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SKILLS REVIEW HANDBOOK
Logical Argument
A logical argument has two given statements, called premises, and a statement,
called a conclusion, that follows from the premises. Below is an example.
Premise 1
Premise 2
Conclusion
If a triangle has a right angle, then it is a right triangle.
In nABC, ∠B is a right angle.
nABC is a right triangle.
Letters are often used to represent the statements of a logical argument and
to write a pattern for the argument. The table below gives five types of logical
arguments. In the examples, p, q, and r represent the following statements.
p: a figure is a square
Type of Argument
q: a figure is a rectangle
Pattern
r: a figure is a parallelogram
Example
Direct Argument
If p is true, then q is true.
p is true.
Therefore, q is true.
If ABCD is a square, then it is a rectangle.
ABCD is a square.
Therefore, ABCD is a rectangle.
Indirect Argument
If p is true, then q is true.
q is not true.
Therefore, p is not true.
If ABCD is a square, then it is a rectangle.
ABCD is not a rectangle.
Therefore, ABCD is not a square.
Chain Rule
If p is true, then q is true.
If q is true, then r is true.
Therefore, if p, then r.
If ABCD is a square, then it is a rectangle. If ABCD is
a rectangle, then it is a parallelogram. Therefore, if
ABCD is a square, then it is a parallelogram.
Or Rule
p is true or q is true.
p is not true.
Therefore, q is true.
ABCD is a square or a rectangle.
ABCD is not a square.
Therefore, ABCD is a rectangle.
And Rule
p and q are not both true.
q is true.
Therefore, p is not true.
ABCD is not both a square and a rectangle.
ABCD is a rectangle.
Therefore, ABCD is not a square.
An argument that follows one of these patterns correctly has a valid conclusion.
EXAMPLE
State whether the conclusion is valid or invalid. If the
conclusion is valid, name the type of logical argument used.
a. If it is raining at noon, Peter’s family will not have a picnic lunch. Peter’s
family had a picnic lunch. Therefore, it was not raining at noon.
c The conclusion is valid. This is an example of indirect argument.
b. If a triangle is equilateral, then it is an acute triangle. Triangle XYZ is an acute
triangle. Therefore, triangle XYZ is equilateral.
c The conclusion is invalid.
c. If x 5 4, then 2x 2 7 5 1. If 2x 2 7 5 1, then 2x 5 8. x 5 4. Therefore, if x 5 4, then
2x 5 8.
c The conclusion is valid. This is an example of the chain rule.
d. If it is at least 808F outside today, you will go swimming. It is 858F outside today.
Therefore, you will go swimming.
c The conclusion is valid. This is an example of direct argument.
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A compound statement has two or more parts joined by or or and.
• For an or compound statement to be true, at least one part must be true.
EXAMPLE
State whether the compound statement is true or false.
a. 12 < 20 and 212 > 220
True
b. 2 < 4 and 4 < 3
True
True
c True, because each part is true.
c. 10 > 0 or 210 > 0
True
False
c False, because one part is false.
d. 28 > 27 or 27 > 26 or 26 > 25
False
False
c True, because at least one part is true.
False
SKILLS REVIEW HANDBOOK
• For an and compound statement to be true, each part must be true.
False
c False, because every part is false.
PRACTICE
State whether the conclusion is valid or invalid. If the conclusion is valid, name
the type of logical argument used.
1. If Scott goes to the store, then he will buy sugar. If he buys sugar, then he will
bake cookies. Scott goes to the store. Therefore, he will bake cookies.
2. If a triangle has at least two congruent sides, then it is isosceles. Triangle MNP
has sides 5 in., 6 in., and 5 in. long. Therefore, triangle MNP is isosceles.
3. If a horse is an Arabian, then it is less than 16 hands tall. Andrea’s horse is
13 hands tall. Therefore, Andrea’s horse is an Arabian.
4. If a figure is a rhombus, then it has four sides. Figure WXYZ has four sides.
Therefore, WXYZ is a rhombus.
5. Jeff cannot buy both a new coat and new boots. Jeff decides to buy new
boots. Therefore, Jeff cannot buy a new coat.
6. If x 5 0, then y 5 4. If y 5 4, then z 5 7. Therefore, if z 5 7, then x 5 0.
7. Kate will order either tacos or burritos for lunch. Kate does not order tacos for
lunch. Therefore, Kate orders burritos for lunch.
8. If a triangle is equilateral, then it is equiangular. Triangle ABC is not
equiangular. Therefore, triangle ABC is not equilateral.
9. An animal cannot be both a fish and a bird. Courtney’s pet is not a fish.
Therefore, Courtney’s pet must be a bird.
State whether the compound statement is true or false.
10. 27 < 25 and 25 < 26
11. 6 > 2 or 8 < 4
12. 0 ≤ 21 or 5 ≥ 5
13. 4 ≤ 3 or 12 ≥ 13
14. 3 < 5 and 23 < 25
15. 1 5 21 or 1 5 1 or 1 5 0
16. 7 < 8 and 8 < 12
17. 22 < 2 and 3 ≥ 2
18. 3(24) 5 12 or 23(4) 5 12
19. 28 > 8 or 28 5 8 or 28 ≥ 0
20. 140 Þ 145 or 140 > 2145 or 2140 < 2145
21. 28(9) 5 272 and 8(29) 5 272
22. 22 ≤ 23 and 222 < 223 and 23 > 22
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SKILLS REVIEW HANDBOOK
Conditional Statements and
Counterexamples
A conditional statement has two parts, a hypothesis and a conclusion. When
a conditional statement is written in if-then form, the “if” part contains the
hypothesis and the “then” part contains the conclusion. An example of a
conditional statement is shown below.
If a triangle is equiangular, then each angle of the triangle measures 608.
Hypothesis
Conclusion
The converse of a conditional statement is formed by switching the hypothesis
and the conclusion. The converse of the statement above is as follows:
If each angle of a triangle measures 608, then the triangle is equiangular.
EXAMPLE
Rewrite the conditional statement in if-then form. Then write
its converse and tell whether the converse is true or false.
a. Bob will earn $20 by mowing the lawn.
If-then form: If Bob mows the lawn, then he will earn $20.
Converse: If Bob earns $20, then he mowed the lawn. False
b. x 5 8 when 5x 1 1 5 41.
If-then form: If 5x 1 1 5 41, then x 5 8.
Converse: If x 5 8, then 5x 1 1 5 41. True
A biconditional statement is a statement that has the words “if and only if.” You
can write a conditional statement and its converse together as a biconditional
statement.
A triangle is equiangular if and only if each angle of the triangle measures 608.
A biconditional statement is true only when the conditional statement and its
converse are both true.
EXAMPLE
Tell whether the biconditional statement is true or false.
Explain.
a. An angle measures 90° if and only if it is a right angle.
Conditional: If an angle is a right angle, then it measures 908. True
Converse: If an angle measures 908, then it is a right angle. True
c The biconditional statement is true because the conditional and its converse
are both true.
b. Bonnie has $.50 if and only if she has two quarters.
Conditional: If Bonnie has two quarters, then she has $.50. True
Converse: If Bonnie has $.50, then she has two quarters. False
c The biconditional statement is false because the converse is not true.
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A counterexample is an example that shows that a statement is false.
SKILLS REVIEW HANDBOOK
EXAMPLE
Tell whether the statement is true or false. If false,
give a counterexample.
a. If a polygon has four sides and opposite sides are parallel, then
it is a rectangle.
c False. A counterexample is the parallelogram shown.
b. If x2 5 49, then x 5 7.
c False. A counterexample is x 5 27, because (27)2 5 49.
PRACTICE
Rewrite the conditional statement in if-then form. Then write its converse and
tell whether the converse is true or false.
1. The graph of the equation y 5 mx 1 b is a line.
2. You will earn $35 for working 5 hours.
3. Abby can go swimming if she finishes her homework.
4. In a right triangle, the sum of the squares of the lengths of the
legs equals the square of the length of the hypotenuse.
5. x 5 5 when 4x 1 8 5 28.
6. The sum of two even numbers is an even number.
Tell whether the biconditional statement is true or false. Explain.
7. Two lines are perpendicular if and only if they intersect to form a right angle.
8. x 3 5 27 if and only if x 5 3.
9. A vegetable is a carrot if and only if it is orange.
10. A rhombus is a square if and only if it has four right angles.
11. The graph of a function is a parabola if and only if the function is y 5 x2.
12. An integer is odd if and only if it is not even.
Tell whether the statement is true or false. If false, give a counterexample.
13. If an integer is not negative, then it is positive.
14. If you were born in the summer, then you were born in July.
15. If a polygon has exactly 5 congruent sides, then the polygon is a pentagon.
16. If x 5 26, then x2 5 36.
17. If B is 6 inches from A and 8 inches from C, then A is 14 inches from C.
18. If a triangle is isosceles, then it is obtuse.
19. If Charlie has $1.00 in coins, then he has four quarters.
20. If you are in Montana, then you are in the United States.
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SKILLS REVIEW HANDBOOK
Venn Diagrams
A Venn diagram uses shapes to show how sets are related.
EXAMPLE
Draw a Venn diagram of the positive integers less than
13 where set A consists of factors of 12 and set B consists
of even numbers.
Positive integers less than 13:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Positive integers less than 13
A
Set A (factors of 12): 1, 2, 3, 4, 6, 12
1
Set B (even numbers): 2, 4, 6, 8, 10, 12
Both set A and set B: 2, 4, 6, 12
B
2
4
3
6
8
12
10
5
11
7
9
Neither set A nor set B: 5, 7, 9, 11
EXAMPLE
Use the Venn diagram above to decide if the statement is true
or false. Explain your reasoning.
a. If a positive integer less than 13 is not even, then it is not a factor of 12.
c False. 1 and 3 are not even, but they are factors of 12.
b. All positive integers less than 13 that are even are factors of 12.
c False. 8 and 10 are even, but they are not factors of 12.
PRACTICE
Draw a Venn diagram of the sets described.
1. Of the positive integers less than 11, set A consists of factors of 10 and set B
consists of odd numbers.
2. Of the positive integers less than 10, set A consists of prime numbers and
set B consists of even numbers.
3. Of the positive integers less than 25, set A consists of multiples of 3 and
set B consists of multiples of 4.
Use the Venn diagrams you drew in Exercises 1–3 to decide if the statement is
true or false. Explain your reasoning.
4. The only factors of 10 less than 11 that are not odd are 2 and 10.
5. If a number is neither a multiple of 3 nor a multiple of 4, then it is odd.
6. All prime numbers less than 10 are not even.
7. If a positive odd integer less than 11 is a factor of 10, then it is 5.
8. There are 2 positive integers less than 25 that are both a multiple of 3 and a
multiple of 4.
9. If a positive even integer less than 10 is prime, then it is 2.
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Mean, Median, Mode, and Range
The mean of a data
set is the sum of the
values divided by the
number of values.
The mean is also
called the average.
EXAMPLE
The median of a data set is the
middle value when the values
are written in numerical order. If
a data set has an even number
of values, the median is the
mean of the two middle values.
The mode of a data
set is the value that
occurs most often. A
data set can have no
mode, one mode, or
more than one mode.
The range of a
data set is the
difference between
the greatest value
and the least value.
SKILLS REVIEW HANDBOOK
Mean, median, and mode are measures of central tendency; they measure the
center of data. Range is a measure of dispersion; it measures the spread of data.
Find the mean, median, mode(s), and range of the data.
Daily High Temperatures, Week of June 21–27
Day
Sunday
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
76
74
70
69
70
75
78
Temperature (8F)
Mean
Add the values. Then divide by the number of values.
76 1 74 1 70 1 69 1 70 1 75 1 78 5 512
mean 5 512 4 7 ø 73
The mean of the data is about 738F.
Median Write the values in order from least to greatest. Find the middle value(s).
69, 70, 70, 74, 75, 76, 78
median 5 74
Mode
Find the value that occurs most often.
mode 5 70
Range
The median of the data is 748F.
The mode of the data is 708F.
Subtract the least value from the greatest value.
range 5 78 2 69 5 9
The range of the data is 98F.
PRACTICE
Find the mean, median, mode(s), and range of the data.
1. Apartment rents: $650, $800, $700, $525, $675, $750, $500, $650, $725
2. Ages of new drivers: 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 18, 18
3. Monthly cell-phone minutes: 581, 713, 423, 852, 948, 337, 810, 604, 897
4. Prices of a CD: $12.98, $14.99, $13.49, $12.98, $13.89, $16.98, $11.98
5. Cookies in a batch: 36, 60, 52, 44, 48, 45, 48, 41, 60, 45, 38, 55, 60, 48, 40
6. Ages of family members: 41, 45, 8, 10, 40, 44, 3, 5, 42, 42, 13, 14, 67, 70
7. Hourly rates of pay: $8.80, $6.50, $10.85, $7.90, $9.50, $9, $8.70, $12.35
8. Weekly quiz scores: 8, 9, 8, 10, 10, 7, 9, 8, 9, 9, 10, 7, 8, 6, 10, 9, 9, 8, 8, 10
9. People on a bus: 9, 14, 5, 22, 18, 30, 6, 25, 18, 12, 15, 10, 8, 22, 10, 11, 20
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SKILLS REVIEW HANDBOOK
Graphing Statistical Data
There are many ways to display data. An appropriate
graph can help you analyze data. The table at the
right summarizes how data are shown in some
statistical graphs.
EXAMPLE
Bar Graph
Compares data in categories.
Circle Graph
Compares data as parts of a whole.
Line Graph
Shows data change over time.
Use the bar graph to answer the questions.
a. On which day of the week were the greatest
Cars Parked in Student Lot
number of cars parked in the student lot?
120
b. How many cars were parked in the student
lot on Monday?
80
Cars
c The tallest bar on the graph is for Friday.
So, the answer is Friday.
40
c The bar for Monday shows that about
70 cars were parked in the student lot.
EXAMPLE
0
Tu
W
Th
F
Use the circle graph to answer the questions.
a. Which type of transportation is used almost half the
Transportation to School
time?
Car 45%
Bus 20%
c Almost half of the total area of the circle is labeled
“Car 45%.” So, a car is used almost half the time.
Walk or bike
35%
b. Which type of transportation is used the least often?
c The smallest part of the circle is labeled “Bus 20%.”
So, a bus is used the least often.
EXAMPLE
M
Use the line graph to answer the questions.
a. In which month(s) was Jamie’s balance
Jamie’s Savings Account Balance
$250?
b. Between which two consecutive months
did Jamie’s balance increase the most?
c Of the graph’s line segments that have
positive slope, the graph is steepest from
June to July. So, Jamie’s balance increased
the most between June and July.
400
300
Dollars
c The points on the graph to the right of
$250 show that Jamie’s balance was $250
in May and December.
200
100
0
J F M A M J J A S O N D
Month
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PRACTICE
Friday at Ferraro’s Restaurant
1. At which hour did Ferraro’s have 22 diners?
3. How many diners were at Ferraro’s at 11 P.M.?
Were they gone by midnight?
Diners
30
2. At which hour did Ferraro’s have the most diners?
20
10
0
4. Between which two consecutive hours did the
5
number of diners at Ferraro’s change the most?
6
7
8
9
10
11
Time (hours since noon)
12
5. How many fewer diners were at Ferraro’s at
10 P.M. than at 6 P.M.?
Use the bar graph to answer Exercises 6–8.
SKILLS REVIEW HANDBOOK
Use the line graph to answer Exercises 1–5.
Seasons of Students’ Birthdays
6. In which season were the fewest students born?
12
Students
7. In which season(s) were 7 students born?
8. How many more students were born in spring
than in summer?
8
4
0
Use the circle graph to answer Exercises 9–11.
Fall
Winter
Spring
Summer
Heat Sources for U.S. Homes
9. What is the heat source of more than half the
Natural gas 52%
Electricity 22%
homes in the United States?
10. What percent of homes in the United States are
Fuel oil 10%
heated with electricity?
Other 16%
11. If you randomly selected 500 U.S. homes, about
how many would be heated with fuel oil?
12. The table below shows the high temperatures in degrees Fahrenheit for one
week. Display the data in a line graph.
Mon.
Tues.
Wed.
Thurs.
Fri.
Sat.
Sun.
83
89
79
73
69
67
71
13. A high school conducted a survey to determine the numbers of students
involved in various school activities. Display the survey results
in a bar graph.
Computer
club
Music
club
Yearbook
club
Drama
club
Student
council
Chess
club
34
75
16
57
28
12
14. The table below shows the items sold at a café in one day. Display the data in
a circle graph.
Juice
Soda
Water
Muffin
Cookie
95
180
100
55
40
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SKILLS REVIEW HANDBOOK
Organizing Statistical Data
Because it is difficult to analyze unorganized data, it is helpful to organize data
using a line plot, stem-and-leaf plot, histogram, or box-and-whisker plot.
EXAMPLE
Sydney’s math test scores are 90, 85, 88, 95, 100, 77, 85, 100,
80, 77, and 90.
a. Draw a line plot to display the data.
Make a number line from 75 to 100. Each time a value is listed in the data set,
draw an X above the value on the number line.
75
3
3
3
3
3
3
3
3
3
3
3
77
80
85
88
90
95
100
b. Draw a stem-and-leaf plot to display the data.
First write the leaves next to their stems.
7
7
7
8
5
8
5
9
0
5
0
10
0
Then order the leaves from least to greatest.
7
7
8
0
5
5
9
0
0
5
10
0
0
0
Key: 7 | 7 5 77
0
7
8
Key: 7 | 7 5 77
c. Draw a histogram to display the data.
First make a frequency table. Use equal
intervals.
Then make a histogram.
Sydney’s Math Test Scores
Tally
Frequency
71–80
3
3
81–90
5
5
91–100
3
3
6
Frequency
Score
4
2
0
71–80
81– 90
Score
91–100
d. Draw a box-and-whisker plot to display the data.
Write the data in order from least to greatest. Ordered data are divided into a
lower half and an upper half by the median. The median of the lower half is
the lower quartile, and the median of the upper half is the upper quartile.
77
77
Low
value
80
Lower
quartile
85
85
88
90
90
Median
Plot the median, quartiles, and low
and high values below a number
line. Draw a box between quartiles
with a vertical line through the
median as shown. Draw whiskers
to the low and high values.
95
100
100
Upper
quartile
High
value
Sydney’s Math Test Scores
75
80
77
80
85
90
88
95
100
95
100
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PRACTICE
SKILLS REVIEW HANDBOOK
Use the following list of ticket prices to answer Exercises 1–4: $50, $42, $65,
$54, $70, $65, $59, $30, $67, $49, $54, $30, $73, $47, and $54.
1. Draw a line plot to display the data.
2. How many ticket prices are $50 or less?
3. Draw a stem-and-leaf plot to display the data.
4. What is the range of ticket prices costs?
Use the following list of hourly wages of employees to answer Exercises 5–8:
$8.50, $6, $10, $14.25, $5.75, $7, $6.50, $14, $10, $9, $6.50, $8.25, $8.50,
$11.25, $7, $16, $12, $6, $6.75.
5. Draw a histogram to display the data. Begin with the interval $5.00 to $6.99.
6. Copy and complete: The greatest number of employees earn from ? to ?
per hour.
7. Draw a box-and-whisker plot to display the data.
8. Copy and complete: About half of the employees have an hourly wage of ?
or less.
Use the line plot, which shows the results of a survey
asking people the average number of e-mails they
receive daily, to answer Exercises 9 and 10.
9. Copy and complete: Most people surveyed receive
an average of ? e-mails per day.
3
3 3
3 3
4
5
3
3
3
3
3
3
3
7
10
12
15
17
10. How many people receive an average of more than 10 e-mails per day?
Use the stem-and-leaf plot, which shows the weights
(in pounds) of dogs at an animal shelter, to answer
Exercises 11–13.
11. How many dogs were at the shelter?
12. Find the median of the data.
2
2
5 5 9
3
1
3 5 8
4
0
0 1 2 2 5 6 7
5
0
3 5 8 9
6
4
5
Key: 2 | 2 5 22
13. Find the range of the data.
Use the histogram to answer Exercises 14–16.
Baseball Game Attendance
9
9
–6
60
–5
9
50
–4
40
–3
9
30
20
oldest group?
–2
9
0
–1
9
16. Which age group had the same attendance as the
20
9
the baseball game?
40
10
15. How many children up to the age of 9 years attended
0–
baseball game? Which had the least?
People
14. Which age group had the greatest attendance at the
Age (years)
Use the box-and-whisker plot to answer Exercises 17–19.
17. What is the median number of songs on Sam’s CDs?
Number of Songs on Sam’s CDs
10
12
14
10 11 12
14
16
18
18. What is the upper quartile of songs on Sam’s CDs?
19. What is the least number of songs on one of Sam’s
CDs? What is the greatest number?
18
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E xt
xtrra P ra
racc tice
Chapter 1
1.1 Graph the numbers on a number line.
5 , 0.2, 2Ï}
5
1. 22, }
2 , 2}
4
3
4 , 1, 21.2, Ï 3 , 1.9
2. 2}
3
}
1 , 4, Ï}
3. 3.7, 2Ï 7 , 2}
15
2
}
1.1 Perform the indicated conversion.
EXTRA PRACTICE
4. 18 feet to inches
5. 20 ounces to pounds
6. 3 years to hours
1.2 Evaluate the expression for the given value of the variable.
8. 3x 2 2 x 1 7 when x 5 21
7. 22p 1 5 when p 5 25
9. 8z3 2 6z when z 5 2
1.2 Simplify the expression.
10. 2y 2 2 3y 1 5y 2
11. 4r 2 2 5r 1 2r 2 1 12
12. 2w 3 1 w 2 2 7w 2 2 8w 3
13. 2(b 1 5) 1 3(2b 2 10)
14. 27(t 2 1 2) 1 9(t 2 2)
15. 4(m 2 3) 2 5(m2 2 m)
1.3 Solve the equation. Check your solution.
16. 3a 1 2 5 11
17. 29 5 b 2 14
18. 8 2 0.5c 5 1
19. 23n 2 7 5 2n 1 17
20. 12m 5 15m 2 7.5
21. 6p 1 1 5 21 2 4p
22. 6(x 1 1) 5 2x 2 10
23. 4(y 2 3) 5 2(y 1 8)
24. 11(z 2 5) 5 2(z 1 6) 2 13
1.4 Solve the equation for y. Then find the value of y for the given value of x.
25. 6y 2 x 5 18; x 5 2
26. 2x 1 3y 5 12; x 5 26
27. 4y 2 9x 5 230; x 5 6
28. 3x 2 xy 5 20; x 5 8
29. 4y 1 6xy 5 10; x 5 22
30. 5x 1 8y 1 4xy 5 0; x 5 21
1.5 Look for a pattern in the table. Then write an equation that represents the
table.
31.
32.
x
0
1
2
3
y
25
22
19
16
x
0
1
2
3
y
1.5
4
6.5
9
1.6 Solve the inequality. Then graph the solution.
33. x 1 2 > 9
34. 213 2 3x < 11
35. 4x 2 9 ≤ 2x 1 1
36. 23x 2 8 ≥ 29x 1 10
37. 27 < x 1 3 ≤ 1
38. 24 ≤ 3x 2 7 ≤ 4
39. 29 ≤ 5 2 2x < 7
40. x 1 3 < 22 or x 2 7 > 0
41. 2x 1 9 ≥ 3 or 25x 1 1 ≤ 0
1.7 Solve the equation. Check for extraneous solutions.
42. g 1 5 5 4
43.
1
2
}3 q 2 }3  5 1
44. 10 2 3t 5 t 1 4
45. 3z 1 1 5 26z
1.7 Solve the inequality. Then graph the solution.
46. a < 2
47. 2c > 14
48. g 1 11 ≥ 2
49. 4j 2 7 ≤ 9
50. 0.25m 1 3 ≥ 1
51. 10 2 2p > 9
52. 0.6r 1 8 ≤ 17
53. 5t 2 9 1 9 < 10
1010 Student Resources
n2pe-9030.indd 1010
10/17/05 12:26:30 PM
Chapter 2
2.1 Tell whether the relation is a function. Explain.
1.
2.
Input
Output
1
21
21
2
2
0
Input
3
3
1
4
5
2
Output
3.
Input
4.
Output
3
6
21
22
Input
Output
27
14
8
4
28
9
6
12
4
2.2 Find the slope of the line passing through the given points. Then tell whether
the line rises, falls, is horizontal, or is vertical.
6. (2, 21), (8, 21)
7. (3, 5), (3, 212)
8. (1, 8), (21, 24)
2.2 Tell whether the lines are parallel, perpendicular, or neither.
9. Line 1: through (5, 24) and (24, 2)
10. Line 1: through (0, 24) and (22, 2)
Line 2: through (25, 24) and (22, 22)
Line 2: through (4, 23) and (5, 26)
2.3 Graph the equation using any method.
11. y 5 2x 2 2
12. y 5 2x 1 2
2x 2 1
13. f(x) 5 }
3
14. x 1 2y 5 26
15. 24x 1 5y 5 10
16. y 2 2 5 0
17. 22x 5 6y 1 5
18. 2y 1 10 5 22.5x
EXTRA PRACTICE
5. (23, 0), (5, 24)
2.4 Write an equation of the line that satisfies the given conditions.
19. m 5 7, b 5 23
1, b 5 4
20. m 5 }
3
21. m 5 0, passes through (7, 22)
1 , passes through (3, 6)
22. m 5 2}
4
23. passes through (21, 23) and (2, 7)
24. passes through (4, 22) and (0, 4)
2.5 The variables x and y vary directly. Write an equation that relates x and y. Then
find y when x 5 22.
25. x 5 2, y 5 4
26. x 5 21, y 5 3
27. x 5 228, y 5 27
28. x 5 6, y 5 24
2.6 In Exercises 29 and 30, (a) draw a scatter plot of the data, (b) approximate the
best-fitting line, and (c) estimate y when x 5 12.
29.
x
1
2
3
4
5
y
8
11
13
16
18
30.
x
1
2
3
4
5
y
50
41
37
22
20
2.7 Graph the function. Compare the graph with the graph of y 5 x.
31. y 5 x 1 3
32. y 5 22x 2 5
33. y 5 3x 1 1 2 2
1 x12 13
34. y 5 2}


2
2.8 Graph the inequality in a coordinate plane.
35. x < 4
36. y ≥ 22
37. y ≤ 2x 2 1
38. x 1 2y > 8
39. 2x 2 4y ≤ 6
40. 3x 1 4y > 12
41. y < x 1 1
42. y ≥ 3x 2 2 2 1
Extra Practice
n2pe-9030.indd 1011
1011
10/17/05 12:26:34 PM
Chapter 3
3.1 Graph the linear system and estimate the solution. Then check the solution
algebraically.
1. y 5 2x 2 1
2. y 5 2x 1 3
y5x24
3. x 1 2y 5 6
y 5 24x
4. 22x 1 7y 5 27
25x 1 6y 5 22
4x 2 14y 5 14
3.2 Solve the system using any algebraic method.
5. 25x 2 y 5 23
6. 4x 2 2y 5 26
7. 4x 1 3y 5 25
23x 1 y 5 23
12x 1 4y 5 10
x 2 4y 5 9
8. 3x 1 2y 5 4
27x 2 5y 5 27
EXTRA PRACTICE
3.3 Graph the system of inequalities.
9. x > 4
10. x 1 y < 22
11. x ≤ 5
x 2 3y > 6
y>3
y>x
y ≥ 21
12. x > 23
x≤2
2x 1 3y < 10
y > 24x
3.4 Solve the system using any algebraic method.
13. 3x 1 y 2 z 5 26
14. x 1 y 2 z 5 7
2x 1 2y 1 3z 5 21
5x 2 2y 1 6z 5 54
15. 2x 1 y 2 2z 5 1.5
16. 26x 1 y 1 9z 5 4
4x 2 y 1 5z 5 26
2x 1 y 2 2z 5 6
2x 2 3y 2 z 5 26
8x 1 5y 2 4z 5 10
2x 2 3y 1 z 5 2
4x 1 2y 2 2z 5 20
3.5 Perform the indicated operation.
17.
F G F G
26 7
1
0 3
F
26 2
28 1
G
3
2 29
18. 2}
3
4 21
19.
F
10 17 29
26 4 11
3.6 Find the product. If the product is not defined, state the reason.
20.
F GF
4 1
23 0
G
27
5
7 23
21.
F GF G
216
2
4
15
3.7 Evaluate the determinant of the matrix.
23.
F G
5 8
22 10
24.
F
G
13
7
211 24
25.
22.
F
1 23 22
7
4
0
27
2
3
F
5 21 0
4 22 9
2
GF
26
8 22
24 29
4
G
G
12
27
3
G F
26.
G F
6 0
5
24 2
1
1 0 0.5
G
3.7 Use Cramer’s rule to solve the linear system.
27. 2x 1 y 5 28
28. 8x 1 3y 5 1
25x 2 2y 5 13
29. 2x 2 2y 2 3z 5 9
7x 1 3y 5 21
30. 2x 1 y 1 3z 5 4
3x 1 z 5 10
x1y50
28x 1 4y 1 z 5 27
x 1 2y 1 3z 5 21
3.8 Find the inverse of the matrix.
31.
F G
3 7
3 8
32.
F G
1 4
0 5
33.
F
22 25
3
8
G
34.
F G
9 2
18 5
3.8 Use an inverse matrix to solve the linear system.
35. x 1 3y 5 24
22x 1 y 5 234
36. 2x 1 3y 5 6
2x 2 6y 5 29
37. 3x 2 8y 5 0
2x 1 y 5 219
38. x 1 y 5 7
25x 1 3y 5 23
1012 Student Resources
n2pe-9030.indd 1012
10/17/05 12:26:35 PM
Chapter 4
4.1 Graph the function. Label the vertex and axis of symmetry.
1. y 5 3x 2 1 5
2. y 5 2x2 2 4x 2 4
3. y 5 22x2 1 4x 1 1
4. y 5 2x 2 1 5x 1 6
4.2 Graph the function. Label the vertex and axis of symmetry.
5. y 5 4(x 2 2)2 1 1
6. y 5 2(x 1 3)2 2 2
7. y 5 3(x 2 1)(x 2 5)
1 (x 1 3)(x 1 2)
8. y 5 }
2
4.2 Write the quadratic function in standard form.
9. y 5 7(x 1 2)(x 1 4)
10. y 5 2(x 1 5)(x 2 3)
11. y 5 (x 2 7)2 1 7
12. y 5 2(x 1 1)2 2 4
13. x2 2 4x 1 4
14. t 2 2 11t 2 26
15. x2 1 21x 1 108
16. b2 2 400
18. x2 2 11x 1 24 5 0
19. c 2 1 6c 5 55
20. n2 5 5n
4.3 Solve the equation.
17. x2 1 5x 2 14 5 0
4.4 Factor the expression. If the expression cannot be factored, say so.
21. 2x2 1 x 2 15
22. 10a2 2 19a 1 7
23. 3r 2 1 9r 2 4
EXTRA PRACTICE
4.3 Factor the expression. If the expression cannot be factored, say so.
24. 4t 2 1 8t 1 3
4.4 Find the zeros of the function by rewriting the function in intercept form.
25. y 5 81x 2 2 16
26. y 5 2x 2 2 9x 2 5
27. y 5 4x 2 1 18x 1 18
28. y 5 23x 2 2 30x 2 27
4.5 Simplify the expression.
}
29. Ï 56
}
}
Î 47
}
30. 3Ï 2 p Ï 50
31.
34. p2 1 6 5 127
35. (x 2 5)2 5 10
6
32. }
}
1 1 Ï2
}
4.5 Solve the equation.
33. b2 5 8
36. 3(x 1 2)2 2 4 5 11
4.6 Write the expression as a complex number in standard form.
37. (5 1 2i) 1 (6 2 5i)
38. 23i(7 1 i)
1 1 2i
39. }
3 2 8i
(3 2 2i) 1 2i
40. }
(21 1 7i) 2 (2 1 3i)
43. 2c 2 2 12c 1 6 5 0
44. 3z2 2 3z 1 9 5 0
47. 4s 2 1 3s 5 12
48. 22r 2 5 r 1 17
51. 2x2 1 7x 1 6 > 1
52. 3x 2 1 16x 1 2 ≤ 3x
4.7 Solve the equation by completing the square.
41. x2 1 6x 5 10
42. x2 2 9x 2 2 5 0
4.8 Use the quadratic formula to solve the equation.
45. x2 1 10x 2 10 5 0
46. x2 2 x 2 1 5 0
4.9 Solve the inequality using any method.
49. x2 2 10x ≥ 0
50. x2 2 8x 1 12 < 0
4.10 Write a quadratic function in standard form for the parabola that passes
through the given points.
53. (21, 26), (0, 27), (2, 9)
54. (22, 21), (1, 2), (3, 26)
55. (23, 36), (0, 36), (2, 16)
Extra Practice
n2pe-9030.indd 1013
1013
10/17/05 12:26:37 PM
Chapter 5
5.1 Write the answer in scientific notation.
1. (3.4 3 103)(2.8 3 108)
4.6 3 1027
3. }
9.2 3 1029
2. (5.8 3 1026)
4
5.1 Simplify the expression. Tell which properties of exponents you used.
214x23y 5
4. }
35xy 3
5. (4a5b22)23
xy21 7x 3
7. }
p}
y24
x 2y
6. (2r 3s 3)(r27s5)
EXTRA PRACTICE
5.2 Graph the polynomial function.
8. f(x) 5 x4
9. f(x) 5 x 3 1 x 1 4
10. f(x) 5 2x 3 1 3x
11. f(x) 5 x5 1 2x 3
5.3 Perform the indicated operation.
12. (4z3 1 9) 1 (3z2 2 4z 2 2)
13. (x2 1 3x 2 1) 2 (4x2 1 7)
14. (3x 2 4) 3
5.4 Factor the polynomial completely using any method.
15. 3x4 1 18x 3 1 27x2
16. 343x 3 1 1000
17. 2x 3 1 x2 2 8x 2 4
5.4 Find the real-number solutions of the equation.
18. 3x 3 1 18x2 5 48x
19. x4 1 32 5 14x2
20. 2x 3 1 48 5 3x2 1 32x
5.5 Divide using polynomial long division or synthetic division.
21. (2x 3 1 4x2 2 5x 1 16) 4 (x 2 3)
22. (x4 1 2x 3 2 7x2 2 14) 4 (x 1 2)
5.6 Find all real zeros of the function.
23. f(x) 5 2x 3 1 3x2 2 8x 1 3
24. f(x) 5 2x4 1 x 3 2 53x2 2 14x 1 20
5.7 Determine the possible numbers of positive real zeros, negative real zeros, and
imaginary zeros of the function.
25. f(x) 5 2x 3 1 2x2 2 11x 2 1
26. f(x) 5 4x5 1 3x2 2 8x 2 10
27. f(x) 5 x4 2 3x 3 2 7x 2 13
5.8 Estimate the coordinates of each turning point and state whether each
corresponds to a local maximum or a local minimum. Then estimate all real
zeros and determine the least degree the function can have.
28.
1
29.
y
1
x
1
30.
y
y
x
1
1
3
x
5.9 Use finite differences and a system of equations to find a polynomial function
that fits the data in the table.
31.
x
1
2
3
4
5
6
y
2.5
11
27.5
55
96.5
155
32.
x
1
2
3
4
5
6
y
27
26
39
188
525
1158
1014 Student Resources
n2pe-9030.indd 1014
10/17/05 12:26:39 PM
Chapter 6
6.1 Find the indicated real nth root(s) of a.
1. n 5 4, a 5 81
2. n 5 3, a 5 512
3. n 5 5, a 5 2243
6.1 Evaluate the expression without using a calculator.
5} 4
3 } 22
6. (Ï 216 )
5. 645/6
4. 3621/2
7. (Ï 232 )
6.1 Solve the equation. Round the result to two decimal places when appropriate.
8. x 3 5 28
9. x4 1 9 5 90
10. (x 2 3) 5 5 60
11. 24x6 5 2400
12. 45/2 p 421/2
5}
5}
16. 5Ï 7 2 7Ï 7
173/7
13. }
174/7
3}
Ï135
15. }
3}
Ï5
3241/4
18. }
421/4
19. 4Ï 108 p 2Ï 4
4}
3}
17. Ï 2 1 2Ï 128
3}
14. (Ï 5 p Ï 5 )
}
4
3}
3}
6.2 Write the expression in simplest form. Assume all variables are positive.
}
20.
Ï20x6y7
21.
Î
}
5}
Ï18x3y14z20
22.
4
x5
y
23.
}
16
3}
EXTRA PRACTICE
6.2 Simplify the expression.
3}
Ï16x7y 2 p Ï6xy 5
x . Perform the indicated operation and
6.3 Let f(x) 5 2x 1 4, g(x) 5 x3, and h(x) 5 }
4
state the domain.
24. f(x) 1 g(x)
25. g(x) 2 f(x)
26. g(x) p h(x)
f (x)
27. }
g(x)
28. f(g(x))
29. g(h(x))
30. h(f(x))
31. f(f(x))
6.4 Verify that f and g are inverse functions.
x21
33. f(x) 5 3x2 1 1, x ≥ 0; g(x) 5 }
3
1x 1 2
32. f(x) 5 2x 2 4, g(x) 5 }
2
1
1/2
2
6.4 Find the inverse of the function.
34. f(x) 5 5x 2 3
4x 1 2
35. f(x) 5 }
3
1 x 2, x ≥ 0
36. f(x) 5 }
2
37. f(x) 5 2x6 1 2, x ≤ 0
4x4 2 1 , x ≥ 0
38. f(x) 5 }
18
39. f(x) 5 32x5 1 4
6.5 Graph the function. Then state the domain and range.
1 Ï}
40. y 5 2}
x
3
3}
44. y 5 22Ï x 2 1 1 2
2 3}
41. y 5 }
Ïx
5
3}
45. f(x) 5 3Ï x
5 Ï}
42. y 5 }
x
6
43. y 5 Ï x 1 2 2 3
1 Ï}
46. g(x) 5 2}
x22
2
47. h(x) 5 2Ï x 1 3 1 4
}
}
6.6 Solve the equation. Check your solution.
}
48. Ï 2x 1 3 5 7
3}
51. 2Ï 8x 1 9 5 5
}
54. x 2 8 5 Ï 18x
}
3}
49. 25Ï x 1 1 1 12 5 2
50. Ï 5x 2 1 1 6 5 10
52. 7x4/3 5 175
53. (x 2 2) 3/4 5 1
}
55. x 5 Ï 4x 2 3
}
}
56. Ï 2x 1 1 1 5 5 Ï x 1 12 2 8
Extra Practice
n2pe-9030.indd 1015
1015
10/17/05 12:26:40 PM
Chapter 7
7.1 Graph the function. State the domain and range.
4
1. y 5 }
3
1 2
x
2. y 5 22 p 2x
3. y 5 3x 2 3 2 2
1 p 3x 1 1 1 2
4. y 5 }
4
7. y 5 (0.8) x 2 3 2 2
2
8. y 5 2 }
3
7.2 Graph the function. State the domain and range.
3
5. y 5 }
5
1 2
x
1
6. y 5 22 }
4
1 2
x
1 2
x
11
EXTRA PRACTICE
7.3 Simplify the expression.
9. e23 p e28
10.
28e 3x
12. }
21e2x
}
(2e 2x)25
11.
Ï81e 8x
7.3 Graph the function. State the domain and range.
13. y 5 0.5e 3x
15. y 5 1.5e x 1 1 1 3
14. y 5 2e2x 2 2
16. y 5 e 3(x 2 2) 1 1
7.4 Evaluate the logarithm without using a calculator.
1
17. log4 }
16
18. log 6 6
19. log5 125
64
20. log3/4 }
27
22. 10log 9
23. log4 16x
24. eln 5
26. y 5 log1/2 (x 2 4)
27. y 5 log5 x 1 3
28. y 5 log3 (x 2 2) 1 1
100x 2
30. log }
y
31. ln 20x 3y 2
32. log 2 Ï 8x4
7.4 Simplify the expression.
21. 5log5 x
7.4 Graph the function. State the domain and range.
25. y 5 log 7 x
7.5 Expand the expression.
2x
29. log5 }
5
3}
7.5 Condense the expression.
33. log4 20 1 4 log4 x
34. log 7 1 2 log x 2 5 log y
35. 0.5 ln 100 2 2 ln x 1 8 ln y
7.5 Use the change-of-base formula to evaluate the logarithm.
36. log 2 5
37. log4 80
38. log5 100
39. log 7 27
7.6 Solve the equation. Check for extraneous solutions.
x23
40. 24x 1 2 5 8x 1 2
1
41. }
9
43. ln (3x 1 7) 5 ln (x 2 1)
44. log5 (3x 1 2) 5 3
1 2
5 33x 1 1
42. 79x 5 18
45. log 6 (x 1 9) 1 log6 x 5 2
7.7 Write an exponential function y 5 ab x whose graph passes through the given
points.
46. (1, 8), (2, 32)
47. (1, 3), (3, 12)
48. (2, 29), (5, 2243)
49. (1, 4), (2, 4)
7.7 Write a power function y 5 axb whose graph passes through the given points.
50. (2, 2), (5, 16)
51. (3, 27), (6, 432)
52. (1, 4), (8, 17)
53. (5, 36), (10, 220)
1016 Student Resources
n2pe-9030.indd 1016
10/17/05 12:26:42 PM
Chapter 8
8.1 The variables x and y vary inversely. Use the given values to write an equation
relating x and y. Then find y when x 5 25.
1. x 5 2, y 5 210
1 , y 5 24
2. x 5 }
3
3. x 5 23, y 5 25
2
4. x 5 25, y 5 2}
5
8.1 Determine whether x and y show direct variation, inverse variation, or neither.
5.
6.
