 471, 534, 546,
766, 801
Technology Lab
Connect Function Rules, Tables, and
Graphs, 263
Create a Table to Evaluate Expressions,
12–13
Explore Roots, Zeros, and x-intercepts,
648–649
Families of Linear Functions, 368
Families of Quadratic Functions, 632
Graph Linear Functions, 359
Graph Radical Functions, 828
Graph Rational Functions, 893
Graphing to Solve Equations, AT9
Interpret Scatter Plots and Trend Lines,
274
Solve Equations by Graphing, 91
Solve Linear Equations by Using a
Spreadsheet, 396
Solve Systems of Linear Inequalities,
441
Use a Graph to Factor Polynomials,
575
Use Random Numbers, 743
Use Technology to Make Graphs,
724–725
Index
IN13
Technology Link, 142, 801
Telephone numbers, 765
Temperature, 116, 215, 217
Ten, powers of, 466–467
Terminating decimals, 33
Terms, 47, 276, 496, 841
Test Prep
Test Prep questions are found in every
exercise set. Some examples: 11, 19,
25, 31, 37; see also Assessment
Tests, see Assessment
Test Tackler, see also Assessment
Any Question Type
Read the Problem for
Understanding, 450–451
Spatial Reasoning, 780–781
Translate Words to Math, 600–601
Use a Diagram, 536–537
Extended Response
Explain Your Reasoning, 690–691
Understand the Scores, 290–291
Gridded Response
Fill in Answer Grids Correctly, 68–69
Multiple Choice
Choose Combinations of Answers,
936–937
Eliminate Answer Choices, 160–161
None of the Above or All of the
Above, 862–863
Recognize Distracters, 388–389
Short Response
Understand Short Response Scores,
226–227
Theoretical probability, 744–747
Third quartile, 718
Tip (amount of money), 140
Tolkowsky, Marcel, 881
Transcontinental railroad, 910
Transformations
of absolute-value functions, 379–381
of linear functions, 369–372
of quadratic functions, 633–636
Transitive Property of Equality, 187
Translations, 369, 379
Transportation, 96, 124, 179, 209, 254,
261, 271, 304, 305, 341, 500, 851
Transportation Link, 254, 500
Trapezoid, area of, 335, 654
Travel, 18, 36, 90, 104, 185, 278, 279,
308, 319, 334, 630, 875, 909, 910,
924, AT8
Travel Link, 319, 630, 910
Tree diagram, 760
Trend lines
finding equations for, 360
interpreting, 274, 360
scatter plots and, 266–269, 360
Trial, 737
Trial-and-error, AT10
IN14
Index
Triangle(s)
area of, 83
Pythagorean Theorem, 331, 661, 831
Triangle Inequality, 211
Triathlon, 46
Trigonometric ratios, 928–929
Trinomials, see also Factoring
polynomials, Polynomial(s)
definition of, 497
difference of two squares, 580–581,
585
perfect-square, 521, 578
Truth tables, 203
Tsunamis, 826
Turning points, 680
Two-step equations, solving, 92–95
Two-step inequalities, solving,
190–192
U
Undefined slope, 316
Understanding
the problem, 459
read a lesson for, 543
read the problem for, 450–451
Unfactorable polynomials, 587
Union, 206
Unit conversions, 121
Unit rate, 120
Unlike denominators, 907
Upper quartile, see Third quartile
Use logical reasoning, PS9
Use a Venn diagram, PS10
V
Van Dyk, Ernst, 107
Variable(s), 6
on both sides
modeling equations with, 99
solving equations with, 100–103
solving inequalities with, 196–199
dependent, 250–254
expressions and, 6–8
independent, 250–254
solving for, 107–109
Variable relationships, model, 248
Variation
constant of, 336, 871