7.
y
2.5
11
3.5
8.75
16
5
6.4
12.5
8
10
y
32
1
4
20
5
y
2.5
8.
x
y
30
1
12
14
61
3
4
12.5
16
85
8
1.5
8
20
24
92
12
1
9
22.5
27
105
15
0.8
EXTRA PRACTICE
x
x
x
8.2 Graph the function. State the domain and range.
6
9. y 5 }
x
22 1 3
10. y 5 }
x
5 22
11. y 5 }
x21
4x 1 19
12. y 5 }
x13
x2 1 1
14. y 5 }
2
x 1 4x 1 3
2
1 2x 2 3
15. y 5 x}
x12
2x 2 2 8
16. f(x) 5 }
x 2 2 2x
2
2 5x 2 84
19. x}
2x 2 2 98
2
1 7x 1 10
20. x}
x 2 2 7x 1 10
8.3 Graph the function.
x
13. y 5 }
x2 2 4
8.4 Simplify the rational expression, if possible.
x2 1 x 2 6
17. }
x 2 1 9x 1 18
x 3 2 100x
18. }
4
x 1 20x 3 1 100x 2
8.4 Multiply or divide the expressions. Simplify the result.
6x 2y 2y
21. }
p}
xy 2 9x 3
2x 2 2 x 2 6 p x 2 1 x
22. }
}
2x 2 1 5x 1 3 x 2 2 4
3x 2 1 15x p (x 2 2 x 2 30)
23. }
2
x 2 12x 1 36
12x 8y
3y 2
24. }
4
}
5y 5
x2
6x 2 1 x 2 1 4 6x 2 2 2x
25. }
}
4x 3 1 4x 2
x 2 2 4x 2 5
x 2 2 4x 2 32 4
x
26. }
}
2x 2 2 13x 2 24
4x 2 2 9
8.5 Add or subtract the expressions. Simplify the result.
x2 2 1
27. }
}
x11
x11
x15 1 1
28. }
}
x16
x22
5 1
35
29. }
}
x12
x 2 2 3x 2 10
x
3
31. }
1
}13
x
3
x 24
32. }
x11
2
}2}
x12
x2 2 x 2 6
8.5 Simplify the complex fraction.
x
2x 1 1
30. }
3
51}
x
}
}
2
}12
8.6 Solve the equation. Check for extraneous solutions.
7
14
33. }
5}
3x 2 7
x11
1 1 2 523
34. }
}
}
3
x
x2
4 52
35. 2 2 }
}
x12
x
4 1 6x 2 5 3x
36. }
}
}
x22
x12
x2 2 4
Extra Practice
n2pe-9030.indd 1017
1017
10/17/05 12:26:43 PM
Chapter 9
9.1 Find the distance between the two points. Then find the midpoint of the line
segment joining the two points.
1. (25, 0), (5, 4)
2. (2, 1), (3, 7)
3. (212, 12), (14, 24)
4. (12, 21), (18, 29)
9.2 Graph the equation. Identify the focus, directrix, and axis of symmetry of the
parabola.
5. y 2 5 2x
6. x2 5 24y
7. 14x 2 5 221y
8. 12y 2 1 3x 5 0
9.3 Graph the equation. Identify the radius of the circle.
EXTRA PRACTICE
9. x2 1 y 2 5 4
10. x2 1 y 2 5 14
11. 3x 2 1 3y 2 5 75
12. 16x2 1 16y 2 5 4
9.3 Write the standard form of the equation of the circle that passes through the
given point and whose center is at the origin.
13. (8, 0)
14. (0, 29)
15. (7, 21)
16. (25, 211)
9.4 Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse.
2
x2 1 y 5 1
17. }
}
81
16
y2
18. x2 1 } 5 1
9
19. 9x2 1 4y 2 5 576
20. 49x 2 1 64y 2 5 12,544
9.4 Write an equation of the ellipse with the given characteristics and center
at (0, 0).
21. Vertex: (4, 0)
22. Vertex: (0, 25)
Co-vertex: (0, 2)
23. Vertex: (9, 0)
Co-vertex: (4, 0)
24. Co-vertex: (0, 10)
Focus: (23, 0)
Focus: (8, 0)
9.5 Graph the equation. Identify the vertices, foci, and asymptotes of the
hyperbola.
2
x2 2 y 5 1
25. }
}
36
16
26. x2 2 y 2 5 4
27. 49y 2 2 81x2 5 3969
9.5 Write an equation of the hyperbola with the given foci and vertices.
28. Foci: (0, 28), (0, 8)
Vertices: (0, 26), (0, 6)
29. Foci: (22, 0), (2, 0)
Vertices: (21, 0), (1, 0)
30. Foci: (0, 25), (0, 5)
}
}
Vertices: (0, 23Ï 2 ), (0, 3Ï2 )
9.6 Graph the equation. Identify the important characteristics of the graph.
y2
(x 2 3)2
31. } 1 } 5 1
25
9
32. (x 1 2)2 1 (y 2 1)2 5 4
(x 1 1)2
33. (y 2 4)2 2 } 5 1
16
9.6 Classify the conic section and write its equation in standard form. Then graph
the equation.
34. x2 1 y 2 1 2x 1 2y 2 7 5 0
35. 9x2 1 4y 2 2 72x 1 16y 1 16 5 0
36. 9x2 2 4y 2 1 16y 2 52 5 0
37. x2 2 6x 2 4y 1 17 5 0
9.7 Solve the system.
38. x2 1 y 2 5 4
2
2
9x 2 4y 5 36
39. y 5 x 2 2
2
2
x 1 y 2 6x 2 4y 2 12 5 0
40. y 2 5 x 2 5
9x2 2 25y 2 5 225
1018 Student Resources
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Chapter 10
10.1 For the given password configuration, determine how many passwords are
possible if (a) digits and letters can be repeated, and (b) digits and letters
cannot be repeated.
1. 8 digits
2. 8 letters
3. 5 letters followed by 1 digit
4. 2 digits followed by 2 letters
10.1 Find the number of permutations.
5.
P
6.
5 2
P
7.
6 1
P
8.
9 9
P
12 4
10.1 Find the number of distinguishable permutations of the letters in the word.
10. CHOCOLATE
11. STRAWBERRY
EXTRA PRACTICE
9. VANILLA
12. COFFEE
10.2 Find the number of combinations.
13. 7C3
14.
C
15.
4 1
C
16.
10 9
C
15 6
10.2 Use the binomial theorem to write the binomial expansion.
17. (x 2 3) 3
18. (2x 1 3y)4
20. (x 3 1 y 2)6
19. (p2 1 4) 5
10.3 You have an equally likely chance of choosing any integer from 1 through 25.
Find the probability of the given event.
21. An odd number is chosen.
22. A multiple of 3 is chosen.
10.3 Find the probability that a dart thrown at the given target will hit the shaded
region. Assume the dart is equally likely to hit any point inside the target.
23.
24.
25.
8
10
4
4
8
20
10.4 Events A and B are disjoint. Find P(A or B).
26. P(A) 5 0.4, P(B) 5 0.15
27. P(A) 5 0.3, P(B) 5 0.5
28. P(A) 5 0.7, P(B) 5 0.21
10.4 Find the indicated probability. State whether A and B are disjoint events.
29. P(A) 5 0.25
P(B) 5 0.55
P(A or B) 5 ?
P(A and B) 5 0.2
30. P(A) 5 0.52
P(B) 5 0.15
P(A or B) 5 0.67
P(A and B) 5 ?
31. P(A) 5 0.54
32. P(A) 5 0.5
P(B) 5 0.28
P(A or B) 5 0.65
P(A and B) 5 ?
P(B) 5 0.4
P(A or B) 5 ?
P(A and B) 5 0.3
10.5 Find the probability of drawing the given cards from a standard deck of
52 cards (a) with replacement and (b) without replacement.
33. A jack, then a 3
34. A club, then another club
35. A black ace, then a red card
10.6 Calculate the probability of tossing a coin 15 times and getting the given
number of heads.
36. 1
37. 4
38. 7
39. 15
Extra Practice
n2pe-9030.indd 1019
1019
10/17/05 12:26:46 PM
Chapter 11
11.1 Find the mean, median, mode, range, and standard deviation of the data set.
1. 5, 5, 6, 9, 11, 12, 14, 16, 16, 16
2. 16, 18, 29, 30, 34, 35, 35, 38, 46
3. 24, 23, 23, 4, 1, 0, 0, 23, 22, 10, 11
4. 1.7, 2.2, 1.8, 3.0, 0.4, 1.2, 2.8, 2.9
5. 4.5, 5.7, 4.3, 6.9, 22.1, 5.7, 21.2, 3.8
6. 27.2, 3.9, 2.6, 29.1, 2.5, 27.2, 3.9, 27.2
11.2 Find the mean, median, mode, range, and standard deviation of the given
data set and of the data set obtained by adding the given constant to each data
value.
EXTRA PRACTICE
7. 33, 36, 36, 39, 49, 56; constant: 2
8. 10, 12, 14, 16, 16, 18, 19; constant: 21
11.2 Find the mean, median, mode, range, and standard deviation of the given data
set and of the data set obtained by multiplying each data value by the given
constant.
9. 22, 22, 5, 4, 2, 22, 8, 3; constant: 1.5
10. 52, 52, 76, 56, 67, 89, 70; constant: 3
11.3 A normal distribution has a mean of 2.7 and a standard deviation of 0.3. Find
the probability that a randomly selected x-value from the distribution is in the
given interval.
11. Between 2.4 and 2.7
12. At least 3.0
13. At most 2.1
11.4 Identify the type of sample described. Then tell if the sample is biased. Explain
your reasoning.
14. The owner of a movie rental store wants to know how often her customers
rent movies. She asks every tenth customer how many movies the customer
rents each month.
15. A school wants to consult parents about updating its attendance policy. Each
student is sent home with a survey for a parent to complete. The school uses
only surveys that are returned within one week.
11.4 Find the margin of error for a survey that has the given sample size. Round
your answer to the nearest tenth of a percent.
16. 100
17. 600
18. 2900
19. 5000
11.4 Find the sample size required to achieve the given margin of error. Round your
answer to the nearest whole number.
20. 61%
21. 62%
22. 65.5%
23. 66.2%
11.5 Use a graphing calculator to find a model for the data. Then graph the model
and the data in the same coordinate plane.
24.
25.
x
0
2
4
6
8
10
12
14
y
210
23
4
10
14
20
21
36
x
1
2
3
4
5
6
7
8
y
0.5
0.8
1.1
3
9
30
90
280
1020 Student Resources
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10/17/05 12:26:47 PM
Chapter 12
12.1 For the sequence, describe the pattern, write the next term, and write a rule for
the nth term.
1 , 2 , 1, 4 , . . .
2. }
}
}
3 3
3
1. 9, 16, 25, 36, . . .
3. 12.5, 7, 1.5, 24, . . .
12.1 Write the series using summation notation.
1 1 2 1 3 1 4 1 1 1...
5. }
}
}
}
}
7
6
8
9
2
4. 16 1 32 1 48 1 64 1 . . . 1 144
12.1 Find the sum of the series.
5
5
∑ (3i 1 2)
7.
i51
∑ 4i2
6
8.
i50
8
n
∑}
n54 n 1 3
∑ k3
9.
k56
12.2 Write a rule for the nth term of the arithmetic sequence. Then graph the first
six terms of the sequence.
10. a5 5 15, d 5 6
11 , d 5 2 2
12. a 6 5 2}
}
5
5
11. a10 5 278, d 5 210
EXTRA PRACTICE
6.
12.2 Write a rule for the nth term of the arithmetic sequence. Then find a15.
13. 11, 20, 29, 38, . . .
7 , 5 , 1, . . .
15. 3, }
}
3 3
14. 28, 215, 222, 229, . . .
12.2 Write a rule for the nth term of the arithmetic sequence that has the two given
terms.
16. a2 5 9, a7 5 37
14 , a 5 2 42
18. a 3 5 2}
}
10
5
5
17. a 8 5 10.5, a16 5 18.5
12.3 Write a rule for the nth term of the geometric sequence. Then find a10.
1 , 1 , 1 , 1, . . .
19. }
} }
27 9 3
16 , 64 , 256 , . . .
21. 4, }
} }
3 9 27
20. 5, 4, 3.2, 2.56, . . .
12.3 Find the sum of the geometric series.
4
22.
∑ 3(4)i 2 1
i51
7
23.
∑ 0.5(23)i 2 1
i51
5
24.
∑ 10 1 }35 2
7
i21
25.
i51
∑ 2(1.2)i 2 1
i51
12.4 Find the sum of the infinite geometric series, if it exists.
26. 8 1 4 1 2 1 1 1 . . .
27. 2 2 4 1 8 2 16 1 . . .
28. 26.75 1 4.5 2 3 1 2 2 . . .
12.4 Write the repeating decimal as a fraction in lowest terms.
29. 0.333. . .
30. 0.898989. . .
31. 0.212121. . .
32. 1.50150150. . .
12.5 Write a recursive rule for the sequence. The sequence may be arithmetic,
geometric, or neither.
33. 2.5, 5, 10, 20, . . .
34. 2, 22, 26, 210, . . .
35. 1, 2, 2, 4, 8, 32, . . .
12.5 Find the first three iterates of the function for the given initial value.
36. f(x) 5 2x 2 5, x0 5 3
4 x 2 2, x 5 210
37. f(x) 5 }
0
5
38. f(x) 5 3x2 1 x, x0 5 21
Extra Practice
n2pe-9030.indd 1021
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10/17/05 12:26:48 PM
Chapter 13
13.1 Let u be an acute angle of a right triangle. Find the values of the other five
trigonometric functions of u.
3
1. sin u 5 }
5
}
8
2. tan u 5 }
15
Ï7
4. cos u 5 }
4
3. sec u 5 2
EXTRA PRACTICE
13.1 Solve n ABC using the diagram and the given measurements.
5. A 5 218, c 5 8
6. B 5 668, a 5 14
7. B 5 608, c 5 20
8. A 5 298, b 5 6
9. A 5 188, c 5 18
10. B 5 568, c 5 7
B
c
A
a
b
C
13.2 Convert the degree measure to radians or the radian measure to degrees.
11. 1008
3p
13. }
4
12. 268
p
14. 2}
6
13.2 Find the arc length and area of a sector with the given radius r and central
angle u.
15. r 5 5 ft, u 5 908
16. r 5 2 in., u 5 3008
17. r 5 12 cm, u 5 π
13.3 Sketch the angle. Then find its reference angle.
18. 2508
19. 2308
8p
20. }
3
11p
21. 2}
6
7p
24. tan }
4
5p
25. cos 2}
4
13.3 Evaluate the function without using a calculator.
22. sin (2608)
23. csc 2408
1
2
13.4 Evaluate the expression without using a calculator. Give your answer in both
radians and degrees.
26. sin21 0
}
Ï3
27. cos21 2}
2
1
2
28. cos21 3
29. tan21 1
13.4 Solve the equation for u.
30. sin u 5 0.25; 908 < u < 1808
31. cos u 5 0.9; 2708 < u < 3608
32. tan u 5 2; 1808 < u < 2708
13.5 Solve n ABC. (Hint: Some of the “triangles” may have no solution and some
may have two solutions.)
33. A 5 348, a 5 6, b 5 7
34. A 5 508, C 5 658, b 5 60
35. B 5 868, b 5 13, c 5 11
13.5 Find the area of n ABC with the given side lengths and included angle.
36. A 5 358, b 5 50, c 5 120
37. B 5 358, a 5 7, c 5 12
38. C 5 208, a 5 10, b 5 16
40. C 5 508, a 5 12, b 5 14
41. A 5 808, b 5 7, c 5 5
13.6 Solve n ABC.
39. a 5 16, b 5 23, c 5 17
13.6 Find the area of n ABC with the given side lengths.
42. a 5 6, b 5 3, c 5 4
43. a 5 14, b 5 30, c 5 27
44. a 5 16, b 5 16, c 5 20
1022 Student Resources
n2pe-9030.indd 1022
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Chapter 14
14.1 Graph the function.
1x
1. y 5 cos }
4
2. y 5 3 sin x
3. y 5 sin 2πx
4. y 5 2 tan 2x
14.2 Graph the sine or cosine function.
p 11
5. y 5 sin 2 1 x 2 }
42
p
6. y 5 2sin 1 x 1 }
42
7. y 5 2 cos x 1 3
14.2 Graph the tangent function.
p 21
10. y 5 tan 1 x 2 }
22
14.3 Simplify the expression.
p 2 x 1 cos2 (2x)
11. cos2 1 }
2
2
(sec x 2 1)(sec x 1 1)
12. }
tan x
p 2 x cot x 2 csc2 x
13. tan 1 }
2
2
cos2 x 1 sin2 x 5 cos2 x
15. }
tan2 x 1 1
16. 2 2 sec2 x 5 1 2 tan 2 x
14.3 Verify the identity.
cos (2x)
14. } 5 sec x 1 tan x
1 1 sin (2x)
EXTRA PRACTICE
1 tan 2x
9. y 5 2}
4
8. y 5 2 tan x 1 2
14.4 Find the general solution of the equation.
17. 12 tan 2 x 2 4 5 0
19. tan 2 x 2 2 tan x 5 21
18. 3 sin x 5 22 sin x 1 3
14.4 Solve the equation in the given interval. Check your solutions.
20. cos2 x sin x 5 5 sin x; 0 ≤ x < 2π
21. 2 2 2 cos2 x 5 3 1 5 sin x; 0 ≤ x < 2π
22. 8 cos x 5 4 sec x; 0 ≤ x < π
23. cos2 x 2 4 cos x 1 1 5 0; 0 ≤ x < π
14.5 Write a function for the sinusoid.
24.
y
25.
sπ4 , 3d
3
y
(0, 3.5)
(1, 2.5)
π
4
π
1
x
s 3π4 , 21d
1
x
14.6 Find the exact value of the expression.
26. sin (2158)
27. cos 1658
11p
28. tan }
12
p
29. cos }
12
14.7 Find the exact values of sin 2a, cos 2a, and tan 2a.
2 , π < a < 3p
30. tan a 5 }
}
3
2
9 ,0<a< p
31. cos a 5 }
}
10
2
3 , 3p < a < 2π
32. sin a 5 2}
}
5 2
14.7 Find the general solution of the equation.
33. cos 2x 2 cos x 5 0
x 5 sin x
34. cos }
2
35. sin 2x 5 21
Extra Practice
n2pe-9030.indd 1023
1023
10/17/05 12:26:52 PM
Tables
Symbols
Symbol
Meaning
Page
Symbol
...
and so on
2
e
ø
is approximately equal to
2
p
multiplication, times
Page
irrational number ø 2.718
492
log b y
log base b of y
499
3
log x
log base 10 of x
500
opposite of a
4
ln x
log base e of x
500
}
a
1
reciprocal of a, a Þ 0
4
n!
n factorial; number of
permutations of n objects
684
b1
b sub 1
P
n r
number of permutations
of r objects from n distinct
objects
685
C
n r
number of combinations
of r objects from n distinct
objects
690
2a
26
π
pi; irrational number ø 3.14
26
<
is less than
41
>
is greater than
41
≤
is less than or equal to
41
≥
is greater than or equal to
41
P(A)
probability of event A
698
absolute value of x
51
}
P(A)
probability of the
complement of event A
709
is not equal to
52
<
union of two sets
715
ordered pair
72
ù
intersection of two sets
715
f of x, or the value of f at x
75
0⁄
empty set
slope
82
715
m
i
#
84
is a subset of
716
is parallel to
⊥
is perpendicular to
84
P(BA)
probability of event B given
that event A has occurred
718
(x, y, z)
ordered triple
178
}x
x-bar; the mean of a data set
744
F G
matrix
187
s
sigma; the standard
deviation of a data set
745
A
determinant of matrix A
203
∑
summation
796
A21
inverse of matrix A
210
u
theta
852
Ïa
the nonnegative square root
of a
266
sin
sine
852
cos
cosine
852
i
imaginary unit equal to Ï21
275
tan
tangent
852
absolute value of complex
number z
279
csc
cosecant
852
x approaches positive
infinity
sec
secant
852
339
cot
cotangent
852
nth root of a
414
sin21
inverse sine
875
438
21
inverse cosine
875
21
inverse tangent
875
x
Þ
TABLES
Meaning
(x, y)
f(x)
1
0
0
1
}
z
x → 1`
n}
Ïa
f
21
}
inverse of function f
cos
tan
1024 Student Resources
n2pe-9040.indd 1024
10/14/05 11:05:58 AM
Measures
Time
60 seconds (sec) 5 1 minute (min)
60 minutes 5 1 hour (h)
24 hours 5 1 day
7 days 5 1 week
4 weeks (approx.) 5 1 month
365 days
52 weeks (approx.) 5 1 year
12 months
10 years 5 1 decade
100 years 5 1 century
Metric
United States Customary
Length
Length
10 millimeters (mm) 5 1 centimeter (cm)
12 inches (in.) 5 1 foot (ft)
100 cm
1000 mm 5 1 meter (m)
36 in. 5 1 yard (yd)
3 ft
1000 m 5 1 kilometer (km)
5280 ft 5 1 mile (mi)
1760 yd
Area
100 square millimeters 5 1 square centimeter
(mm2)
(cm2)
2
10,000 cm 5 1 square meter (m2 )
10,000 m2 5 1 hectare (ha)
144 square inches (in.2 ) 5 1 square foot (ft2 )
9 ft2 5 1 square yard (yd2 )
Volume
Volume
1000 cubic millimeters 5 1 cubic centimeter
(mm3)
(cm3)
3
1,000,000 cm 5 1 cubic meter (m3)
1728 cubic inches (in.3) 5 1 cubic foot (ft3)
27 ft3 5 1 cubic yard (yd3)
Liquid Capacity
43,560 ft2 5 1 acre (A)
4840 yd2
Liquid Capacity
1000 milliliters (mL)
5 1 liter (L)
1000 cubic centimeters (cm3)
1000 L 5 1 kiloliter (kL)
Mass
8 fluid ounces (fl oz) 5 1 cup (c)
2 c 5 1 pint (pt)
2 pt 5 1 quart (qt)
4 qt 5 1 gallon (gal)
Weight
1000 milligrams (mg) 5 1 gram (g)
1000 g 5 1 kilogram (kg)
1000 kg 5 1 metric ton (t)
Temperature Degrees Celsius (°C)
0°C 5 freezing point of water
37°C 5 normal body temperature
100°C 5 boiling point of water
16 ounces (oz) 5 1 pound (lb)
2000 lb 5 1 ton
Temperature Degrees Fahrenheit (°F)
32°F 5 freezing point of water
98.6°F 5 normal body temperature
212°F 5 boiling point of water
Tables
n2pe-9040.indd 1025
TABLES
Area
1025
10/14/05 11:06:01 AM
Formulas
Formulas from Coordinate Geometry
y 2y
2
1
m5}
x 2 x where m is the slope of the nonvertical line through
Slope of a line (p. 82)
2
1
points (x1, y1) and (x 2, y 2)
Parallel and perpendicular
lines (p. 84)
If line l1 has slope m1 and line l2 has slope m2, then:
l1 i l2 if and only if m1 5 m2
1
l1 ⊥ l2 if and only if m1 5 2}
m , or m1m2 5 21
2
}}
d 5 (x2 2 x1)2 1 (y2 2 y1)2 where d is the distance between
Ï
Distance formula (p. 615)
points (x1, y1) and (x 2, y 2)
1
x 1x
y 1y
2
2
2
1
2
1
2
M }
,}
is the midpoint of the line segment joining
Midpoint formula (p. 615)
points (x1, y1) and (x 2, y 2).
TABLES
Formulas from Matrix Algebra
Determinant of a
2 3 2 matrix
(p. 203)
Determinant of a
3 3 3 matrix
(p. 203)
Area of a triangle
(p. 204)
det
F G
a b
c d
5
a
b
) c d ) 5 ad 2 cb
F G)
a b
det d e
g h
c
a b
f 5 d e
i
g h
c
f 5 (aei 1 bfg 1 cdh) 2 (gec 1 hfa 1 idb)
i
)
The area of a triangle with vertices (x1, y1), (x 2, y 2), and (x 3, y 3) is given by
1
x1 y 1 1
)
Area 5 6}
2 x2 y 2 1
x3 y 3 1
)
where the appropriate sign (6) should be chosen to yield a positive value.
Cramer’s rule
(p. 205)
Let A 5
F G
a b
c d
be the coefficient matrix of this linear system:
ax 1 by 5 e
cx 1 dy 5 f
If det A Þ 0, then the system has exactly one solution.
e
b
a
) f d)
e
)c f)
The solution is x 5 } and y 5 }.
det A
Inverse of a
2 3 2 matrix
The inverse of the matrix A 5
(p. 210)
F
d 2b
2c
a
A
1
A21 5 }
G
det A
F G
F G
1
5}
a b
c d
is
d 2b
a
ad 2 cb 2c
provided ad 2 cb Þ 0.
1026 Student Resources
n2pe-9040.indd 1026
10/14/05 11:06:02 AM
Formulas and Theorems from Algebra
Quadratic formula (p. 292)
The solutions of ax 2 1 bx 1 c 5 0 are
}
2b 6 Ï b2 2 4ac
x5 }
2a
where a, b, and c are real numbers such that a Þ 0.
Discriminant of a quadratic
equation (p. 294)
The expression b2 2 4ac is called the discriminant of the associated equation
ax 2 1 bx 1 c 5 0. The value of the discriminant can be positive, zero, or
negative, which corresponds to an equation having two real solutions, one real
solution, or two imaginary solutions, respectively.
Special product patterns
Sum and difference:
(a 1 b)(a 2 b) 5 a2 2 b2
(p. 347)
Square of a binomial:
(a 1 b)2 5 a2 1 2ab 1 b2
(a 2 b)2 5 a2 2 2ab 1 b2
Cube of a binomial:
(a 1 b) 3 5 a 3 1 3a2b 1 3ab2 1 b3
(a 2 b) 3 5 a 3 2 3a2b 1 3ab2 2 b3
Sum of two cubes:
a 3 1 b3 5 (a 1 b)(a2 2 ab 1 b2)
(p. 354)
Difference of two cubes:
a 3 2 b3 5 (a 2 b)(a2 1 ab 1 b2)
Remainder theorem (p. 363)
If a polynomial f(x) is divided by x 2 k, then the remainder is r 5 f(k).
Factor theorem (p. 364)
A polynomial f(x) has a factor x 2 k if and only if f(k) 5 0.
Rational zero theorem (p. 370)
If f(x) 5 anx n 1 . . . 1 a1x 1 a 0 has integer coefficients, then every rational zero
of f has this form:
p
TABLES
Special factoring patterns
factor of constant term a0
}
q 5 }}}
factor of leading coefficient an
Fundamental theorem of
algebra (p. 379)
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) 5 0 has at
least one solution in the set of complex numbers.
Corollary to the fundamental
theorem of algebra (p. 379)
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) 5 0
has exactly n solutions provided each solution repeated twice is counted as
2 solutions, each solution repeated three times is counted as 3 solutions,
and so on.
Complex conjugates theorem
If f is a polynomial function with real coefficients, and a 1 bi is an imaginary
zero of f, then a 2 bi is also a zero of f.
(p. 380)
Irrational conjugates
theorem (p. 380)
Suppose f is a polynomial function
with rational coefficients,
and a and b are}
}
}
rational numbers such that Ï b is irrational. If a 1 Ïb is a zero of f, then a 2 Ïb
is also a zero of f.
Descartes’ rule of signs
Let f(x) 5 anx n 1 an 2 1x n 2 1 1 . . . 1 a2x 2 1 a1x 1 a 0 be a polynomial function
with real coefficients.
(p. 381)
• The number of positive real zeros of f is equal to the number of changes in
sign of the coefficients of f(x) or is less than this by an even number.
• The number of negative real zeros of f is equal to the number of changes in
sign of the coefficients of f(2x) or is less than this by an even number.
Tables
n2pe-9040.indd 1027
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10/14/05 11:06:04 AM
Formulas and Theorems from Algebra (continued)
Discriminant of a general
second-degree equation
(p. 653)
Any conic can be described by a general second-degree equation in x
and y: Ax 2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0. The expression B2 2 4AC is
the discriminant of the conic equation and can be used to identify it.
Discriminant
Type of Conic
B2 2 4AC < 0, B 5 0, and A 5 C
Circle
2
B 2 4AC < 0, and either B Þ 0 or A Þ C
Ellipse
B2 2 4AC 5 0
Parabola
2
B 2 4AC > 0
Hyperbola
If B 5 0, each axis of the conic is horizontal or vertical.
Formulas from Combinatorics
Fundamental counting principle
(p. 682)
If one event can occur in m ways and another event can occur in n
ways, then the number of ways that both events can occur is m p n.
Permutations of n objects taken r at
a time (p. 685)
The number of permutations of r objects taken from a group of n
distinct objects is denoted by nPr and is given by:
n!
P 5}
TABLES
n r
Permutations with repetition
(p. 685)
(n 2 r)!
The number of distinguishable permutations of n objects where one
object is repeated s1 times, another is repeated s 2 times, and so on is:
n!
s1! p s2! p . . . p sk!
}}
Combinations of n objects taken r at
a time (p. 690)
The number of combinations of r objects taken from a group of n
distinct objects is denoted by nCr and is given by:
n!
C 5}
(n 2 r)! p r!
n r
Pascal’s triangle (p. 692)
If you arrange the values of nCr in a triangular pattern in which each
row corresponds to a value of n, you get what is called Pascal’s triangle.
C
1
0 0
C
C
1 0
C
C
2 0
C
C
4 0
C
2 1
C
3 0
C
C
1
C
3 2
4 2
1
2 2
C
3 1
4 1
1
1 1
3 3
C
4 3
1
C
4 4
1
2
3
4
1
3
6
1
4
1
The first and last numbers in each row are 1. Every number other than
1 is the sum of the closest two numbers in the row directly above it.
Binomial theorem (p. 693)
The binomial expansion of (a 1 b)n for any positive integer n is:
(a 1 b) n 5 C anb 0 1 C an 2 1b1 1 C an 2 2b2 1 . . . 1 C a 0bn
n 0
n
5
∑
r50
n 1
n 2
n n
C a n 2 r br
n r
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Formulas from Probability
Theoretical probability of
an event (p. 698)
When all outcomes are equally likely, the theoretical probability that an
event A will occur is:
Number of outcomes in A
P(A) 5 }}}
Total number of outcomes
Odds in favor of an event
When all outcomes are equally likely, the odds in favor of an event A are:
(p. 699)
Number of outcomes in A
Number of outcomes not in A
}}}
Odds against an event
When all outcomes are equally likely, the odds against an event A are:
(p. 699)
Number of outcomes not in A
}}}
Number of outcomes in A
Experimental probability
of an event (p. 700)
When an experiment is performed that consists of a certain number of trials,
the experimental probability of an event A is given by:
of trials where A occurs
P(A) 5 Number
}}}
Total number of trials
Probability of compound
events (p. 707)
If A and B are any two events, then the probability of A or B is:
P(A or B) 5 P(A) 1 P(B) 2 P(A and B)
If A and B are disjoint events, then the probability of A or B is:
P(A or B) 5 P(A) 1 P(B)
The probability of the complement of event A, denoted }
A, is:
P(}
A) 5 1 2 P(A)
Probability of independent
events (p. 717)
If A and B are independent, the probability that both A and B occur is:
P(A and B) 5 P(A) p P(B)
Probability of dependent
events (p. 718)
If A and B are dependent, the probability that both A and B occur is:
P(A and B) 5 P(A) p P(BA)
Binomial probabilities
For a binomial experiment consisting of n trials where the probability of
success on each trial is p, the probability of exactly k successes is:
P(k successes) 5 nCk p k (1 2 p) n 2 k
(p. 725)
TABLES
Probability of the complement
of an event (p. 709)
Formulas from Statistics
...
Mean of a data set
(p. 744)
Standard deviation of a
data set (p. 745)
1 x2 1
1 xn
1
}x 5 x}}
where }
x (read “x-bar”) is the mean of the data x1, x 2, . . . , xn
n
s5
Î
}}}}
} )2 1 (x2 2 x} )2 1 . . . 1 (xn 2 x} )2
(x1 2 x
}} where s (read “sigma”) is the standard
n
deviation of the data x1, x 2, . . . , xn
Areas under a normal
curve (p. 757)
A normal distribution with mean }
x and standard deviation s has these properties:
z-score (p. 758)
2x
}
z 5 x}
s where x is a data value, x is the mean, and s is the standard deviation
•
•
•
•
The total area under the related normal curve is 1.
About 68% of the area lies within 1 standard deviation of the mean.
About 95% of the area lies within 2 standard deviations of the mean.
About 99.7% of the area lies within 3 standard deviations of the mean.
}
Tables
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Formulas for Sequences and Series
n
Formulas for sums of special series
n
∑15n
n
1 1)
∑ i 5 n(n
}
2
∑
i51
n(n 1 1)(2n 1 1)
6
i2 5 }}
(p. 797)
i51
Explicit rule for an arithmetic
sequence (p. 802)
The nth term of an arithmetic sequence with first term a1 and common
difference d is:
an 5 a1 1 (n 2 1)d
Sum of a finite arithmetic series
The sum of the first n terms of an arithmetic series is:
(p. 804)
i51
1
a 1a
1
n
Sn 5 n }
2
2
Explicit rule for a geometric
sequence (p. 810)
The nth term of a geometric sequence with first term a1 and
common ratio r is:
an 5 a1r n 2 1
Sum of a finite geometric series
The sum of the first n terms of a geometric series with common ratio
r Þ 1 is:
(p. 812)
1 2 rn
Sn 5 a1 }
1 12r 2
Sum of an infinite geometric series
(p. 821)
The sum of an infinite geometric series with first term a1 and common
ratio r is
a
TABLES
1
S5}
12r
provided r  < 1. If r  ≥ 1, the series has no sum.
Recursive equation for an
arithmetic sequence (p. 827)
an 5 an 2 1 1 d where d is the common difference
Recursive equation for a geometric
sequence (p. 827)
an 5 r p an 2 1 where r is the common ratio
Formulas and Identities from Trigonometry
Conversion between degrees
and radians (p. 860)
p radians .
To rewrite a degree measure in radians, multiply by }
180°
180° .
To rewrite a radian measure in degrees, multiply by }
p radians
Definition of trigonometric
functions (p. 866)
Let u be an angle in standard position and (x, y) be any point
}
(except the origin) on the terminal side of u. Let r 5 Ïx 2 1 y 2 .
y
Law of sines (p. 882)
y
sin u 5 }r
cos u 5 }xr
tan u 5 }x , x Þ 0
csc u 5 }yr , y Þ 0
sec u 5 }xr , x Þ 0
cot u 5 }xy , y Þ 0
If n ABC has sides of length a, b, and c, then:
sin A
a
sin B
b
sin C
c
}5}5}
Area of a triangle (given two
sides and the included angle)
(p. 885)
If n ABC has sides of length a, b, and c, then its area is:
1
bc sin A
Area 5 }
2
1
Area 5 }
ac sin B
2
1
Area 5 }
ab sin C
2
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Formulas and Identities from Trigonometry (continued)
Law of cosines (p. 889)
If n ABC has sides of length a, b, and c, then:
a2 5 b2 1 c 2 2 2bc cos A
b2 5 a2 1 c 2 2 2ac cos B
c 2 5 a2 1 b2 2 2ab cos C
Heron’s area formula (p. 891)
The area of the triangle with sides of length a, b, and c is
}}
Area 5 Ïs(s 2 a)(s 2 b)(s 2 c)
1
(a 1 b 1 c).
where s 5 }
2
Reciprocal identities (p. 924)
1
csc u 5 }
sin u
1
sec u 5 }
cos u
Tangent and cotangent identities
sin u
tan u 5 }
cos u
cos u
cot u 5 }
sin u
Pythagorean identities (p. 924)
sin 2 u 1 cos2 u 5 1
1 1 tan 2 u 5 sec2 u
1 1 cot 2 u 5 csc2 u
Cofunction identities (p. 924)
π
sin 1 }
2 u 2 5 cos u
π
cos 1 }
2 u 2 5 sin u
π
tan 1 }
2 u 2 5 cot u
Negative angle identities (p. 924)
sin (2u) 5 2sin u
cos (2u) 5 cos u
tan (2u) 5 2tan u
(p. 924)
2
2
TABLES
Sum formulas (p. 949)
2
1
cot u 5 }
tan u
sin (a 1 b) 5 sin a cos b 1 cos a sin b
cos (a 1 b) 5 cos a cos b 2 sin a sin b
tan a 1 tan b
tan (a 1 b) 5 }}
1 2 tan a tan b
Difference formulas (p. 949)
sin (a 2 b) 5 sin a cos b 2 cos a sin b
cos (a 2 b) 5 cos a cos b 1 sin a sin b
tan a 2 tan b
tan (a 2 b) 5 }}
1 1 tan a tan b
Double-angle formulas (p. 955)
cos 2a 5 cos2 a 2 sin 2 a
sin 2a 5 2 sin a cos a
cos 2a 5 2 cos2 a 2 1
2 tan a
tan 2a 5 }
2
1 2 tan a
cos 2a 5 1 2 2 sin 2 a
}
Half-angle formulas (p. 955)
a
2 cos a
sin }
5 6 1}
2
Ï
2
}
a
1 cos a
cos }
5 6 1}
2
Ï
2
a
2 cos a
tan }
5 1}
2
sin a
a
sin a
tan }
5}
2
1 1 cos a
a
a
a
and cos }
depend on the quadrant in which }
lies.
The signs of sin }
2
2
2
Tables
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Formulas from Geometry
Basic geometric figures
Area of an equilateral
triangle
See pages 991–993 for area formulas for basic two-dimensional geometric figures.
}
s
Ï3 2
s where s is the length of a side
Area 5 }
s
4
s
Arc length and area of
a sector
Arc length 5 ru where r is the radius and u is
the radian measure of the central angle
that intercepts the arc
1 2
Area 5 }
r u
2
Area of an ellipse
sector
r
arc
length
s
central
angle u
Area 5 πab where a and b are half the lengths
of the major and minor axes of the ellipse
b
a
TABLES
Volume and surface area
of a right rectangular
prism
Volume 5 lwh where l is the length, w is the
width, and h is the height
h
Surface area 5 2(lw 1 wh 1 lh)
w
l
Volume and surface
area of a right cylinder
Volume 5 πr 2h where r is the base radius and h is the height
Lateral surface area 5 2πrh
h
Surface area 5 2πr 2 1 2πrh
Volume and surface
area of a right regular
pyramid
1
Bh where B is the area of the base and h is the height
Volume 5 }
3
l
1
nsl where n is the number
Lateral surface area 5 }
2
of sides of the base, s is the length of a side of the
base, and l is the slant height
1
nsl
Surface area 5 B 1 }
2
Volume and surface area
of a right circular cone
r
h
s
B
s
1 2
πr h where r is the base radius and h is the height
Volume 5 }
3
Lateral surface area 5 πrl where l is the slant height
2
Surface area 5 πr 1 πrl
l
h
r
Volume and surface area
of a sphere
4 3
Volume 5 }
πr where r is the radius
3
Surface area 5 4πr 2
r
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Properties
Properties of Real Numbers
Let a, b, and c be real numbers.
Addition
Multiplication
Closure Property (p. 3)
a 1 b is a real number.
ab is a real number.
Commutative Property (p. 3)
a1b5b1a
ab 5 ba
Associative Property (p. 3)
(a 1 b) 1 c 5 a 1 (b 1 c)
(ab)c 5 a(bc)
Identity Property (p. 3)
a 1 0 5 a, 0 1 a 5 a
a p 1 5 a, 1 p a 5 a
Inverse Property (p. 3)
a 1 (2a) 5 0
1
ap}
a 5 1, a Þ 0
Distributive Property (p. 3)
The distributive property involves both addition and multiplication:
a(b 1 c) 5 ab 1 ac
Zero Product Property (p. 253)
Let A and B be real numbers or algebraic expressions.
If AB 5 0, then A 5 0 or B 5 0.
Properties of Matrices
Let A, B, and C be matrices, and let k be a scalar.
(A 1 B) 1 C 5 A 1 (B 1 C)
Commutative Property of Addition (p. 188)
A1B5B1A
Distributive Property of Addition (p. 188)
k(A 1 B) 5 kA 1 kB
Distributive Property of Subtraction (p. 188)
k(A 2 B) 5 kA 2 kB
Associative Property of Matrix Multiplication (p. 197)
(AB)C 5 A(BC)
Left Distributive Property of Matrix Multiplication (p. 197)
A(B 1 C) 5 AB 1 AC
Right Distributive Property of Matrix Multiplication (p. 197)
(A 1 B)C 5 AC 1 BC
Associative Property of Scalar Multiplication (p. 197)
TABLES
Associative Property of Addition (p. 188)
k(AB) 5 (kA)B 5 A(kB)
Multiplicative Identity (p. 210)
An n 3 n matrix with 1’s on the main diagonal
and 0’s elsewhere is an identity matrix, denoted I.
For any n 3 n matrix A, AI 5 IA 5 A.
Inverse Matrices (p. 210)
If the determinant of an n 3 n matrix A is
nonzero, then A has an inverse, denoted A21, such
that AA21 5 A21 A 5 I.
Properties of Exponents
Let a and b be real numbers, and let m and n be integers.
Product of Powers Property (p. 330)
am p an 5 am 1 n
Power of a Power Property (p. 330)
(am ) n 5 amn
Power of a Product Property (p. 330)
(ab) m 5 ambm
Negative Exponent Property (p. 330)
1 ,aÞ0
a2m 5 }
m
a
0
Zero Exponent Property (p. 330)
a 5 1, a Þ 0
Quotient of Powers Property (p. 330)
}
n 5a
Power of a Quotient Property (p. 330)
1 }b 2
am
a
a m
m2n
,aÞ0
m
a
5}
m, b Þ 0
b
Tables
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Properties of Radicals and Rational Exponents
Number of Real nth Roots
Let n be an integer greater than 1, and let a be a real number.
(p. 414)
Radicals and Rational
Exponents (p. 415)
•
•
•
•
n}
If n is odd, then a has one real nth root: Ï a 5 a1/n
n}
If n is even and a > 0, then a has two real nth roots: 6 Ï a 5 6a1/n
}
n
If n is even and a 5 0, then a has one nth root: Ï0 5 01/n 5 0
If n is even and a < 0, then a has no real nth roots.
Let a1/n be an nth root of a, and let m be a positive integer.
n}
• am/n 5 (a1/n ) m 5 1 Ï a 2m
1
1
1
• a2m/n 5 }
5}
5}
,aÞ0
n} m
m/n
1 a1/n 2m
a
1 Ïa 2
Properties of Rational
Exponents (p. 420)
All of the properties of exponents listed on the previous page apply to rational
exponents as well as integer exponents.
Product and Quotient
Properties of Radicals (p. 421)
Let n be an integer greater than 1, and let a and b be positive real
n}
n}
n}
numbers. Then Ïa p b 5 Ï a p Ïb and
n
}
n}
Ïa
a
}5}
n }.
Ïb
Ïb
Properties of Logarithms
TABLES
Let a, b, c, m, n, x, and y be positive real numbers such that
b Þ 1 and c Þ 1.
Logarithms and Exponents (p. 499)
log b y 5 x if and only if b x 5 y
Special Logarithm Values (p. 499)
log b 1 5 0 because b 0 5 1 and log b b 5 1 because b1 5 b
Common and Natural Logarithms (p. 500)
log10 x 5 log x and loge x 5 ln x
Product Property of Logarithms (p. 507)
log b mn 5 log b m 1 log b n
Quotient Property of Logarithms (p. 507)
m
log b }
n 5 log b m 2 log b n
Power Property of Logarithms (p. 507)
log b mn 5 n log b m
Change of Base (p. 508)
logc a 5 }
logb a
logb c
Properties of Functions
Operations on Functions
(pp. 428, 430)
Let f and g be any two functions. A new function h can be defined using
any of the following operations.
Addition:
Subtraction:
Multiplication:
h(x) 5 f(x) 1 g(x)
h(x) 5 f(x) 2 g(x)
h(x) 5 f(x) p g(x)
Division:
h(x) 5 }
Composition:
h(x) 5 g(f(x))
f(x)
g(x)
For addition, subtraction, multiplication, and division, the domain
of h consists of the x-values that are in the domains of both f and g.
Additionally, the domain of the quotient does not include x-values for
which g(x) 5 0.
For composition, the domain of h is the set of all x-values such that x is in
the domain of f and f(x) is in the domain of g.
Inverse Functions (p. 438)
Functions f and g are inverses of each other provided:
f(g(x)) 5 x and g(f(x)) 5 x
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English–Spanish Glossary
A
2
2
2
2
absolute value (p. 51) The absolute value of a number x,
represented by the symbol x, is the distance the number is
from 0 on a number line.
}3 5 }3, 24.3 5 4.3, and 0 5 0.
valor absoluto (pág. 51) El valor absoluto de un número x,
representado por el símbolo x, es la distancia a la que está
el número de 0 en una recta numérica.
}3 5 }3, 24.3 5 4.3 y 0 5 0.
absolute value function (p. 123) A function that contains
an absolute value expression.
y 5 x, y 5 x 2 3, and y 5 4x 1 8 – 9 are
absolute value functions.
función de valor absoluto (pág. 123) Función que contiene
una expresión de valor absoluto.
y 5 x, y 5 x 2 3e y 5 4x 1 8 – 9 son
funciones de valor absoluto.
absolute value of a complex number (p. 279) If z 5 a 1 bi,
then the absolute value of z, denoted z, is a nonnegative
}
real number defi ned as z 5 Ïa2 1 b2 .
}
}
2
2
24 1 3i 5 Ï (24) 1 3 5 Ï25 5 5
valor absoluto de un número complejo (pág. 279) Si
z 5 a 1 bi, entonces el valor absoluto de z, denotado por z,
}
es un número real no negativo defi nido como z 5 Ï a2 1 b2 .
expresión algebraica (pág. 11) Expresión formada por
números, variables, operaciones y signos de agrupación.
2
3
8
72r
2
}p, }, k – 5, and n + 2n are algebraic
expressions.
2
3
8
72r
2
}p, }, k – 5 y n 1 2n son expresiones
algebraicas.
amplitude (p. 908) The amplitude of the graph of a sine or
4
1
cosine function is }
(M – m), where M is the maximum value
2
y
M54
of the function and m is the minimum value of the function.
π
2
amplitud (pág. 908) La amplitud de la grafica de una
3π
2
1
función seno o coseno es }
(M – m), donde M es el valor
2
x
m 5 24
máximo de la función y m es el valor mínimo de la función.
The graph of y 5 4 sin x has an amplitude of
ENGLISH-SPANISH GLOSSARY
algebraic expression (p. 11) An expression that consists of
numbers, variables, operations, and grouping symbols. Also
called variable expression.
1
2
}(4 – (–4)) 5 4.
La gráfica de y 5 4 sen x tiene una amplitud
de }1 (4 – (–4)) 5 4.
2
angle of depression (p. 855) The angle by which an
observer’s line of sight must be depressed from the
horizontal to the point observed.
See angle of elevation.
ángulo de depresión (pág. 855) El ángulo con el que se debe
bajar la línea de visión de un observador desde la horizontal
hasta el punto observado.
Ver ángulo de elevación.
English-Spanish Glossary
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angle of elevation (p. 855) The angle by which an observer’s
line of sight must be elevated from the horizontal to the
point observed.
angulo
´
de
depresión
ángulo de elevación (pág. 855) El ángulo con el que se debe
elevar la línea de visión de un observador desde la horizontal
hasta el punto observado.
´
angulo
de
elevación
angle of
depression
angle of
elevation
arithmetic sequence (p. 802) A sequence in which the
difference of consecutive terms is constant.
24, 1, 6, 11, 16, . . . is an arithmetic sequence
with common difference 5.
progresión aritmética (pág. 802) Progresión en la que la
diferencia entre los términos consecutivos es constante.
24, 1, 6, 11, 16, . . . es una progresión
aritmética con una diferencia común de 5.
arithmetic series (p. 804) The expression formed by adding
the terms of an arithmetic sequence.
serie aritmética (pág. 804) La expresión formada al sumar
los términos de una progresión aritmética.
5
∑ 2i 5 2 1 4 1 6 1 8 1 10
i=1
asymptote (p. 478) A line that a graph approaches more and
more closely.
asíntota (pág. 478) Recta a la que se aproxima una gráfica
cada vez más.
y
asymptote
asíntota
1
ENGLISH-SPANISH GLOSSARY
1
x
The asymptote for the graph shown is the line
y 5 3.
La asíntota para la gráfica que se muestra es la
recta y 5 3.
axis of symmetry of a parabola (pp. 236, 620) The line
perpendicular to the parabola’s directrix and passing
through its focus.
See parabola.
eje de simetría de una parábola (págs. 236, 620) La recta
perpendicular a la directriz de la parábola y que pasa por su
foco.
Ver parábola.
B
base of a power (p. 10) The number or expression that is
used as a factor in a repeated multiplication.
In the power 25, the base is 2.
base de una potencia (pág. 10) El número o la expresión
que se usa como factor en la multiplicación repetida.
En la potencia 25, la base es 2.
1036 Student Resources
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best-fitting line (p. 114) The line that lies as close as
possible to all the data points in a scatter plot.
mejor recta de regresión (pág. 114) La recta que se ajusta lo
más posible a todos los puntos de datos de un diagrama de
dispersión.
best-fitting quadratic model (p. 311) The model given by
using quadratic regression on a set of paired data.
600
500
modelo cuadrático con mejor ajuste (pág. 311) El modelo
dado al realizar una regresión cuadrática sobre un conjunto
de pares de datos.
400
300
200
100
0
0
10
20
30
40
50
60
70
80
“Would you rather see an exciting laser show
or a boring movie?” is a biased question.
pregunta capciosa (pág. 772) Pregunta que no refleja con
exactitud las opiniones o acciones de los encuestados.
“¿Preferirías ver un emocionante
espectáculo de láser o una película
aburrida?” es una pregunta capciosa.
biased sample (p. 767) A sample that overrepresents or
underrepresents part of a population.
The members of a school’s basketball team
would form a biased sample for a survey
about whether to build a new gym.
muestra sesgada (pág. 767) Muestra que representa de
forma excesiva o insuficiente a parte de una población.
Los miembros del equipo de baloncesto de
una escuela formarían una muestra sesgada
si participaran en una encuesta sobre si
quieren que se construya un nuevo gimnasio.
binomial (p. 252) The sum of two monomials.
3x 2 1 and t 3 2 4t are binomials.
binomio (pág. 252) La suma de dos monomios.
3x 2 1 y t 3 2 4t son binomios.
distribución binomial (pág. 725) La distribución de
probabilidades asociada a un experimento binomial.
0.30
Probability
Probabilidad
binomial distribution (p. 725) The probability distribution
associated with a binomial experiment.
0.20
ENGLISH-SPANISH GLOSSARY
biased question (p. 772) A question that does not accurately
reflect the opinions or actions of the people surveyed.
0.10
0
0 1 2 3 4 5 6 7 8
Number of successes
Número de éxitos
Binomial distribution for 8 trials with p 5 0.5.
Distribución binomial de 8 ensayos con p 5 0.5.
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binomial experiment (p. 725) An experiment that meets
the following conditions. (1) There are n independent trials.
(2) Each trial has only two possible outcomes: success and
failure. (3) The probability of success is the same for each
trial.
experimento binomial (pág. 725) Experimento que
satisface las siguientes condiciones. (1) Hay n pruebas
independientes. (2) Cada prueba tiene sólo dos resultados
posibles: éxito y fracaso. (3) La probabilidad de éxito es igual
para cada prueba.
binomial theorem (p. 693) The binomial expansion of
(a 1 b) n for any positive integer n:
(a 1 b) n 5 nC 0 anb 0 1 nC1an – 1b1 1 nC2an – 2b2 1 . . . 1 nCna 0bn .
teorema binomial (pág. 693) La expansión binomial de
(a 1 b) n para cualquier número entero positivo n:
(a 1 b) n 5 nC 0 anb 0 1 nC1an – 1b1 1 nC2an – 2b2 1 . . . 1 nCna 0bn .
A fair coin is tossed 12 times. The probability
of getting exactly 4 heads is as follows:
Una moneda normal se lanza 12 veces. La
probabilidad de sacar exactamente 4 caras
es la siguiente:
C p k (1 – p) n – k
5 12C4 (0.5)4 (1 – 0.5) 8
5 495(0.5)4 (0.5) 8
ø 0.121
P(k 5 4) 5
n k
(x 2 1 y) 3
5 3C 0 (x 2 ) 3y 0 1 3C1(x 2 )2y1 1 3C 2 (x 2 )1y 2 1
C (x 2 ) 0y 3
3 3
5 (1)(x 6 )(1) 1 (3)(x 4)(y) 1 (3)(x 2 )(y 2 ) 1
(1)(1)(y 3)
5 x 6 1 3x 4y 1 3x 2y 2 1 y 3
ENGLISH-SPANISH GLOSSARY
C
center of a circle (p. 626) See circle.
The circle with equation (x – 3)2 1 (y 1 5)2 5
36 has its center at (3, –5). See also circle.
centro de un círculo (pág. 626) Ver círculo.
El círculo con la ecuación (x – 3)2 1 (y 1 5)2 5
36 tiene el centro en (3, –5). Ver también
círculo.
center of a hyperbola (p. 642) The midpoint of the
transverse axis of a hyperbola.
See hyperbola.
centro de una hipérbola (pág. 642) El punto medio del eje
transverso de una hipérbola.
Ver hipérbola.
center of an ellipse (p. 634) The midpoint of the major axis
of an ellipse.
See ellipse.
centro de una elipse (pág. 634) El punto medio del eje
mayor de una elipse.
Ver elipse.
central angle (p. 861) An angle formed by two radii of a
circle.
See sector.
ángulo central (pág. 861) Ángulo formado por dos radios de
un círculo.
Ver sector.
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circle (p. 626) The set of all points (x, y) in a plane that are of
distance r from a fi xed point, called the center of the circle.
círculo (pág. 626) El conjunto de todos los puntos (x, y) de un
plano que están a una distancia r de un punto fijo, llamado
centro del círculo.
y
center
centro
r
(x, y)
x
circle
círculo
x2 1 y2 5 r 2
coefficient (p. 12) When a term is the product of a number
and a power of a variable, the number is the coefficient of the
power.
In the algebraic expression
2x 2 1 (24x) 1 (21), the coefficient of 2x 2 is 2
and the coefficient of 24x is 24.
coeficiente (pág. 12) Cuando un término es el producto de
un número y una potencia de una variable, el número es el
coeficiente de la potencia.
En la expresión algebraica
2x 2 1 (24x) 1 (21), el coeficiente de 2x 2 es 2
y el coeficiente de 24x es 24.
coefficient matrix (p. 205) The coefficient matrix of the
linear system ax 1 by 5 e, cx 1 dy 5 f is a b .
c d
F G
matriz coeficiente (pág. 205) La matriz coeficiente del
sistema lineal ax 1 by 5 e, cx 1 dy 5 f es a b .
c d
F G
n!
where nCr 5 }
.
coefficient matrix:
matriz coeficiente:
F 93 254 G
matrix of constants:
matriz de constantes:
26
F 221
G
matrix of variables:
matriz de variables:
Fxy G
There are 6 combinations of the n 5 4 letters
A, B, C, and D selected r 5 2 at a time: AB, AC,
AD, BC, BD, and CD.
(n 2 r)! p r!
combinación (pág. 690) Selección de r objetos de un grupo
de n objetos en el que el orden no importa, denotado nCr,
n!
donde nCr 5 }
.
Hay 6 combinaciones de las letras n 5 4 A, B,
C y D seleccionadas r 5 2 cada vez: AB, AC,
AD, BC, BD y CD.
(n 2 r)! p r!
common difference (p. 802) The constant difference of
consecutive terms of an arithmetic sequence.
See arithmetic sequence.
diferencia común (pág. 802) La diferencia constante entre
los términos consecutivos de una progresión aritmética.
Ver progresión aritmética.
common logarithm (p. 500) A logarithm with base 10. It is
denoted by log10 or simply by log.
log10 100 5 log 100 5 2 because 102 5 100.
logaritmo común (pág. 500) Logaritmo con base 10. Se
denota por log10 ó simplemente por log.
log10 100 5 log 100 5 2 ya que 102 5 100.
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combination (p. 690) A selection of r objects from a group of
n objects where the order is not important, denoted nCr
9x 1 4y 5 26
3x 2 5y 5 221
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common ratio (p. 810) The constant ratio of consecutive
terms of a geometric sequence.
See geometric sequence.
razón común (pág. 810) La razón constante entre los
términos consecutivos de una progresión geométrica.
Ver progresión geométrica.
complement of a set (p. 715) The complement of a set A,
}
written A , is the set of all elements in the universal set U that
are not in A.
Let U be the set of all integers from 1 to 10
}
and let A 5 {1, 2, 4, 8}. Then A 5 {3, 5, 6, 7, 9,
10}.
complemento de un conjunto (pág. 715) El complemento
}
de un conjunto A, escrito A, es el conjunto de todos los
elementos del conjunto universal U que no están en A.
Sea U el conjunto de todos los números
enteros entre 1 y 10 y sea A 5 {1, 2, 4, 8}. Por
}
lo tanto, A 5 {3, 5, 6, 7, 9, 10}.
completing the square (p. 284) The process of adding a
term to a quadratic expression of the form x2 1 bx to make it
a perfect square trinomial.
To complete the square for x 2 1 16x, add
completar el cuadrado (pág. 284) El proceso de sumar un
término a una expresión cuadrática de la forma x2 1 bx, de
modo que sea un trinomio cuadrado perfecto.
Para completar el cuadrado para x 2 1 16x,
complex conjugates (p. 276) Two complex numbers of the
form a 1 bi and a 2 bi.
ENGLISH-SPANISH GLOSSARY
números complejos conjugados (pág. 276) Dos números
complejos de la forma a 1 bi y a 2 bi.
complex fraction (p. 584) A fraction that contains a fraction
in its numerator or denominator.
fracción compleja (pág. 584) Fracción que tiene una
fracción en su numerador o en su denominador.
complex number (p. 276) A number a 1 bi where a and b
are real numbers and i is the imaginary unit.
número complejo (pág. 276) Un número a 1 bi, donde a y b
son números reales e i es la unidad imaginaria.
complex plane (p. 278) A coordinate plane in which
each point (a, b) represents a complex number a 1 bi. The
horizontal axis is the real axis and the vertical axis is the
imaginary axis.
plano complejo (pág. 278) Plano de coordenadas en el
que cada punto (a, b) representa un número complejo
a 1 bi. El eje horizontal es el eje real, y el eje vertical es el eje
imaginario.
composition of functions (p. 430) The composition of a
function g with a function f is h(x) 5 g(f(x)).
composición de funciones (pág. 430) La composición de
una función g con una función f es h(x) 5 g(f(x)).
2
16
1}
2 2
5 64: x 2 1 16x 1 64 5 (x 1 8)2 .
2
16
suma 1 }
2 5 64: x2 1 16x 1 64 5 (x 1 8)2.
2
2 1 4i, 2 2 4i
5
x14
}
1
},}
1 1
6x
}
}
p1}
q
2
3x
}
0, 2.5, Ï3 , π, 5i, 2 2 i
imaginary
imaginario
22 1 4i
3i
i
1
real
real
3 2 2i
24 2 3i
f(x) 5 5x 2 2, g(x) 5 4x21
4
2
g(f(x)) 5 g(5x 2 2) 5 4(5x 2 2) 21 5 }
,xÞ}
5x 2 2
5
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compound event (p. 707) The union or intersection of two
events.
When you roll a six-sided die, the event “roll a
2 or an odd number” is a compound event.
suceso compuesto (pág. 707) La unión o la intersección de
dos sucesos.
Cuando lanzas un cubo numerado de seis
lados, el suceso “salir el 2 ó un número impar”
es un suceso compuesto.
compound inequality (p. 41) Two simple inequalities
joined by “and” or “or.”
2x > 0 or x 1 4 < 21 is a compound inequality.
desigualdad compuesta (pág. 41) Dos desigualdades
simples unidas por “y” u “o”.
2x > 0 ó x 1 4 < 21 es una desigualdad
compuesta.
conditional probability (p. 718) The conditional probability
of B given A, written P(B | A), is the probability that event B
will occur given that event A has occurred.
Two cards are randomly selected from a
standard deck of 52 cards. Let event A be
“the fi rst card is a club” and let event B be
“the second card is a club.” Then
12
4
5}
because there are 12 (out of
P(B | A) 5 }
51
17
13) clubs left among the remaining 51 cards.
probabilidad condicional (pág. 718) La probabilidad
condicional de B dado A, escrito P(B | A), es la probabilidad
de que ocurra el suceso B dado que ha ocurrido el suceso A.
Dos cartas se seleccionan al azar de una
baraja normal de 52 cartas. Sea el suceso
A “la primera carta es de tréboles” y sea el
suceso B “la segunda carta es de tréboles”.
12
4
Entonces P(B | A) 5 }
5}
ya que quedan
51
17
12 (del total de 13) cartas de tréboles entre
las 51 cartas restantes.
See conic section.
cónica (pág. 650) Ver sección cónica.
Ver sección cónica.
conic section (p. 650) A curve formed by the intersection of
a plane and a double-napped cone. Conic sections are also
called conics.
See circle, ellipse, hyperbola, and parabola.
sección cónica (pág. 650) Una curva formada por la
intersección de un plano y un cono. Las secciones cónicas
también se llaman cónicas.
Ver círculo, elipse, hipérbola y parábola.
}
}
conjugates (p. 267) The expressions a 1 Ïb and a 2 Ïb
where a and b are rational numbers.
}
}
}
ENGLISH-SPANISH GLOSSARY
conic (p. 650) See conic section.
}
The conjugate of 7 1 Ï2 is 7 2 Ï2 .
}
}
El conjugado de 7 1 Ï2 es 7 2 Ï2 .
conjugados (pág. 267) Las expresiones a 1 Ïb y a 2 Ï b
cuando a y b son números racionales.
consistent system (p. 154) A system of equations that has at
least one solution.
sistema compatible (pág. 154) Sistema de ecuaciones que
tiene al menos una solución.
y 5 2 1 3x
6x 1 2y 5 4
The system above is consistent, with solution
(0, 2).
El sistema de arriba es compatible, con la
solución (0, 2).
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constant of variation (pp. 107, 551, 553) The nonzero
constant a in a direct variation equation y 5 ax, an inverse
a , or a joint variation equation
variation equation y 5 }
x
z 5 axy.
constante de variación (págs. 107, 551, 553) La constante
distinta de cero a de una ecuación de variación directa
ENGLISH-SPANISH GLOSSARY
a o de una
y 5 ax, de una ecuación de variación inversa y 5 }
x
ecuación de variación conjunta z 5 axy.
5
In the direct variation equation y 5 – }
x, the
2
5
constant of variation is – }
.
2
5
En la ecuación de variación directa y 5 – }
x,
2
5
.
la constante de variación es – }
2
constant term (pp. 12, 337) A term that has a number part
but no variable part.
The constant term of the algebraic
expression 3x 2 1 5x 1 (27) is 27.
término constante (págs. 12, 337) Término que tiene una
parte numérica pero sin variable.
El término constante de la expresión
algebraica 3x 2 1 5x 1 (27) es 27.
constraints (p. 174) In linear programming, the linear
inequalities that form a system.
See linear programming.
restricciones (pág. 174) En la programación lineal, las
desigualdades lineales que forman un sistema.
Ver programación lineal.
continuous function (p. 80) A function whose graph is
unbroken.
Any linear function, such as y 5 2x 1 4, is a
continuous function.
función continua (pág. 80) Función que tiene una gráfica
no interrumpida.
Cualquier función lineal, como y 5 2x 1 4, es
una función continua.
control group (p. 773) A group that does not undergo a
procedure or treatment when an experiment is conducted.
See also experimental group.
See experimental group.
grupo de control (pág. 773) Grupo que no se somete a
ningún procedimiento o tratamiento durante la realización
de un experimento. Ver también grupo experimental.
Ver grupo experimental.
correlation coefficient (p. 114) A measure, denoted by r
where –1 ≤ r ≤ 1, of how well a line fits a set of data pairs (x, y).
A data set that shows a strong positive
correlation has a correlation coefficient of
r ≈ 1. See also positive correlation and
negative correlation.
coeficiente de correlación (pág. 114) Medida denotada
por r, donde –1 ≤ r ≤ 1, y que describe el ajuste de una recta
a un conjunto de pares de datos (x, y).
Un conjunto de datos que muestra una
correlación positiva fuerte tiene un
coeficiente de correlación de r ≈ 1. Ver
también correlación positiva y correlación
negativa.
cosecant function (p. 852) If θ is an acute angle of a right
triangle, the cosecant of θ is the length of the hypotenuse
divided by the length of the side opposite θ.
See sine function.
función cosecante (pág. 852) Si θ es un ángulo agudo de un
triángulo rectángulo, la cosecante de θ es la longitud de la
hipotenusa dividida por la longitud del lado opuesto a θ.
Ver función seno.
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cosine function (p. 852) If θ is an acute angle of a right
triangle, the cosine of θ is the length of the side adjacent to θ
divided by the length of the hypotenuse.
See sine function.
función coseno (pág. 852) Si θ es un ángulo agudo de un
triángulo rectángulo, el coseno de θ es la longitud del lado
adyacente a θ dividida por la longitud de la hipotenusa.
Ver función seno.
cotangent function (p. 852) If θ is an acute angle of a right
triangle, the cotangent of θ is the length of the side adjacent
to θ divided by the length of the side opposite θ.
See sine function.
función cotangente (pág. 852) Si θ es un ángulo agudo de un
triángulo rectángulo, la cotangente de θ es la longitud del lado
adyacente a θ dividida por la longitud del lado opuesto a θ.
Ver función seno.
coterminal angles (p. 860) Angles in standard position with
terminal sides that coincide.
y
1408
ángulos coterminales (pág. 860) Ángulos en posición
normal cuyos lados terminales coinciden.
x
5008
The angles with measures 500° and 140° are
coterminal.
Los ángulos que miden 500° y 140° son
coterminales.
See ellipse.
puntos extremos del eje menor de una elipse (pág. 634)
Los puntos de intersección de una elipse y la recta
perpendicular al eje mayor en el centro.
Ver elipse.
Cramer’s rule (p. 205) A method for solving a system of
linear equations using determinants: For the linear system
ax 1 by 5 e, cx 1 dy 5 f, let A be the coefficient matrix. If
det A Þ 0, the solution of the system is as follows:
e b
f d



a e
c f

x 5 }, y 5 }
det A
det A
regla de Cramer (pág. 205) Método para resolver un
sistema de ecuaciones lineales usando determinantes:
Para el sistema lineal ax 1 by 5 e, cx 1 dy 5 f, sea A la
matriz coeficiente. Si det A Þ 0, la solución del sistema es la
siguiente:
e b
a e
det A
det A
9x 1 4y 5 26
3x 2 5y 5 221;
9
3
4 5 257
25 
Applying Cramer’s rule gives the following:
ENGLISH-SPANISH GLOSSARY
co-vertices of an ellipse (p. 634) The points of intersection
of an ellipse and the line perpendicular to the major axis at
the center.
Al aplicar la regla de Cramer se obtiene lo
siguiente:
26 4

221 25 
114
5 22
x5}5}
257
257
9 26
3 221  2171 5 3
y5}5}
257
257
f d  c f 
x 5 }, y 5 }
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cross multiplying (p. 589) A method for solving a simple
rational equation for which each side of the equation is a
single rational expression.
3
9
To solve }
5}
, cross multiply.
4x 1 5
3
9
, multiplica en
Para resolver } 5 }
x 1 1 4x 1 5
x11
cruz.
multiplicar en cruz (pág. 589) Método para resolver una
ecuación racional simple en la que cada miembro es una sola
expresión racional.
3(4x 1 5) 5 9(x 1 1)
12x 1 15 5 9x 1 9
3x 5 26
x 5 22
cycle (p. 908) The shortest repeating portion of the graph of
a periodic function.
See periodic function.
ciclo (pág. 908) En una función periódica, la parte más corta
de la gráfica que se repite.
Ver función periódica.
ENGLISH-SPANISH GLOSSARY
D
decay factor (p. 486) The quantity b in the exponential
decay function y 5 abx with a . 0 and 0 , b , 1.
The decay factor for the function y 5 3(0.5) x
is 0.5.
factor de decrecimiento (pág. 486) La cantidad b de la
función de decrecimiento exponencial y 5 abx, con a . 0 y
0 , b , 1.
El factor de decrecimiento de la función
y 5 3(0.5) x es 0.5.
degree of a polynomial function (p. 337) The exponent
in the term of a polynomial function where the variable is
raised to the greatest power.
See polynomial function.
grado de una función polinómica (pág. 337) En una
función polinómica, el exponente del término donde la
variable se eleva a la mayor potencia.
Ver función polinómica.
dependent events (p. 718) Two events such that the
occurrence of one event affects the occurrence of the other
event.
Two cards are drawn from a deck without
replacement. The events “the fi rst is a 3” and
“the second is a 3” are dependent.
sucesos dependientes (pág. 718) Dos sucesos tales que la
ocurrencia de uno de ellos afecta a la ocurrencia del otro.
Se sacan dos cartas de una baraja y no se
reemplazan. Los sucesos “la primera es un
3” y “la segunda es un 3” son dependientes.
dependent system (p. 154) A consistent system of equations
that has infi nitely many solutions.
sistema dependiente (pág. 154) Sistema compatible de
ecuaciones que tiene infi nitas soluciones.
2x 2 y 5 3
4x 2 2y 5 6
Any ordered pair (x, 2x 2 3) is a solution of
the system above.
Cualquier par ordenado (x, 2x 2 3) es una
solución del sistema que figura arriba.
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dependent variable (p. 74) The output variable in an
equation in two variables.
See independent variable.
variable dependiente (pág. 74) La variable de salida de una
ecuación con dos variables.
Ver variable independiente.
determinant (p. 203) A real number associated with any
square matrix A, denoted by det A or A.
determinante (pág. 203) Número real asociado a toda
matriz cuadrada A, denotada por det A o A.
F 41 G 5 5(1) 2 3(4) 5 27
det 5
3
det a
c
F db G 5 ad 2 cb
dimensions of a matrix (p. 187) The dimensions of a matrix
with m rows and n columns are m 3 n.
A matrix with 2 rows and 3 columns has the
dimensions 2 3 3 (read “2 by 3”).
dimensiones de una matriz (pág. 187) Las dimensiones de
una matriz con m fi las y n columnas son m 3 n.
Una matriz con 2 fi las y 3 columnas tiene por
dimensiones 2 3 3 (leído “2 por 3”).
direct variation (p. 107) Two variables x and y show direct
variation provided that y 5 ax where a is a nonzero constant.
The equation 5x 1 2y 5 0 represents direct
variation because it is equivalent to the
5
equation y 5 – }
x.
2
variación directa (pág. 107) Dos variables x e y indican
una variación directa siempre que y 5 ax, donde a es una
constante distinta de cero.
La ecuación 5x 1 2y 5 0 representa una
variación directa ya que es equivalente a la
5
ecuación y 5 – }
x.
2
directrix of a parabola (p. 620) See parabola.
See parabola.
directriz de una parábola (pág. 620) Ver parábola.
Ver parábola.
función discreta (pág. 80) Función cuya gráfica consiste en
puntos aislados.
discriminant of a general second-degree equation
(p. 653) The expression B2 – 4AC for the equation
Ax2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0. Used to identify which
type of conic the equation represents.
y
x
For the equation 4x 2 1 y 2 – 8x – 8 5 0,
A 5 4, B 5 0, and C 5 1.
B2 – 4AC 5 02 – 4(4)(1) 5 –16
ENGLISH-SPANISH GLOSSARY
discrete function (p. 80) A function whose graph consists of
separate points.
Because B2 – 4AC < 0, B 5 0, and A Þ C, the
conic is an ellipse.
discriminante de una ecuación general de segundo
grado (pág. 653) La expresión B2 – 4AC para la ecuación
Ax2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0. Se usa para identificar
qué tipo de cónica representa la ecuación.
Para la ecuación 4x 2 1 y 2 – 8x – 8 5 0,
A 5 4, B 5 0 y C 5 1.
B2 – 4AC 5 02 – 4(4)(1) 5 –16
Debido a que B2 – 4AC < 0, B 5 0 y A Þ C, la
cónica es un elipse.
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discriminant of a quadratic equation (p. 294) The
expression b2 – 4ac for the quadratic equation
ax2 1 bx 1 c 5 0; also the expression under the radical
sign in the quadratic formula.
The value of the discriminant of
2x 2 2 3x 2 7 5 0 is b2 2 4ac 5
(23)2 2 4(2)(27) 5 65.
discriminante de una ecuación cuadrática (pág. 294) La
expresión b2 – 4ac para la ecuación cuadrática
ax2 1 bx 1 c 5 0; es también la expresión situada bajo
el signo radical de la fórmula cuadrática.
El valor del discriminante de
2x 2 2 3x 2 7 5 0 es b2 2 4ac 5
(23)2 2 4(2)(27) 5 65.
disjoint events (p. 707) Events A and B are disjoint if
they have no outcomes in common; also called mutually
exclusive events.
When you randomly select a card from a
standard deck of 52 cards, selecting a club
and selecting a heart are disjoint events.
sucesos disjuntos (pág. 707) Los sucesos A y B son disjuntos
si no tienen casos en común; también se llaman sucesos
mutuamente excluyentes.
Al seleccionar al azar una carta de una
baraja normal de 52 cartas, sacar una
de tréboles y sacar una de corazones son
sucesos disjuntos.
distance formula (p. 614) The distance d between any two
The distance between (–3, 5) and (4, –1) is
}}
2
2
}
}
Ï(4 2 (23))2 1 (21 2 5)2 5 Ï49 1 36 5 Ï85 .
fórmula de la distancia (pág. 614) La distancia d entre
dos puntos cualesquiera (x1, y1) y (x2, y 2) es
La distancia entre (–3, 5) y (4, –1) es
}}
2
2
d 5 Ï(x2 2 x1) 1 (y2 2 y1) .
ENGLISH-SPANISH GLOSSARY
}}}
points (x1, y1) and (x2, y 2) is d 5 Ï(x2 2 x1) 1 (y2 2 y1) .
}}}
}
}
Ï(4 2 (23))2 1 (21 2 5)2 5 Ï49 1 36 5 Ï85 .
domain (p. 72) The set of input values of a relation.
See relation.
dominio (pág. 72) El conjunto de valores de entrada de una
relación.
Ver relación.
E
(x 1 4)2
36
(y 2 2)2
16
eccentricity of a conic section (p. 665) The eccentricity e
For the ellipse } 1 } 5 1,
of a hyperbola or an ellipse is }c where c is the distance from
c 5 Ï36 2 16 5 2Ï 5 , so the eccentricity is
a
}
}
}
}
each focus to the center and a is the distance from each
vertex to the center. The eccentricity of a circle is e 5 0. The
eccentricity of a parabola is e 5 1.
c 5 2Ï5 5 Ï5 ≈ 0.745.
e5}
}
}
}
excentricidad de una sección cónica (pág. 665) La
Para la elipse } 1 } 5 1,
excentricidad e de una hipérbola o de una elipse es }c , donde
c 5 Ï36 2 16 5 2Ï 5 , por lo tanto la
c es la distancia entre cada foco y el centro y a es la distancia
entre cada vértice y el centro. La excentricidad de un círculo
es e 5 0. La excentricidad de una parábola es e 5 1.
Ï5
2Ï 5
c
excentricidad es e 5 }
} 5 } ≈ 0.745.
a 5}
element of a matrix (p. 187) Each number in a matrix.
See matrix.
elemento de una matriz (pág. 187) Cada número de una
matriz.
Ver matriz.
a
a
Ï36
}
3
(x 1 4)2
36
}
(y 2 2)2
16
}
Ï36
}
3
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element of a set (p. 715) Each object in a set; also called a
member of the set.
The elements of the set A 5 {1, 2, 3, 4} are 1, 2,
3, and 4.
elemento de un conjunto (pág. 715) Cada objeto de un
conjunto; también se llama miembro del conjunto.
Los elementos del conjunto A 5 {1, 2, 3, 4}
son 1, 2, 3 y 4.
elimination method (p. 161) A method of solving a system
of equations by multiplying equations by constants, then
adding the revised equations to eliminate a variable.
To use the elimination method to solve the
system with equations 3x 2 7y 5 10 and
6x 2 8y 5 8, multiply the fi rst equation by 22
and add the equations to eliminate x.
método de eliminación (pág. 161) Método para resolver un
sistema de ecuaciones en el que se multiplican ecuaciones
por constantes y se agregan luego las ecuaciones revisadas
para eliminar una variable.
Para usar el método de eliminación a fi n de
resolver el sistema con las ecuaciones
3x 2 7y 5 10 y 6x 2 8y 5 8, multiplica
la primera ecuación por 22 y suma las
ecuaciones para eliminar x.
ellipse (p. 634) The set of all points P in a plane such that the
sum of the distances between P and two fi xed points, called
the foci, is a constant.
elipse (pág. 634) El conjunto de todos los puntos P de un
plano tales que la suma de las distancias entre P y dos puntos
fijos, llamados focos, es una constante.
y
center
centro
vertex
vértice
(2a, 0)
co-vertex
puntos extremos
(0, b)
P vertex
d1
vértice
d2
(a, 0)
x
(c, 0)
focus
major axis
foco
eje mayor
minor axis
(0, 2b)
eje menor
co-vertex
constant
puntos extremos d1 1 d2 5 constante
(2c, 0)
focus
foco
The set of positive integers less than 0 is the
empty set, Ø.
conjunto vacío (pág. 715) El conjunto que no tiene
elementos, indicado Ø.
El conjunto de los números enteros positivos
menores que 0 es el conjunto vacío, Ø.
end behavior (p. 339) The behavior of the graph of a
function as x approaches positive infi nity (1`) or negative
infi nity (2`).
comportamiento (pág. 339) El comportamiento de la
gráfica de una función al aproximarse x a infi nito positivo
(1`) o a infi nito negativo (2`).
ENGLISH-SPANISH GLOSSARY
empty set (p. 715) The set with no elements, denoted Ø.
f (x) → 1` as x → 2` or as x → 1`.
f (x) → 1` según x → 2` o según x → 1`.
equal matrices (p. 187) Matrices that have the same
dimensions and equal elements in corresponding positions.
matrices iguales (pág. 187) Matrices que tienen las
mismas dimensiones y elementos iguales en posiciones
correspondientes.
F G
6
0
4
2}
4
3
}
4
5
F
3p2
21 1 1
21
0.75
G
English-Spanish Glossary
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equation (p. 18) A statement that two expressions are equal.
2x 2 3 5 7, 2x 2 5 4x
ecuación (pág. 18) Enunciado que establece la igualdad de
dos expresiones.
equation in two variables (p. 74) An equation that contains
two variables.
y 5 3x – 5, d 5 –16t 2 1 64
ENGLISH-SPANISH GLOSSARY
ecuación con dos variables (pág. 74) Ecuación que tiene
dos variables.
equivalent equations (p. 18) Equations that have the same
solution(s).
x 1 8 5 3 and 4x 5 220 are equivalent
because both have the solution 25.
ecuaciones equivalentes (pág. 18) Ecuaciones que tienen la
misma solución o soluciones.
x 1 8 5 3 y 4x 5 220 son equivalentes porque
tienen ambas la solución 25.
equivalent expressions (p. 12) Two algebraic expressions
that have the same value for all values of their variable(s).
8x 1 3x and 11x are equivalent expressions,
as are 2(x 2 3) and 2x 2 6.
expresiones equivalentes (pág. 12) Dos expresiones
algebraicas que tienen el mismo valor para todos los valores
de la variable o variables.
8x 1 3x y 11x son expresiones equivalentes,
como también lo son 2(x 2 3) y 2x 2 6.
equivalent inequalities (p. 42) Inequalities that have the
same solution.
3n 2 1 ≤ 8 and n 1 1.5 ≤ 4.5 are equivalent
inequalities because the solution of both
inequalities is all numbers less than or equal
to 3.
desigualdades equivalentes (pág. 42) Desigualdades que
tienen la misma solución.
3n 2 1 ≤ 8 y n 1 1.5 ≤ 4.5 son desigualdades
equivalentes ya que la solución de ambas son
todos los números menores o iguales a 3.
experimental group (p. 773) A group that undergoes some
procedure or treatment when an experiment is conducted.
See also control group.
One group of headache sufferers, the
experimental group, is given pills
containing medication. Another group, the
control group, is given pills containing no
medication.
grupo experimental (pág. 773) Grupo que se somete a
algún procedimiento o tratamiento durante la realización de
un experimento. Ver también grupo de control.
Un grupo de personas que sufren de dolores
de cabeza, el grupo experimental, recibe
píldoras que contienen el medicamento.
Otro grupo, el grupo de control, recibe
píldoras sin el medicamento.
experimental probability (p. 700) A probability based on
performing an experiment, conducting a survey, or looking
at the history of an event.
You roll a six-sided die 100 times and get
a 4 nineteen times. The experimental
probability of rolling a 4 with the die
19
is }
5 0.19.
100
probabilidad experimental (pág. 700) Probabilidad basada
en la realización de un experimento o una encuesta o en el
estudio de la historia de un suceso.
Lanzas 100 veces un dado de seis caras y
sale diecinueve veces el 4. La probabilidad
experimental de que salga el 4 al lanzar el
19
dado es }
5 0.19.
100
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explicit rule (p. 827) A rule for a sequence that gives the nth
term an as a function of the term’s position number n in the
sequence.
The rules an 5 211 1 4n and an 5 3(2) n 2 1
are explicit rules for sequences.
regla explícita (pág. 827) Regla de una progresión que
expresa el término enésimo an en función del número de
posición n del término en la progresión.
Las reglas an 5 211 1 4n y an 5 3(2) n 2 1 son
reglas explícitas de progresiones.
exponent (p. 10) The number or variable that represents the
number of times the base of a power is used as a factor.
In the power 25, the exponent is 5.
exponent (pág. 10) El número o la variable que representa
la cantidad de veces que la base de una potencia se usa como
factor.
En la potencia 25, el exponente es 5.
exponential decay function (p. 486) If a > 0 and 0 < b < 1,
then the function y 5 ab x is an exponential decay function
with decay factor b.
y
y52
función de decrecimiento exponencial (pág. 486) Si a > 0
y 0 < b < 1, entonces la función y 5 ab x es una función de
decrecimiento exponencial con factor de decrecimiento b.
x 14 cx
1
x
1
exponential equation (p. 515) An equation in which a
variable expression occurs as an exponent.
1
4x 5 1 }
2
x23
ecuación exponencial (pág. 515) Ecuación que tiene como
exponente una expresión algebraica.
1
4x 5 1 }
2
x23
exponential function (p. 478) A function of the form
y 5 abx, where a Þ 0, b > 0, and b Þ 1.
See exponential growth function and
exponential decay function.
función exponencial (pág. 478) Función de la forma
y 5 abx, donde a Þ 0, b > 0 y b Þ 1.
Ver función de crecimiento exponencial y
función de decrecimiento exponencial.
función de crecimiento exponencial (pág. 478) Si a > 0 y
b > 1, entonces la función y 5 abx es una función de
crecimiento exponencial con factor de crecimiento b.
2
is an exponential equation.
es una ecuación exponencial.
y
y5
3
1
1
2
? 4x
x
extraneous solution (p. 51) An apparent solution that must
be rejected because it does not satisfy the original equation.
Solving 2x 1 12 5 4x gives the apparent
solutions x 5 6 and x 5 22. The apparent
solution 22 is extraneous because it does not
satisfy the original equation.
solución extraña (pág. 51) Solución aparente que debe
rechazarse ya que no satisface la ecuación original.
Al resolver 2x 1 12 5 4x se obtienen las
soluciones aparentes x 5 6 y x 5 22. La
solución aparente 22 es extraña ya no
satisface la ecuación original.
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ENGLISH-SPANISH GLOSSARY
exponential growth function (p. 478) If a > 0 and b > 1,
then the function y 5 abx is an exponential growth function
with growth factor b.
2
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F
factor by grouping (p. 354) To factor a polynomial with four
terms by grouping, factor common monomials from pairs of
terms, and then look for a common binomial factor.
x 3 2 3x 2 2 16x 1 48
5 x 2 (x 2 3) 2 16(x 2 3)
5 (x 2 2 16)(x 2 3)
5 (x 1 4)(x 2 4)(x 2 3)
factorizar por grupos (pág. 354) Para factorizar por
grupos un polinomio con cuatro términos, factoriza unos
monomios comunes a partir de los pares de términos y luego
busca un factor binómico común.
factored completely (p. 353) A factorable polynomial with
integer coefficients is factored completely if it is written
as a product of unfactorable polynomials with integer
coefficients.