direct, 336–339
inverse, 871–874, 896
Vertex
of absolute-value functions, 378
of a parabola
axis of symmetry through, 618
finding the, 621
Vertical line(s), 879
Vertical-line test, 247
Vertical method for multiplication
of polynomials, 515
Vertical translations
of linear functions, 369
of parabolas, 635
Vocabulary, 9, 17, 23, 29, 35, 43, 49,
57, 80, 103, 109, 123, 130, 136, 141,
147, 173, 208, 237, 243, 253, 270,
279, 304, 310, 318, 333, 339, 365,
373, 400, 423, 432, 438, 469, 491,
499, 525, 547, 614, 623, 645, 667,
675, 705, 712, 720, 728, 740, 747,
754, 764, 793, 800, 809, 825, 832,
837, 850, 875, 883, 890, 923
Vocabulary Connections, 4, 74, 168,
232, 298, 394, 458, 542, 608, 696,
788, 868
Volume
of a cone, 520
of a cylinder, 520, 661
of a prism, 520
of a pyramid, 520
of a sphere, 493, AT4
Voluntary response sample, 733
Vomit Comet, 822
W
Wadlow, Robert P., 88
Wages, 305, 339
War Admiral, 238
Waterfalls, 645
Weather, 23, 712, 713, 721
What if...?, 16, 18, 22, 28, 30, 55, 79,
86, 90, 98, 104, 137, 178, 272, 280,
303, 341, 354, 372, 374, 402, 414,
424, 430, 498, 552, 580, 644, 652,
673, 723, 901, 908
Whole numbers, 33
Width of a parabola, 633
Wildlife refuge, 271
Wind turbines, 844
Winter Sports, 876
Winter Sports Link, 876
Work backward, PS5
Write About It
Write About It questions are found in
every exercise set. Some examples:
9, 19, 24, 29, 31
Writing Math, 6, 32, 33, 55, 77, 102,
112, 170, 241, 258, 344, 407, 466,
482, 488, 505, 551, 709, 790, 879,
AT19
Writing Strategies, Write
a Convincing Argument/
Explanation, 395
X
x-axis, 54
x-coordinate, 54
x-intercept, 307, 648–649
x-values, 240
Y
y-axis, 54
y-coordinate, 54
y-intercept, 307, 344–347
Yosemite Falls, 645
y-values, 240
Z
Zero(s)
division by, 21–22
exploring, 648–649
finding the axis of symmetry of a
parabola by using, 620
of a function, 619
multiplication by, 21–22
properties of, 21
of quadratic functions, 619
Zero exponent, 460
Zero Product Property, 650, 886
Zero slope, 316
Index
IN15
Credits
Abbreviations used: (t) top, (c) center, (b) bottom, (l) left,
(r) right, (bkgd) background
Photo
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Chapter Three: Page 164–165 (all), Sam Dudgeon/HRW; 168 (tr), © Charles Crust;
169 (bl), Digital Vision/gettyimages; 172 (bl), © Creatas; 174 (tr), © Mingasson/
Getty/HRW; 174 (cellphone), ©Mingasson/Getty/HRW; 175 (bc, br), © Mingasson/
Getty/HRW; 178 (tl), Buzz Orr/The Gazette/AP/Wide World Photos; 178 (cr),
PhotoDisc/gettyimages; 178 (bl), © Creatas; 180 (tr), on-page credit; 182 (cl), Sam
Dudgeon/HRW; 184 (cl), © Reuters/CORBIS; 184 (bl), © Creatas; 186 (br), Scott
Vallance/VIP Photographic/HRW; 187 (tl), © Creatas; 188 (tr), © Peter Beck/CORBIS;
190 (tr), Paul Sakuma/AP/Wide World Photos; 192 (bl), © Brand X Pictures; 194 (tr),
© Ariel Skelley/CORBIS; 198 (cl), Brad Mitchell/Alamy; 198 (bl), © Brand X Pictures;
200 (bl), © Jose Luis Pelaez, Inc./CORBIS; 202 (tr), © Michele Westmorland/CORBIS;
CR2
Credits
202 (br), Sam Dudgeon/HRW; 207 (tl), © Brand X Pictures; 207 (cl), Richard Megna/
Fundamental Photographs; 210 (tl), © Brand X Pictures; 210 (br), Peter Beavis/Taxi/
Getty Images; 210 (cr, bl), PhotoDisc/Getty Images.