3x(x 2 5) is factored completely.
completamente factorizado (pág. 353) Un polinomio
que puede factorizarse y que tiene coeficientes enteros está
completamente factorizado si está escrito como producto
de polinomios que no pueden factorizarse y que tienen
coeficientes enteros.
3x(x 2 5) está completamente factorizado.
(x 1 2)(x 2 2 6x 1 8) is not factored
completely because x 2 2 6x 1 8 can be
factored as (x 2 2)(x 2 4).
(x 1 2)(x 2 2 6x 1 8) no está completamente
factorizado ya que x 2 2 6x 1 8 puede
factorizarse como (x 2 2)(x 2 4).
factorial (p. 684) For any positive integer n, the expression
n!, read “n factorial,” is the product of all the integers from
1 to n. Also, 0! is defi ned to be 1.
6! 5 6 · 5 · 4 · 3 · 2 · 1 5 720
ENGLISH-SPANISH GLOSSARY
factorial (pág. 684) Para cualquier número entero positivo
n, la expresión n!, leída “factorial de n”, es el producto de
todos los números enteros entre 1 y n. También, 0! se defi ne
como 1.
feasible region (p. 174) In linear programming, the graph of
the system of constraints.
See linear programming.
región factible (pág. 174) En la programación lineal, la
gráfica del sistema de restricciones.
Ver programación lineal.
finite differences (p. 393) When the x-values in a data set
are equally spaced, the differences of consecutive y-values
are called fi nite differences.
diferencias finitas (pág. 393) Cuando los valores de x de
un conjunto de datos están a igual distancia entre sí, las
diferencias entre los valores de y consecutivos se llaman
diferencias fi nitas.
f (x) 5 x 2
f(1)
1
f(2)
4
42153
f(3)
9
92455
f (4)
16
16 2 9 5 7
The first-order finite differences are 3, 5, and 7.
Las diferencias fi nitas de primer orden son
3, 5 y 7.
foci of a hyperbola (p. 642) See hyperbola.
See hyperbola.
focos de una hipérbola (pág. 642) Ver hipérbola.
Ver hipérbola.
foci of an ellipse (p. 634) See ellipse.
See ellipse.
focos de una elipse (pág. 634) Ver elipse.
Ver elipse.
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focus of a parabola (p. 620) See parabola.
See parabola.
foco de una parábola (pág. 620) Ver parábola.
Ver parábola.
formula (p. 26) An equation that relates two or more
quantities, usually represented by variables.
The formula P 5 2l 1 2w relates the length
and width of a rectangle to its perimeter.
fórmula (pág. 26) Ecuación que relaciona dos o más
cantidades que generalmente se representan por variables.
La fórmula P 5 2l 1 2w relaciona el largo y el
ancho de un rectángulo con su perímetro.
frequency of a periodic function (p. 910) The reciprocal
of the period. Frequency is the number of cycles per unit of
time.
2p
1
P 5 2 sin 4000pt has period }
5}
,
frecuencia de una función periódica (pág. 910) El
recíproco del período. La frecuencia es el número de ciclos
por unidad de tiempo.
4000p
2000
so its frequency is 2000 cycles per second
(hertz) when t represents time in seconds.
2p
1
P 5 2 sen 4000pt tiene período }
5}
,
4000p
2000
por lo que su frecuencia es de 2000 ciclos por
segundo (hertzios) cuando t representa el
tiempo en segundos.
The relation (–4, 6), (3, –9), and (7, –9) is
a function. The relation (0, 3), (0, 6), and
(10, 8) is not a function because the input 0 is
mapped onto both 3 and 6.
función (pág. 73) Relación para la que cada entrada tiene
exactamente una salida.
La relación (–4, 6), (3, –9) y (7, –9) es una
función. La relación (0, 3), (0, 6) y (10, 8) no
es una función ya que la entrada 0 se hace
corresponder tanto con 3 como con 6.
function notation (p. 75) Using f(x) (or a similar symbol
such as g(x) or h(x)) to represent the dependent variable of a
function.
The linear function y 5 mx 1 b can be
written using function notation as
f(x) 5 mx 1 b.
notación de función (pág. 75) Usar f(x) (o un símbolo
semejante como g(x) o h(x)) para representar la variable
dependiente de una función.
La función lineal y 5 mx 1 b escrita en
notación de función es f(x) 5 mx 1 b.
G
general second-degree equation in x and y (p. 653) The
form Ax2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0.
16x 2 – 9y 2 – 96x 1 36y – 36 5 0 and
4x 2 1 y 2 – 8x – 8 5 0 are second-degree
equations in x and y.
ecuación general de segundo grado en x e y (pág. 653) La
forma Ax2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0.
16x 2 – 9y 2 – 96x 1 36y – 36 5 0 y
4x 2 1 y 2 – 8x – 8 5 0 son ecuaciones de
segundo grado en x e y.
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ENGLISH-SPANISH GLOSSARY
function (p. 73) A relation for which each input has exactly
one output.
1051
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geometric probability (p. 701) A probability found by
calculating a ratio of two lengths, areas, or volumes.
14
probabilidad geométrica (pág. 701) Probabilidad hallada al
calcular una razón entre dos longitudes, áreas o volúmenes.
14
The probability that a dart that hits the square
p7
at random lands inside the circle is p
< 0.785.
}
2
2
14
La probabilidad de que un dardo que da con el
blanco cuadrado, dé al azar en el interior del
p7
círculo es p
< 0.785.
}
2
2
14
geometric sequence (p. 810) A sequence in which the ratio
of any term to the previous term is constant.
219, 38, 276, 152 is a geometric sequence
with common ratio 22.
progresión geométrica (pág. 810) Progresión en la que la
razón entre cualquier término y el término precedente es
constante.
219, 38, 276, 152 es una progresión
geométrica con una razón común de 22.
geometric series (p. 812) The expression formed by adding
the terms of a geometric sequence.
serie geométrica (pág. 812) La expresión formada al sumar
los términos de una progresión geométrica.
5
∑ 4(3)i 2 1 5 4 1 12 1 36 1 108 1 324
i51
ENGLISH-SPANISH GLOSSARY
graph of a linear inequality in two variables (p. 132) The
set of all points in a coordinate plane that represent solutions
of the inequality.
y
y > 4x 2 3
1
2x
gráfica de una desigualdad lineal con dos variables
(pág. 132) El conjunto de todos los puntos de un plano
de coordenadas que representan las soluciones de la
desigualdad.
graph of a system of linear inequalities (p. 168) The graph
of all solutions of the system.
y
graph of system
gráfica del sistema
y≥x23
1
gráfica de un sistema de desigualdades lineales (pág. 168)
La gráfica de todas las soluciones del sistema.
x
1
y < 22x 1 3
graph of an equation in two variables (p. 74) The set of all
points (x, y) that represent solutions of the equation.
gráfica de una ecuación con dos variables (pág. 74) El
conjunto de todos los puntos (x, y) que representan
soluciones de la ecuación.
y
y 5 2 12 x 1 4
1
x
1
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graph of an inequality in one variable (p. 41) All points on
a number line that represent solutions of the inequality.
gráfica de una desigualdad con una variable (pág. 41)
Todos los puntos de una recta numérica que representan
soluciones de la desigualdad.
0
21
1
2
3
4
x<3
growth factor (p. 478) The quantity b in the exponential
growth function y 5 abx with a > 0 and b > 1.
The growth factor for the function
y 5 8(3.4) x is 3.4.
factor de crecimiento (pág. 478) La cantidad b de la función
de crecimiento exponencial y 5 abx, con a > 0 y b > 1.
El factor de crecimiento de la función
y 5 8(3.4) x es 3.4.
H
half-planes (p. 132) The two regions into which the
boundary line of a linear inequality divides the coordinate
plane.
The solution of y , 3 is the half-plane
consisting of all the points below the line
y 5 3.
semiplanos (pág. 132) Las dos regiones en que la recta
límite de una desigualdad lineal divide al plano de
coordenadas.
La solución de y , 3 es el semi-plano que
consta de todos los puntos que se encuentran
debajo de la recta y 5 3.
hyperbola (pp. 558, 642) The set of all points P in a plane
such that the difference of the distances from P to two fi xed
points, called the foci, is constant.
center
centro
vertex
vértice
(2a, 0)
y
(0, b)
d2
P
d1
vertex
vértice
(a, 0)
(c, 0)
focus
foco
(2c, 0)
focus
foco
x
(0, 2b)
transverse axis
eje transverso
constant
d2 2 d1 5 constante
I
identity (p. 12) A statement that equates two equivalent
expressions.
8x 1 3x 5 11x and 2(x 2 3) 5 2x 2 6 are
identities.
identidad (pág. 12) Enunciado que hace iguales a dos
expresiones equivalentes.
8x 1 3x 5 11x y 2(x 2 3) 5 2x 2 6 son
identidades.
identity matrix (p. 210) The n 3 n matrix that has 1’s on the
main diagonal and 0’s elsewhere.
The 2 3 2 identity matrix is
matriz identidad (pág. 210) La matriz n 3 n que tiene los
1 en la diagonal principal y los 0 en las otras posiciones.
La matriz identidad 2 3 2 es
F G
F G
1 0
0 1
1 0
0 1
.
.
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ENGLISH-SPANISH GLOSSARY
hipérbola (págs. 558, 642) El conjunto de todos los puntos P
de un plano tales que la diferencia de distancias entre P y dos
puntos fijos, llamados focos, es constante.
branches of hyperbola
ramas de una hypérbola
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imaginary number (p. 276) A complex number a 1 bi where
b Þ 0.
5i and 2 2 i are imaginary numbers.
número imaginario (pág. 276) Un número complejo a 1 bi,
donde b Þ 0.
5i y 2 2 i son números imaginarios.
}
imaginary unit i (p. 275) i 5 Ï21 , so i 2 5 21.
}
}
2
unidad imaginaria i (pág. 275) i 5 Ï21 , por lo que i 5 21.
inconsistent system (p. 154) A system of equations that has
no solution.
sistema incompatible (pág. 154) Sistema de ecuaciones que
no tiene solución.
}
Ï 23 5 i Ï 3
x1y54
x1y51
The system above has no solution because
the sum of two numbers cannot be both 4
and 1.
ENGLISH-SPANISH GLOSSARY
El sistema de arriba no tiene ninguna
solución porque la suma de dos números no
puede ser 4 y 1.
independent events (p. 717) Two events such that the
occurrence of one event has no effect on the occurrence of
the other event.
If a coin is tossed twice, the outcome of the
fi rst toss (heads or tails) and the outcome of
the second toss are independent events.
sucesos independientes (pág. 717) Dos sucesos tales que la
ocurrencia de uno de ellos no afecta a la ocurrencia del otro.
Al lanzar una moneda dos veces, el resultado
del primer lanzamiento (cara o cruz) y el
resultado del segundo lanzamiento son
sucesos independientes.
independent system (p. 154) A consistent system that has
exactly one solution.
The system consisting of 4x 1 y 5 8 and
2x 2 3y 5 18 has exactly one solution, (3, 24).
sistema independiente (pág. 154) Sistema compatible que
tiene exactamente una solución.
El sistema que consiste de 4x 1 y 5 8 y
2x 2 3y 5 18 tiene exactamente una solución,
(3, 24).
independent variable (p. 74) The input variable in an
equation in two variables.
In y 5 3x – 5, the independent variable is
x. The dependent variable is y because the
value of y depends on the value of x.
variable independiente (pág. 74) La variable de entrada de
una ecuación con dos variables.
En y 5 3x – 5, la variable independiente es x.
La variable dependiente es y ya que el valor
de y depende del valor de x.
index of a radical (p. 414) The integer n, greater than 1, in
n}
the expression Ïa .
The index of Ï2216 is 3.
índice de un radical (pág. 414) El número entero n, que es
n}
mayor que 1 y aparece en la expresión Ïa .
El índice de Ï2216 es 3.
initial side of an angle (p. 859) See terminal side of an
angle.
See standard position of an angle.
lado inicial de un ángulo (pág. 859) Ver lado terminal de un
ángulo.
Ver posición normal de un ángulo.
3}
3}
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intercept form of a quadratic function (p. 246) The form
y 5 a(x 2 p)(x 2 q), where the x-intercepts of the graph are p
and q.
The function y 5 2(x 1 3)(x 2 1) is in
intercept form.
forma de intercepto de una función cuadrática (pág. 246)
La forma y 5 a(x 2 p)(x 2 q), donde los interceptos en x de la
gráfica son p y q.
La función y 5 2(x 1 3)(x 2 1) está en la forma
de intercepto.
intersection of sets (p. 715) The intersection of two sets A
and B, written A ∩ B, is the set of all elements in both A and B.
If A 5 {1, 2, 4, 8} and B 5 {2, 4, 6, 8, 10}, then
A ∩ B 5 {2, 4, 8}.
intersección de conjuntos (pág. 715) La intersección de
dos conjuntos A y B, escrita A ∩ B, es el conjunto de todos los
elementos que están tanto en A como en B.
Si A 5 {1, 2, 4, 8} y B 5 {2, 4, 6, 8, 10}, entonces
A ∩ B 5 {2, 4, 8}.
inverse cosine function (p. 875) If 21 ≤ a ≤ 1, then the
inverse cosine of a is an angle θ, written θ 5 cos21 a, where
cos θ 5 a and 0 ≤ θ ≤ π (or 08 ≤ θ ≤ 1808).
When 08 ≤ θ ≤ 1808, the angle θ whose cosine
1
1
is }
is 608, so θ 5 cos21 }
5 608
2
2
p
1 }
(or θ 5 cos21 }
5 3 ).
2
función inversa del coseno (pág. 875) Si 21 ≤ a ≤ 1,
entonces el coseno inverso de a es un ángulo θ, escrito
θ 5 cos21 a, donde cos θ 5 a y 0 ≤ θ ≤ π (ó 08 ≤ θ ≤ 1808 ).
Cuando 08 ≤ θ ≤ 1808, el ángulo θ cuyo coseno
1
1
es }
es de 608, por lo que θ 5 cos21 }
5 608
2
2
p
1 }
(ó θ 5 cos21 }
5 3 ).
2
inverse function (p. 438) An inverse relation that is a
function. Functions f and g are inverses provided that
f(g(x)) 5 x and g(f(x)) 5 x.
inverse matrices (p. 210) Two n 3 n matrices are inverses
of each other if their product (in both orders) is the n 3 n
identity matrix. See also identity matrix.
matrices inversas (pág. 210) Dos matrices n 3 n son
inversas entre sí si su producto (de ambos órdenes) es la
matriz identidad n 3 n. Ver también matriz identidad.
f(g(x)) 5 (x 2 5) 1 5 5 x
g(f(x)) 5 (x 1 5) 2 5 5 x
So, f and g are inverse functions.
Entonces, f y g son funciones inversas.
F G F G
F GF G F G
F GF G F G
21
25
8
2
23
5
3 8
25
8
2 5
2
23
25
8
3 8
2
23
2 5
3 8 because
2 5 ya que
5
5
1 0 and
0 1 y
1 0
0 1
.
inverse relation (p. 438) A relation that interchanges the
input and output values of the original relation. The graph of
an inverse relation is a reflection of the graph of the original
relation, with y 5 x as the line of reflection.
To fi nd the inverse of y 5 3x 2 5, switch x
and y to obtain x 5 3y 2 5. Then solve for y
relación inversa (pág. 438) Relación en la que se
intercambian los valores de entrada y de salida de la relación
original. La gráfica de una relación inversa es una reflexión
de la gráfica de la relación original, con y 5 x como eje de
reflexión.
Para hallar la inversa de y 5 3x 2 5,
intercambia x e y para obtener x 5 3y 2 5.
Luego resuelve para y para obtener la
5
1
to obtain the inverse relation y 5 }
x1}
.
3
3
5
1
relación inversa y 5 }
x1}
.
3
3
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ENGLISH-SPANISH GLOSSARY
función inversa (pág. 438) Relación inversa que es una
función. Las funciones f y g son inversas siempre que
f(g(x)) 5 x y g(f(x)) 5 x.
f(x) 5 x 1 5; g(x) 5 x 2 5
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inverse sine function (p. 875) If 21 ≤ a ≤ 1, then the inverse
sine of a is an angle θ, written θ 5 sin21 a, where sin θ 5 a
π
π
≤θ≤}
(or 2908 ≤ θ ≤ 908).
and 2}
2
2
función inversa del seno (pág. 875) Si 21 ≤ a ≤ 1, entonces
el seno inverso de a es un ángulo θ, escrito θ 5 sen21 a,
inverse tangent function (p. 875) If a is any real number,
then the inverse tangent of a is an angle θ, written
π
θ 5 tan21 a, where tan θ 5 a and – }
<θ<}
(or 2908 < θ < 908).
2
2
función inversa de la tangente (pág. 875) Si a es un
número real cualquiera, entonces la tangente inversa de a
π
π
es un ángulo θ, escrito θ 5 tan21 a, donde tan θ 5 a y – }
<θ<}
2
2
(ó 2908 < θ < 908 ).
inverse variation (p. 551) The relationship of two variables
a
x and y if there is a nonzero number a such that y 5 }
x.
variación inversa (pág. 551) La relación entre dos variables
a
x e y si hay un número a distinto de cero tal que y 5 }
x.
ENGLISH-SPANISH GLOSSARY
1
1
is }
is 308, so θ 5 sin21 }
5 308
2
2
p
21 1
(or θ 5 sin } 5 }).
6
2
Cuando –908 ≤ θ ≤ 908, el ángulo θ cuyo seno
1
1
es }
es de 308, por lo que θ 5 sen21 }
5 308
2
2
p
1 }
(ó θ 5 sen21 }
5 ).
6
2
π
π
donde sen θ 5 a y 2}
≤θ≤}
(ó 2908 ≤ θ ≤ 908 ).
2
2
π
When –908 ≤ θ ≤ 908, the angle θ whose sine
iteration (p. 830) The repeated composition of a function
with itself. The result of one iteration is f(f(x)), and of two
iterations is f(f(f(x))).
iteración (pág. 830) La composición repetida de una función
usando la función misma. El resultado de una iteración es
f(f(x)), y el de dos iteraciones es f(f(f(x))).
When –908 < θ < 908, the angle θ whose
}
tangent is 2Ï3 is 260°, so
}
21
θ 5 tan (2Ï3 ) 5 2608
}
p
21
(or θ 5 tan (2Ï3 ) 5 2}).
3
Cuando –908 < θ < 908, el ángulo θ cuya
}
tangente es 2Ï3 es de 260°, por lo que
}
21
θ 5 tan (2Ï3 ) 5 2608
}
p
(ó θ 5 tan21 (2Ï 3 ) 5 2}).
3
3 represent
The equations xy 5 7 and y 5 – }
x
inverse variation.
3 representan la
Las ecuaciones xy 5 7 e y 5 – }
x
variación inversa.
f(x) 5 23x 1 1; x 0 5 2
x1 5 f(x 0 ) 5 f(2) 5 23(2) 1 1 5 25
x 2 5 f(x1) 5 f(25) 5 23(25) 1 1 5 16
x 3 5 f(x 2 ) 5 f(16) 5 23(16) 1 1 5 247
J
joint variation (p. 553) A relationship that occurs when
a quantity varies directly with the product of two or more
other quantities.
The equation z 5 5xy represents joint
variation.
variación conjunta (pág. 553) Relación producida cuando
una cantidad varía directamente con el producto de dos o
más otras cantidades.
La ecuación z 5 5xy representa la variación
conjunta.
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L
law of cosines (p. 889) If TABC has sides
of length a, b, and c as shown, then a2 5 b2 1
c 2 2 2bc cos A, b2 5 a2 1 c 2 2 2ac cos B,
and c 2 5 a2 1 b2 2 2ab cos C.
ley de los cosenos (pág. 889) Si TABC
tiene lados de longitud a, b y c como se
indica, entonces a2 5 b2 1 c 2 2 2bc cos A,
b2 5 a2 1 c 2 2 2ac cos B y c 2 5 a2 1 b2 2
2ab cos C.
c 5 14
A
348
a
c
b
a 5 11
A
C
b
C
b2 5 a2 1 c 2 2 2ac cos B
b2 5 112 1 142 2 2(11)(14) cos 348
b2 ≈ 61.7
b ≈ 7.85
law of sines (p. 882) If TABC has sides of
length a, b, and c as shown, then
sin A sin B sin C
5}
}
a 5}
c .
b
B
B
B
b 5 15
a
c
C
1078
a
258
A
ley de los senos (pág. 882) Si TABC tiene
lados de longitud a, b y c como se indica, entonces
b
A
C
c
sin 258
}5
15
sen 258
}5
15
sen A sen B sen C
5}
}
a 5}
c .
b
B
sin 1078
c
sen 1078 → c < 33.9
}
c
} → c < 33.9
leading coefficient (p. 337) The coefficient in the term of a
polynomial function that has the greatest exponent.
See polynomial function.
coeficiente inicial (pág. 337) En una función polinómica, el
coeficiente del término con el mayor exponente.
Ver función polinómica.
like radicals (p. 422) Radical expressions with the same
index and radicand.
Ï10 and 7Ï10 are like radicals.
radicales semejantes (pág. 422) Expresiones radicales con
el mismo índice y el mismo radicando.
Ï10 y 7Ï10 son radicales semejantes.
like terms (p. 12) Terms that have the same variable parts.
Constant terms are also like terms.
In the algebraic expression
5x 2 1 (23x) 1 7 1 4x 1 (22),
23x and 4x are like terms, and 7 and 22 are
like terms.
términos semejantes (pág. 12) Términos que tienen las
mismas variables. Los términos constantes también son
términos semejantes.
En la expresión algebraica
5x 2 1 (23x) 1 7 1 4x 1 (22),
23x y 4x son términos semejantes, y 7 y 22
también lo son.
linear equation in one variable (p. 18) An equation that
can be written in the form ax 1 b 5 0 where a and b are
constants and a Þ 0.
4
The equation }
x 1 8 5 0 is a linear equation
ecuación lineal con una variable (pág. 18) Ecuación que
puede escribirse en la forma ax 1 b 5 0, donde a y b son
constantes y a Þ 0.
4
La ecuación }
x 1 8 5 0 es una ecuación
4}
4}
5
in one variable.
5
lineal con una variable.
English-Spanish Glossary
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ENGLISH-SPANISH GLOSSARY
4}
4}
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linear equation in three variables (p. 178) An equation of
the form ax 1 by 1 cz 5 d where a, b, and c are not all zero.
2x 1 y 2 z 5 5 is a linear equation in three
variables.
ecuación lineal con tres variables (pág. 178) Ecuación de la
forma ax 1 by 1 cz 5 d, donde a, b y c no son todos cero.
2x 1 y 2 z 5 5 es una ecuación lineal con tres
variables.
linear function (p. 75) A function that can be written in the
form y 5 mx 1 b where m and b are constants.
The function y 5 –2x – 1 is a linear function
with m 5 –2 and b 5 –1.
función lineal (pág. 75) Función que puede escribirse en la
forma y 5 mx 1 b, donde m y b son constantes.
La función y 5 –2x – 1 es una función lineal
con m 5 –2 y b 5 –1.
linear inequality in one variable (p. 41) An inequality that
can be written in one of the following forms:
ax 1 b < 0, ax 1 b ≤ 0, ax 1 b > 0, or ax 1 b ≥ 0.
5x 1 2 > 0 is a linear inequality in one
variable.
desigualdad lineal con una variable (pág. 41) Desigualdad
que puede escribirse de una de las siguientes formas:
ax 1 b < 0, ax 1 b ≤ 0, ax 1 b > 0 ó ax 1 b ≥ 0.
5x 1 2 > 0 es una desigualdad lineal con una
variable.
linear inequality in two variables (p. 132) An inequality
that can be written in one of the following forms:
Ax 1 By < C, Ax 1 By ≤ C, Ax 1 By > C, or Ax 1 By ≥ C.
5x – 2y ≥ –4 is a linear inequality in two
variables.
ENGLISH-SPANISH GLOSSARY
desigualdad lineal con dos variables (pág. 132)
Desigualdad que puede escribirse de una de las siguientes
formas:
Ax 1 By < C, Ax 1 By ≤ C, Ax 1 By > C o Ax 1 By ≥ C.
5x – 2y ≥ –4 es una desigualdad lineal con dos
variables.
linear programming (p. 174) The process of maximizing or
minimizing a linear objective function subject to a system
of linear inequalities called constraints. The graph of the
system of constraints is called the feasible region.
programación lineal (pág. 174) El proceso de maximizar o
minimizar una función objetivo lineal sujeta a un sistema de
desigualdades lineales llamadas restricciones. La gráfica del
sistema de restricciones se llama región factible.
y
(4, 5)
feasible region
región factible
1
1
(4, 0)
(8, 0)
x
To maximize the objective function P 5 35x 1 30y
subject to the constraints x ≥ 4, y ≥ 0, and
5x 1 4y ≤ 40, evaluate P at each vertex. The
maximum value of 290 occurs at (4, 5).
Para maximizar la función objetivo P 5 35x 1 30y
sujeta a las restricciones x ≥ 4, y ≥ 0 y 5x 1 4y ≤ 40,
evalúa P en cada vértice. El valor máximo de 290
ocurre en (4, 5).
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local maximum (p. 388) The y-coordinate of a turning point
of a function if the point is higher than all nearby points.
máximo local (pág. 388) La coordenada y de un punto
crítico de una función si el punto está situado más alto que
todos los puntos cercanos.
Maximum Maximo
X=0
Y=6
The function f(x) 5 x 3 2 3x 2 1 6 has a local
maximum of y 5 6 when x 5 0.
La función f(x) 5 x 3 2 3x 2 1 6 tiene un máximo
local de y 5 6 cuando x 5 0.
local minimum (p. 388) The y-coordinate of a turning point
of a function if the point is lower than all nearby points.
mínimo local (pág. 388) La coordenada y de un punto crítico
de una función si el punto está situado más bajo que todos
los puntos cercanos.
Minimum Minimo
X=-.56971
Y=-6.50858
The function f(x ) 5 x 4 2 6x 3 1 3x 2 1 10x 2 3 has a
local minimum of y < 26.51 when x < 20.57.
La función f(x ) 5 x 4 2 6x 3 1 3x 2 1 10x 2 3 tiene un
mínimo local de y < 26.51 cuando x < 20.57.
log 2 8 5 3 because 23 5 8.
logaritmo de y con base b (pág. 499) Sean b e y números
positivos, con b Þ 1. El logaritmo de y con base b, denotado
por log b y y leído “log base b de y”, se defi ne de esta manera:
log b y 5 x si y sólo si bx 5 y.
log 2 8 5 3 ya que 2 3 5 8.
logarithmic equation (p. 517) An equation that involves a
logarithm of a variable expression.
log5 (4x 2 7) 5 log5 (x 1 5) is a logarithmic
equation.
ecuación logarítmica (pág. 517) Ecuación en la que aparece
el logaritmo de una expresión algebraica.
log5 (4x 2 7) 5 log5 (x 1 5) es una ecuación
logarítmica.
21
1
log1/4 4 5 21 because 1 }
2
4
1
log1/4 4 5 21 ya que 1 }
2
4
21
5 4.
5 4.
ENGLISH-SPANISH GLOSSARY
logarithm of y with base b (p. 499) Let b and y be positive
numbers with b Þ 1. The logarithm of y with base b, denoted
log b y and read “log base b of y,” is defi ned as follows:
log b y 5 x if and only if bx 5 y.
M
major axis of an ellipse (p. 634) The line segment joining
the vertices of an ellipse.
See ellipse.
eje mayor de una elipse (pág. 634) El segmento de recta
que une los vértices de una elipse.
Ver elipse.
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margin of error (p. 768) The margin of error gives a limit
on how much the response of a sample would be expected to
differ from the response of the population.
If 40% of the people in a poll prefer
candidate A, and the margin of error is
±4%, then it is expected that between 36%
and 44% of the entire population prefer
candidate A.
margen de error (pág. 768) El margen de error indica un
límite acerca de cuánto se prevé que diferirían las respuestas
obtenidas en una muestra de las obtenidas en la población.
Si el 40% de los encuestados prefiere al
candidato A y el margen de error es ±4%,
entonces se prevé que entre el 36% y el 44%
de la población total prefiere al candidato A.
matrix, matrices (p. 187) A rectangular arrangement of
numbers in rows and columns. Each number in a matrix is
an element.
matriz, matrices (pág. 187) Disposición rectangular de
números colocados en fi las y columnas. Cada numero de la
matriz es un elemento.
A5
F
4 21 5
0
3
6
G
Matrix A has 2 rows and 3 columns. The
element in the second row and fi rst column
is 0.
La matriz A tiene 2 fi las y 3 columnas. El
elemento en la segunda fi la y en la primera
columna es 0.
matrix of constants (p. 212) The matrix of constants of the
linear system ax 1 by 5 e, cx 1 dy 5 f is e .
f
See coefficient matrix.
matriz de constantes (pág. 212) La matriz de constantes del
sistema lineal ax 1 by 5 e, cx 1 dy 5 f es e .
f
Ver matriz coeficiente.
matrix of variables (p. 212) The matrix of variables of the
linear system ax 1 by 5 e, cx 1 dy 5 f is xy .
See coefficient matrix.
matriz de variables (pág. 212) La matriz de variables del
sistema lineal ax 1 by 5 e, cx 1 dy 5 f es xy .
Ver matriz coeficiente.
FG
ENGLISH-SPANISH GLOSSARY
FG
FG
FG
maximum value of a quadratic function (p. 238) The
y-coordinate of the vertex for y 5 ax2 1 bx 1 c when a < 0.
valor máximo de una función cuadrática (pág. 238) La
coordenada y del vértice para y 5 ax2 1 bx 1 c cuando a < 0.
2
y
y 5 2x 2 1 2x 2 1
(1, 0)
3
x
The maximum value of y 5 2x 2 1 2x 2 1 is 0.
El valor máximo de y 5 2x 2 1 2x 2 1 es 0.
mean (p. 744) For the data set x1, x2, . . . , xn, the mean is
_ x1 1 x2 1 . . . 1 xn
x 5 }}
. Also called average.
n
See measure of central tendency.
media (pág. 744) Para el conjunto de datos x1, x2, . . . , xn, la
_ x1 1 x2 1 . . . 1 xn
media es x 5 }}
. También se llama promedio.
n
Ver medida de tendencia central.
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measure of central tendency (p. 744) A number used to
represent the center or middle of a set of data values. Mean,
median, and mode are three measures of central tendency.
14, 17, 18, 19, 20, 24, 24, 30, 32
1 17 1 18 1 . . . 1 32
198
The mean is 14
}} 5 } 5 22.
9
9
The median is the middle number, 20.
The mode is 24 because 24 occurs the most
frequently.
medida de tendencia central (pág. 744) Número usado
para representar el centro o la posición central de un
conjunto de valores de datos. La media, la mediana y la moda
son tres medidas de tendencia central.
1 17 1 18 1 . . . 1 32
198
La media es 14
}} 5 } 5 22.
measure of dispersion (p. 745) A statistic that tells you
how dispersed, or spread out, data values are. Range and
standard deviation are measures of dispersion.
See range and standard deviation.
medida de dispersión (pág. 745) Estadística que te indica
cómo se dispersan, o distribuyen, los valores de datos. El
rango y la desviación típica son medidas de dispersión.
Ver rango y desviación típica.
median (p. 744) The median of n numbers is the middle
number when the numbers are written in numerical order.
If n is even, the median is the mean of the two middle
numbers.
See measure of central tendency.
mediana (pág. 744) La mediana de n números es el número
central cuando los números se escriben en orden numérico.
Si n es par, la mediana es la media de los dos números
centrales.
Ver medida de tendencia central.
midpoint formula (p. 615) The midpoint M of the line
The midpoint of the line segment joining
2
segment joining A(x1, y1) and B(x2, y 2) is M } , } .
2
2
fórmula del punto medio (pág. 615) El punto medio M del
segmento de recta que une A(x1, y1) y B(x2, y 2) es
1
x1 1 x2 y1 1 y2
2
22 1 8 3 1 6
9
(–2, 3) and (8, 6) is 1 }
, } 2 5 1 3, }
2.
2
2
minimum value of a quadratic function (p. 238) The
y-coordinate of the vertex for y 5 ax2 1 bx 1 c when a > 0.
2
2
El punto medio del segmento de recta que
22 1 8 3 1 6
9
une (–2, 3) y (8, 6) es 1 }
, } 2 5 1 3, }
2.
2
M },} .
2
9
2
2
y
x
2
valor mínimo de una función cuadrática (pág. 238) La
coordenada y del vértice para y 5 ax2 1 bx 1 c cuando a > 0.
y 5 x 2 2 6x 1 5
ENGLISH-SPANISH GLOSSARY
1
x1 1 x2 y1 1 y2
9
La mediana es el número central, 20.
La moda es 24 ya que 24 ocurre más veces.
(3, 24)
The minimum value of y 5 x 2 2 6x 1 5 is 24.
El valor mínimo de y 5 x 2 2 6x 1 5 es 24.
minor axis of an ellipse (p. 634) The line segment joining
the co-vertices of an ellipse.
See ellipse.
eje menor de una elipse (pág. 634) El segmento de recta
que une los puntos extremos de una elipse.
Ver elipse.
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mode (p. 744) The mode of n numbers is the number or
numbers that occur most frequently.
See measure of central tendency.
moda (pág. 744) La moda de n números es el número o
números que ocurren más veces.
Ver medida de tendencia central.
monomial (p. 252) An expression that is either a number,
a variable, or the product of a number and one or more
variables with whole number exponents.
1
6, 0.2x, }
ab, and –5.7n 4 are monomials.
monomio (pág. 252) Expresión que es un número, una
variable o el producto de un número y una o más variables
con exponentes enteros.
1
6, 0.2x, }
ab y –5.7n 4 son monomios.
mutually exclusive events (p. 707) See disjoint events.
See disjoint events.
sucesos mutuamente excluyentes (pág. 707) Ver sucesos
disjuntos.
Ver sucesos disjuntos.
2
2
N
natural base e (p. 492) An irrational number defi ned as
See natural logarithm.
1 n
follows: As n approaches 1`, 1 1 1 }
n 2 approaches
e ≈ 2.718281828.
base natural e (pág. 492) Número irracional defi nido de
Ver logaritmo natural.
1 n
esta manera: Al aproximarse n a 1`, 1 1 1 }
n 2 se aproxima a
ENGLISH-SPANISH GLOSSARY
e ≈ 2.718281828.
natural logarithm (p. 500) A logarithm with base e. It can
be denoted loge, but is more often denoted by ln.
ln 0.3 ≈ 21.204 because
e21.204 ≈ (2.7183) 21.204 ≈ 0.3.
logaritmo natural (pág. 500) Logaritmo con base e. Puede
denotarse loge, pero es más frecuente que se denote ln.
ln 0.3 ≈ 21.204 ya que
e21.204 ≈ (2.7183) 21.204 ≈ 0.3.
negative correlation (p. 113) The paired data (x, y) have a
negative correlation if y tends to decrease as x increases.
y
correlación negativa (pág. 113) Los pares de datos (x, y)
presentan una correlación negativa si y tiende a disminuir al
aumentar x.
x
normal curve (p. 757) A smooth, symmetrical, bell-shaped
curve that can model normal distributions and approximate
some binomial distributions.
See normal distribution.
curva normal (pág. 757) Curva lisa, simétrica y con forma
de campana que puede representar distribuciones normales
y aproximar a algunas distribuciones binomiales.
Ver distribución normal.
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normal distribution (p. 757) A probability distribution with
mean }
x and standard deviation σ modeled by a bell-shaped
curve with the area properties shown at the right.
95%
x1 σ
2
x1 σ
3σ
x
x1
3
x2 σ
2σ
x2
σ
99.7%
x2
distribución normal (pág. 757) Una distribución
de probabilidad con media }
x y desviación normal σ
representada por una curva en forma de campana y que
tiene las propiedades vistas a la derecha.
68%
3}
nth root of a (p. 414) For an integer n greater }
than 1, if
n
bn 5 a, then b is an nth root of a. Written as Ïa.
Ï2216 5 26 because (26) 3 5 2216.
raíz enésima de a (pág. 414) Para un número entero n
mayor que 1,}si bn 5 a, entonces b es una raíz enésima de a.
n
Se escribe Ïa .
Ï2216 5 26 ya que (26) 3 5 2216.
numerical expression (p. 10) An expression that consists of
numbers, operations, and grouping symbols.
24(23)2 2 6(23) 1 11 is a numerical
expression.
expresión numérica (pág. 10) Expresión formada por
números, operaciones y signos de agrupación.
24(23)2 2 6(23) 1 11 es una expresión
numérica.
3}
O
See linear programming.
función objetivo (pág. 174) En la programación lineal, la
función lineal que se maximiza o minimiza.
Ver programación lineal.
odds against (p. 699) When all outcomes are equally likely,
The odds against rolling a 4 using a
Odds against
Number of outcomes not in A
5 }}}
.
event A
Number of outcomes in A
5
standard six-sided die are }
, or 5 : 1, because
probabilidad en contra (pág. 699) Cuando todos los casos
son igualmente posibles,
La probabilidad en contra de sacar el 4 al
Número de casos no del A
Probabilidad en contra 5 }}}
.
del suceso A
Número de casos del A
1
5 outcomes correspond to not rolling a 4 and
only 1 outcome corresponds to rolling a 4.
5
lanzar un dado normal de seis caras es }
,ó
1
5 : 1, ya que 5 casos corresponden a un
número que no sea el 4 y sólo 1 caso
corresponde al 4.
odds in favor (p. 699) When all outcomes are equally likely,
The odds in favor of rolling a 4 using a
Number of outcomes in A
Odds in favor 5 }}}
.
of event A
Number of outcomes not in A
1
standard six-sided die are }
, or 1 : 5, because
probabilidad a favor (pág. 699) Cuando todos los casos son
igualmente posibles,
La probabilidad a favor de sacar el 4 al lanzar
Número de casos del A
Probabilidad a favor 5 }}}
.
del suceso A
Número de casos no del A
5
only 1 outcome corresponds to rolling a 4
and 5 outcomes correspond to not rolling a 4.
1
un dado normal de seis caras es }
, ó 1 : 5, ya
5
que sólo 1 caso corresponde al 4 y 5 casos
corresponden a un número que no sea el 4.
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objective function (p. 174) In linear programming, the
linear function that is maximized or minimized.
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opposite (p. 4) The opposite, or additive inverse, of any
number b is 2b.
6.2 and 26.2 are opposites.
opuesto (pág. 4) El opuesto, o inverso aditivo, de cualquier
número b es 2b.
6.2 y 26.2 son opuestos.
ordered triple (p. 178) A set of three numbers of the form
(x, y, z) that represents a point in space.
The ordered triple (2, 1, 23) is a solution of
the equation 4x 1 2y 1 3z 5 1.
terna ordenada (pág. 178) Un conjunto de tres números de
la forma (x, y, z) que representa un punto en el espacio.
La terna ordenada (2, 1, 23) es una solución
de la ecuación 4x 1 2y 1 3z 5 1.
outlier (p. 746) A value that is much greater than or much
less than most of the other values in a data set.
3 is an outlier in the data set 3, 11, 12, 13, 13,
14, 15, 15, 15, 15, 17.
valor extremo (pág. 746) Valor que es mucho mayor o
mucho menor que la mayoría de los otros valores de un
conjunto de datos.
3 es un valor extremo del conjunto de datos
3, 11, 12, 13, 13, 14, 15, 15, 15, 15, 17.
P
parabola (pp. 236, 620) The set of all points equidistant from
a point called the focus and a line called the directrix. The
graph of a quadratic function y 5 ax2 1 bx 1 c is a parabola.
ENGLISH-SPANISH GLOSSARY
parábola (págs. 236, 620) El conjunto de todos los puntos
equidistantes de un punto, llamado foco, y de una recta,
llamada directriz. La gráfica de una función cuadrática
y 5 ax2 1 bx 1 c es una parábola.
axis of symmetry
eje de simetría
focus
foco
vertex
vértice
directrix
directriz
parallel lines (p. 84) Two lines in the same plane that do not
intersect.
y
rectas paralelas (pág. 84) Dos rectas del mismo plano que
no se cortan.
x
parent function (p. 89) The most basic function in a family
of functions.
The parent function for the family of all
linear functions is y 5 x.
función básica (pág. 89) La función más fundamental de
una familia de funciones.
La función básica de la familia de todas las
funciones lineales es y 5 x.
partial sum (p. 820) The sum Sn of the fi rst n terms of an
infi nite series.
suma parcial (pág. 820) La suma Sn de los n primeros
términos de una serie infi nita.
1
2
1
4
1
8
1
16
1
32
}1}1}1}1}1...
The series above has the partial sums
S1 5 0.5, S2 5 0.75, S3 ≈ 0.88, S4 ≈ 0.94, . . . .
La serie de arriba tiene las sumas parciales
S1 5 0.5, S2 5 0.75, S3 ≈ 0.88, S4 ≈ 0.94, . . . .
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Pascal’s triangle (p. 692) An arrangement of the values of
C in a triangular pattern in which each row corresponds to
n r
a value of n.
C
0 0
C
C
1 0
C
triángulo de Pascal (pág. 692) Disposición de los valores de
C en un patrón triangular en el que cada fi la corresponde a
n r
un valor de n.
C
2 0
C
C
C
C
C
5 2
C
período (pág. 908) La longitud horizontal de cada ciclo de
una función periódica.
Ver función periódica.
C
4 3
5 3
See periodic function.
función periódica (pág. 908) Función cuya gráfica tiene un
patrón que se repite.
3 3
C
4 2
period (p. 908) The horizontal length of each cycle of a
periodic function.
periodic function (p. 908) A function whose graph has a
repeating pattern.
C
3 2
C
4 1
5 1
2 2
C
3 1
C
5 0
C
2 1
C
3 0
4 0
1 1
4 4
C
5 4
C
5 5
y
2
1
π
4
π
π
2
3π
2
x
2π
period: π
período: π
The graph shows 3 cycles of y 5 tan x, a periodic
function with a period of p.
permutation (p. 684) An ordering of objects. The number of
permutations of r objects taken from a group of n distinct
n!
objects is denoted nPr where nPr 5 }
.
There are 6 permutations of the n 5 3 letters
A, B, and C taken r 5 3 at a time: ABC, ACB,
BAC, BCA, CAB, and CBA.
(n 2 r)!
permutación (pág. 684) Ordenación de objetos. El número
de permutaciones de r objetos tomados de un grupo de
n!
n objetos diferenciados se indica nPr , donde nPr 5 }
.
Hay 6 permutaciones de las letras n 5 3 A, B
y C tomadas r 5 3 cada vez: ABC, ACB, BAC,
BCA, CAB y CBA.
(n 2 r)!
perpendicular lines (p. 84) Two lines in the same plane that
intersect to form a right angle.
y
rectas perpendiculares (pág. 84) Dos rectas del mismo
plano que al cortarse forman un ángulo recto.
x
piecewise function (p. 130) A function defi ned by at least
two equations, each of which applies to a different part of the
function’s domain.
función definida a trozos (pág. 130) Función defi nida por
al menos dos ecuaciones, cada una de las cuales se aplica a
una parte diferente del dominio de la función.
g(x) 5
5
3x 2 1, if x , 1
0, if x 5 1
2x 1 4, if x . 1
g(x) 5
5
3x 2 1, si x , 1
0, si x 5 1
2x 1 4, si x . 1
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La gráfica muestra 3 ciclos de y 5 tan x, función
periódica con período p.
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point-slope form (p. 98) An equation of a line written in
the form y – y1 5 m(x – x1) where the line passes through the
point (x1, y1) and has a slope of m.
The equation y 1 2 5 –4(x – 5) is in pointslope form.
forma punto-pendiente (pág. 98) Ecuación de una recta
escrita en la forma y – y1 5 m(x – x1), donde la recta pasa por
el punto (x1, y1) y tiene pendiente m.
La ecuación y 1 2 5 –4(x – 5) está en la forma
punto-pendiente.
polynomial (p. 337) A monomial or a sum of monomials,
each of which is called a term of the polynomial. See also
monomial.
1 2
214, x 4 2 }
x 1 3, and 7b 2 Ï3 1 pb2
polinomio (pág. 337) Monomio o suma de monomios,
cada uno de los cuales se llama término del polinomio. Ver
también monomio.
1 2
x 1 3 y 7b 2 Ï3 1 pb2 son
214, x 4 2 }
are polynomials.
}
4
polinomios.
polynomial function (p. 337) A function of the form
f(x) 5 anxn 1 a n 2 1xn 2 1 1 . . . 1 a1x 1 a 0 where an Þ 0, the
exponents are all whole numbers, and the coefficients are all
real numbers.
f(x) 5 11x 5 2 0.4x 2 1 16x 2 7 is a polynomial
function. The degree of f(x) is 5, the leading
coefficient is 11, and the constant term is 27.
función polinómica (pág. 337) Función de la forma
f(x) 5 anxn 1 a n 2 1xn 2 1 1 . . . 1 a1x 1 a 0 donde an Þ 0, los
exponentes son todos números enteros y los coeficientes son
todos números reales.
f(x) 5 11x 5 2 0.4x 2 1 16x 2 7 es una
función polinómica. El grado de f(x) es
5, el coeficiente inicial es 11 y el término
constante es 27.
polynomial long division (p. 362) A method used to divide
polynomials similar to the way you divide numbers.
ENGLISH-SPANISH GLOSSARY
}
4
división desarrollada polinómica (pág. 362) Método
utilizado para dividir polinomios semejante a la manera en
que divides números.
x 2 1 7x 1 7
x 2 2 q x3 1 5x2 2 7x 1 2
x 3 2 2x 2
7x2 2 7x
7x 2 2 14x
7x 1 2
7x 2 14
16
}}
x3 1 5x2 2 7x 1 2
x22
16
x22
2
}} 5 x 1 7x 1 7 1 }
population (p. 766) A group of people or objects that you
want information about.
A sportswriter randomly selects 5% of
college baseball coaches for a survey. The
population is all college baseball coaches.
The 5% of coaches selected is the sample.
población (pág. 766) Grupo de personas u objetos acerca del
cual deseas informarte.
Un periodista deportiva selecciona al azar
al 5% de los entrenadores universitarios de
béisbol para que participe en una encuesta.
La población son todos los entrenadores
universitarios de béisbol. El 5% de los
entrenadores que resultó seleccionado es la
muestra.
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positive correlation (p. 113) The paired data (x, y) have a
positive correlation if y tends to increase as x increases.
y
correlacion positiva (pág. 113) Los pares de datos (x, y)
presentan una correlación positiva si y tiende a aumentar al
aumentar x.
x
power (p. 10) An expression that represents repeated
multiplication of the same factor.
32 is the fi fth power of 2 because
32 5 2 p 2 p 2 p 2 p 2 5 25.
potencia (pág. 10) Expresión que representa la
multiplicación repetida del mismo factor.
32 es la quinta potencia de 2 ya que
32 5 2 p 2 p 2 p 2 p 2 5 25.
power function (p. 428) A function of the form y 5 axb,
where a is a real number and b is a rational number.
f(x) 5 4x 3/2 is a power function.
función potencial (pág. 428) Función de la forma y 5 axb,
donde a es un número real y b es un número racional.
f(x) 5 4x 3/2 es una función potencial.
probability distribution (p. 724) A function that gives the
probability of each possible value of a random variable. The
sum of all the probabilities in a probability distribution must
equal 1.
Let the random variable X represent the
number showing after rolling a standard sixsided die.
distribución de probabilidades (pág. 724) Función que
indica la probabilidad de cada valor posible de una variable
aleatoria. La suma de todas las probabilidades de una
distribución de probabilidades debe ser igual a 1.
Sea la variable aleatoria X el número que
salga al lanzar un dado normal de seis caras.
Probability Distribution for Rolling a Die
Distribución de probabilidad al lanzar
un dado
1
2
3
4
5
6
P(X)
1
}
6
1
}
6
1
}
6
1
}
6
1
}
6
}
1
6
probability of an event (p. 698) A number from 0 to 1 that
indicates the likelihood that the event will occur.
See experimental probability, geometric
probability, and theoretical probability.
probabilidad de un suceso (pág. 698) Número entre 0 y 1
que indica la probabilidad de que ocurra el suceso.
Ver probabilidad experimental, probabilidad
geométrica y probabilidad teórica.
pure imaginary number (p. 276) A complex number a 1 bi
where a 5 0 and b Þ 0.
24i and 1.2i are pure imaginary numbers.
número imaginario puro (pág. 276) Número complejo
a 1 bi, donde a 5 0 y b Þ 0.
24i y 1.2i son números imaginarios puros.
ENGLISH-SPANISH GLOSSARY
X
Q
quadrantal angle (p. 867) An angle in standard position
whose terminal side lies on an axis.
ángulo cuadrantal (pág. 867) Ángulo en posición normal
cuyo lado terminal se encuentra en un eje.
y
u
x
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quadratic equation in one variable (p. 253) An equation
that can be written in the form ax2 1 bx 1 c 5 0 where a Þ 0.
The equation x 2 2 5x 5 36 is a quadratic
equation in one variable because it can be
written in the form x 2 – 5x – 36 5 0.
ecuación cuadrática con una variable (pág. 253) Ecuación
que puede escribirse en la forma ax2 1 bx 1 c 5 0, donde
a Þ 0.
La ecuación x 2 2 5x 5 36 es una ecuación
cuadrática con una variable ya que puede
escribirse en la forma x 2 – 5x – 36 5 0.
quadratic form (p. 355) The form au2 1 bu 1 c, where u is
any expression in x.
The expression 16x 4 2 8x 2 2 8 is in quadratic
form because it can be written as u22 2u 2 8
where u 5 4x 2 .
forma cuadrática (pág. 355) La forma au2 1 bu 1 c, donde u
es cualquier expresión en x.
La expresión 16x 4 2 8x 2 2 8 está en la forma
cuadrática ya que puede escribirse
u22 2u 2 8, donde u 5 4x 2 .
}
2b 6 Ï b2 2 4ac
quadratic formula (p. 292) The formula x 5 }}
2a
used to fi nd the solutions of the quadratic equation
ax2 1 bx 1 c 5 0 when a, b, and c are real numbers and a Þ 0.
fórmula cuadrática (pág. 292) La fórmula
}
2
x 5 2b 6 Ïb 2 4ac que se usa para hallar las soluciones de
}}
2a
2
ENGLISH-SPANISH GLOSSARY
la ecuación cuadrática ax 1 bx 1 c 5 0 cuando a, b y c son
números reales y a Þ 0.
To solve 3x 2 1 6x 1 2 5 0, substitute 3 for a,
6 for b, and 2 for c in the quadratic formula.
Para resolver 3x 2 1 6x 1 2 5 0, sustituye
a por 3, b por 6 y c por 2 en la fórmula
cuadrática.
}}
26 6 62 2 4(3)(2)
2(3)
}
Ï
23 6 Ï 3
5}
x 5 }}
3
quadratic function (p. 236) A function that can be written
in the form y 5 ax2 1 bx 1 c where a Þ 0.
The functions y 5 3x 2 2 5 and y 5 x 2 2 4x 1 6
are quadratic functions.
función cuadrática (pág. 236) Función que puede escribirse
en la forma y 5 ax2 1 bx 1 c, donde a Þ 0.
Las funciones y 5 3x 2 2 5 e y 5 x 2 2 4x 1 6
son funciones cuadráticas.
quadratic inequality in one variable (p. 302) An inequality
that can be written in the form ax2 1 bx 1 c < 0,
ax2 1 bx 1 c ≤ 0, ax2 1 bx 1 c > 0, or ax2 1 bx 1 c ≥ 0.
x 2 1 x ≤ 0 and 2x 2 1 x 2 4 > 0 are quadratic
inequalities in one variable.
desigualdad cuadrática con una variable (pág. 302)
Desigualdad que se puede escribir en la forma ax2 1 bx 1 c < 0,
ax2 1 bx 1 c ≤ 0, ax2 1 bx 1 c > 0 ó ax2 1 bx 1 c ≥ 0.
x 2 1 x ≤ 0 y 2x 2 1 x 2 4 > 0 son desigualdades
cuadráticas con una variable.
quadratic inequality in two variables (p. 300) An
inequality that can be written in the form y < ax2 1 bx 1 c,
y ≤ ax2 1 bx 1 c, y > ax2 1 bx 1 c, or y ≥ ax2 1 bx 1 c.
y > x 2 1 3x 2 4 is a quadratic inequality in
two variables.
desigualdad cuadrática con dos variables (pág. 300)
Desigualdad que se puede escribir en la forma y < ax2 1 bx 1 c,
y ≤ ax2 1 bx 1 c, y > ax2 1 bx 1 c ó y ≥ ax2 1 bx 1 c.
y > x 2 1 3x 2 4 es una desigualdad cuadrática
con dos variables.
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quadratic system (p. 658) A system of equations that
includes one or more equations of conics.
sistema cuadrático (pág. 658) Sistema de ecuaciones que
incluye una o más ecuaciones de cónicas.
y 2 – 7x 1 3 5 0
2x – y 5 3
x 2 1 4y 2 1 8y 5 16
2x 2 – y 2 – 6x – 4 5 0
The systems above are quadratic systems.
Los sistemas de arriba son sistemas
cuadráticos.
R
radian (p. 860) In a circle with radius r and center at the
origin, one radian is the measure of an angle in standard
position whose terminal side intercepts an arc of length r.
y
r
radián (pág. 860) En un círculo con radio r y cuyo centro
está en el origen, un radián es la medida de un ángulo en
posición normal cuyo lado terminal intercepta un arco de
longitud r.
}
1 radian
1 radián
x
n}
radical (p. 266) An expression of the form Ïs or Ïs where s
is a number or an expression.
}
r
}
3}
Ï 5 , Ï 2x 1 1
n}
radical (pág. 266) Expresión de la forma Ï s o Ï s, donde s es
un número o una expresión.
radical equation (p. 452) An equation with one or more
radicals that have variables in their radicands.
3}
ecuación radical (pág. 452) Ecuación con uno o más
radicales en cuyo radicando aparecen variables.
radical function (p. 446) A function that contains a radical
with a variable in its radicand.
}
3}
1
Ïx, g(x) 5 23Ïx 1 5
f(x) 5 }
2
función radical (pág. 446) Función que tiene un radical con
una variable en su radicando.
}
radicand (p. 266) The number or expression beneath a
radical sign.
The radicand of Ï5 is 5, and the radicand of
radicando (pág. 266) El número o la expresión que aparece
bajo el signo radical.
El radicando de Ï5 es 5, y el radicando de
}
Ï8y2 is 8y2.
}
}
Ï8y2 es 8y2.
radius of a circle (p. 626) The distance from the center of
a circle to a point on the circle. Also, a line segment that
connects the center of a circle to a point on the circle. See
also circle.
The circle with equation (x – 3)2 1
}
(y 1 5)2 5 36 has radius Ï36 5 6. See also
circle.
radio de un círculo (pág. 626) La distancia desde el centro
de un círculo hasta un punto del círculo. También, es un
segmento de recta que une el centro de un círculo con un
punto del círculo. Ver también círculo.
El círculo con la ecuación (x – 3)2 1
}
(y 1 5)2 5 36 tiene el radio Ï 36 5 6. Ver
también círculo.
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random variable (p. 724) A variable whose value is
determined by the outcomes of a random event.
The random variable X representing the
number showing after rolling a six-sided die
has possible values of 1, 2, 3, 4, 5, and 6.
variable aleatoria (pág. 724) Variable cuyo valor viene
determinado por los resultados de un suceso aleatorio.
La variable aleatoria X que representa el
número que sale al lanzar un dado de seis
caras tiene como valores posibles
1, 2, 3, 4, 5 y 6.
range of a relation (p. 72) The set of output values of a
relation.
See relation.
rango de una relación (pág. 72) El conjunto de los valores
de salida de una relación.
Ver relación.
range of data values (p. 745) A measure of dispersion equal
to the difference between the greatest and least data values.
rango de valores de datos (pág. 745) Medida de dispersión
igual a la diferencia entre el valor máximo y el valor mínimo
de los datos.
rate of change (p. 85) A comparison of how much one
quantity changes, on average, relative to the change in
another quantity.
14, 17, 18, 19, 20, 24, 24, 30, 32
The range of the data set above is 32 – 14 5 18.
El rango del conjunto de datos de arriba es
32 – 14 5 18.
The temperature rises from 75°F at 8 A .M. to
91°F at 12 P.M. The average rate of change in
918F 2 758F 5 16°F 5 4°/h.
temperature is }
}
12 P.M. 2 8 A.M.
ENGLISH-SPANISH GLOSSARY
relación de cambio (pág. 85) Comparación entre el cambio
producido, por término medio, en una cantidad y el cambio
producido en otra cantidad.
4h
La temperatura sube de 75°F a las 8 de la
mañana a 91°F a las 12 del mediodía. La
relación de cambio media en la temperatura
918F 2 758F 5 16°F 5 4°/h.
es }
}
12 P.M. 2 8 A.M.
4h
rational function (p. 558) A function of the form
6 and y 5 2x 1 1 are
The functions y 5 }
}
p(x)
f(x) 5 }, where p(x) and q(x) are polynomials and q(x) Þ 0.
q(x)
rational functions.
función racional (pág. 558) Función de la forma
6 e y 5 2x 1 1 son
Las funciones y 5 }
}
p(x)
q(x)
x
x
x23
x23
f(x) 5 }, donde p(x) y q(x) son polinomios y q(x) Þ 0.
funciones racionales.
rationalizing the denominator (p. 267) The process
of eliminating a radical expression in the denominator
of a fraction by multiplying both the numerator and
denominator by an appropriate radical expression.
Ï5
To rationalize the denominator of }
},
racionalizar el denominador (pág. 267) El proceso
de eliminar una expresión radical del denominador de
una fracción al multiplicar tanto el numerador como el
denominador por una expresión radical adecuada.
Ï5
Para racionalizar el denominador de }
},
reciprocal (p. 4) The reciprocal, or multiplicative inverse, of
1 5 2} are reciprocals.
22 and }
2
}
Ï2
multiply the numerator and denominator
}
by Ï2 .
}
1
any nonzero number b is }
.
b
recíproco (pág. 4) El recíproco, o inverso multiplicativo, de
1
cualquier número b distinto de cero es }
.
Ï2
multiplica el numerador y el denominador
}
por Ï2 .
1
22
1
1 5 2} son recíprocos.
22 y }
2
22
b
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recursive rule (p. 827) A rule for a sequence that gives
the beginning term or terms of the sequence and then a
recursive equation that tells how the nth term an is related to
one or more preceding terms.
The recursive rule a 0 5 1, an 5 an 2 1 1 4 gives
the arithmetic sequence 1, 5, 9, 13, … .
regla recursiva (pág. 827) Regla de una progresión que
da el primer término o términos de la progresión y luego
una ecuación recursiva que indica qué relación hay
entre el término enésimo an y uno o más de los términos
precedentes.
La regla recursiva a 0 5 1, an 5 an 2 1 1 4 da la
progresión aritmética 1, 5, 9, 13, … .
reference angle (p. 868) If θ is an angle in standard
position, its reference angle is the acute angle θ´ formed by
the terminal side of θ and the x-axis.
y
u
u'
ángulo de referencia (pág. 868) Si θ es un ángulo en
posición normal, su ángulo de referencia es el ángulo agudo
θ´ formado por el lado terminal de θ y el eje de x.
x
The acute angle θ´ is the reference angle for
angle θ.
El ángulo agudo θ´ es el ángulo de referencia para
el ángulo θ.
reflection (p. 124) A transformation that fl ips a graph or
figure in a line.
(2, 3)
(5, 3)
f (x)
1
x
1
g(x)
(2, 23) (5, 23)
The graph of g(x) is the reflection of the graph of
f(x) in the x-axis.
La gráfica de g(x) es la reflexión de la gráfica de
f(x) en el eje de x.
relation (p. 72) A mapping, or pairing, of input values with
output values.
The ordered pairs (22, 22), (22, 2), (0, 1),
and (3, 1) represent the relation with inputs
(domain) of –2, 0, and 3 and outputs (range)
of –2, 1, and 2.
relación (pág. 72) Correspondencia entre los valores de
entrada y los valores de salida.
Los pares ordenados (22, 22), (22, 2), (0, 1)
y (3, 1) representan la relación con entradas
(dominio) de –2, 0 y 3 y salidas (rango) de
–2, 1 y 2.
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ENGLISH-SPANISH GLOSSARY
reflexión (pág. 124) Transformación que vuelca una gráfica
o una figura en una recta.
y
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repeated solution (p. 379) For the polynomial equation
f(x) 5 0, k is a repeated solution if and only if the factor
x – k has an exponent greater than 1 when f(x) is factored
completely.
21 is a repeated solution of the equation
(x 1 1)2 (x 2 2) 5 0.
solución repetida (pág. 379) Para la ecuación polinómica
f(x) 5 0, k es una solución repetida si y sólo si el factor
x – k tiene un exponente mayor que 1 cuando f(x) está
completamente factorizado.
21 es una solución repetida de la ecuación
(x 1 1)2 (x 2 2) 5 0.
root of an equation (p. 253) The solutions of a quadratic
equation are its roots.
The roots of the quadratic equation
x 2 2 5x 2 36 5 0 are 9 and 24.
raíz de una ecuación (pág. 253) Las soluciones de una
ecuación cuadrática son sus raíces.
Las raíces de la ecuación cuadrática
x 2 2 5x 2 36 5 0 son 9 y 24.
sample (p. 766) A subset of a population.
See population.
muestra (pág. 766) Subconjunto de una población.
Ver población.
scalar (p. 188) A real number by which you multiply a
matrix.
See scalar multiplication.
escalar (pág. 188) Número real por el que se multiplica una
matriz.
Ver multiplicación escalar.
scalar multiplication (p. 188) Multiplication of each
element of a matrix by a real number, called a scalar.
F GF
28
2
4 21
0
22 1 0 5 22
24 214
2 7
multiplicación escalar (pág. 188) Multiplicación de cada
elemento de una matriz por un número real llamado escalar.
scatter plot (p. 113) A graph of a set of data pairs (x, y) used
to determine whether there is a relationship between the
variables x and y.
diagrama de dispersión (pág. 113) Gráfica de un conjunto
de pares de datos (x, y) que sirve para determinar si hay una
relación entre las variables x e y.
Test scores
Resultados
de las pruebas
ENGLISH-SPANISH GLOSSARY
S
G
y
90
70
50
0
0
2
4
6
8
Hours of studying
Horas de estudio
scientific notation (p. 331) The representation of a number
in the form c 3 10n where 1 ≤ c < 10 and n is an integer.
0.693 is written in scientific notation as
6.93 3 1021.
notación científica (pág. 331) La representación de un
número de la forma c 3 10n, donde 1 ≤ c < 10 y n es un
número entero.
0.693 escrito en notación científica es
6.93 3 1021.
x
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secant function (p. 852) If θ is an acute angle of a right
triangle, the secant of θ is the length of the hypotenuse
divided by the length of the side adjacent to θ.
See sine function.
función secante (pág. 852) Si θ es un ángulo agudo de un
triángulo rectángulo, la secante de θ es la longitud de la
hipotenusa dividida por la longitud del lado adyacente a θ.
Ver función seno.
sector (p. 861) A region of a circle that is bounded by two
radii and an arc of the circle. The central angle θ of a sector is
the angle formed by the two radii.
sector (pág. 861) Región de un círculo delimitada por dos
radios y un arco del círculo. El ángulo central θ de un sector
es el ángulo formado por dos radios.
sector
sector
r
arc length s
longitud de un arco s
central angle u
ángulo central u
For the domain n 5 1, 2, 3, and 4, the
sequence defi ned by an 5 2n has the terms
2, 4, 6, and 8.
progresión (pág. 794) Función cuyo dominio es un conjunto
de números enteros consecutivos. El dominio da la posición
relativa de cada término de la secuencia. El rango da los
términos de la secuencia.
Para el dominio n 5 1, 2, 3 y 4, la secuencia
defi nida por an 5 2n tiene los términos 2, 4,
6 y 8.
series (p. 796) The expression formed by adding the terms of
a sequence. A series can be fi nite or infi nite.
Finite series: 2 1 4 1 6 1 8
Infinite series: 2 1 4 1 6 1 8 1 . . .
serie (pág. 796) La expresión formada al sumar los términos
de una progresión. La serie puede ser fi nita o infi nita.
Serie fi nita: 2 1 4 1 6 1 8
Serie infinita: 2 1 4 1 6 1 8 1 . . .
set (p. 715) A collection of distinct objects.
If A is the set of positive integers less than 5,
then A 5 {1, 2, 3, 4}.
conjunto (pág. 715) Colección de objetos diferenciados.
Si A es el conjunto de números enteros positivos
menores que 5, entonces A 5 {1, 2, 3, 4}.
sigma notation (p. 796) See summation notation.
See summation notation.
notación sigma (pág. 796) Ver notación de sumatoria.
Ver notación de sumatoria.
simplest form of a radical (p. 422) A radical with index n is
in simplest form if the radicand has no perfect nth powers as
factors and any denominator has been rationalized.
Ï 135 in simplest form is 3Ï5 .
forma más simple de un radical (pág. 422) Un radical con
índice n está escrito en la forma más simple si el radicando
no tiene como factor ninguna potencia enésima perfecta y el
denominador ha sido racionalizado.
3}
5}
Ï7
Ï8
3}
5}
Ï28
2
in simplest form is }.
}
5}
3}
3}
Ï 135 en la forma más simple es 3Ï5 .
5}
Ï7
Ï8
5}
Ï28
2
en la forma más simple es }.
}
5}
English-Spanish Glossary
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ENGLISH-SPANISH GLOSSARY
sequence (p. 794) A function whose domain is a set of
consecutive integers. The domain gives the relative position
of each term of the sequence. The range gives the terms of
the sequence.
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simplified form of a rational expression (p. 573) A rational
expression in which the numerator and denominator have
no common factors other than 61.
x2 2 2x 2 15
x 29
(x 1 3)(x 2 5)
(x 1 3)(x 2 3)
↑
Simplified form
Forma simplificada
forma simplificada de una expresión racional (pág. 573)
Expresión racional en la que el numerador y el denominador
no tienen factores comunes además de 61.
sine function (p. 852) If θ is an acute angle of a right
triangle, the sine of θ is the length of the side opposite θ
divided by the length of the hypotenuse.
función seno (pág. 852) Si θ es un ángulo agudo de un
triángulo rectángulo, el seno de θ es la longitud del lado
opuesto a θ dividida por la longitud de la hipotenusa.
13
5
u
12
hyp
opp
13
hyp
adj
12
cos θ 5 } 5 }
13
hyp
opp
5
tan θ 5 } 5 }
12
adj
5
sin θ 5 } 5 }
13
csc θ 5 }
opp 5 }
5
hyp
13
sec θ 5 } 5 }
12
adj
adj
12
cot θ 5 }
opp 5 }
5
op
13
hip
ady
12
cos θ 5 } 5 }
13
hip
op
5
tan θ 5 } 5 }
12
ady
hip
5
sen θ 5 } 5 }
sinusoids (p. 941) Graphs of sine and cosine functions.
y
sinusoides (pág. 941) Gráficas de funciones seno y coseno.
13
cosec θ 5 }
op 5 }
5
hip
13
sec θ 5 } 5 }
12
ady
ady
12
cot θ 5 }
op 5 }
5
y 5 2 sin 4x 1 3
y 5 2 sen 4x 1 3
1
π
4
skewed distribution (p. 727) A probability distribution that
is not symmetric. See also symmetric distribution.
distribución asimétrica (pág. 727) Distribución de
probabilidades que no es simétrica. Ver también distribución
simétrica.
Probability
Probabilidad
ENGLISH-SPANISH GLOSSARY
x25
x23
5}5}
}
2
π x
2
0.40
0.20
0
0 1 2 3 4 5 6 7 8
Number of successes
Número de éxitos
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slope (p. 82) The ratio of vertical change (the rise) to
horizontal change (the run) for a nonvertical line. For a
nonvertical line passing through the points (x1, y1) and
y 2y
2
1
(x2, y 2), the slope is m 5 }
x 2x .
2
1
pendiente (pág. 82) Para una recta no vertical, la razón
entre el cambio vertical (distancia vertical) y el cambio
horizontal (distancia horizontal). Para una recta no vertical
que pasa por los puntos (x1, y1) y (x2, y 2), la pendiente es
The slope of the line that passes through the
points (23, 0) and (3, 4) is:
La pendiente de la recta que pasa por los
puntos (23, 0) y (3, 4) es:
y 2y
420
2
1
4
2
m5}
x 2 x 5 }5 } 5 }
2
3 2 (23)
1
6
3
y 2y
2
1
m5}
x 2x .
2
1
2
slope-intercept form (p. 90) A linear equation written in
the form y 5 mx 1 b where m is the slope and b is the
y-intercept of the equation’s graph.
The equation y 5 2}x 2 1 is in slope3
intercept form.
forma pendiente-intercepto (pág. 90) Ecuación lineal
escrita en la forma y 5 mx 1 b, donde m es la pendiente y b es
el intercepto en y de la gráfica de la ecuación.
La ecuación y 5 2}x 2 1 está en la forma
3
pendiente-intercepto.
solution of a linear inequality in two variables (p. 132) An
ordered pair (x, y) that produces a true statement when the
values of x and y are substituted into the inequality.
The ordered pair (1, 2) is a solution of
3x 1 4y > 8 because 3(1) 1 4(2) 5 11, and
11 > 8.
solución de una desigualdad lineal con dos variables
El par ordenado (1, 2) es una solución de
3x 1 4y > 8 ya que 3(1) 1 4(2) 5 11, y 11 > 8.
(pág. 132) Par ordenado (x, y) que produce una expresión
2
verdadera cuando x e y se sustituyen por sus valores en la
desigualdad.
solución de un sistema de ecuaciones lineales en
tres variables (pág. 178) Terna ordenada (x, y, z) cuyas
coordenadas hacen que cada ecuación del sistema sea
verdadera.
solution of a system of linear equations in two
variables (p. 153) An ordered pair (x, y) that satisfies each
equation of the system.
solución de un sistema de ecuaciones lineales en dos
variables (pág. 153) Par ordenado (x, y) que satisface cada
ecuación del sistema.
solution of a system of linear inequalities in two
variables (p. 168) An ordered pair (x, y) that is a solution of
each inequality in the system.
solución de un sistema de desigualdades lineales en dos
variables (pág. 168) Par ordenado (x, y) que es una solución
de cada desigualdad del sistema.
4x 1 2y 1 3z 5 1
2x 2 3y 1 5z 5 214
6x 2 y 1 4z 5 21
(2, 1, 23) is the solution of the system above.
(2, 1, 23) es la solución del sistema de arriba.
4x 1 y 5 8
2x 2 3y 5 18
(3, 24) is the solution of the system above.
(3, 24) es la solución del sistema de arriba.
y > 22x 2 5
y≤x13
(–1, 1) is a solution of the system above.
(–1, 1) es una solución del sistema de arriba.
English-Spanish Glossary
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ENGLISH-SPANISH GLOSSARY
solution of a system of linear equations in three
variables (p. 178) An ordered triple (x, y, z) whose
coordinates make each equation in the system true.
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ENGLISH-SPANISH GLOSSARY
solution of an equation in one variable (p. 18) A number
that produces a true statement when substituted for the
variable in the equation.
4 x 1 8 5 20
The solution of the equation }
5
is 15.
solución de una ecuación con una variable (pág. 18)
Número que produce una expresión verdadera al sustituir la
variable por él en la ecuación.
4 x 1 8 5 20 es 15.
La solución de la ecuación }
solution of an equation in two variables (p. 74) An
ordered pair (x, y) that produces a true statement when the
values of x and y are substituted in the equation.
(–2, 3) is a solution of y 5 –2x – 1.
solución de una ecuación con dos variables (pág. 74) Par
ordenado (x, y) que produce una expresión verdadera al
sustituir x e y por sus valores en la ecuación.
(–2, 3) es una solución de y 5 –2x – 1.
solution of an inequality in one variable (p. 41) A number
that produces a true statement when substituted for the
variable in the inequality.
21 is a solution of the inequality
5x 1 2 > 7x 2 4.
solución de una desigualdad con una variable (pág. 41)
Número que produce una expresión verdadera al sustituir la
variable por él en la desigualdad.
21 es una solución de la desigualdad
5x 1 2 > 7x 2 4.
solve for a variable (p. 26) Rewrite an equation as an
equivalent equation in which the variable is on one side and
does not appear on the other side.
When you solve the circumference formula
resolver para una variable (pág. 26) Escribir una ecuación
como ecuación equivalente que tenga la variable en uno de
sus miembros pero no en el otro.
Al resolver para r la fórmula de
circunferencia C 5 2p r, el resultado es
5
C
C 5 2p r for r, the result is r 5 }
.
2p
C
r5}
.
2p
square root (p. 266)
If b2 5 a, then b is a square root of a. The
}
radical symbol Ï represents a nonnegative square root.
The square roots of 9 are 3 and 23 because
}
32 5 9 and (23)2 5 9. So, Ï9 5 3 and
}
2Ï9 5 23.
raíz cuadrada (pág. 266) Si b2 5 a, entonces
b es una
}
raíz cuadrada de a. El signo radical Ï representa una raíz
cuadrada no negativa.
Las raíces cuadradas de 9 son 3 y 23 ya que
}
32 5 9 y (23)2 5 9. Así pues, Ï9 5 3 y
}
2Ï9 5 23.
standard deviation (p. 745) The typical difference (or
deviation) between a data value and the mean. The standard
deviation s of a numerical data set x1, x2, . . . , xn is given by
the following formula:
}}}}
s5
Ï
(x1 2 }
x)2 1 (x2 2 }
x)2 1 . . . 1 (xn 2 }
x)2
}}
n
desviación típica (pág. 745) La diferencia (o desviación)
más común entre un valor de los datos y la media. La
desviación típica s de un conjunto de datos numéricos
x1, x2, . . . , xn viene dada por la siguiente fórmula:
14, 17, 18, 19, 20, 24, 24, 30, 32
Because the mean of the data set is 22, the
standard deviation is:
Como la media del conjunto de datos es 22, la
desviación típica es:
Î
}}}}
s5
Î
(14 2 22)2 1 (17 2 22)2 1 . . . 1 (32 2 22)2
9
}}
}
290 < 5.7
5 }
9
}}}
s5
Ï
(x1 2 }
x)2 1 (x2 2 }
x)2 1 . . . 1 (xn 2 }
x)2
}}}
n
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standard form of a complex number (p. 276) The form
a 1 bi where a and b are real numbers and i is the imaginary
unit.
The standard form of the complex number
i(1 1 i) is 21 1 i.
forma general de un número complejo (pág. 276) La
forma a 1 bi, donde a y b son números reales e i es la unidad
imaginaria.
La forma general del número complejo
i(1 1 i) es 21 1 i.
standard form of a linear equation (p. 91) A linear
equation written in the form Ax 1 By 5 C where A and B are
not both zero.
The linear equation y 5 –3x 1 4 can be
written in standard form as 3x 1 y 5 4.
forma general de una ecuación lineal (pág. 91) Ecuación
lineal escrita en la forma Ax 1 By 5 C, donde A y B no son
ambos cero.
La ecuación lineal y 5 –3x 1 4 escrita en la
forma general es 3x 1 y 5 4.
standard form of a polynomial function (p. 337) The
form of a polynomial function that has terms written in
descending order of exponents from left to right.
The function g(x) 5 7x 2 Ï 3 1 p x 2 can
be written in standard form as
}
g(x) 5 p x 2 1 7x 2 Ï3 .
forma general de una función polinómica (pág. 337) La
forma de una función polinómica en la que los términos
se ordenan de tal modo que los exponentes disminuyen de
izquierda a derecha.
La función g(x) 5 7x 2 Ï3 1 p x 2 escrita en
}
la forma general es g(x) 5 p x 2 1 7x 2 Ï3 .
standard form of a quadratic equation in one variable
(p. 253) The form ax 2 1 bx 1 c 5 0 where a Þ 0.
The quadratic equation x 2 2 5x 5 36 can be
written in standard form as x 2 2 5x 2 36 5 0.
forma general de una ecuación cuadrática con una
variable (pág. 253) La forma ax2 1 bx 1 c 5 0, donde a Þ 0.
La ecuación cuadrática x 2 2 5x 5 36 escrita
en la forma general es x 2 2 5x 2 36 5 0.
standard form of a quadratic function (p. 236) The form
y 5 ax2 1 bx 1 c where a Þ 0.
The quadratic function y 5 2(x 1 3)(x 2 1)
can be written in standard form as
y 5 2x 2 1 4x 2 6.
forma general de una función cuadrática (pág. 236) La
forma y 5 ax2 1 bx 1 c, donde a Þ 0.
La función cuadrática y 5 2(x 1 3)(x 2 1)
escrita en la forma general es
y 5 2x 2 1 4x 2 6.
}
}
2
z5 3
2
z5 2
2
1
z5
0
z5
1
z5
2
z5
3
z5
distribución normal típica (pág. 758) La distribución
normal con media 0 y desviación típica 1. Ver también
puntuación z.
ENGLISH-SPANISH GLOSSARY
standard normal distribution (p. 758) The normal
distribution with mean 0 and standard deviation 1. See also
z-score.
English-Spanish Glossary
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standard position of an angle (p. 859) In a coordinate
plane, the position of an angle whose vertex is at the origin
and whose initial side lies on the positive x-axis.
908 y
terminal side
lado terminal
08
posición normal de un ángulo (pág. 859) En un plano de
coordenadas, la posición de un ángulo cuyo vértice está en el
origen y cuyo lado inicial se sitúa en el eje de x positivo.
1808
vertex
vértice
x
initial side
lado inicial 3608
2708
statistics (p. 744) Numerical values used to summarize and
compare sets of data.
See mean, median, mode, range, and
standard deviation.
estadística (pág. 744) Valores numéricos utilizados para
resumir y comparar conjuntos de datos.
Ver media, mediana, moda, rango y
desviación típica.
step function (p. 131) A piecewise function defi ned by
a constant value over each part of its domain. Its graph
resembles a series of stair steps.
5
función escalonada (pág. 131) Función defi nida a trozos
y por un valor constante en cada parte de su dominio. Su
gráfica parece un grupo de escalones.
ENGLISH-SPANISH GLOSSARY
1, if 0 ≤ x , 1
f(x) 5 2, if 1 ≤ x , 2
3, if 2 ≤ x , 3
5
1, si 0 ≤ x , 1
f(x) 5 2, si 1 ≤ x , 2
3, si 2 ≤ x , 3
subset (p. 716) If every element of a set A is also an element
of a set B, then A is a subset of B. This is written as A ⊆ B. For
any set A, ∅ ⊆ A and A ⊆ A.
If A 5 {1, 2, 4, 8} and B is the set of all positive
integers, then A is a subset of B, or A ⊆ B.
subconjunto (pág. 716) Si cada elemento de un conjunto
A es también un elemento de un conjunto B, entonces A es
un subconjunto de B. Esto se escribe A ⊆ B. Para cualquier
conjunto A, ∅ ⊆ A y A ⊆ A.
Si A 5 {1, 2, 4, 8} y B es el conjunto de todos
los números enteros positivos, entonces A es
un subconjunto de B, o A ⊆ B.
substitution method (p. 160) A method of solving a system
of equations by solving one of the equations for one of the
variables and then substituting the resulting expression in
the other equation(s).
método de sustitución (pág. 160) Método para resolver
un sistema de ecuaciones mediante la resolución de una
de las ecuaciones para una de las variables seguida de
la sustitución de la expresión resultante en la(s) otra(s)
ecuación (ecuaciones).
summation notation (p. 796) Notation for a series that
uses the uppercase Greek letter sigma, o. Also called sigma
notation.
notación de sumatoria (pág. 796) Notación de una serie
que usa la letra griega mayúscula sigma, o. También se llama
notación sigma.
2x 1 5y 5 25
x 1 3y 5 3
Solve equation 2 for x: x 5 23y 1 3.
Substitute the expression for x in equation
1 and solve for y: y 5 11. Use the value of y to
fi nd the value of x: x 5 230.
Resuelve la ecuación 2 para x: x 5 23y 1 3.
Sustituye la expresión para x en la ecuación
1 y resuelve para y: y 5 11. Usa el valor de y
para hallar el valor de x: x 5 230.
5
∑ 7i 5 7(1) 1 7(2) 1 7(3) 1 7(4) 1 7(5)
i51
5 7 1 14 1 21 1 28 1 35
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distribución simétrica (pág. 727) Distribución de
probabilidad representada por un histograma en la que se
puede trazar una recta vertical que divida al histograma en
dos partes; éstas son imágenes especulares entre sí.
0.30
Probability
Probabilidad
symmetric distribution (p. 727) A probability distribution,
represented by a histogram, in which you can draw a vertical
line that divides the histogram into two parts that are mirror
images.
0.20
0.10
0
synthetic division (p. 363) A method used to divide a
polynomial by a divisor of the form x – k.
23
0 1 2 3 4 5 6 7 8
Number of successes
Número de éxitos
2
división sintética (pág. 363) Método utilizado para dividir
un polinomio por un divisor en la forma x – k.
2
1
28
5
26
15
221
25
7
216
2x 3 1 x 2 2 8x 1 5 5 2x 2 2 5x 1 7 2 16
}
}
x13
x13
synthetic substitution (p. 338) A method used to evaluate a
polynomial function.
sustitución sintética (pág. 338) Método utilizado para
evaluar una función polinómica.
3
2
2
25
0
24
8
6
3
9
15
1
3
5
23
The synthetic substitution above indicates that for
f(x) 5 2x 4 2 5x 3 2 4x 1 8, f(3) 5 23.
La sustitución sintética de arriba indica que para
f(x) 5 2x 4 2 5x 3 2 4x 1 8, f(3) 5 23.
sistema de desigualdades lineales con dos variables
(pág. 168) Sistema que consiste de dos o más desigualdades
lineales con dos variables. Ver también desigualdad lineal
con dos variables.
system of three linear equations in three variables
(p. 178) A system consisting of three linear equations in three
variables. See also linear equation in three variables.
sistema de tres ecuaciones lineales en tres variables
(pág. 178) Sistema formado por tres ecuaciones lineales con
tres variables. Ver también ecuación lineal con tres variables.
x1y≤8
4x 2 y > 6
2x 1 y 2 z 5 5
3x 2 2y 1 z 5 16
4x 1 3y 2 5z 5 3
English-Spanish Glossary
ENGLISH-SPANISH GLOSSARY
system of linear inequalities in two variables (p. 168) A
system consisting of two or more linear inequalities in two
variables. See also linear inequality in two variables.
1079
system of two linear equations in two variables
(p. 153) A system consisting of two equations that can be
written in the form Ax 1 By 5 C and Dx 1 Ey 5 F where x and
y are variables, A and B are not both zero, and D and E are
not both zero.
sistema de dos ecuaciones lineales con dos variables
(pág. 153) Un sistema que consiste en dos ecuaciones que se
pueden escribir de la forma Ax 1 By 5 C y Dx 1 Ey 5 F, donde
x e y son variables, A y B no son ambos cero, y D y E tampoco
son ambos cero.
4x 1 y 5 8
2x 2 3y 5 18
ENGLISH-SPANISH GLOSSARY
T
tangent function (p. 852) If θ is an acute angle of a right
triangle, the tangent of θ is the length of the side opposite θ
divided by the length of the side adjacent to θ.
See sine function.
función tangente (pág. 852) Si θ es un ángulo agudo de un
triángulo rectángulo, la tangente de θ es la longitud del lado
opuesto a θ dividida por la longitud del lado adyacente a θ.
Ver función seno.
terminal side of an angle (p. 859) In a coordinate plane, an
angle can be formed by fi xing one ray, called the initial side,
and rotating the other ray, called the terminal side, about the
vertex.
See standard position of an angle.
lado terminal de un ángulo (pág. 859) En un plano de
coordenadas, un ángulo puede formarse al fijar un rayo,
llamado lado inicial, y al girar el otro rayo, llamado lado
terminal, en torno al vértice.