Chapter Four: Page 226–227 (all), © David McGlynn/Taxi/Getty Images; 230 (tr),
© Royalty-Free/CORBIS; 234 (cl), © Bettmann/CORBIS; 234 (bl), Sam Dudgeon/HRW;
235 (bc, bl), Sam Dudgeon/HRW; 236 (tr), Aflo Foto Agency; 238 (cl), © Comstock
Images/Alamy Photos; 241 (tl), Sam Dudgeon/HRW; 245 (tr), RubberBall/Alamy;
250 (tl), © Bettmann/CORBIS; 250 (bl), Sam Dudgeon/HRW; 257 (bl), Sam Dudgeon/
HRW; 260 (br), Big Cheese Photo/Alamy Photos; 260 (tl), Sam Dudgeon/HRW;
262 (tr), David Welling/Animals Animals; 267 (bl), Roy Toft; 268 (bl), comstock.com;
272 (tr), © COMSTOCK, Inc.; 276 (tl), Digital Vision/gettyimages; 276 (bl),
comstock.com; 278 (tl), comstock.com; 278 (bl), AP Photo/Daniel Hulshizer; 290 (tr),
© Dennis Cox/WorldViews, All Rights Reserved.; 291 (br), Ed Reschke/Peter Arnold,
Inc./Alamy; 291 (bl), © Randy M. Ury/CORBIS; 291 (tc), Sam Dudgeon/HRW.
Chapter Five: Page 292–293 (all), © Dennis Hallinan/Alamy Photos; 296 (tr),
© Aldo Torelli/Getty Images; 301 (bl), Victoria Smith/HRW; 304 (tl), © LWADann Tardif/CORBIS; 304 (cr), © Buddy Mays/CORBIS; 307 (tl), © John and Lisa
Merrill/CORBIS; 308 (tl), Victoria Smith/HRW; 310 (c), © elaineitalia/Alamy; (b), ©
imageZebra/Alamy; 315 (bl), © Dave G. Houser/CORBIS; 316 (tl), Victoria Smith/HRW;
320 (tr), © Patrick Eden/Alamy Photos; 325 (tl), © Patrick Eden/Alamy Photos; 326
(tr), © Royalty-Free/Corbis; 328 (bl), Genevieve Vallee/Alamy; 330 (cl), Courtesy
NASA/JPL-Caltech; 330 (bl), Victoria Smith/HRW; 332 (tl), Victoria Smith/HRW; 332
(br), © Rick Gomez/CORBIS; 332 (tr), © age fotostock/SuperStock; 339 (bl), © Brand
X Pictures; 341 (tr), © Pixtal/SuperStock; 346 (tl), Jake Norton; 346 (bl), © Brand X
Pictures; 347 (bl), image100/Alamy; 349 (tr), © Owen Franken/CORBIS; 354 (bl),
© Brand X Pictures; 357 (tr), © Jim Cummins/CORBIS; 360 (c), Victoria Smith/HRW;
362 (bl), © Brand X Pictures; 362 (cl), AP Photo/Ted S. Warren; 364 (tl), © Brand X
Pictures; 364 (b), © Tim Pannell/CORBIS; 364 (tr), Andy Christiansen/HRW.
Chapter Six: Page 378–379 (all), Glenn James/NBAE/Getty Images; 383 (tr),
© Kwame Zikomo/SuperStock; 386 (bl), Victoria Smith/HRW; 387 (tl), © Lee Snider/
Photo Images/CORBIS; 388 (bl), © R. Holz/CORBIS; 390 (tr), Cartoon copyrighted by
Mark Parisi, printed with permission.; 392 (bl), © Michael Pole/SuperStock; 395 (bl),
Victoria Smith/HRW; 397 (tr), Photofusion Picture Library/Alamy; 400 (tl), Victoria
Smith/HRW; 402 (cl), © UNICOVER CORPORATION 1986; 402 (bl), Victoria Smith/
HRW; 410 (cl), David R. Frazier Photolibrary, Inc./Alamy; 410 (bl), Victoria Smith/HRW;
412 (tl), Victoria Smith/HRW; 412 (cr), David Madison © 2005; 412 (b), © Les Stone/
CORBIS; 414 (tr), © Susan Werner/Getty Images; 419 (tl), Max Gibbs/photolibrary.com;
419 (bl), Nathan Kaey/HRW; 419 (cl), PhotoDisc Blue/Getty Images; 421 (tr),
© Justin Pumfrey/Taxi/Getty Images; 425 (cl), © Reuters/CORBIS; 425 (bl), Nathan
Kaey/HRW; 428 (all), Nathan Kaey/HRW; 440 (bl), © Bob Krist/CORBIS; 440 (tr),
© Tom Till; 441 (b), Hope Ryden/National Geographic Image Collection; 441 (tr),
Image Source/Alamy.