Ver posición normal de un ángulo.
terms of a sequence (p. 794) The values in the range of a
sequence.
The fi rst 4 terms of the sequence 1, 23, 9,
227, 81, 2243, . . . are 1, 23, 9, and 227.
términos de una progresión (pág. 794) Los valores del
rango de una progresión.
Los 4 primeros términos de la progresión 1,
23, 9, 227, 81, 2243, . . . son 1, 23, 9 y 227.
terms of an expression (p. 12) The parts of an expression
that are added together.
The terms of the algebraic expression
3x 2 1 5x 1 (27) are 3x 2, 5x, and 27.
términos de una expresión (pág. 12) Las partes de una
expresión que se suman.
Los términos de la expresión algebraica
3x 2 1 5x 1 (27) son 3x 2, 5x y 27.
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theoretical probability (p. 698) When all outcomes are
equally likely, the theoretical probability that an event A will
The theoretical probability of rolling an even
number using a standard six-sided die
Number of outcomes in event A
occur is P(A) 5 }}}
.
3 5 1 because 3 outcomes correspond
is }
}
Total number of outcomes
6
2
to rolling an even number out of 6 total
outcomes .
probabilidad teórica (pág. 698) Cuando todos los casos son
igualmente posibles, la probabilidad teórica de que ocurra
La probabilidad teórica de sacar un número
par al lanzar un dado normal de seis caras
Número de casos del suceso A
.
un suceso A es P(A) 5 }}}
3 5 1 ya que 3 casos corresponden a un
es }
}
Número total de casos
6
2
número par del total de 6 casos.
transformation (p. 123) A transformation changes a graph’s
size, shape, position, or orientation.
Translations, vertical stretches and
shrinks, reflections, and rotations are
transformations.
transformación (pág. 123) Una transformación cambia el
tamaño, la forma, la posición o la orientación de una gráfica.
Las traslaciones, las expansiones y
contracciones verticales, las reflexiones y las
rotaciones son transformaciones.
translation (p. 123) A transformation that shifts a graph
horizontally and/or vertically, but does not change its size,
shape, or orientation.
traslación (pág. 123) Transformación que desplaza una
gráfica horizontal o verticalmente, o de ambas maneras,
pero que no cambia su tamaño, forma u orientación.
y
y5zx14z22
y 5 zx z
24
21 22
x
24
La gráfica de y 5x 1 4 2 2 es la gráfica de
y 5x al trasladar ésta 2 unidades hacia abajo
y 4 unidades hacia la izquierda.
transverse axis of a hyperbola (p. 642) The line segment
joining the vertices of a hyperbola.
See hyperbola.
eje transverso de una hipérbola (pág. 642) El segmento de
recta que une los vértices de una hipérbola.
Ver hipérbola.
trigonometric identity (p. 924) A trigonometric equation
that is true for all domain values.
sin (– θ) 5 –sin θ
sin2 θ 1 cos2 θ 5 1
identidad trigonométrica (pág. 924) Ecuación
trigonométrica que es verdadera para todos los valores del
dominio.
sen (– θ) 5 –sen θ
sen2 θ 1 cos2 θ 5 1
trinomial (p. 252) The sum of three monomials.
4x 2 1 3x 2 1 is a trinomial.
trinomio (pág. 252) La suma de tres monomios.
4x 2 1 3x 2 1 es un trinomio.
English-Spanish Glossary
n2pe-9050.indd 1081
ENGLISH-SPANISH GLOSSARY
The graph of y 5x 1 4 2 2 is the graph of
y 5x translated down 2 units and left 4 units.
1081
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U
unbiased sample (p. 767) A sample that is representative of
the population you want information about.
You want to poll members of the senior class
about where to hold the prom. If every senior
has an equal chance of being polled, then the
sample is unbiased.
muestra no sesgada (pág. 767) Muestra que es
representativa de la población acerca de la cual deseas
informarte.
Quieres encuestar a algunos estudiantes de
último curso sobre el lugar donde organizar
el baile de fi n de año. Si cada estudiante de
último curso tiene iguales posibilidades de
ser encuestado, entonces es una muestra no
sesgada.
union of sets (p. 715) The union of two sets A and B, written
A ∪ B, is the set of all elements in either A or B.
If A 5 {1, 2, 4, 8} and B 5 {2, 4, 6, 8, 10}, then
A ∪ B 5 {1, 2, 4, 6, 8, 10}.
unión de conjuntos (pág. 715) La unión de dos conjuntos
A y B, escrita A ∪ B, es el conjunto de todos los elementos que
están en A o B.
Si A 5 {1, 2, 4, 8} y B 5 {2, 4, 6, 8, 10}, entonces
A ∪ B 5 {1, 2, 4, 6, 8, 10}.
unit circle (p. 867) The circle x2 1 y 2 5 1, which has center
(0, 0) and radius 1. For an angle θ in standard position, the
terminal side of θ intersects the unit circle at the point
(cos θ, sin θ ).
ENGLISH-SPANISH GLOSSARY
círculo unidad (pág. 867) El círculo x2 1 y 2 5 1, que tiene
centro (0, 0) y radio 1. Para un ángulo θ en posición normal,
el lado terminal de θ corta al círculo unidad en el punto
(cos θ, sen θ ).
y
u
x
(cos u, sin u )
(cos u, sen u)
r51
universal set (p. 715) The set of all elements under
consideration; denoted U.
See complement of a set.
conjunto universal (pág. 715) El conjunto de todos los
elementos tenidos en cuenta; se indica U.
Ver complemento de un conjunto.
V
variable (p. 11) A letter that is used to represent one or more
numbers.
In the expressions 6x, 3x 2 1 1, and
12 2 5x, the letter x is the variable.
variable (pág. 11) Letra utilizada para representar uno o
más números.
En las expresiones 6x, 3x 2 1 1 y
12 2 5x, la letra x es la variable.
variable term (p. 12) A term that has a variable part.
The variable terms of the algebraic
expression 3x 2 1 5x 1 (27) are 3x 2 and 5x.
término algebraico (pág. 12) Término que tiene variable.
Los términos algebraicos de la expresión
algebraica 3x 2 1 5x 1 (27) son 3x 2 y 5x.
verbal model (p. 34) A word equation that represents a reallife problem.
Distance
(miles)
5
Rate
p
Time
(miles/hour)
(hours)
modelo verbal (pág. 34) Ecuación expresada mediante
palabras que representa un problema de la vida real.
Distancia
(millas)
5
Velocidad
p Tiempo
(millas/hora)
(horas)
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1
vertex form of a quadratic function (p. 245) The form
y 5 a(x 2 h)2 1 k, where the vertex of the graph is (h, k) and
the axis of symmetry is x 5 h.
The quadratic function y 5 2}
(x 1 2)2 1 5
4
is in vertex form.
forma de vértice de una función cuadrática (pág. 245) La
forma y 5 a(x 2 h)2 1 k, donde el vértice de la gráfica es (h, k)
y el eje de simetría es x 5 h.
(x 1 2)2 1 5
La función cuadrática y 5 2}
4
está en la forma de vértice.
vertex of a parabola (pp. 236, 620) The point on a parabola
that lies on the axis of symmetry.
See parabola.
vértice de una parábola (págs. 236, 620) El punto de una
parábola que se encuentra en el eje de simetría.
Ver parábola.
1
vertex of an absolute value graph (p. 123) The highest or
lowest point on the graph of an absolute value function.
y
vértice de una gráfica de valor absoluto (pág. 123) El
punto más alto o más bajo de la gráfica de una función de
valor absoluto.
(4, 3)
1
x
1
The vertex of the graph of y 5 x 2 4 1 3 is the
point (4, 3).
El vértice de la gráfica de y 5 x 2 4 1 3 es el
punto (4, 3).
See hyperbola.
vértices de una hipérbola (pág. 642) Los puntos de
intersección de una hipérbola y la recta que pasa por los
focos de la hipérbola.
Ver hipérbola.
vertices of an ellipse (p. 634) The points of intersection of
an ellipse and the line through the foci of the ellipse.
See ellipse.
vértices de una elipse (pág. 634) Los puntos de intersección
de una elipse y la recta que pasa por los focos de la elipse.
Ver elipse.
ENGLISH-SPANISH GLOSSARY
vertices of a hyperbola (p. 642) The points of intersection
of a hyperbola and the line through the foci of the hyperbola.
X
x-intercept (p. 91) The x-coordinate of a point where a
graph intersects the x-axis.
intercepto en x (pág. 91) La coordenada x de un punto
donde una gráfica corta al eje de x.
y
(0, 3)
x 1 2y 5 6
1
(6, 0) x
1
The x-intercept is 6.
El intercepto en x es 6.
English-Spanish Glossary
n2pe-9050.indd 1083
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Y
y-intercept (p. 89) The y-coordinate of a point where a
graph intersects the y-axis.
y
(0, 3)
x 1 2y 5 6
intercepto en y (pág. 89) La coordenada y de un punto
donde una gráfica corta al eje de y.
1
(6, 0) x
1
The y-intercept is 3.
El intercepto en y es 3.
Z
zero of a function (p. 254) A number k is a zero of a
function f if f(k) 5 0.
The zeros of the function f(x) 5 2(x 1 3)(x 2 1)
are 23 and 1.
cero de una función (pág. 254) Un número k es un cero de
una función f si f(k) 5 0.
Los ceros de la función f(x) 5 2(x 1 3)(x 2 1)
son 23 y 1.
z-score (p. 758) The number z of standard deviations that a
data value lies above or below the mean of the data set:
A normal distribution has a mean of 76 and
a standard deviation of 9. The z-score for
x2x
z5}
.
64 2 76
x 5 64 is z 5 } 5 }
≈ 21.3.
puntuación z (pág. 758) El número z de desviaciones típicas
que un valor se encuentra por encima o por debajo
Una distribución normal tiene una
media de 76 y una desviación típica de 9.
La puntuación z para x 5 64 es
_
σ
_
x2x
de la media del conjunto de datos: z 5 }
.
σ
ENGLISH-SPANISH GLOSSARY
_
x 2x
σ
9
_
x 2 x 5 64 2 76 ≈ 21.3.
z5}
}
σ
9
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Index
A
Alternative method, See Another
Way; Problem Solving
Workshop
Ambiguous case, 883–884
Amplitude, 908, 909
Analyze, exercises, 8, 185, 696, 712,
748, 749
And rule, 1000–1001
Angle(s)
Brewster’s, 930
central, 861
complementary, 994
coterminal, 860, 863
degree measure of, 859–864
of depression, 855
of elevation, 855
initial side of, 859
quadrantal, 867
radian measure of, 860–864
reference, 868, 871
of repose, 879
special, 963
in standard position, 859
supplementary, 994
terminal side of, 859
Angle bisector, 994
Animated Algebra, Throughout. See
for example 1, 71, 151, 235,
329, 413, 477, 549, 613, 681,
743, 793, 851, 907
Another Way, 18, 48–49, 91, 92, 105,
179, 189, 218–219, 272–273,
284, 293, 360–361, 396,
460–461, 493, 523–525, 575,
596–597, 640, 660, 714, 720,
781, 834–835, 867, 889, 895,
938–939, 950
Applications
advertising, 170, 181, 220
apparel, 410, 461, 681, 720
archaeology, 105, 358, 470, 619, 890
archery, 743, 748
art, 200, 264, 358, 373, 400, 404,
676, 729, 808
astronomy, 8, 270, 332, 369, 426,
459, 477, 519, 524, 525, 535,
632, 638, 639, 641, 648, 654,
666, 673, 677, 800, 864, 888,
904
aviation, 34, 215, 398, 426, 570, 587,
631, 639, 667, 790, 879, 894, 961
baseball, 54, 57, 69, 78, 173, 233,
287, 315, 327, 345, 442, 475,
571, 595, 663, 766, 790, 846,
847, 862, 896
Index
n2pe-9060.indd 1085
INDEX
Absolute value, 50, 51
of a complex number, 279, 280
equations, 50–58, 60, 64
functions, 121–129, 144
graphing, 50, 51
inequalities, 50–58, 60, 64, 135, 136
in a system, 169–173
ACT, See Standardized Test
Preparation
Activities, See also Geometry
software; Graphing calculator
absolute value equations and
inequalities, 50
collect and model trigonometric
data, 948
completing the square, 283
end behavior of polynomial
functions, 336
exploring inverse functions, 437
exploring recursive rules, 826
exploring transformations,
121–122
fitting a line to data, 112
fitting a model to data, 774
graphing linear equations in three
variables, 177
infinite geometric series, 819
intersections of planes and cones,
649
inverse trigonometric functions,
874
inverse variation, 550
using the Location Principle, 378
modeling data with an exponential
function, 528
probability using Venn diagrams,
706
solve linear systems using tables,
152
trigonometric identities, 923
Addition
of complex numbers, 276–277, 279
with fractions, 979
of functions, 428–435
integer, 975
matrix, 187, 189–192, 194
as opposite of subtraction, 4
of polynomials, 346–352, 403
properties, 3, 18
for matrices, 188
of rational expressions, 582–588,
602, 605
Addition property of equality, 18
Additive inverse, 4
of a complex number, 280
Algebra, formulas and theorems
from, 1027–1028
Algebra tiles
to model binomial products, 985
to model completing the square,
283
Algorithm
for adding or subtracting a rational
expression, 582, 583
for evaluating a trigonometric
function, 868
for graphing an absolute value
function, 124, 125
for graphing an equation of a
circle, 626
translated, 650
for graphing an equation of an
ellipse, 635
translated, 652
for graphing an equation of a
hyperbola, 643
translated, 651
for graphing an equation of a
parabola, 621
translated, 651
for graphing an equation in slopeintercept form, 90
for graphing an equation in
standard form, 91
for graphing an equation in two
variables, 74
for graphing a horizontal
translation, 916
for graphing a linear inequality, 133
for graphing a quadratic function
in intercept form, 247
for graphing a quadratic inequality
in two variables, 300
for graphing a rational function,
558
for graphing a system of linear
inequalities, 168
for graphing a vertical translation,
915
for order of operations, 10
for solving an absolute value
equation, 52
for solving a radical equation, 452
for translating a trigonometric
graph, 915
for writing an exponential
function, 529
for writing a power function, 531
1085
9/28/05 9:28:03 AM
INDEX
basketball, 57, 74, 215, 241, 368,
458, 606, 679, 691, 721, 729,
740, 770, 846, 962, 973
bicycling, 78, 351, 472, 601, 683,
756, 879, 940, 946
biology, 44, 88, 91, 108, 186, 250,
298, 319, 335, 344, 369, 400,
416, 421, 426, 429, 433, 458,
485, 491, 494, 497, 505, 512,
532, 533, 534, 536, 541, 624,
711, 722, 759, 761, 782, 832
botany, 38, 46, 85, 762, 779, 782
bowling, 94, 418, 436, 469
business, 32, 103, 117, 128, 137,
165, 166, 174, 175, 176, 185,
186, 192, 194, 201, 202, 209,
213, 218, 219, 220, 262, 264,
274, 290, 348, 365, 367, 385,
398, 400, 405, 464, 475, 498,
563, 631, 679, 689, 732, 769,
779, 824, 849
camping, 94, 368
chemistry, 59, 69, 206, 209, 491,
497, 504, 506, 521, 588, 589,
601, 818, 832, 872, 904
computers and Internet, 15, 47, 65,
106, 139, 145, 274, 480, 484,
535, 547, 553, 562, 607, 657,
713, 741, 767, 769, 782, 787,
791, 841, 847
construction, 39, 145, 306, 323, 377,
392, 458, 557, 594, 754, 818,
896, 901, 902, 904
consumer economics, 16, 23, 33,
40, 49, 68, 103, 111, 155, 157,
185, 223, 230, 316, 431, 433,
473, 611, 679, 784
contests, 46, 269, 272, 273, 303,
696, 699, 942
crafts, 137, 148, 174, 176, 186, 216,
226, 261, 264, 274, 291, 326,
843, 905
design, 24, 31, 39, 258, 316, 326,
358, 360, 361, 369, 376, 377,
560, 570, 618, 667, 688, 849
diving, 38, 76, 110, 270, 597
earth science, 3, 6, 57, 945
education, 119, 201, 230, 351, 392,
485, 537, 546, 611, 688, 692,
695, 705, 708, 737, 754, 773,
775, 817
electricity, 144, 277, 281, 282, 556,
587, 749, 849, 945
employment, 7, 19, 23, 33, 49, 157,
172, 436, 735, 753, 755, 764,
770, 818, 864, 922
engineering, 129, 246, 250
entertainment, 59, 103, 120, 200,
208, 227, 351, 475, 579, 699,
808, 843
1086 Student Resources
equipment, 58, 198, 484, 504, 552,
556, 572
exercising, 15, 439
fairs, 11, 42, 165, 176, 258, 809
finance, 15, 33, 62, 66, 144, 219,
227, 352, 386, 407, 418, 475,
481, 483, 484, 485, 488, 489,
490, 491, 495, 497, 498, 506,
521, 525, 526, 537, 543, 547,
570, 588, 679, 817, 832, 838,
847, 849
fitness, 94, 105, 157, 184, 705
food, 7, 33, 62, 137, 149, 176, 184,
193, 216, 251, 351, 361, 403,
521, 547, 572, 697, 705, 709,
712, 730, 735, 765, 873, 940
football, 59, 106, 247, 250, 298, 314,
441, 485, 638, 723, 741, 748,
787, 791, 871, 962, 970
fundraising, 81, 104, 129, 139, 162,
166, 475, 600, 717, 849
games, 128, 172, 231, 290, 316, 326,
592, 596, 687, 696, 701, 702,
703, 704, 732, 738, 746, 800,
865
gardening, 67, 103, 139, 208, 230,
258, 264, 274, 522, 543, 631,
676, 678, 872, 887, 895
geography, 204, 208, 209, 320, 704,
858, 872, 951, 961
geology, 8, 334, 335, 419, 505, 521,
543, 616, 619, 631, 663
golf, 149, 756, 893, 961
government, 79, 106, 146, 147, 149,
352, 407, 410, 703, 723, 737,
740, 769, 771, 772
gymnastics, 54, 55, 57, 233
history, 139, 148, 158, 233, 359, 385,
392, 400, 407, 410, 598, 600,
611, 679, 779, 846
hockey, 33, 198, 231
law enforcement, 157
manufacturing, 57, 176, 189, 225,
358, 367, 368, 369, 376, 433,
475, 567, 572, 574, 601, 732,
761, 780
measurement, 31, 33, 58, 59, 68,
81, 104, 257, 265, 343, 383,
389, 391, 400, 408, 411, 436,
444, 450, 451, 569, 601, 611,
647, 664, 679, 855, 857, 901,
927
medicine, 490, 498, 639, 723, 729
movies, 15, 29, 59, 134, 148, 159,
271, 343, 398, 399, 433, 547,
696, 705, 790, 813
music, 101, 224, 242, 316, 367, 537,
552, 556, 594, 688, 718, 767,
808, 829, 844, 864, 913, 948,
963, 972
nutrition, 216
oceanography, 932, 933, 938, 939
Olympics, 119, 201, 684, 705, 748,
749, 752, 780, 864
parabolic reflectors, 622–625, 641
pets, 219, 713, 749
photography, 13, 106, 242, 426, 512,
585, 644, 683, 688, 735, 755,
880, 896, 954
physics, 119, 125, 143, 272, 273,
306, 311, 340, 418, 444, 447,
450, 451, 453, 457, 458, 464,
469, 475, 497, 509, 511, 512,
513, 519, 522, 524, 535, 547,
585, 654, 673, 864, 913, 929,
953, 957, 972
physiology, 57, 78, 173, 306, 385,
444, 536, 540, 572, 765, 782,
790, 832, 910, 921
population, 79, 106, 119, 385, 410,
484, 546, 842
racing, 38, 451, 463, 718
recreation, 8, 35, 36, 46, 68, 79, 95,
137, 138, 166, 186, 239, 299,
556, 562, 656, 669, 679, 737,
815, 855, 870, 871, 873, 894,
916, 921, 922, 940
repairs, 23, 65
running, 39, 314, 579
safety, 105, 662, 720
skateboarding, 82, 290, 343, 564
soccer, 68, 307, 322, 676, 726, 815,
904, 957, 961
softball, 200, 290
space travel, 243, 396, 570, 656,
751, 752
structures, 87, 88, 104, 242, 246,
250, 257, 258, 289, 299, 311,
314, 315, 356, 358, 359, 386,
419, 498, 631, 716, 808, 847,
851, 864, 873, 887, 888, 918,
921, 953
swimming, 46, 158, 243, 392
telephone, 94, 113, 137, 298, 369,
497, 595, 600, 628, 630, 954
television, 138, 184, 323, 407, 600,
607, 731, 768, 770, 788
temperature, 32, 44, 57, 59, 69, 444,
521, 535, 562, 570, 754, 756, 781,
943, 946, 947, 963, 972, 1005
tennis, 325, 335, 679, 723, 904
track and field, 32, 184, 418, 689,
695, 756, 864
travel, 5, 7, 15, 33, 62, 63, 81, 88,
158, 159, 243, 601, 641, 703,
857
vehicles, 7, 36, 38, 76, 95, 115, 128,
166, 192, 227, 233, 298, 307,
410, 444, 445, 469, 472, 530,
537, 563, 572, 618, 619, 641,
Associative property, 3
for matrix operations, 188, 197
Asymptote
for exponential decay functions,
486, 487
for exponential growth functions,
478, 479
horizontal, 558
of a hyperbola, 642
for logarithmic functions, 502
for rational functions, 558, 565
vertical, 558
@Home Tutor, Throughout. See for
example xxiv, 8, 15, 17, 23, 25,
31, 38, 46, 57, 61, 63, 70, 78,
94, 97, 103, 110, 122, 141, 143
Augmented matrices, 218–219
Average, See Mean
Avoid Errors, 12, 20, 28, 43, 52, 73, 82,
108, 115, 126, 133, 153, 163,
187, 205, 238, 247, 252, 260,
267, 277, 287, 292, 330, 338,
347, 354, 355, 362, 370, 371,
415, 416, 423, 430, 441, 480,
488, 507, 553, 573, 584, 599,
644, 654, 658, 683, 691, 693,
700, 708, 719, 726, 744, 795,
797, 803, 811, 821, 828, 862,
877, 890
Axis (Axes)
coordinate, 987
of an ellipse, 634
of symmetry for a conic section,
652, 655
of symmetry for a parabola, 236,
620
B
Bar graph, 1006–1007
Base
of a logarithm, 499
of a power, 10
Best-fitting line, 112–120, 143
correlation coefficient, 114
linear regression and, 116
Best-fitting quadratic model, 311
Bias in sampling, 767, 769, 770–773
Biased question, 772–773
Biased sample, 767
Biconditional statement, 1002–1003
Big Ideas, 1, 60, 71, 140, 151, 221, 235,
317, 329, 401, 413, 465, 477,
538, 549, 602, 613, 668, 681,
733, 743, 783, 793, 839, 851,
964
Binomial(s), See also Polynomial(s),
252
cube of, 347
multiplying, 347–351, 985
square of, 347
Binomial distribution, 725–731, 733,
736
approximating, 763–765
calculating, 731
skewed, 727, 728
symmetric, 727, 728
Binomial expansion, 693–696
Pascal’s triangle and, 693, 695
Binomial experiment, 725
Binomial theorem, 693
using, 693–696, 735
Bisector
angle, 994
perpendicular, 615–617
Bounded region, 174
Boundary line, for an inequality, 132
Box-and-whisker plot, 1008–1009
Branches, of a hyperbola, 558
Brewster’s angle, 930
C
Calculator, See also Graphing
calculator
approximating roots, 415, 417
calculating compound
interest, 481
entering negative numbers, 17
evaluating expressions, 17
evaluating inverse trigonometric
functions, 876
evaluating logarithms, 500–501
evaluating permutations, 685
evaluating trigonometric
functions, 854
simplifying natural base
expressions, 492
Calculator Based Laboratory (CBL),
948
Center
of a circle, 626, 992
of a conic section, 650–652
of a hyperbola, 642
of rotation, 988
Central angle, of a sector, 861
Central tendency, measures of,
744–750, 783, 784, 1005
Chain rule, 1000–1001
Challenge, exercises, Throughout. See
for example 7, 9, 15, 16, 23, 24,
31, 32, 38, 39, 45, 47, 56, 58,
78, 79, 87, 88, 94, 96
Change-of-base formula, 508–509,
511
Chapter Review, 61–64, 141–144,
222–226, 318–322, 402–406,
466–468, 539–542, 603–606,
669–672, 734–736, 784–786,
840–842, 898–900, 965–968
Index
INDEX
646, 660, 662, 667, 722, 777,
780, 857, 869, 877, 879, 898,
901, 904, 905, 921, 969
volleyball, 64, 290, 321, 327, 594
volunteering, 95, 696
weather, 105, 108, 110, 458, 460,
500, 513, 562, 636, 712, 719,
755, 785, 791, 936, 943, 953,
969
wildlife, 8, 9, 16, 44, 108, 109, 111,
172, 204, 242, 251, 273, 306,
331, 344, 385, 400, 429, 485,
630, 648, 664, 729, 922
winter sports, 343, 556, 682, 734
Approximation, See also Estimation;
Prediction
of the area of an ellipse, 640
of best-fitting line, 115–120, 146
of binomial distribution, 763–765
of correlation, 113, 114, 117
exercises, 335, 557, 904
of real zeros of a function, 382–383,
384
of roots, 415, 417
Arc length, of a sector, 861–865
Archimedes, 857
Area, See also Formulas, 991
using determinants to find, 204,
208, 209, 217
of an ellipse, 636, 638, 640
Heron’s area formula, 891
of a parallelogram, 991
of a rectangle, 991
of a sector, 861–865
of a trapezoid, 991
of a triangle, 885, 887, 888, 891, 991
Area model
for completing the square, 283
for a quadratic equation, 254, 257,
258, 261
Arithmetic sequence, 802–809, 839,
841
recursive rules and, 827–833, 839,
842
Arithmetic series, 804–809, 839, 841
Assessment, See also Online Quiz;
State Test Practice
Chapter Test, 65, 145, 227, 323, 407,
469, 543, 607, 673, 737, 787,
843, 901, 969
Quiz, Throughout. See for example
40, 58, 96, 120, 138, 167, 193,
217, 265, 291, 315, 352, 377,
399
Standardized Test Practice, 68–69,
148–149, 230–231, 326–327,
410–411, 472–473, 546–547,
610–611, 676–677, 740–741,
790–791, 846–847, 904–905,
972–973
1087
INDEX
Chapter Summary, 60, 140, 221, 317,
401, 465, 538, 602, 668, 733,
783, 839, 897, 964
Chapter Test, See Assessment
Checking solutions
using a calculator, 876
using end behavior, 393
by graphing, 255, 311, 440, 518, 931
using a graphing calculator, 161,
285, 292, 293, 462, 518, 591,
659, 958
using inverse operations, 362
using logical reasoning, 373, 763
using slope-intercept form, 154
using substitution, 18, 19, 20, 36,
52, 91, 133, 153, 160, 179, 205,
267, 285, 381, 452, 454, 455,
468, 517, 518, 591, 934
using unit analysis, 5, 7, 34
Choosing a method
exercises, 94, 164, 183, 305, 357,
892, 893
for solving linear systems, 163
Choosing a model, for data, 774–781,
786
Circle, 626, 992
area of, 992
center of, 626, 992
central angle of, 861
circumference of, 992
degree measure of, 860
diameter of, 992
eccentricity of, 665–666
equation of, 626
graphing, 626–633, 668, 670
translated, 650, 652–657
writing, 627–632, 668, 670
finding the center given three
points, 616, 619
inequalities and, 628, 630–632
radian measure of, 860
radius of, 626, 992
sector of, 861
translated, 650
unit, 867
Circle graph, 1006–1007
Circular function, 866
Circular model, 628, 630–632
Circular motion, modeling, 916, 921,
922
Circular permutation, 689
Circumference, of a circle, 992
Classifying
conics, 653, 654, 656
functions, 75, 80–81, 337, 479, 487,
489
inverse and direct variation, 551
linear systems, 154–157
lines by slope, 83
numbers, 2
1088 Student Resources
parallel and perpendicular lines,
84, 86
probability distributions, 727, 728
samples, 766, 769
series, 805
triangles using the distance
formula, 614–615, 617
zeros of a polynomial function,
381–382, 384, 385
Closure property, 3
Coefficient
leading, 337
of a power, 12
Coefficient matrix, 205–206
Cofunction identities, 924
Combination(s), 690–697, 733, 735
formula, 690
Pascal’s triangle and, 692, 695
probability and, 699, 702
Combinatorics, formulas from, 1028
Common difference, 802
Common factors, 978
Common logarithm, 500
change-of-base formula and,
508–509, 511
Common misconceptions, See Error
Analysis
Common multiple, 978
Common ratio, 810
Communication
describing in words, 7, 13, 17, 22,
30, 38, 45, 50, 56, 68, 77, 86,
93, 94, 102, 117, 118, 119, 122,
127, 128, 136, 149, 156, 164,
171, 180, 186, 190, 199, 207,
209, 214, 216, 240, 256, 263,
270, 280, 281, 289, 296, 297,
304, 308, 313, 316, 334, 344,
349, 357, 366, 375, 390, 397,
418, 424, 436, 442, 450, 456,
483, 489, 496, 503, 505, 510,
511, 520, 534, 555, 557, 562,
568, 570, 578, 586, 623, 624,
629, 637, 645, 646, 655, 694,
702, 710, 722, 747, 748, 753,
754, 769, 770, 778, 806, 814,
823, 829, 831, 835, 856, 863,
878, 886, 893, 898, 905, 913,
919, 920, 928, 935, 942, 944,
952, 959, 960
reading math, 54, 83, 174, 277, 339,
830, 854, 861, 868
writing in math, 6, 13, 21, 25, 37,
44, 49, 55, 61, 76, 86, 93, 101,
109, 117, 127, 128, 135, 136,
156, 164, 171, 182, 190, 199,
207, 214, 240, 249, 255, 263,
269, 279, 288, 296, 304, 312,
318, 333, 341, 349, 356, 374,
383, 390, 392, 397, 402, 417,
424, 432, 442, 449, 456, 466,
482, 489, 510, 519, 525, 533,
539, 555, 561, 568, 577, 586,
592, 617, 624, 629, 654, 661,
686, 694, 710, 714, 721, 734,
747, 753, 760, 769, 778, 798,
806, 807, 814, 823, 840, 856,
862, 870, 878, 886, 892, 898,
912, 919, 927, 935, 939, 944,
945, 952, 959
Commutative property, 3
for matrix operations, 188, 196
Compare, exercises, 8, 39, 61, 135,
137, 208, 243, 251, 271, 289,
327, 458, 472, 528, 572, 580,
619, 623, 645, 649, 702, 714,
740, 749, 756, 787, 791, 800,
807, 818, 872, 896, 912, 948
Comparing
graphs of absolute value functions,
121–125
graphs of quadratic functions,
236–237, 240
independent and dependent
events, 719
standard and translated equations,
650, 651
types of variation, 554
Complement
of an event, 709–713, 718
of a set, 715–716
Complementary angles, 994
Completing the square, 283–291, 317,
321, 653
using algebra tiles, 283
Complex conjugates, 278, 380
Complex conjugates theorem, 380
Complex fraction, 584
simplifying, 584–588
Complex number(s), 276
absolute value of, 279, 280
additive inverse of, 280
Julia set and, 282
Mandelbrot set and, 281
multiplicative inverse of, 280
operations with, 276–282, 317,
320–321
plotting, 278
standard form of, 276
Complex plane, 278
Julia set on, 282
Mandelbrot set on, 281
Composite number, 978
Composition, of a function, 430–435,
465, 467
Compound event, 707
probability of, 707–713
Compound inequality, 41–47
absolute value form of, 53
Compound interest, 481, 483–485
graphing, 874, 908–914, 964, 965
reflections, 917, 920
translations, 915–917, 919–922,
966
half-angle formula for, 955
using, 955–962
inverse of, 874–879, 897
sinusoids, and, 941–948
sum formula for, 949
using, 949–954
Cosines, law of, 889–895, 897, 900
Cotangent function, See also
Trigonometric function(s)
evaluating for any angle, 866–872
evaluating for right triangles,
852–858
Cotangent identities, 924, 966
Coterminal angle, 860, 863
Counterexample, 1003
Co-vertices, of an ellipse, 634
Cramer’s rule, 205–209, 221, 226
Critical x-values, 303, 599
Cross multiplication, to solve
rational equations, 589–590
Cube root function, 447–451, 465, 468
parent, 446, 465
Cubes
difference of, 354
sum of, 354
Cubic function, See also Polynomial
function(s), 337
inverse of, 441
writing, 393–399
Cubic regression, 396
Cumulative Review, 232–233,
474–475, 678–679, 848–849
Cycle, of a function, 908
Cylinder
surface area of, 63, 567, 572, 580,
993
volume of, 334, 350, 567, 572, 580,
993
D
Data, See also Graphs; Modeling;
Statistics
analyzing
using best-fitting line, 112–120
choosing a model for, 774–781,
786
finite differences, 393–399
fitting a model to, 774–781
geometric mean, 749
hypothesis testing, 764–765
margin of error, 768–771
measures of central tendency,
744–750, 783, 784, 1005
measures of dispersion,
744–750, 783, 784, 1005
negative correlation, 113, 114,
117
normal distribution, 757–762,
783, 785
outlier, 746, 747
positive correlation, 113, 114,
117
quartiles, 1008–1009
range, 745–750
standard deviation, 745–750
applying transformations to,
751–755
collecting, 112, 550
biased question, 772–773
biased sample, 767
control group, 773
convenience sample, 766
from an experiment, 308, 528,
772–773, 774
population, 766
random sample, 766
sampling, 766–771, 783, 786
self-selected sample, 766
using simulation, 714
from a survey, 763, 764, 766–771,
772–773, 786
systematic sample, 766
unbiased sample, 767
displaying
in a bar graph, 1006–1007
in a circle graph, 1006–1007
in a line graph, 1006–1007
in a scatter plot, 113–120
organizing
in a box-and-whisker plot,
1008–1009
in a histogram, 724, 726–731,
1008–1009
in a line plot, 1008–1009
using matrices, 189, 192
in a stem-and-leaf plot,
1008–1009
in a table, 112, 528
in a Venn diagram, 706
Decay factor, 486
Decay function
exponential, 486–491, 538, 540
involving e, 493–498
Decimal exponents, 425
Decimals, fractions, percents, and,
976
Degree
converting between radians and,
860–864, 899
measure of a circle, 860
of a polynomial function, 337, 339
Dependent events, 718–723
Dependent linear system, 154–157
Dependent variable, 74
Depression, angle of, 855
Index
INDEX
Compound statement, 1001
Concept Summary, 188, 197, 387,
861
Concepts, See Big Ideas; Concept
Summary; Key Concept
Condensing a logarithmic
expression, 508, 510, 541
Conditional probability, 718–723
Cone, intersected by a plane, See
Conics
Congruent figures, 996–997
Conics, See also Circle; Ellipse;
Hyperbola; Parabola, 649–657
classifying, 653, 656
degenerate, 657
discriminant of, 653, 656
eccentricity of, 665–666
lines of symmetry of, 652, 655
translated, equations of, 650–657,
672
Conjugates, 267
complex, 278
Connections, See Applications
Consistent linear system, 154–157
Constant
adding to data values, 751–755
common difference, 802
Constant ratio, 810
Constant term, 12
Constant of variation, 107, 551
Constraints, 174
Continuous function, 80–81
Continuously compounded interest,
494–495, 497
Control group, 773
Convenience sample, 766
Converse, of a conditional statement,
1002–1003
Coordinate geometry, formulas from,
1026
Coordinate plane, 987
Corollary to the fundamental
theorem of algebra, 379
Correlation, describing, 113–114
Correlation coefficient, 114
Cosecant function, See also
Trigonometric function(s)
evaluating for any angle, 866–872
evaluating for right triangles,
852–858
Cosine function, See also
Trigonometric equation(s);
Trigonometric function(s)
difference formula for, 949
using, 949–954
double-angle formula for, 955
using, 955–962
evaluating for any angle, 866–872
evaluating for right triangles,
852–858
1089
INDEX
Derivation
of Snell’s law, 930
of a trigonometric model, 957
Descartes, René, 381
Descartes’ Rule of Signs, 381
using, 381–382, 384, 385
Determinant, of a matrix, 203–209,
226
Diagram
drawing, problem solving strategy,
35, 37, 39
interpreting, 324, 326, 608, 609,
610, 844, 846, 847, 937
mapping, 72, 73, 77
Pascal’s triangle, 692, 695
tree, 682, 686, 720, 978
Venn, 2, 430, 706–708, 715–716,
1004
Diameter, of a circle, 992
Difference
of two cubes, 354
of two squares, 253
Difference formulas, 949, 964
using, 949–954, 968
Dilation, on the coordinate plane,
989
Dimensions, of a matrix, 187, 195
Direct argument, 1000–1001
Direct substitution, for evaluating
polynomial functions, 338
Direct variation, 107–111, 140, 143
Directrix, of a parabola, 620
Discrete function, 80–81
Discrete mathematics
counting methods, 682–689
discrete functions, 80–81
finite differences, 393–399
greatest common factor (GCF),
978–979
least common denominator (LCD),
979
least common multiple (LCM),
978–979
matrices, 187–219
mutually exclusive events, 707
Pascal’s triangle, 692, 695
scatter plots, 113–120, 143
sequences, 794–816
set theory, 715–716
tree diagram, 682, 686, 720, 978
triangular numbers, 394
triangular pyramidal numbers, 395
Discriminant, 294, 296
of a conic equation, 653, 656
Disjoint event, 707, 736
Dispersion, measures of, 744–750,
783, 784, 1005
Distance formula, 614, 619, 669
Distribution
binomial, 763–765
1090 Student Resources
normal, 757–762, 783, 785
standard normal, 758–762
Distributive property, 3
to add and subtract like radicals,
422
for matrix operations, 188, 197
for solving linear equations, 20,
22–24
Division
of complex numbers, 278, 280
of functions, 429–435
inequalities and, 42–47
integer, 975
as opposite of multiplication, 4
polynomial, 362–368
properties, 18
with rational expressions, 576–580,
602, 605
synthetic, 363–368
Division property of equality, 18
Domain
of a function, 73, 76, 391, 428–430,
446–447, 463, 479, 482, 485,
487, 489, 491
of a relation, 72
of a sequence, 794
Doppler effect, 563
Double-angle formulas, 955, 968
using, 955–962, 968
Draw angles in standard position,
859, 860, 863
Draw conclusions
examples, 132
exercises, 50, 112, 122, 152, 177,
283, 308, 336, 437, 528, 550,
649, 688, 706, 774, 819, 826,
874, 881, 923, 948
from samples, 766–771
Draw a diagram
exercises, 887, 895
problem solving strategy, 35, 37,
39
Draw a graph
exercises, 57, 95, 104, 119, 129, 242,
290, 306, 314, 343, 344, 451,
631, 729, 816, 914, 929, 937
problem solving strategy, 49, 273
E
Eccentricity of conic sections,
665–666
Efficiency, 574, 580
Element
of a matrix, 187
of a set, 715–716
Elevation, angle of, 855
Eliminate choices, test-taking
strategy, 3, 228, 229, 286, 544,
545, 590, 627, 788, 789, 933
Elimination method
for solving linear systems, 161–167,
179–185, 221, 223
for solving quadratic systems,
660–664, 668
Ellipse, 634
area of, 636, 638, 640
co-vertices of, 634
eccentricity of, 665–666
equation of, 634
graphing, 634–639, 668, 671
translated, 650, 652–657
writing, 635–639, 668, 671
foci of, 634
major axis of, 634
minor axis of, 634
vertices of, 634
Empty set, 715
End behavior, for a polynomial
function, 336, 339–344
Equation(s), See also Formulas;
Function(s); Inequalities;
Linear equation(s);
Polynomial(s); Quadratic
equation(s), 18
absolute value, 50–58, 60, 64
of circles, 626–633
of conic sections, 650–657, 668,
670–672
direct variation, 107–111
of ellipses, 634–639
equivalent, 18
exponential, 515–516, 519–525, 542
general second-degree, 653
of hyperbolas, 642–648, 650–657,
668, 671, 672
inverse variation, 551–557
joint variation, 553–557
logarithmic, 499–501, 503–505
matrix, 190–192
of parabolas, 620–625
radical, 452–461, 465, 468
rational, 589–597, 602, 606
with rational exponents, 453, 456,
458–459
recursive, 826–833
rewriting, 26–32, 63
for sequences, 794–795, 798–801
for translated conics, 650–657
trigonometric, 876–880, 931–939,
964, 967
in two variables, 74–79
Equivalent equations, 18
Equivalent expressions, 12
Equivalent inequalities, 42
Error analysis
Avoid Errors, 12, 20, 28, 43, 52, 73,
82, 108, 115, 126, 133, 153,
163, 187, 238, 247, 252, 260,
267, 277, 287, 292, 330, 338,
Exponential equation(s), 515
modeling with, 516, 521–522
property of equality for, 515
solving, 515–516, 519–525, 538, 542
Exponential function(s), 478
decay, 486–491, 538, 540, 776
graphing, 478–485, 486–491, 538,
539, 776
growth, 478–485, 538, 539
as inverse of logarithmic functions,
501
involving e, 493–498, 540
modeling with, 480–481, 483–485,
488–491, 528, 530, 534–536
natural base, 493–498
writing, 529–531, 533–536, 542
Exponential inequalities, 526, 527
Exponential regression, 528, 530
Exponentiating an equation, 517
Expression(s)
combining like terms in, 12–17
equivalent, 12
evaluating, 10–17, 60, 62, 330–335
exponential, 330–335
factorial, 684
logarithmic, 499–501, 503–505,
507–512, 541
natural base, 492–493, 495–496
numerical, 10–17, 330–331, 333
in quadratic form, 355
rational, 573–588, 602, 605
with rational exponents, 415–419,
420–427, 467
simplifying, 10–17, 60, 62, 330–335,
420–427
terms of, 12
trigonometric, 925–926, 928,
955–956, 959, 960
writing, 984
Extended response questions,
146–148, 470–472, 738–740,
970–972
practice, Throughout. See for
example 8, 32, 33, 47, 59, 69,
88, 95, 106, 119, 139, 158, 166,
173, 185, 186
Extensions
approximate binomial
distributions, 763–765
design surveys and experiments,
772–773
determine eccentricity of conic
sections, 665–666
discrete and continuous functions,
80–81
linear programming, 174–176
piecewise functions, 130–131
prove statements using
mathematical induction,
836–837