Chapter Seven: Page 442–423 (all), © Charles Gupton/CORBIS; 446 (tr),
Advertising Archive; 449 (cr), B. G. Thomson/Photo Researchers, Inc.; 450 (cl),
© PHOTOTAKE Inc./Alamy; 452 (tr), Arscimed/Photo Researchers, Inc.; 454 (tr),
Courtesy NASA/JPL-Caltech; 455 (br), © Mediscan/Corbis; 456 (tl), Cern/Photo
Researchers, Inc.; 460 (tr), D. Parker/Photo Researchers, Inc.; 461 (bl),
© Brand X Pictures; 462 (bl), © BananaStock Ltd.; 474 (br), Robert Flusic/PhotoDisc/
Getty Images; 476 (tr), © Jeff Hunter/Getty Images; 480 (tl), © Don Johnston/Stone/
Getty Images; 480 (bl), © Comstock, Inc.; 484 (tr), www.cartoonstock.com; 487
(frame), ©1998 Image Farm Inc.; 487 (girls), Artville/Getty Images; 488 (bl),
© Comstock, Inc.; 492 (tr), © Pat and Chuck Blackley; 496 (cr), Royalty-Free/Corbis;
498 (cl), PHILIPPE DESMAZES/AFP/Getty Images; 498 (tl), © Comstock, Inc.; 501 (tr),
Juniors Bildarchiv/Alamy; 506 (tr), Molly Eckler–puzzle, photo by Victoria Smith/HRW;
506 (bl), © Ruggero Vanni/CORBIS; 507 (tl), © Comstock, Inc.; 508 (tl), © Comstock,
Inc.; 508 (tr), SuperStock; 508 (b), Ryan McVay/PhotoDisc/Getty Images.
Chapter Eight: Page 520–521 (all), © Hugh Sitton/Getty Images; 523 Sam
Dudgeon/HRW; 527 (tr), Sam Dudgeon/HRW; 527 (b), Brian Hagiwara/FoodPix; 528
(tr), Photo by Jolesch Photography. © 2004 Drum Corps International. All rights
reserved.; 528 (tl), Photo by Jolesch Photography. © 2004 Drum Corps International.
All rights reserved.; 529 (t), © Royalty-Free/Corbis; 531 (tr), PhotoDisc/Getty Images;
532 (br, bc), Victoria Smith/HRW; 535 (tr), Tony Freeman/Photoedit; 536 (bl),
© Royalty-Free/Corbis; 537 (cr), Sam Dudgeon/HRW; 540 (tr), Travel-Shots/Alamy;
545 (cl), Composition 17, 1919 (oil on canvas) by Doesburg, Theo van (1883–1931)
© Haags Gemeentemuseum, The Hague, Netherlands/The Bridgeman Art Library
Nationality/copyright status: Dutch/out of copyright; 547 (bl), Janine Wiedel
Photolibrary/Alamy; 548 (tr), AP Photo/Peter Cosgrove; 551 (tl), © Stockbyte;
553 (bl), © Royalty-Free/Corbis; 556 (tl), © Royalty-Free/Corbis; 556 (br), © Peter
Adams/Getty Images; 558 (tr), © Lee Snider/Photo Images/CORBIS; 559 (cr), © Yann
Arthus-Bertrand/CORBIS; 563 (tl), © Stockbyte; 566 (tr), © Robbie Jack/CORBIS;
570 (tl), © Stockbyte; 570 (cl), Sheila Terry/Science Photo Library/Photo Researchers,
Inc.; 572 (tl), © Stockbyte; 572 (br), David Young Wolff/Photoedit; 584 (tr), Eric
Horan; 584 (br), Eric Horan; 584 (bl), PhotoDisc/Getty Images; 585 (tc), Courtesy
Mike Brown/University of South Carolina; 585 (br), Mike Brown/University of South
Carolina, University Publications; 585 (cr), John Sanford/Science Photo Library/Photo
Researchers, Inc.