set theory, 715–716
solve exponential inequalities, 526,
527
solve logarithmic inequalities, 527
solve radical inequalities, 462–463
solve rational inequalities, 598–600
Extra Practice, 1010–1023
Extraneous solutions
for absolute value equations, 52
for logarithmic equations, 518
for radical equations, 454
for rational equations, 591
for trigonometric equations, 934
F
Factor(s), 978
common, 978
conversion, 981
decay, 486
growth, 478
scale, 989
Factor theorem, 364, 404
Factor tree, 978
Factorial, See also Combination(s);
Permutation(s), 684
Factoring
completely, 353
difference of two squares, 253
patterns, 354
perfect square trinomials, 253
polynomials, 353–359, 364–368,
404
by grouping, 354, 357
quadratic equations, 252–265, 317,
319–320
quadratic expressions, 252–253,
255–256, 259–260, 263
with special patterns, 253, 256, 260,
263
the sum or difference of cubes, 354
trinomials, 252–265
zeros and, 262–265
Factorization, prime, 978–979
Feasible region, 174
Fibonacci sequence, 828, 832
recursive rule for, 828
Find the error, See Error analysis
Finite differences, 393–399
first-order differences, 393
properties of, 394
second-order differences, 394
third-order differences, 395
Finite sequence, 794
First-order differences, 393
Focal length, 585, 624
Focus (Foci)
of an ellipse, 634
of a hyperbola, 642
of a parabola, 620
Index
INDEX
347, 354, 355, 362, 370, 371,
415, 416, 423, 430, 441, 480,
488, 507, 553, 573, 584, 599,
644, 654, 658, 683, 691, 693,
700, 708, 719, 726, 744, 795,
797, 803, 811, 821, 828, 862,
877, 890
exercises, Throughout. See for
example 7, 13, 17, 22, 30, 38,
45, 56, 77, 86, 93, 102, 110,
118, 128, 136
Estimation, See also Approximation;
Prediction
of best-fitting line, 113–120, 143
of coordinates of turning points,
390
of correlation coefficients, 114–120
using exponential decay models,
488–491
using exponential growth models,
480, 483–485
using linear graphs, 91, 153
using natural base functions, 494,
497–498
using nth roots, 416
of solutions of linear systems,
153–158
using transformed data, 781
Euler, Leonhard, 492
Euler number e, 492
Even function, 928
Event(s)
complement of, 709
compound, 707, 733, 736
dependent, 718–723, 733, 736
disjoint, 707, 736
independent, 717–719, 721–723,
733, 736
mutually exclusive, 707
overlapping, 707, 733, 736
probability, 698
Expanding a logarithmic expression,
508, 510, 541
Experiment, 308, 528
binomial, 725, 728, 729
control groups and, 773
designing, 772–773
Experimental group, 773
Experimental probability, 700, 702
Explicit rule, 827, 839
Exponent(s)
decimal, 425
evaluating, 10–17
irrational, 425
properties of, 330, 402, 420, 465,
1034
using, 330–335, 402, 420–427,
467
rational, 415–419, 420–427, 465,
466
1091
INDEX
FOIL method, 248, 985
Formulas, 26
area
of a circle, 26, 992
of a parallelogram, 991
of a rectangle, 26, 991
of a trapezoid, 26, 991
of a triangle, 26, 885, 891, 991
Beaufort number, 458
change-of-base, 508
circumference, 26, 992
combinations, 690
degrees/radians, 860
distance, 26, 34
to the horizon, 450
between points, 614, 669
double angle, 955
Fahrenheit/Celsius, 26, 44, 69
half angle, 955
interest
compound, 481
continuously compounded, 494
interior angle of a regular polygon,
799
Kelvin/Celsius, 450
margin of error, 768
midpoint, 615
Newton’s law of cooling, 516
nth pentagonal number, 394
nth triangular number, 394
perimeter, of a rectangle, 26, 27,
991
permutation, 685
probability, 698, 700
of the complement of an event,
709
of compound events, 707
of dependent events, 718
of disjoint events, 707
of independent events, 717
rewriting, 26–32, 63
slant height, of a truncated
pyramid, 459
slope, 82
standard deviation, 748
standard normal distribution, 758
sum of first n positive integers,
797
sum of squares of first n positive
integers, 797
surface area
of a cone, 451
of a cylinder, 63, 567, 572, 580,
993
of a hemisphere, 472
of a rectangular prism, 993
of a sphere, 427
table of, 1026–1032
trigonometric difference, 949
trigonometric sum, 949
1092 Student Resources
volume
of a cone, 65
of a cube, 350, 601
of a cylinder, 334, 350, 567, 572,
580, 993
of a dodecahedron, 419
of an icosahedron, 419
of an octahedron, 419
of a pyramid, 350, 373
of a rectangular prism, 68, 334,
350, 993
of a sphere, 332, 409, 427, 436,
475, 601
of a tetrahedron, 419
Forty-five degree angle,
trigonometric values for, 853
Fractal geometry
fractal tree, 838
Julia set, 282
Mandelbrot set, 281
Sierpinski carpet, 816
Sierpinski triangle, 825
Fraction(s)
adding, 979
complex, 584
decimals, percents, and, 976
subtracting, 979
writing repeating decimals as, 822
Fraction bars, as grouping symbols,
14
Frequency, of a periodic function,
910
Function(s), See also Graphs; Linear
function(s); Parent function;
Quadratic function(s), 73, 140,
141
absolute value, 121–129
classifying, 75, 80–81, 479, 487, 489
composition of, 430–435, 465, 467
continuous, 80–81
cosine, 852–858, 866–872, 949–962
cube root, 446–451, 465, 468
discrete, 80–81
domain of, 73, 76, 428–430
even, 928
exponential growth and decay,
478–491, 528–531, 533–536
family, 89
greatest integer, 131
inverse, 438–445, 465, 467, 501
horizontal line test for, 440
iterating, 830, 831, 833
linear, 75–79, 89–97, 438–439,
442–444
logarithmic, 502–505
logistic, 522
natural base, 493–498
objective, 174
odd, 928
operations on, 428–435, 465, 467
piecewise, 130–131
power, 428–435, 531–535
properties of, 1034
quadratic, 236–243, 245–251,
310–315, 322
radical, 446–451, 465, 468
range of, 73
rational, 548–607, 602, 604
recursive rules and, 827–835
representing, 73–79
rounding, 131
sine, 852–858, 866–872, 949–962
square root, 446–451, 465, 468
step, 131
tangent, 852–858, 866–872,
911–914, 949–962
vertical line test for, 73–74, 77
Function notation, 75
Fundamental counting principle,
682
and permutations, 684–689,
734–735
with repetition, 683, 687–689
using, 682–683, 686–689, 720
Fundamental theorem of algebra,
379
applying, 379–386, 405
corollary to, 379
G
Gauss, Karl Friedrich, 379
General rational functions, 565–571,
602, 604
General second-degree equation, 653
Generalize, exercises, 703, 705, 712
Geometric mean, 749
Geometric probability, 701, 703, 704,
738–739
Geometric sequence, 810–816, 839,
841
recursive rules and, 827–835, 839,
842
Geometric series, 812–816, 839, 841
infinite, 819–825, 839, 842
Geometry, See also Angle(s); Circle;
Formulas; Triangle(s);
Trigonometric function(s)
congruent figures, 996–997
conics, 649–657, 665–666
formulas from, 1032
golden rectangle, 594
line symmetry, 990
parallel lines, 84–86, 99, 102
perpendicular bisector, 615–619
Pythagorean theorem, 995
reflection, 988–989
similar figures, 996–997
transformation, 988–989
triangle relationships, 995
modeling natural base
functions, 494
modeling the period of a
pendulum, 447
sinusoidal regression, 943
solving an equation with two
radicals, 455
solving an exponential
inequality, 526
solving a linear quadratic
system, 658
solving a logarithmic inequality,
527
solving radical inequalities,
462–463
solving rational equations,
596–597
solving rational inequalities, 598
solving trigonometric equations,
938–939
exercises, 118, 243, 251, 305, 434,
444, 483, 489, 491, 496, 497,
521, 534–536, 569–571, 638,
662, 712, 761, 816, 888, 914,
920, 937
exponential regression feature,
528, 530, 776, 781
graphing feature, 49, 97, 121, 122,
336, 361, 455, 461, 514, 523,
525, 526, 527, 567, 597, 598,
633, 658, 775, 776, 777, 801,
834, 923, 939, 943
intersect feature, 159, 305, 361, 455,
461, 463, 523, 525, 526, 527,
597, 598, 658, 939, 950
linear regression feature, 116, 775
list feature, 116, 308, 311, 731, 781,
943
LN feature, 514
LOG feature, 514
matrix feature, 194, 211, 213
maximum feature, 244
minimum feature, 244, 567
power regression feature, 533, 535
quadratic regression feature, 308,
311, 777, 786
random number feature, 714
root feature, 382
sequence mode, 801, 834
sinusoidal regression feature, 943
sort feature, 714
statistics feature, 750
STATPLOT feature, 116, 308, 311,
396, 528, 530, 774, 775, 781,
786, 943
summation feature, 801
table feature, 25, 48, 49, 152, 360,
460, 462, 496, 523, 524, 526,
527, 530, 581, 596, 598, 801,
938
table setup feature, 25, 460, 462,
596
test feature, 49
trace feature, 49, 244, 292, 293, 447,
494
window settings, 97, 159, 345, 361,
463, 923
zero feature, 243, 382
zoom feature, 121, 122, 633
Graphs
of absolute value, 50, 51
of absolute value functions,
121–129
of absolute value inequalities,
50, 53, 54, 56, 135, 136, 169,
172
bar, 1006–1007
of best-fitting lines, 115–120
circle, 1006–1007
of continuous functions, 80–81
of cosine functions, 874, 908–914,
964, 965
translations, 915–917, 919–922,
966
of cube root functions, 446–451,
465, 468
of discrete functions, 80–81
of equations of circles, 626–633,
668, 670
translated, 650
of equations of ellipses, 634–639,
668, 671
translated, 652
of equations of hyperbolas,
642–648, 668, 671
translated, 651
of equations of parabolas, 620–625,
668, 670
translated, 651
of equations in two variables,
74–79
of exponential decay functions,
486–491, 538, 539
of exponential growth functions,
478–485, 538, 539
histogram, 724, 726–731
of horizontal and vertical lines,
92, 94
of horizontal and vertical
translation, 916
of infinite geometric series, 820
interpreting, 325–326
of inverse functions, 437, 438, 440,
443, 445
line, 1006–1007
of linear equations, 89–97, 142
in three variables, 177
of linear inequalities
in one variable, 41–49, 64, 133
in two variables, 132–138, 144
Index
INDEX
Geometry software activity,
explore the law of sines, 881
Golden ratio, 594
Golden rectangle, 594
Graphing calculator
activities
calculate a binomial
distribution, 731
calculate one-variable statistics,
750
end behavior of functions, 336
evaluate expressions, 17
find maximum and minimum
values, 244
function operations, 435
graph equations of circles, 633
graph linear equations, 97
graph logarithmic functions, 514
graph rational functions, 564
graph systems of equations, 159
use matrix operations, 194
modeling data with a quadratic
function, 308
operations with functions, 435
operations with sequences, 801
set a good viewing window, 345
solving linear systems using
tables, 152
use tables to solve equations, 25
transformations, 121–122
trigonometric identities, 923
verify operations with rational
expressions, 581
binomial probability feature, 731
checking solutions with, 161, 285,
292, 293, 462, 518, 591, 659,
958
connected mode, 564, 581
cubic regression feature, 396
dot mode, 564
entering equations, 25, 48, 49, 97,
121, 122, 152, 159, 345, 396,
455, 460, 461, 462, 463, 494,
514, 523, 524, 525, 564, 581,
596, 597, 633, 834, 923, 939
examples
approximating real zeros of a
polynomial function, 382–383
computing inverse matrices,
211–213
drawing a histogram, 731
finding a best-fitting line, 116
finding an exponential model,
530
finding a polynomial model, 396
finding a power model, 532
finding turning points of a
polynomial function, 388
maximizing a polynomial
model, 589
1093
INDEX
of linear-quadratic systems, 658,
661, 662–664
of linear systems, 153–159, 221,
222
of logarithmic functions, 502–505,
514, 541
of natural base functions, 493–494,
496–498
of parallel and perpendicular lines,
84–86
of piecewise functions, 130–131
of polynomial functions, 336,
339–344, 387–392, 401, 403,
406
of quadratic functions
in intercept form, 246–251, 317,
319
in standard form, 236–243, 317,
318
in vertex form, 245–246,
249–251, 317, 319
of quadratic inequalities, 300–307
of quadratic systems, 658–664, 668
of radical functions, 446–451, 465,
468
of rational functions, 558–571, 602,
604
of real numbers, 2
of relations, 72, 76
scatter plots, 112–120, 143
of sequences, 795, 798, 800, 801
of sine functions, 874, 908–914,
964, 965
translations, 915–917, 919–922,
966
of square root functions, 446–451,
465, 468
of systems of constraints, 174–176
of systems of linear inequalities,
168–173, 221, 223
of systems of quadratic
inequalities, 301, 304, 305
of tangent functions, 911–914, 964,
965
translations, 918, 920, 921
of trigonometric functions, 874,
908–922, 964, 965
vertical shrinking of, 479
vertical stretching of, 479
Greatest common factor (GCF),
978–979
Greatest integer function, 131
Gridded-answer questions,
Throughout. See for example
33, 59, 69, 106, 139, 149, 186,
220, 231, 274, 316, 327, 369,
400, 411
Grouping symbols
fraction bars, 14
parentheses, 15
Growth factor, for an exponential
growth function, 478
Growth function
exponential, 478–485, 538, 539
involving e, 493–498, 540
Guess, check, and revise, problem
solving strategy, 998–999
H
Half-angle formulas, 955, 964
using, 955–962, 968
Half plane, 132
Heron’s area formula, 891
Hexagonal number, 837
Histogram, 1008–1009
on a graphing calculator, 731
probability distribution, 724,
726–731, 733
Hooke’s law, 444
Horizontal asymptote, 558
Horizontal line, graph of, 92
Horizontal line test, 440, 443
Horizontal translation, graphing, 916
Hyperbola, 558, 642
asymptotes of, 642
branches of, 642
center of, 642
eccentricity of, 665–666
equation of
graphing, 642–648, 668, 671
translated, 650, 651, 652–657,
672
writing, 643–648, 668, 671
foci of, 642
transverse axis of, 642
vertices of, 642
Hypotenuse, 995
Hypothesis, 1002–1003
Hypothesis testing, 764–765
I
Identity, 12
Identity matrix, 210
Identity property, 3
If-then form, of a conditional
statement, 1002–1003
Imaginary number, 276
Imaginary unit i, 275
Inconsistent linear system, 154–157
Independent events, 717–719,
721–723
Independent linear system, 154–157
Independent variable, 74
Index of a radical, 414
Index of refraction, 879, 930, 963
Index of summation, 796, 797
Indirect argument, 1000–1001
Indirect measurement, 855, 857–858
Induction, mathematical, 836–837
Inequalities, See also Linear
inequalities
absolute value, 50–58, 135, 136,
169, 172
compound, 41–47
equivalent, 42
exponential, 526, 527
linear, 41–49, 132–138
systems of, 168–176, 221, 223
logarithmic, 527
quadratic, 300–307, 322
systems of, 301, 304, 305
radical, 462–463
rational, 598–600
Infinite geometric series, 819–825,
839, 842
Infinite sequence, 794
Infinite series, 796
Infinity, positive and negative, 336,
339
Initial side, of an angle, 859
Integers, 2
operations with, 975
Intercept form, of a quadratic
function, 246–251, 317, 319
Interest
compound, 481, 483, 484, 485
continuously compounded,
494–495, 497
Interpret
examples, 91, 124, 132, 429, 494,
560, 869
exercises, 88, 306, 307, 335, 458,
535, 555, 762, 809, 872, 888,
930
probability distributions, 725, 726
Intersection
of graphs of linear-quadratic
systems, 658
of graphs of quadratic systems, 659
of sets, 707–713, 715–716
Inverse cosine, 874–879
Inverse function(s), 437–445, 465,
467
logarithmic and exponential, 501
trigonometric, 874–880, 897, 899
Inverse matrices, 210–217, 221, 226
Inverse property, 3, 501
Inverse relation(s), 438, 442
Inverse sine, 874–879
Inverse tangent, 874–879
Inverse variation, 550–557, 603
equations, 551–557
modeling with, 550, 552, 556–557
Investigating Algebra, See Activities
Irrational conjugates theorem, 380
Irrational exponent, 425
Irrational number, 2
Iteration, 830, 831, 833
1094 Student Resources
n2pe-9060.indd 1094
9/28/05 9:28:10 AM
J
Joint variation, 553–557, 603
Julia set, 282
Justify results, Throughout. See for
example 4, 6, 7, 68, 271, 281,
391, 398, 418, 490, 512, 550,
556, 748, 819, 832
K
Kepler’s second law, 904
Key Concept, 2, 3, 10, 12, 18, 42, 51,
52, 53, 72, 73, 74, 80, 82, 83,
84, 89, 90, 91, 92, 98, 107, 115,
123, 126, 133, 154, 160, 161,
168, 174, 179, 187, 195, 203,
204, 205, 206, 210, 212, 218,
236, 237, 245, 246, 248, 253,
266, 275, 276, 279, 284, 292,
294, 330, 339, 353, 354, 363,
364, 370, 379, 380, 381, 388,
394, 414, 415, 420, 421, 428,
438, 440, 446, 452, 478, 481,
486, 492, 493, 494, 499, 502,
507, 508, 515, 517, 551, 553,
558, 559, 565, 573, 575, 576,
582, 583, 584, 614, 615, 621,
626, 634, 642, 650, 653, 655,
682, 685, 690, 692, 693, 698,
699, 700, 707, 717, 718, 724,
725, 744, 745, 751, 752, 757,
763, 764, 768, 794, 796, 797,
802, 804, 810, 812, 821, 827,
852, 853, 859, 860, 861, 866,
867, 868, 875, 882, 883, 885,
889, 891, 908, 909, 911, 915,
924, 949, 955
L
number of solutions of, 154–157
solution of, 152, 153, 178
solving
algebraically, 160–166, 223
using augmented matrices,
218–219
using Cramer’s rule, 205–206,
208–209, 221, 226
using the elimination method,
161–167, 179–185, 221, 223
by graphing, 153–159, 221, 222
using inverse matrices, 210–217,
221, 226
using the substitution method,
160–167, 181–185, 221
using tables, 152
in three variables, 178–185, 224
Lines
classifying by slope, 83
horizontal line test, 440, 443
parallel
equations for, 99, 102
slope of, 84–86
perpendicular
equations for, 99, 102
slope of, 84–86
of reflection, 988–989
slope of, 83, 100
of symmetry, 652, 655, 990
vertical, slope of, 73–74, 77
List, making to solve problems,
998–999
Local maximum, of a polynomial
function, 388
Local minimum, of a polynomial
function, 388
Location Principle, 378
Logarithm(s), 499
change-of-base formula and,
508–509, 511
common, 500
natural, 500
properties of, 507, 1034
using, 507–513, 541
Logarithmic equation(s), 517
evaluating, 499–501, 503–505, 541
modeling with, 519, 521–522,
524–525
property of equality for, 517
solving, 517–525, 538, 542
Logarithmic expression(s)
condensing, 508, 510, 541
evaluating, 499–501, 503–505,
507–512
expanding, 508, 510, 541
Logarithmic function(s)
graphing, 502–505, 514, 541
as inverse of exponential functions,
501
modeling with, 500, 504–505
Index
n2pe-9060.indd 1095
INDEX
Law of cosines, 889–895, 897, 900
Law of sines, 881–888, 897, 900
Law of universal gravitation, 557
Leading coefficient, of a polynomial
function, 337
Least common denominator (LCD),
979
of a rational expression, 583, 986
for solving a rational equation, 590
Least common multiple (LCM),
978–979
for a rational expression, 583
Left distributive property, 197
Legs, of a triangle, 995
Like radicals, 422
adding and subtracting, 422–427
Like terms, 12
combining, 12
Likelihood, of an event, 698
Line graph, 1006–1007
Line plot, 1008–1009
Line of reflection, 988–989
Line symmetry, 990
Line of symmetry
for a conic section, 652, 655
for a plane figure, 990
Linear equation(s), See also Linear
systems, 18
for best-fitting line, 112–120, 143
direct variation, 107–111
forms of, 140
graphing, 89–97, 177
using slope-intercept form,
90–97, 140, 142
using standard form, 91–96
in three variables, 177
with no solutions, 23
point-slope form of, 98–99, 101,
140
rewriting, 28, 30–32
slope-intercept form of, 90–97, 98,
100, 140, 142
solving, 18–25, 60, 62
using the distributive property,
20–24
standard form of, 91–96
in three variables, 177–185
writing, 19, 20, 23–24, 98–104, 142
Linear function(s), 75–79
graphing, 89–97
inverse, 438–439, 442–444
linear programming and, 174–176
Linear inequalities
constraints, 174–176
forms of, 41
graphing
in one variable, 41–49, 64, 133
systems of constraints, 174–176,
221, 223
in two variables, 132–138, 144
reading, 41
solution of, 41
solving, 41–49, 64
systems of, 168–173
three or more, 170–173
in two variables, 132–138, 144
Linear programming, 174–176
constraints, 174
feasible region, 174
objective function, 174
Linear-quadratic systems, 658–664
Linear regression, 116
Linear systems, 152, 153
classifying, 154–157
coefficient matrix of, 205–206
with infinitely many solutions, 154,
163, 178, 180
with no solutions, 154, 163, 178,
180
1095
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Logarithmic inequalities, 527
Logical reasoning, See Reasoning
Logistic function, 522
Lower limit of summation, 796
Lower quartile, 1008–1009
INDEX
M
Major axis, of an ellipse, 634
Make a list, problem solving strategy,
998–999
Make a table, problem solving
strategy, 48–49, 272–273,
998–999
Mandelbrot set, 281
Manipulatives, See also Calculator;
Graphing calculator
algebra tiles, 283
Calculator Based Laboratory, 948
coins, 308, 528
compass, 308
flashlight, 649
index cards, 774
measuring tools, 112, 437, 550
musical instruments, 948
Mapping diagram, 72, 73, 77, 140, 141
Margin of error, 768–771
formula, 768
Mathematical induction, 836–837
Mathematical modeling, formulas
from, 1031
Matrix (Matrices), 187
adding and subtracting, 187,
189–193, 194
augmented, 218–219
coefficient, 205–206
Cramer’s rule and, 205–209
describing products, 195
determinant of, 203–209
dimensions of, 187, 195
elements of, 187
equal, 187
equations, 190–192
identity, 210
inverse, 210–217
multiplying, 195–202, 937
order of operations, 188
properties, 188, 197, 1033
row operations, 218–219
scalar multiplication, 188–192
for solving linear systems, 205–209,
210–219
total cost, 198
transition, 201
triangular form, 218–219
Matrix algebra, formulas from, 1026
Maximum value
of a polynomial function, 388–392
of a quadratic function, 238–239,
241, 244, 287
of sine and cosine functions, 909
Mean, 744, 746–750, 783, 784, 1005
geometric, 749
transformation and, 751–755
Measurement
converting measurements, 5, 7
converting units of, 981
Measures, table of, 1025
Median, 744, 746–750, 783, 784,
1005
transformation and, 751–755
of a triangle, 618
Midline, of a trigonometric graph,
915
Midpoint
formula, 615, 669
of a line segment, 615, 617–619
Minimum value
of a polynomial function, 388–392
of a quadratic function, 238–239,
241, 244
of sine and cosine functions, 909
Minor axis, of an ellipse, 634
Mixed Review, Throughout. See for
example 9, 16, 24, 32, 40, 47,
58, 79, 88, 96, 104, 111, 120,
129, 138, 158
Mixed Review of Problem Solving,
33, 59, 106, 139, 186, 220, 274,
316, 369, 400, 436, 464, 506,
537, 572, 601, 641, 667, 705,
732, 756, 782, 818, 838, 873,
896, 940, 963
Mode, 744, 746–749, 783, 784, 1005
transformation and, 751–755
Modeling, See also Expression(s);
Formulas; Graphing
calculator; Graphs; Linear
equation(s); Polynomial(s);
Polynomial function(s);
Quadratic function(s)
absolute value, 51
using algebraic expressions, 13,
15–16
using area models, 254, 257, 258,
261, 283
using best-fitting quadratic
models, 311
choosing a model, 774–781, 786
using circular models, 628, 630–632
circular motion, 916, 921, 922
completing the square, 283
using conic sections, 654, 656–657
direct variation, 107–111, 143
dropped objects, 268–270, 272–273
using eccentricity, 666
using equations of circles, 628,
630–632
using equations of ellipses, 636,
638–639, 640
using equations of hyperbolas, 644,
646–648
using equations of parabolas, 622,
624–625
exercises, 15, 39, 271, 557, 580, 639,
809, 888, 946
exponential decay, 488–491, 776
with exponential equations, 516,
521–523
using exponential functions,
480–481, 483–485, 488–491,
528, 530, 534–536
exponential growth, 480–481,
483–485
factors using a tree, 978
finite differences, 395, 398
fitting a model to data, 774–781
using hundreds squares, 976
infinite geometric series, 819, 822,
824–825
using inverse of a power functions,
441–442, 444–445
using inverse variation, 550, 552,
556–557
launched objects, 295, 298
using linear equations, 19, 20,
23–24, 29, 31–32, 98–105, 775
using linear inequalities, 44, 46–47
using logarithmic equations, 519,
521–522, 524–525
using logarithmic functions, 500,
504–505
using mapping diagrams, 72, 73, 77
using natural base functions, 494,
495, 497–498
normal distribution, 757–762
using a number line, 2, 41–51, 53,
54, 56, 303, 599, 975
operations on sets, 715–716
pendulum periods, 447
with power functions, 532–533,
535, 542
using quadratic equations, 254,
257–258, 261, 262, 264–265
using quadratic functions, 308,
311, 314
using quadratic inequalities, 303,
306–307
using quadratic regression, 308,
311
using quadratic systems, 660,
662–664
using rational equations, 589,
594–597
using rational expressions, 574,
579–580
real numbers, 2
relations, 72
using scatter plots, 112–120
with sine functions, 910, 913–914
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inequalities and, 42–47
integer, 975
matrix, 195–202
as opposite of division, 4
of polynomials, 347–352, 403
properties, 3, 18
for matrices, 197
of rational expressions, 575–580,
602, 605
scalar, 188–192
Multiplication property of equality,
18
Multiplicative inverse, 4
of a complex number, 280
Multi-step problems
examples, 29, 85, 91, 134, 170, 189,
206, 213, 239, 262, 311, 340,
373, 396, 429, 431, 439, 447,
480, 488, 494, 560, 567, 574,
616, 636, 644, 654, 691, 720,
795, 829, 862, 891, 951
exercises, 8, 23, 32, 33, 39, 47, 57,
59, 78, 88, 95, 103, 106, 137,
139, 184, 186, 200, 209, 216,
220, 250, 257, 274, 298, 307,
314, 316, 335, 344, 351, 358,
369, 376, 400, 418, 433, 436,
444, 458, 464, 484, 491, 497,
505, 506, 535, 537, 556, 570,
572, 580, 601, 619, 625, 631,
641, 647, 663, 667, 688, 696,
705, 712, 732, 748, 756, 761,
782, 800, 818, 838, 864, 872,
873, 888, 893, 896, 921, 940,
954, 961, 963
Mutually exclusive events, 707
N
Natural base e, 492
Natural base expression, 492–493,
495–496
Natural base function, 493–498
Natural logarithm, 500
change-of-base formula and,
508–509, 511
Negative angle identities, 924
Negative correlation, 113, 114, 117
Negative exponent property, 330
Negative number, square root of,
275
Newton’s law of cooling, 516
Normal curve, 757
Normal distribution, 757–762, 783,
785
nth root, 414
evaluating, 414–419, 466
Number line, 2
to add integers, 975
for graphing absolute value, 50, 51
for graphing absolute value
equations, 50
for graphing absolute value
inequalities, 50, 53, 54, 56
for graphing linear inequalities,
41–49
for graphing quadratic inequalities,
303
real numbers on, 2
to show critical x-values, 599
to subtract integers, 975
Numbers
absolute value of, 50, 51
classifying, 2
complex, 276
composite, 978
imaginary, 276
integers, 2, 975
irrational, 2
pentagonal, 394
prime, 978–979
pure imaginary, 276
rational, 2
real, 2
triangular, 394
triangular pyramidal, 395
whole, 2
Numerical expression, 10–17, 60
O
Objective function, 174
Odd function, 928
Odds, for and against an event,
699–700, 702–703
Ohm’s law, 749
Online Quiz, Throughout. See for
example 9, 16, 24, 32, 40, 47,
58, 79, 88, 96, 104, 111, 120,
129, 138
Open-ended problems, 6, 14, 32, 33,
45, 59, 87, 94, 103, 106, 110,
118, 127, 136, 139, 157, 165,
183, 186, 191, 200, 208, 215,
220, 250, 257, 270, 274, 280,
289, 297, 305, 313, 316, 334,
342, 369, 375, 384, 391, 398,
400, 417, 425, 433, 436, 443,
456, 464, 483, 490, 496, 506,
511, 520, 537, 556, 562, 569,
572, 593, 601, 617, 630, 638,
641, 667, 705, 711, 722, 728,
732, 748, 756, 782, 815, 818,
824, 831, 838, 873, 878, 896,
913, 940, 963
Or rule, 1000–1001
Order of operations, 10–17
for matrices, 188
Ordered pair, 987
Ordered triple, 177, 178
Index
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solutions
to absolute value equations, 50
to absolute value inequalities,
50, 53, 54, 56
to linear inequalities, 41, 43, 45,
49
with tangent functions, 918, 921
using tree diagrams, 682, 686, 720,
978
with trigonometric functions, 870,
871–872, 941–948, 967
using Venn diagrams, 2, 430, 698,
706–708, 715–716, 1004
using a verbal model, 13, 19, 20, 29,
34, 35, 36, 42, 54, 63, 66, 100,
101, 134, 155, 162, 181, 239,
254, 261, 262, 356, 373, 389,
560, 589, 829
vertical motion, 295, 298
Monomial, See also Polynomial(s),
252, 985
Moore’s law, 547
Multiples, 978
Multiple choice questions, 228–230,
324–326, 544–546, 608–610,
788–790, 844–846
examples, 3, 19, 36, 82, 132, 155,
162, 268, 286, 332, 339, 355,
365, 430, 453, 508, 518, 575,
590, 614, 627, 708, 717, 745,
769, 805, 821, 853, 877, 933,
956
practice, Throughout. See for
example 6, 14, 24, 30, 37, 38,
45, 46, 56, 69, 77, 86, 87, 93,
101, 109
Multiple representations, See also
Manipulatives; Modeling
examples, 35, 48, 53, 105, 107, 115,
134, 153, 161, 218, 239, 272,
285, 292, 293, 340, 360, 396,
440, 455, 460, 480, 488, 523,
530, 552, 560, 567, 591, 596,
635, 636, 640, 651, 652, 659,
714, 720, 724, 781, 795, 834,
855, 884, 895, 910, 931, 934,
938, 958
exercises, 15, 24, 39, 57, 95, 104,
119, 129, 157, 173, 216, 242,
258, 290, 306, 314, 343, 367,
392, 434, 451, 485, 521, 562,
570, 631, 647, 703, 729, 754,
779, 808, 816, 857, 887, 914,
929, 937
Multiplication
of complex numbers, 277–278, 280
cross multiplying, 589–590
of data by a constant, 752–755, 783,
785
of functions, 429–435
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Ordering, real numbers, 3, 6
Origin, in a coordinate plane, 987
Outcome, 698
Outlier, 746, 747
Overlapping events, 707, 733, 736
INDEX
P
Parabola, 236
axis of symmetry of, 236, 620
directrix of, 620
eccentricity of, 665–666
equation of, 621–625, 668, 670
translated, 650–657
focal length of, 624
focus of, 620
graph of, 620–625, 668, 670
as graph of a quadratic function,
236–243
latus rectum of, 625
vertex of, 236, 620
Parallel lines
equations for, 99, 102
slope of, 84–86
Parent function
for absolute value functions, 121,
123
for cosine functions, 908
for cube root functions, 446, 465
for exponential decay functions,
486
for exponential growth functions,
478
for linear functions, 89
for logarithmic functions, 502
for quadratic functions, 236
for simple rational functions, 558
for sine functions, 908
for square root functions, 446, 465
Partial sum, 820
Pascal, Blaise, 692
Pascal’s triangle, 692, 695
binomial expansion and, 693, 695
Pattern(s)
exercises, 8, 350, 512, 695
exponential function models and,
529–531
factoring, 354
to make a generalization, 283
Pascal’s triangle, 692, 695
power function models and,
531–533
product patterns for binomials,
347
Pentagonal numbers, 394
Percent
calculating with, 977
decrease, 488–489
fractions, decimals, and, 976
increase, 480–481, 483–485
Perfect square, 284
Perfect square trinomial, 253
Perimeter, 991
Period, of a function, 908, 909
Periodic function, See also Cosine
function; Sine function, 908
frequency of, 910
Permutation(s), 684–689, 733,
734–735
circular, 689
formula, 685
probability and, 699, 702
with repetition, 685–689
Perpendicular bisector, 615–619
Perpendicular lines
equations for, 99, 102
slope of, 84–86
Piecewise function, 130–131
Point discontinuity, 579
Point-slope form, 98–99, 101, 140,
530, 532
Polynomial(s), 337
adding, 346–352, 403
dividing, 362–368
rational expressions, 576–577
factoring, 353–359, 364–368, 404
by grouping, 354, 357
multiplying, 347–352, 403
rational expressions, 575–576
in quadratic form, 355, 357
subtracting, 346–352, 403
theorems involving, 363, 364, 379
Polynomial equation(s)
factoring, 353–359, 364–368, 404
solving, 355–359, 404
Polynomial function(s), 337
classifying zeros of, 381–382, 384,
385
degree of, 337, 339
Descartes’ Rule of Signs and,
381–382, 384, 385
end behavior of, 336, 339–344
evaluating, 338–344, 402, 403
by synthetic substitution, 338
finding zeros of, 370–378, 379–386,
401, 405
fundamental theorem of algebra
and, 379–386, 405
graphing, 336, 340, 342–344,
387–392, 401, 403, 406
leading coefficient of, 337
maximum of, 388–392
minimum of, 388–392
standard form of, 337
turning points of, 388, 390
types of, 337
writing, 381, 384, 386, 393–399, 406
Polynomial long division, 362–368
Population, 766
Positive correlation, 113, 114, 117
Power function(s), 428–435
inverse, 440–445
modeling with, 532–533, 535, 538,
542
writing, 531–535
Power of a power property, 330, 414,
420
Power of a product property, 330, 420
Power property of logarithms, 507
Power of quotients property, 330, 420
Power regression, 533, 535
Powers, See also; Exponent(s);
Exponential function(s)
of a binomial difference, 693, 695
of a binomial sum, 693–695
coefficient of, 12
evaluating, 10–17
Practice, See Reviews
Precision, significant digits and, 983
Prediction
using direct variation, 108, 110
exercises, 15, 104, 110, 112, 158,
308, 344, 437, 535, 550, 631,
774, 791, 896, 973
using exponential decay models,
488–491
using exponential growth models,
480, 483–485
using exponential regression, 530
using an inverse function, 442
using line of fit, 116–120, 146–147
using rate of change, 85
using regression, 396
Premise, 1000–1001
Prerequisite Skills, xxiv, 70, 150, 234,
328, 412, 476, 548, 612, 680,
742, 792, 850, 906
Prime factorization, 978–979
Prime number, 978–979
Probability
binomial, 763–765
binomial distribution and,
724–731, 733, 736
combinations and, 699, 702, 733,
735
of the complement of an event,
709–713
of compound events, 707–713
conditional, 718–723
of dependent events, 718–723, 733,
736
of disjoint events, 707
event, 698
experimental, 700, 702
formulas, 698, 1028–1029
fundamental counting principle,
682, 684–689, 734–735
geometric, 701, 703, 704, 738–739
of independent events, 717–719,
721–723, 733, 736
1098 Student Resources
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of square roots, 266
of subtraction, 18, 188
zero product, 253
Proportion, 980
Pure imaginary number, 276
Pythagorean identities, 924
Pythagorean theorem, 995
using, 852–858, 866, 895, 898
Q
Quadrant(s), of a coordinate plane,
987
Quadrantal angle, 867
Quadratic equation(s), See also
Polynomial(s)
with complex solutions, 276–282
discriminant in, 294
to model dropped objects,
268–270, 272–273
to model launched objects, 295,
298
to model vertical motion, 295, 298
roots of, 253
solving
by completing the square,
284–291, 317, 321
by factoring, 252–265, 317, 319,
320
by finding square roots, 266–271,
284, 317, 320
using the quadratic formula,
292–299, 317, 321
standard form of, 253
systems of, 658–664
Quadratic expression(s)
completing the square for, 283
factoring, 252–253, 255–256,
259–260, 263
Quadratic form, of a polynomial, 355,
357
Quadratic formula, 292–299, 317,
321, 933
Quadratic function(s)
best-fitting quadratic model and,
311
graphing
in intercept form, 246–251, 317,
319
in standard form, 236–243, 317,
318
in vertex form, 245–246,
248–251, 287, 317, 319
maximum value, 238–239, 241, 244,
287
minimum value, 238–239, 241, 244
parent function, 236
writing
in intercept form, 309, 312–315,
322
in standard form, 310, 312–315,
322
in vertex form, 309, 312–315, 322
zeros of, 254–256
Quadratic inequality (inequalities)
critical x-values of, 303
graphing, 300–307, 322
in one variable, forms, 302–305
solving, 302–307, 322
system of, 301, 304, 305
Quadratic regression, 308, 311
Quadratic system, 658–664, 668, 672
Quartic function, 337
Quartile, 1008–1009
Quotient of powers property, 330,
420
Quotient property
of logarithms, 507
of square roots, 266
Quotient of powers property,
495–500, 542, 544
Quotient property of radicals,
720–726, 753, 755
Quotient rule, for fractions, 915
R
Radian, 860
converting between degrees and,
860–864, 899
Radian measure, 860–865
Radical(s), 266
index of, 414
like, 422
nth root, 414–419
properties of, 421–427, 507, 1034
simplest form, 422
Radical equation
solving, 452–461, 465, 468
with two radicals, 455, 457
Radical function, graphing, 446–451,
465, 468
Radical inequalities, 462–463
Radical sign, 266
Radicand, 266
Radius, of a circle, 626, 992
Random sample, 766
Random variable, 724
Range
as an absolute value inequality, 54
data, 745–750, 783, 784, 1005
of a function, 73, 446–447, 479,
487
of a relation, 72
of a sequence, 794
transformation and, 751–755
Rate, 85
Rate of change
average, 85, 86
slope and, 82–88, 142
Index
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INDEX
of mutually exclusive events, 707
normal distribution and, 757–762
odds and, 699–700, 702–703
outcome, 698
of overlapping events, 707, 733, 736
permutation and, 699, 702, 733,
734–735
standard normal table and,
759–762
theoretical, 698
Venn diagrams and, 698, 706–708
Probability distribution, 724
binomial, 725–731
skewed, 727, 728
symmetric, 727, 728
Problem solving strategies, See also
Eliminate choices; Problem
Solving Workshop
draw a diagram, 35, 37–39
draw a graph, 49, 273
use a formula, 34, 37–39
guess, check, and revise, 998–999
interpret a diagram, 608, 609, 610
look for a pattern, 35, 37–39
make a list, 998–999
make a table, 48–49, 272–273,
998–999
solve a simpler problem, 998–999
use a verbal model, 36–39
write an equation, 36–39
Problem Solving Workshop, 48–49,
105, 218–219, 272–273,
360–361, 460–461, 523–525,
596–597, 640, 714, 781,
834–835, 895, 938–939
Product of powers property, 330, 420
Product property
of logarithms, 507
of square roots, 266
Proof, See also Reasoning
using mathematical induction,
836–837
of properties of logarithms, 511
Properties
of addition, 3, 18, 188
of division, 18
for exponential equations, 515
of exponents, 330, 420, 1033, 1034
of finite differences, 394
of functions, 1034
inverse, 3, 501
for logarithmic equations, 517
of logarithms, 507, 1034
proofs of, 511
of matrix operations, 188, 197, 1033
of multiplication, 3, 18, 197
of radicals, 1034
of rational exponents, 420, 465,
1034
of real numbers, 2–9, 1033