Van Steen/HRW; 728 (bl, bc), Victoria Smith/HRW; 729 (tr), Victoria Smith/HRW; 730
(cr), Sam Dudgeon/HRW; 731 (bl), © Jose Luis Pelaez, Inc./CORBIS; 732 (tl), Sam
Dudgeon/HRW; 735 (all), Sam Dudgeon/HRW; 736 (all), Sam Dudgeon/HRW; 741 (cr),
Sam Dudgeon/HRW; 741 (cl), © Underwood & Underwood/CORBIS; 742 (tl), Sam
Dudgeon/HRW; 743 (tr), Foodcollection.com/Alamy; 743 (bl), © Comstock Images;
744 (tl), Sam Dudgeon/HRW; 744 (b), © Handout/Hasbro/Ray Stubblebine/Reuters/
CORBIS; 749 (cl), Courtesy of Blue Sky Studios/ZUMA Press; 760 (cr), Blair Seitz;
761 (br), AP Photo/John Heller; 761 (tr, bl), Tom Pawlesh.
Chapter Nine: Page 586–587 (all), © James Randklev/CORBIS; 590 (tr), Aflo Foto
Agency; 596 (bl), Sam Dudgeon/HRW; 599 (tr), Gary Crabbe/Alamy; 604 (b l), ©
Gensaku Izumiya/Sebun Photo/Getty Images; 605 (tl), Sam Dudgeon/HRW; 606
(tr), © Michael S. Yamashita/CORBIS; 610 (tl), AP Photo/Fabio Muzzi; 610 (bl), Sam
Dudgeon/HRW; 613 (tr), © Craig Tuttle/CORBIS; 618 (bl), Sam Dudgeon/HRW; 620
(all), Sam Dudgeon/HRW; 622 (tr), © MedioImages/SuperStock; 624 (tr), © Digital
Vision; 626 (cl), © Theo Allofs/CORBIS; 626 (bl), PhotoDisc/Getty Images; 626 (bc),
© COMSTOCK, Inc.; 630 (tr), Dynamic Graphics Group/IT Stock Free/Alamy; 634 (cr),
imagebroker/Alamy; 634 (bl), PhotoDisc/Getty Images; 634 (bc), © COMSTOCK, Inc.;
636 (t), © Gerald French/CORBIS; 640 (cl), Loren Winters/Visuals Unlimited; 640 (bl),
PhotoDisc/Getty Images; 640 (bc), © COMSTOCK, Inc.; 645 (tr), The Garden Picture
Library/Alamy; 650 (bl), PhotoDisc/Getty Images; 650 (bl), © COMSTOCK, Inc.; 652
(tr), © Powerstock/SuperStock; 656 (bl), Comstock Images/Alamy; 659 (tl), PhotoDisc/
Getty Images; 659 (tc), © COMSTOCK, Inc.; 660 (tl), PhotoDisc/Getty Images; 660
(tc), © COMSTOCK, Inc.; 660 (tr), Stuart Franklin/Getty Images; 660 (b), Robert
Laberge/Getty Images.
Chapter Eleven: Page 762–763 (all), © age fotostock/Superstock; 766 (tr),
Randy Lincks/Masterfile; 777 (tl), Lucidio Studio, Inc./Alamy; 781 (tr), © Wendy
Stone/CORBIS; 786 (cr), Tom Goskar/Wessex Archaeology Ltd.; 789 (tr), © Stan
Liu/Icon SMI/ZUMA Press; 791 (tr), Stephen Dalton/Photo Researchers, Inc.; 792
(cl), age fotostock; 796 (c), Laimute Druskis/Stock Boston/IPN; 796 (bc), © D.
Hurst/Alamy; 796 (br), © Brian Hagiwara/Brand X Pictures/Alamy; 798 (tr), NASA;
802 (tl), AP Photo/Andy Eames; 802 (bl), © Brand X Pictures; 805 (tr), Steve
Gottlieb/PictureQuest; 809 (bl), © Brand X Pictures; 814 (bl), © Brand X Pictures;
816 (tr), ON-PAGE CREDIT; 820 (tl), © Richard Cummins/SuperStock; 820 (bl), ©
Brand X Pictures; 827 (bl), AP Photo/Gregory Smith; 828 (bl), © Brand X Pictures;
830 (cr), Walker/PictureQuest; 830 (tl), © Brand X Pictures; 830 (bc), AP Photo/Anna
Branthwaite.
Chapter Ten: Page 672–673 (all), Frans Lanting/Minden Pictures; 678 (tr), Victoria
Smith/HRW; 680 (all), Sam Dudgeon/HRW; 687 (tr), © Tom Stewart/CORBIS; 691
(cl), © Reuters/CORBIS; 694 (tr), AP Photo/Dave Martin; 696 (tr), Victoria Smith/
HRW; 698 (cr), © Michael Prince/CORBIS; 707 (tl), © Bettmann/CORBIS; 710 (cr),
Stuart Franklin/Getty Images; 710 (bl), Stuart Franklin/Getty Images; 712 (tr), Sam
Dudgeon/HRW; 713 (tl), Sam Dudgeon/HRW; 713 (tc), United States Mint image; 713
(tc), United States Mint image; 713 (tr), Sam Dudgeon/HRW; 713 (c), United States
Mint image; 713 (c), United States Mint image; 713 (bc, br), Sam Dudgeon/HRW;
716 (tl, tc), Sam Dudgeon/HRW; 716 (penny), PhotoDisc/Getty Images; 716 (dime),
PhotoDisc/Getty Images; 716 (nickel), PhotoDisc/Getty Images; 716 (bl), PhotoDisc/
Getty Images; 716 (bc), Sam Dudgeon/HRW; 716 (br), Victoria Smith/HRW; 717
(bl), Sam Dudgeon/HRW; 718 (cr), © European Communities; 720 (tr), © Warren
Faidley/Weatherstock; 721 (bl), Aflo Foto Agency/Alamy; 722 (cr), Victoria Smith/HRW;
724 (bl), Sam Dudgeon/HRW; 727 (tl), David Young-Wolff/Alamy; 728 (tl), Peter
Chapter Twelve: Page 846–847 (all), © Robert Daly/Stone/Getty Images; 849 Sam
Dudgeon/HRW; 851 (tr), © RubberBall/Alamy; 853 (tl), MedioImages/Alamy; 856 (tl),
AP Photo/Robert F. Bukaty; 856 (bl), PhotoDisc/Getty Images; 858 (tr), © Steve
Hamblin/Alamy; 864 (bl), PhotoDisc/Getty Images; 866 (tr), Gerald C. Kelley/Photo
Researchers, Inc.; 868 (bl), © Paul Barton/CORBIS; 869 (tl), © SuperStock; 871 (cl),
AP Photo/Lou Krasky; 871 (bl), PhotoDisc/Getty Images; 876 (tl), PhotoDisc/Getty
Images; 876 (br), AP Photo/Tyler Morning Telegraph, Tom Worner; 876 (sky),
PhotoDisc/Getty Images; 878 (tr), © Stock Connection Distribution / Alamy; 885 (tr),
Philippe Blondel/Photo Researchers, Inc.; 890 (tl), Photo Researchers, Inc.; 893 (tc),
AP Photo/Dolores Ochoa; 893 (tr), © Nigel Francis/Robert Harding/Alamy Photos;
898 (cl), AP Photo /NASA, Nick Galante, PMRF; 900 (tr), © Tom Stewart/CORBIS;
904 (br), ON-PAGE CREDIT; 906 (b), © Dynamic Graphics Group/i2i/Alamy; 906
(movie screen), © Ralph Nelson/Imagine Entertainment/ZUMA/CORBIS; 920 (tr), Sam
Dudgeon/HRW; 920 (bl), © David Bergman/Corbis; 920 (bc), AP Photo/Al Behrman;
921 (br), Bicycle Museum of America/New Bremen, Ohio/www.bicyclemuseum.com;
921 (tc), Courtesy of the Bicycle Museum of America.
Student Handbook: S2 (tl), PhotoDisc/Getty Images; S3 (br), Sam Dudgeon/HRW.
Credits
CR3