1099
9/28/05 9:28:13 AM
INDEX
Ratio(s)
golden, 594
to identify direct variation, 108, 109
percent as, 976
proportions and, 980
simplest form, 980
trigonometric, 852–858
Rational equation, 589–597, 602, 606
Rational exponent(s), 413–419, 465,
466
equations with, 453, 456, 458–459,
468
properties of, 420–427, 465, 467,
1034
Rational expression(s), 986
adding, 582–588, 602, 605
dividing, 576–580, 602, 605
least common denominator of,
583, 986
multiplying, 575–580, 602, 605
point discontinuity and, 579
simplified form of, 573
simplifying, 573–574, 577–580, 602
subtracting, 582–588, 602, 605
verifying operations with, 581
Rational function(s)
graphing
general, 565–571, 602, 604
simple, 558–564, 604
inverse variation, 550–557
joint variation, 553–557
parent function for, 558
Rational inequalities, 598–600
Rational numbers, 2
Rational zero theorem, 370
Rationalizing the denominator, 267
Readiness
Prerequisite Skills, xxiv, 70, 150,
234, 328, 412, 476, 548, 612,
680, 742, 792, 850, 906
Skills Review Handbook, 975–1009
Reading
function notation, 75, 430, 438
graphs, 74
linear inequalities, 41
subscripts, 26
summation notation, 796
Reading math, 54, 83, 174, 277, 339,
830, 854, 861, 868
Real numbers
ordering, 3, 6
properties of, 2–9, 61, 1033
subsets of, 2
Reasoning
and rule, 1000–1001
biconditional statement,
1002–1003
chain rule, 1000–1001
compound statement, 1001
conclusion, 1000–1003
valid, 1000–1001
conditional statement, 1002–1003
converse, 1002–1003
if-then form, 1002–1003
counterexample, 1003
derivations, 930, 957
direct argument, 1000–1001
exercises, 7, 25, 31, 49, 56, 87, 88,
94, 103, 105, 110, 117, 118,
128, 183, 219, 256, 273, 289,
297, 334, 345, 384, 386, 391,
443, 450, 485, 491, 497, 505,
522, 531, 557, 577, 618, 630,
639, 640, 646, 647, 656, 662,
666, 695, 702, 714, 716, 722,
723, 748, 759, 761, 770, 771,
807, 823, 831, 835, 837, 920,
923, 936, 945, 946, 957, 960
hypothesis, 1002–1003
hypothesis testing, 764–765
indirect argument, 1000–1001
inductive reasoning, 836–837
or rule, 1000–1001
using a pattern to make a
generalization, 283
premises, 1000–1001
proof
using mathematical induction,
836–837
for properties of logarithms, 511
Venn diagrams, 1004
Reciprocal, multiplying by, 4
Reciprocal identities, 924
Rectangle, area and perimeter of, 991
Rectangular prism
surface area of, 993
volume of, 68, 334, 350, 993
Recursive rule, 826, 827
for a sequence, 826–835, 839, 842
Reference angle, 868, 871
Reflection
on the coordinate plane, 988–989
of the graph of a parent function
absolute value function, 124–129
cosine function, 917, 920
logarithmic function, 502
sine function, 917, 920
line of, 988–989
Refraction, index of, 879, 930, 963
Regression
cubic, 396
exponential, 528, 530
linear, 116
power, 533, 535
quadratic, 308, 311
Relation, 72, 140, 141
inverse, 438, 442
Remainder theorem, 363, 404
Repeated solution, 379
Repeating decimal, 822
Reviews, See Big Ideas; Chapter
Review; Chapter Summary;
Cumulative Review; Mixed
Review; Mixed Review of
Problem Solving; Prerequisite
Skills; Skills Review Handbook
Right distributive property, 197
Right triangle trigonometry,
852–858, 897, 898
Root(s), See also Radical(s)
evaluating nth roots, 414–419
of a quadratic equation, 253
square, 266, 275
Rotation
center of, 988
on the coordinate plane, 988–989
Rounding function, 131
Row equivalent matrices, 218
Row operations, 218–219
Rubric
for scoring extended response
questions, 146, 470, 738, 970
for scoring short response
questions, 66, 408, 674, 902
S
Sample(s)
biased, 767
classifying, 766, 769, 783
selecting, 766
size of, 768
Sampling, 766–771, 783, 786
SAT, See Standardized Test
Preparation
Scalar, 188
Scalar multiplication, 188–192
Scale factor, 989
Scatter plot, 113–120, 143
Science, See Applications
Scientific notation, 331, 982
properties of exponents and, 331,
333, 334
Secant function, See also
Trigonometric function(s)
evaluating for any angle, 866–872
evaluating for right triangles,
852–858
Second-order differences, 394
Sector, 861
arc length, 861–865
area, 861–865
central angle of, 861
Self-selected sample, 766
Sequence(s), 794, 839
arithmetic, 802–809, 839, 841
finite, 794
formulas from, 1029–1030
geometric, 810–816, 839, 841
graphing, 795, 798, 800, 801
1100 Student Resources
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binomial products, 985
expressions, 984
geometry
angle relationships, 994
congruent figures, 996–997
coordinate plane, 987
line symmetry, 990
similar figures, 996–997
transformations, 988–989
triangle relationships, 995
logical reasoning
conditional statements,
1002–1003
counterexamples, 1003
logical argument, 1000–1001
Venn diagrams, 1004
measurement
area, 991
area of a circle, 992
circumference of a circle, 992
converting units of, 981
perimeter, 991
surface area, 993
volume, 993
number sense
factors and multiples, 978–979
fractions, decimals, and
percents, 976
integers, operations with, 975
least common denominators,
986
percent, calculating with, 977
ratios and proportions, 980
scientific notation, 982
significant digits, 983
problem solving strategies,
998–999
statistical data
graphing, 1006–1007
mean, median, mode, and
range, 1005
organizing, 1008–1009
Slope, 82
of best-fitting lines, 114–120
classifying lines by, 83
rate of change and, 82–88, 142
Slope-intercept form, 90–97, 98, 100,
140, 142
Snell’s law, 930
Solution(s)
of an absolute value equation, 50
of an absolute value inequality, 50
of an equation, 18
extraneous, 52, 454, 518, 591, 934
of a linear inequality in two
variables, 132
of a polynomial function, 379, 387
of a quadratic equation, 253, 294
of a system of linear equations,
152, 153
in three variables, 178
of a system of linear inequalities,
168
Solve a simpler problem, problem
solving strategy, 998–999
Special angle, 963
Special patterns, factoring with, 253,
256, 260
Spreadsheet
to evaluate a recursive rule, 826
use the Location Principle, 378
Square(s)
of a binomial, 347
difference of two, 353
Square root(s), 266
imaginary unit i and, 275, 317
of a negative number, 275
properties of, 266
as solutions to quadratic
equations, 267–271, 317, 320
Square root function
graphing, 446–451, 465, 468
parent, 446, 465
Standard deviation, 745–750, 783,
784
transformation and, 751–755
Standard form
of a complex number, 276
of a linear equation, 91–96, 140
of a number, 982
of a polynomial function, 337
of a quadratic equation, 253
of a quadratic function, 236–243,
317, 318
Standard normal distribution,
758–762
formula, 758
Standard normal table, 759
Standard position, for an angle,
859
Standardized Test Practice, See also
Eliminate choices, 68–69,
148–149, 230–231, 326–327,
410–411, 472–473, 546–547,
610–611, 676–677, 740–741,
790–791, 846–847, 904–905,
972–973
examples, 3, 19, 36, 82, 132, 155,
162, 254, 268, 286, 332, 339,
355, 365, 453, 508, 575, 590,
614, 627, 708, 717, 769, 877,
933, 956
exercises, Throughout. See for
example 6, 8, 14, 15, 23, 32,
37, 47, 55
Standardized Test Preparation,
See also Gridded-answer
questions; Multi-step
problems; Open-ended
problems
Index
n2pe-9060.indd 1101
INDEX
infinite, 794
patterns and, 794–795, 797–801
recursive rules and, 826–835, 839,
842
terms of, 794
writing rules for, 795, 798, 799–800
Series
arithmetic, 804–809, 839, 841
finite, 796
formulas from, 797, 1029–1030
geometric, 812–817, 839, 841
infinite, 796
infinite geometric, 819–825, 839,
842
summation notation and, 796–800
Set theory, 715–716
Short response questions, 66–68,
408–410, 674–676, 902–904
practice, Throughout. See for
example 9, 15, 23, 31, 33, 37,
38, 55, 59, 77, 78, 87, 94, 95,
103, 106
Shrink, of the graph of the parent
absolute value function,
124–129
Sierpinski carpet, 816
Sierpinski triangle, 825
Sigma notation, 796
Significant digits, 348, 983
Similar figures, 996–997
Simplest form radical, 422
Simulation, 714
using random numbers, 714
Sine function, See also Trigonometric
equation(s); Trigonometric
function(s)
difference formula for, 949
using, 949–954
double-angle formula for, 955
using, 955–962
evaluating for any angle, 866–872
evaluating for right triangles,
852–858
graphing, 874, 908–914, 964, 965
reflections, 917, 920
translations, 915–917, 919–922,
966
half-angle formula for, 955
using, 955–962
inverse of, 874–879, 897, 899
sinusoids, and, 941–948
sum formula for, 949
Sines, law of, 881–888, 897, 900
Sinusoidal regression, 943
Sinusoids, 941–948, 967
Sixty-degree angle, trigonometric
values for, 853
Skewed distribution, 727, 728
Skills Review Handbook, 975–1009
algebra review
1101
9/28/05 9:28:15 AM
INDEX
Standardized Test Preparation
context-based multiple choice
questions, 324–326, 608–610,
844–846
extended response questions,
146–148, 470–472, 738–740,
970–972
multiple choice questions,
228–230, 544–546, 788–790
short response questions, 66–68,
408–410, 674–676, 902–904
Standing wave, 953
State Test Practice, 33, 59, 69, 139,
149, 186, 220, 231, 274, 316,
327, 369, 400, 411, 436, 464,
473, 506, 537, 547, 572, 601,
641, 667, 677, 732, 741, 782,
818, 838, 847, 873, 896, 940,
963, 973
Statistics, See also Data; Graphs;
Probability
best-fitting line, 114–120
bias in sampling, 767, 769, 771
biased question, 772–773
binomial distribution, 763–765
control group, 773
convenience sample, 766
direct variation, 107–111, 140, 143
experimental group, 773
experiments, designing, 772–773
formulas from, 1029
geometric mean, 749
margin of error, 768–771
measures of central tendency,
744–750, 783, 784, 1005
measures of dispersion, 744–750,
783, 784, 1005
negative correlation, 113, 114, 117
normal distribution, 757–762, 783,
785
outlier, 746, 747
population, 766
positive correlation, 113, 114, 117
random sample, 766
sampling, 766–771, 783, 786
self-selected sample, 766
standard deviation, 745–750, 783,
784
surveys
designing, 772–773
sampling, 766–771
systematic sample, 766
tolerance, 54
unbiased sample, 767
Stem-and-leaf plot, 1008–1009
Step function, 131
Stretch, of the graph of the parent
absolute value function,
124–129
Subscripts, reading, 26
Subset, 716
Substitution, for checking solutions,
18, 19, 20, 36, 52, 91, 133, 153,
160, 179, 205, 267, 285, 381,
452, 454, 455, 468, 517, 518,
591, 934
Substitution method
for evaluating polynomial
functions, 338
for solving linear-quadratic
systems, 659, 661, 662–664
for solving linear systems, 160–167,
181–185, 221
for solving quadratic systems,
660–664, 668, 672
Subtraction
of complex numbers, 276, 279
counting problems and, 691
with fractions, 979
of functions, 428–435
inequalities and, 42–47
integer, 975
matrix, 187, 189–192, 194
as opposite of addition, 4
of polynomials, 346–352, 403
properties, 18
for matrices, 188
with rational expressions, 582–588,
602, 605
Subtraction property of equality, 18
Sum formulas, 949, 964
for special series, 797
using, 949–954, 968
Sum of two cubes, 354
Summation notation, 796–800
Summing rectangles, 640
Supplementary angles, 994
Surface area, 993
Survey
designing, 772–773, 786
probability and, 726, 729, 731, 732,
740, 741
sampling and, 766–771, 786
Symbols
approximately equal to, 2
empty set, 715
factorial, 684
imaginary unit, 275
inequality, 50, 51
infinity, 336, 339
mean, 744
percent, 976
radical sign, 266
standard deviation, 745
subset, 716
summation, 796
table of, 1024
theta, 852
theta prime, 868
universal set, 715
Symmetric distribution, 727, 728
Symmetry
line of
for a conic section, 652, 655
for a plane figure, 990
Synthetic division, 363–368
Synthetic substitution, 363
for evaluating polynomial
functions, 338
System of linear equations, See
Linear systems
System of linear inequalities,
168–173
with no solution, 169
three or more inequalities, 170–173
System of quadratic inequalities,
301, 304, 305
Systematic sample, 766
T
Table(s)
to display data, 8, 9, 47, 57, 59, 69,
108, 110, 111, 112, 115, 117,
118, 119, 120, 206, 400, 421,
426, 472, 552, 553, 570, 580,
777, 779, 780, 781, 782, 787,
824, 904, 946, 969, 973, 1007
to graph cube root functions, 447
to graph equations of parabolas,
621
to graph exponential decay
functions, 486
to graph exponential growth
functions, 478
to graph linear functions, 75, 80
to graph polynomial functions,
340, 342–344
to graph quadratic functions, 236,
237, 240
to graph square root functions, 446
interpreting, 609, 610
for natural base e, 492
for recording experimental data,
308, 819
to represent relations, 72
to solve linear equations, 25
to solve linear systems, 152
to solve problems, exercises, 15, 24,
39, 95, 104, 129, 290, 306, 314,
343, 451, 570, 647, 729, 808,
914, 929, 937
to solve quadratic inequalities, 302
to solve radical inequalities, 462
to solve rational inequalities, 598
spreadsheet, 826
standard normal, 759
Tables of reference
Formulas
from algebra, 1027–1028
1102 Student Resources
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complex conjugates theorem, 380
factor theorem, 364
fundamental theorem of algebra,
379–386, 405
irrational conjugates theorem, 380
Pythagorean theorem, 995
rational zero theorem, 370
remainder theorem, 363
Theoretical probability, 698
Third-order differences, 395
Thirty-degree angle, trigonometric
values for, 853
Tolerance, 54
Total cost matrix, 198
Tower of Hanoi, 800
Transformation, 123
on the coordinate plane, 988–989
data and, 751–755
of exponential data, 529
of general graphs, 126
of the graph of a parent function
absolute value, 121–129, 144
exponential, 479, 487
radical, 448
rational, 558, 559
multiple, 125–129
of power data, 532
producing equivalent inequalities,
42
vertical shrinking of a graph, 479,
487
vertical stretching of a graph, 479,
487
Transition matrix, 201
Translation
of conic sections, 650–657, 672
on the coordinate plane, 988–989
exercises, 39
of the graph of a parent function
absolute value, 121, 122, 123–129
cosine, 915–917, 919–922, 966
exponential, 487
exponential growth, 479
logarithmic, 503
radical, 448
rational, 559
sine, 915–917, 919–922, 966
tangent, 918, 920, 921
horizontal, 916
vertical, 916
Transverse axis, of a hyperbola, 642
Tree diagram
for counting possibilities, 682, 686
for factoring numbers, 978
for finding probability, 720
Triangle(s)
AAS, 882
ambiguous case, 883–884
area of, 885, 887, 888, 991
ASA, 882
classifying, using the distance
formula, 614–615, 617
hypotenuse of, 995
legs of, 995
median of, 618
perimeter of, 991
right, trigonometry and, 852–858,
898
SAS, 889
SSA, 883–884
SSS, 890–891
sum of angle measures of, 995
Triangular numbers, 394
Triangular pyramidal numbers, 395
Trigonometric equation(s), 876–880
identities and, 923–930
solving, 931–939, 964, 967
using double-angle and halfangle formulas, 958, 960–962
in an interval, 932, 935–937
using sum and difference
formulas, 949–954
Trigonometric expressions, 925–926,
928, 955–956, 959, 960
Trigonometric formulas
double-angle and half-angle,
955–962, 964, 968
sum and difference, 949–954, 964,
968
Trigonometric function(s)
of any angle, 866–872, 899
difference formulas for, 949
using, 949–954
double-angle formulas for, 955
using, 955–962
graphing, 874, 908–922, 964, 965
half-angle formulas for, 955
using, 955–962
inverse, 874–880, 897, 899
modeling with, 870, 871–872,
941–948, 967
right angle, 852–858, 898
sum formulas for, 949
using, 949–954
Trigonometric identities, 923–930,
966
verifying, 923, 926–930, 958, 966
Trigonometric ratios, 852–858
Trigonometry
formulas from, 1030–1031
identities from, 1030–1031
Trinomial(s), 252
factoring, 252–265
Turning points, of a polynomial
function, 388, 390
U
Unbiased sample, 767
Unbounded region, 174
Index
n2pe-9060.indd 1103
INDEX
from combinatorics, 1028
from coordinate geometry, 1026
from geometry, 1032
from mathematical modeling,
1031
from matrix algebra, 1026
from probability, 1028–1029
from sequences and series,
1029–1030
from statistics, 1029
from trigonometry, 1030–1031
Identities, from trigonometry,
1030–1031
Measures, 1025
Properties
of exponents, 1033
of functions, 1034
of logarithms, 1034
of matrices, 1033
of radicals, 1034
of rational exponents, 1034
of real numbers, 1033
Symbols, 1024
Theorems, from algebra,
1027–1028
Tangent function, See also
Trigonometric equation(s);
Trigonometric function(s)
difference formula for, 949
using, 949–954
double-angle formula for, 955
using, 955–962
evaluating for any angle, 866–872
evaluating for right triangles,
852–858
graphing, 911–914, 965
translations, 918, 920, 921
half-angle formula for, 955
using, 955–962
inverse, 875–879, 897, 899
sum formula for, 949
using, 949–954
Tangent identities, 924
Technology, See Calculator; Graphing
calculator
Technology support, See Animated
Algebra; @Home Tutor; Online
Quiz; State Test Practice
Term(s)
constant, 12
of an expression, 12
like, 12
of a sequence, 794
variable, 12
Terminal side, of an angle, 859
Test-taking strategies, eliminate
choices, 3, 228, 229, 286, 544,
545, 590, 627, 788, 789, 933
Theorems
binomial theorem, 693
1103
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Undefined slope, 83
Union, of sets, 707–713, 715–716
Unit analysis, 5
for checking solutions, 7, 34
with conversions, 5
with operations, 5
Unit circle, 867
Universal gravitational constant,
557
Universal set, 715
Upper limit of summation, 796
Upper quartile, 1008–1009
INDEX
V
Variable(s), 11
dependent, 74
independent, 74
random, 724
solving for, 26
Variable term, 12
Variation
constant of, 107, 551
direct, 107–111, 140, 143
inverse, 550–557, 603
joint, 553–557, 603
Venn diagram, 1004
classifying numbers, 2
to show composition of functions,
430
to show operations on sets,
715–716
to show probability, 698, 706–708
Verbal model
examples, 13, 19, 20, 29, 34, 35, 36,
42, 54, 63, 66, 100, 101, 134,
155, 162, 181, 239, 254, 261,
262, 356, 373, 389, 560, 589,
829
exercises, 15, 30, 242, 257, 594, 601
writing and evaluating, 11
Verification, of trigonometric
identities, 923, 926–930, 958,
966
Vertex (Vertices)
of an absolute value graph, 123
of an angle, 859
of an ellipse, 634, 650
of a feasible region, 174
of a hyperbola, 642, 650
of a parabola, 236, 620, 650
Vertex form, of quadratic a function,
245–246, 248–251, 287, 317,
319
Vertical asymptote, 558
Vertical line, graph of, 92
Vertical line test, 73–74, 77
Vertical motion problem, 295, 298
Vertical shrinking, of a graph, 479,
487
Vertical stretching, of a graph, 479,
487
Vertical translation, graphing, 916
Visual thinking, See also Graphs;
Manipulatives; Modeling;
Multiple representations;
Transformation
exercises, 15, 102, 185, 281, 297,
342, 624, 862
Vocabulary
overview, 1, 71, 151, 235, 329, 413,
477, 549, 613, 681, 743, 793,
851, 907
prerequisite, xxiv, 70, 150, 234, 328,
412, 476, 548, 612, 680, 742,
792, 850, 906
review, 61, 141, 222, 318, 402, 466,
539, 603, 669, 734, 784, 840,
898, 965
Volume, See Formulas
W
What If? questions, 11, 13, 21, 29, 36,
44, 74, 83, 85, 91, 101, 109,
126, 135, 155, 163, 170, 181,
198, 219, 239, 246, 247, 254,
262, 269, 278, 287, 295, 341,
356, 361, 365, 373, 383, 389,
416, 431, 447, 453, 461, 480,
488, 494, 501, 509, 519, 525,
531, 552, 561, 567, 574, 592,
597, 628, 636, 644, 661, 683,
684, 691, 692, 699, 700, 701,
708, 709, 717, 718, 719, 721,
746, 752, 759, 767, 795, 797,
805, 822, 829, 835, 855, 870,
877, 891, 911, 939, 942, 957
Whole numbers, 2
Work rate problems, 20, 24
Writing, See also Communication;
Verbal model
absolute value functions, 125
algebraic expressions, 984
direct variation equations, 107–111
equations of circles, 627–632
equations of ellipses, 635–639
equations of hyperbolas, 643–648
equations of parabolas, 621–625
exponential functions, 529–531,
533–536, 542
linear equations, 19, 20, 23–24,
98–104, 142
linear systems as matrix equations,
212–213, 215–217
piecewise functions, 131
polynomial functions, 381, 384,
386, 392–399
power functions, 531–535
quadratic functions, 309–315, 322
rational equations, 589, 594–595
rules for nth term of a sequence,
803–809, 810–816
rules for sequences, 795, 798,
799–800
systems of equations, 155, 157–158,
162, 165–166, 181, 184–185
systems of linear inequalities, 170,
172–173
trigonometric equations, 877,
879–880
trigonometric functions, 941–948
X
x-axis, 987
x-coordinate, 987
x-intercept, 91
of the graph of a polynomial
function, 387
x-values, critical, 303, 599
Y
y-axis, 987
y-coordinate, 987
as local maximum of a function,
388
as local minimum of a function,
388
y-intercept, 89
Z
z-intercept, 177
z-score, 758
standard normal table and,
759–762
Zero exponent property, 330
Zero product property, 253
Zero slope, 83
Zeros
of a polynomial function, 364, 365,
366, 367, 370–378, 387, 405
approximating real, 382–383, 384
Descartes’ Rule of Signs and,
381–382, 384, 385
fundamental theorem of algebra
and, 379–386, 405
of a quadratic function, 254–256
average of, 262
of a rational function, 566
1104 Student Resources
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Correz/Getty Images; 383 PhotoDisc/Getty Images; 385 Peter
Yates/SPL/Photo Researchers, Inc.; 387 top right Turner &
de Vries/Getty Images; 393 Terry Renna/AP/Wide World Photos;
398 Tom Stewart/Corbis; 400 Joe McDonald/Corbis; 412–413
David Bergman/Corbis; 414 Javier Soriano/AFP/Getty Images;
416 Azure Computer & Photo Services/Animals Animals; 419
McDougal Littell/Houghton Mifflin Co.; 420 Cameron Heryet/
Getty Images; 426 Lester Lefkowitz/Corbis; 428 Darrell Gulin/
Corbis; 431 StreetStock Images/Brand X Pictures/PictureArts;
434 Courtesy of Professor Tim Pennings, Hope College; 436
Royalty-Free/Corbis; 437 both RMIP/Richard Haynes/McDougal
Littell; 438 Paul A. Souders/Corbis; 441 Gail Burton/AP/Wide
World Photos; 444 Rod Taylor/AP/Wide World Photos; 446
Duomo/Corbis; 447 Navaswan/Getty Images; 451 Brian Erler/
Getty Images; 452–453 Cathrine Wessel/Corbis; 453 Philippe
Giraud/Corbis Sygma; 457 Casey Riffe/Marshfield News-Herald/
AP/Wide World Photos; 458 OSF/Colbeck, M./Animals Animals;
476 background B.A.E. Inc./Alamy; 478 Greer & Associates, Inc./
SuperStock; 484 Bob Daemmrich/Corbis Sygma; 485 Lynda
Richardson/Corbis; 486 age fotostock/SuperStock; 488
PhotoDisc/Getty Images; 491 Richard A. Cooke/Corbis; 492
Cousteau Society/Getty Images; 498 Tim Hursley/SuperStock;
499 Chuck Carlton/Index Stock Imagery; 506 Jim McNee/Index
Stock Imagery; 507 age fotostock/SuperStock; 509 Royalty-Free/
Corbis; 511 center left image100/Alamy; 511 center Pat LaCroix/
Getty Images; 511 center right Tony Arruza/Corbis; 512 center
right Rubberball Productions; 512 center left Cathy Melloan/
PhotoEdit; 512 bottom Rubberball Productions; 515 Roger
Ressmeyer/Corbis; 516 Jeff Sherman/Getty Images; 522 both AM
Corporation/Alamy; 528 both Jay Penni Photography/McDougal
Littell; 529 Mark Chappell/Animals Animals; 530 Nick Dolding/
Getty Images; 532 Gerard Lacz/Animals Animals; 534 Bryn
Colton/Assignments Photographers/Corbis; 536 Bettmann/
Corbis; 548–549 Jeff Hunter/Getty Images; 550 both RMIP/
Richard Haynes/McDougal Littell; 551 Lawrence Manning/
Corbis; 552 age fotostock/SuperStock; 553 Rick Bowmer/AP/
Wide World Photos; 558 Ken Reid/Getty Images; 560 both
Courtesy of Z Corporation; 565 Bobby Model/National
Geographic/Getty Images; 567 Donald C. Johnson/Corbis; 569
The Photolibrary Wales/Alamy; 570 Scaled Composites/SPL/
Photo Researchers, Inc.; 572 Michael Newman/PhotoEdit; 573
Mary Ann Chastain/AP/Wide World Photos; 579 Tony
McConnell/SPL/Photo Researchers, Inc.; 582 Luca DiCecco/
Alamy; 589 Denis Boissavy/Getty Images; 592 don jon red/
Alamy; 594 Fédération Internationale de Volleyball/AP/Wide
World Photos; 601 PhotoDisc/Getty Images; 612–613
Phototake/Getty Images; 614 Nick Vedros & Assoc./Getty Images;
618 Reuters/Corbis; 620 Stephen Frink/Getty Images; 624 Hank
Morgan/Time Life Pictures/Getty Images; 625 Roger Ressmeyer/
Corbis; 626 Royalty-Free/Corbis; 634 Ralph Wetmore/Getty
Images; 638 bottom left NASA/ARC; 638 bottom right Detlev
Van Ravenswaay/SPL/Photo Researchers, Inc.;
1105
9/28/05 10:32:31 AM
CREDITS
641 Tom Uhlman/Visuals Unlimited; 642 Paul A. Souders/
Corbis; 648 Mike Cartwright/AP/Wide World Photos; 649 RMIP/
Richard Haynes/McDougal Littell; 650 Courtesy of
Superdairyboy; 656 Erik S. Lesser/AP/Wide World Photos; 658
Yellow Dog Productions/Getty Images; 676 Fermilab Photo;
680–681 Alexander Walter/Getty Images; 682 Douglas C. Pizac/
AP/Wide World Photos; 684 Oliver Morin/AFP/Getty Images; 690
Robbie Jack/Corbis; 696 Marc Lester/AP/Wide World Photos;
698 PhotoDisc/Getty Images; 699 Ellen Senisi/The Image Works;
705 bottom left Big Cheese Photo/FotoSearch; 707 Jeff
Greenberg/The Image Works; 717 Dennis MacDonald/PhotoEdit;
719 NOAA/AP/Wide World Photos; 724 John Russell/AP/Wide
World Photos; 742–743 Mark E. Gibson/Getty Images; 744 Omar
Torres/AFP/Getty Images; 746 Banana Stock/Alamy; 748 Alan
Diaz/AP/Wide World Photos; 749 Andy Lyons/Getty Images; 751
NASA-HQ-GRIN; 754 bottom Ric Francis/AP/Wide World
Photos; 754 top Phil Cantor/Index Stock Imagery; 756 top age
fotostock/SuperStock; 756 bottom Don Heupel/AP/Wide World
Photos; 756 bottom center ThinkStock/SuperStock; 757
PhotoDisc/Getty Images; 759 Kennan Ward/Corbis; 762 Barbara
Novovitch/Reuters; 765 Royalty-Free/Corbis; 766 David YoungWolff/PhotoEdit; 772 Spencer Grant/PhotoEdit; 774 Jay Penni
Photography/McDougal Littell; 775 Ben Margot/AP/Wide World
Photos; 780 Jay Penni Photography/McDougal Littell; 782
Dynamic Graphics/PictureQuest; 792–793 Steve Gschmeissner/
SPL/Photo Researchers, Inc.; 794 Roger Wood/Corbis; 799 Frank
Chmura/PictureQuest; 802 Richard Cummins/Corbis; 805
Stockbyte/PictureQuest; 808 © 2007 Sol LeWitt/Artist Rights
Society (ARS), New York. Photo Credit: Mary Ann Sullivan,
Bluffton University; 810 PhotoStockFile/Alamy; 813 GDT/Getty
Images; 815 Agence Vandystadt/Photo Researchers, Inc.; 819 all
Jay Penni Photography/McDougal Littell; 820 Courtesy of Gayla
Chandler; 827 Popperfoto/Alamy; 838 Gary S. Settles/Photo
Researchers, Inc.; 850–851 João Paulo/Getty Images; 852 Hugh
Sitton/Getty Images; 859 M. Spencer Green/AP/Wide World
Photos; 864 top Royalty-Free/Corbis; 864 center Courtesy NASA,
Life Sciences Division; 865 both Royalty-Free/Corbis; 866 age
fotostock/SuperStock; 869 Courtesy NASA/JPL-Caltech; 873
bottom right Brand X Pictures/Getty Images; 873 right Richard
Cummins/SuperStock; 873 bottom right Yoshio Tomii/
SuperStock; 875 Greg Ebersole/AP/Wide World Photos; 879
Paolo Curto/The Image Bank; 882 Richard Berenholtz/Corbis;
885 Steve Bein/Corbis; 889 David Madison/Getty Images; 906–
907 Royalty-Free/Corbis; 908 Richard Olseius/National
Geographic/Getty Images; 910 Annabella Bluesky/SPL/Photo
Researchers, Inc.; 913 Peter Arnold, Inc./Alamy; 915 both Craig T.
Lorenz/Photo Researchers, Inc.; 921 Image Source/PunchStock;
924 Lowell Observatory/NOAO/AURA/NSF; 931 Scott
Camazine/Photo Researchers, Inc.; 932 both Christopher
Mackay/Tantramar Interactive; 940 PhotoDisc/Getty Images;
941 Lee Cohen/Corbis; 945 Steve Chenn/Corbis; 948 RMIP/
Richard Haynes/McDougal Littell; 949 Howard Kingsnorth/Getty
Images; 951 left Larry Dunmire/SuperStock; 951 right Ken
Graham/Getty Images; 955 Pascal Rondeau/Getty Images; 963
Bill Ross/Corbis.
Illustrations and Maps
Argosy 1, 51, 70, 151, 235, 329, 413, 477, 549, 557 top, 613,
622 top, 646, 654, 662, 667 top left, 681, 743, 793, 818 bottom
left; 851, 907, 921, 942, 954, 957; Kenneth Batelman 504, 636,
752, 766 , 871, 879, 930, 946; Steve Cowden 15, 32, 39, 95,
241, 243 top center, 247, 268, 298 center, 434, 439, 457, 464,
535; Stephen Durke 290 bottom, 552, 570, 622 center, 660,
689 top, 800, 808 top, 832, 858, 872 center right, 896, 918,
929 center, 929 top right, 940; John Francis 580, 761; Patrick
Gnan/Deborah Wolfe, Ltd. 5, 129, 254, 343 top, 351, 356, 360,
377, 392, 585, 641, 644, 657, 676, 818 top right; 822, 838,
880, 890, 902, 905, 927, 953, 972; Chris Lyons 139, 158, 647
center; Steve McEntee 480, 563, 870, 873 bottom left, 877, 887,
893 bottom, 893 center, 904, 914, 916; Paul Mirocha 497, 574;
Laurie O’Keefe 88, 494, 729; Steve Stankiewicz 258, 505; Doug
Stevens 125; Dan Stuckenschneider 16, 46, 87, 108, 119, 137,
185, 189, 250, 257, 265 top right, 369, 373, 426, 450, 519, 524,
556, 557 center, 688, 795, 824, 936; Matt Zang/American Artists
500; Carol Zuber-Mallison 111, 204, 208, 209, 358, 521, 587,
616, 618, 619, 663, 704, 855, 885, 961 top right, 961 bottom.
All other illustrations © McDougal Littell/Houghton Mifflin
Company.
1106 Student Resources
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Sk ills Re
Rev
v iew Handboo
Handbook
k
To add positive and negative
numbers, you can use a
number line.
To subtract any number,
add its opposite.
EXAMPLE
To add a positive number, move to the right.
To add a negative number, move to the left.
26
25
24
22
0
21
1
2
3
4
5
6
Add or subtract.
a. 1 1 (25)
End
23
SKILLS REVIEW HANDBOOK
Operations with Positive and Negative Numbers
b. 22 2 (25) 5 22 1 5
Move 5 units to the left.
25 24 23 22 21
0
Start
Start
1
Move 5 units to the right.
23 22 21
2
c 1 1 (25) 5 24
The opposite of 25 is 5.
0
1
2
End
3
4
c 22 2 (25) 5 3
To multiply or divide positive and negative numbers, use the following rules.
• The product or quotient of two numbers with the same sign is positive.
• The product or quotient of two numbers with different signs is negative.
EXAMPLE
Multiply or divide.
a. 3 p 7 5 21
b. 23(27) 5 21
c. 18 4 2 5 9
d. 218 4 (22) 5 9
e. 23(7) 5 221
f. 3(27) 5 221
g. 218 4 2 5 29
h. 18 4 (22) 5 29
PRACTICE
Perform the indicated operation.
1. 2 1 (28)
2. 5 2 12
3. 26(10)
4. 230 4 (22)
6. 7(25)
7. 18 2 10
8. 27 1 (212)
9. 11(4)
5. 24 1 6
10. 81 4 (29)
11. 212 4 3
12. 29(28)
13. 21 1 13
14. 45 4 (29)
15. 26(12)
16. 14 2 (29)
17. 232 4 16
18. 223 1 (25)
19. 28 2 (25)
20. 17 2 (218)
21. 29(21)
22. 23 2 (211)
23. 218 4 (23)
24. 14 1 (27)
25. 5(23)
26. 21 1 (28)
27. 22 2 10
28. 29 1 26
29. 220 4 (24)
30. 22 4 (22)
31. 27(26)
32. 1 2 24
33. 215 2 2
34. 0 1 (24)
35. 16 4 8
Skills Review Handbook
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SKILLS REVIEW HANDBOOK
Fractions, Decimals, and Percents
A percent is a ratio with a denominator of 100. The word percent means
“per hundred,” or “out of one hundred.” The symbol for percent is %.
In the model at the right, 71 of the 100 squares are shaded. You can write
the shaded part of the model as a fraction, a decimal, or a percent.
71
Fraction: seventy-one divided by one hundred, or }}
100
Decimal: seventy-one hundredths, or 0.71
Percent: seventy-one percent, or 71%
EXAMPLE
Write as a fraction.
94 5 47
a. 94% 5 }}
}}
100
50
EXAMPLE
3
c. 0.3 5 three tenths 5 }}
10
Write as a decimal.
15 5 0.15
a. 15% 5 }}
100
EXAMPLE
20 5 1
b. 20% 5 }}
}
5
100
106 5 1.06
b. 106% 5 }}
100
5 5 5 4 8 5 0.625
c. }
8
Write as a percent.
41 5 41%
a. 0.41 5 }}
100
8 5 80 5 80%
b. 0.8 5 }}
}}
10
100
5 5 5 p 25 5 125 5 125%
c. }
}}}
}}
4
4 p 25
100
PRACTICE
Write as a fraction.
1. 0.65
2. 0.08
3. 1.5
4. 0.13
5. 0.7
6. 50%
7. 26%
8. 3%
9. 95%
10. 110%
Write as a decimal.
1
11. }
4
9
12. }
10
30
13. }
25
2
14. }
5
3
15. }
8
16. 16%
17. 142%
18. 1%
19. 30%
20. 6.5%
21. 0.6
22. 0.24
23. 1.3
24. 0.07
25. 0.45
1
26. }
10
4
27. }
5
17
28. }
20
5
29. }
2
3
30. }
16
Write as a percent.
976
n2pe-9020.indd 976
Student Resources
11/21/05 10:26:45 AM
Calculating with Percents
EXAMPLE
Word
what
of
is
n
3
5
Symbol
Answer the question.
a. What is 15% of 20?
b. What percent of 8 is 6?
n 5 0.15 3 20
n53
c. 80% of what number is 4?
n3856
0.8 3 n 5 4
n 5 6 4 8 5 0.75 5 75%
3 is 15% of 20.
n 5 4 4 0.8 5 5
75% of 8 is 6.
SKILLS REVIEW HANDBOOK
You can use equations to calculate with percents. Replace
words with symbols as shown in the table at the right. Below
are three types of questions you can answer with percents.
80% of 5 is 4.
Amount of increase or decrease .
To find a percent of change, calculate }}}}}}}}}}}}}}
Original amount
EXAMPLE
Find the percent of change.
a. A class increases from 21 students to 25 students.
25 2 21
4
}}}} 5 }} ø 0.19 5 19% increase
21
21
b. A price decreases from $12 to $9.
12 2 9
3
}}} 5 }} 5 0.25 5 25% decrease
12
12
PRACTICE
Answer the question.
1. What is 98% of 200?
2. What is 25% of 8?
3. What is 30% of 128?
4. What is 5% of 700?
5. What is 100% of 17?
6. What is 150% of 14?
7. What is 0.2% of 500?
8. What is 6.5% of 3000?
9. What percent of 100 is 54?
10. What percent of 18 is 9?
11. What percent of 80 is 8?
12. What percent of 15 is 20?
13. What percent of 30 is 6?
14. What percent of 5 is 8?
15. What percent of 50 is 1?
16. 50% of what number is 6?
17. 55% of what number is 44?
18. 10% of what number is 6?
19. 75% of what number is 45?
20. 1% of what number is 2?
21. 90% of what number is 63?
22. 12% of what number is 60?
23. 200% of what number is 16?
Find the percent of change. Round to the nearest percent if necessary.
24. A class increases from 20 to 28 students.
25. Time decreases from 60 to 45 minutes.
26. A price is reduced from $200 to $180.
27. Votes increase from 200 to 300.
28. A test is shortened from 40 to 32 items.
29. Membership increases from 820 to 1605.
30. A wage rises from $8.75 to $10.00.
31. The temperature drops from 248F to 58F.
Skills Review Handbook
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SKILLS REVIEW HANDBOOK
Factors and Multiples
Factors are numbers or expressions that are multiplied together.
A prime number is a whole number greater than 1 that has exactly
two whole number factors, 1 and itself. The table shows all the prime
numbers less than 100. A composite number is a whole number
greater than 1 that has more than two whole number factors.
Prime Numbers
Less Than 100
2, 3, 5, 7, 11, 13, 17, 19, 23,
29, 31, 37, 41, 43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, 97
When you write a composite number as a product of prime numbers,
you are writing its prime factorization.
EXAMPLE
Write the prime factorization of 60.
Use a factor tree. Write 60 at the top. Then draw two branches
and write 60 as the product of two factors. Continue to draw
branches until all the factors are prime numbers. Two factor
trees for 60 are given at the right. Both show 60 5 2 p 2 p 3 p 5.
60
60
2 p 30
3 p 20
2 p 15
2
c The prime factorization of 60 is 2 p 3 p 5.
3 p 5
4 p 5
2 p 2
A whole number that is a factor of two or more nonzero whole numbers is a
common factor of the numbers. The largest of the common factors is the
greatest common factor (GCF).
EXAMPLE
Find the greatest common factor (GCF) of 18 and 45.
Method 1 List factors.
Method 2 Use prime factorization.
Factors of 18: 1, 2, 3, 6, 9, 18
Prime factorization of 18: 2 p 3 p 3
Factors of 45: 1, 3, 5, 9, 15, 45
Prime factorization of 45: 3 p 3 p 5
The GCF is 9, the greatest of the
common factors.
The GCF is the product of the common
prime factors: 3 p 3 5 9.
A multiple of a whole number is the product of the number and any nonzero
whole number. A common multiple of two or more numbers is a multiple of all of
the numbers. The least common multiple (LCM) is the smallest of the common
multiples.
EXAMPLE
978
n2pe-9020.indd 978
Find the least common multiple (LCM) of 12 and 15.
Method 1 List multiples.
Method 2 Use prime factorization.
Multiples of 12: 12, 24, 36, 48, 60, . . .
Prime factorization of 12: 22 p 3
Multiples of 15: 15, 30, 45, 60, . . .
Prime factorization of 15: 3 p 5
The LCM is 60, the least of the
common multiples.
Form the LCM of the numbers by writing
each prime factor to the highest power it
occurs in either number: 22 p 3 p 5 5 60.
Student Resources
11/21/05 10:26:48 AM
EXAMPLE
3
10
SKILLS REVIEW HANDBOOK
The least common denominator (LCD) of two fractions is the least common
multiple of the denominators. Use the LCD to add or subtract fractions with
different denominators.
5
8
Add: }} 1 }
The least common multiple of the denominators, 10 and 8, is 40.
So, the least common denominator (LCD) of the fractions is 40.
3 5 3 p 4 5 12 and 5 5 5 p 5 5 25
Rewrite the fractions using the LCD of 40: }}
}}}
}}
}
}}}
}}
10
10 p 4
40
8
8p5
40
3 1 5 5 12 1 25 5 37
Add the numerators and keep the same denominator: }}
}
}}
}}
}}
10
8
40
40
40
PRACTICE
Write the prime factorization of the number. If the number is prime,
write prime.
1. 42
2. 104
3. 75
4. 23
5. 70
6. 27
7. 72
8. 180
9. 47
10. 100
11. 88
12. 49
14. 142
15. 32
13. 83
Find the greatest common factor (GCF) of the numbers.
16. 4, 6
17. 24, 40
18. 10, 25
19. 55, 44
20. 28, 35
21. 8, 20
22. 5, 8
23. 15, 12
24. 16, 32
25. 70, 90
26. 2, 18
27. 9, 21
28. 36, 42, 54
29. 7, 12, 17
30. 45, 63, 81
Find the least common multiple (LCM) of the numbers.
31. 4, 16
32. 2, 14
33. 5, 6
34. 16, 24
35. 6, 8
36. 12, 20
37. 3, 6
38. 18, 8
39. 9, 12
40. 9, 5
41. 10, 15
42. 7, 9
43. 40, 4, 5
44. 25, 30, 3
45. 27, 81, 33
Perform the indicated operation(s). Simplify the result.
113
46. }
}
2
8
32 5
47. }
}}
4
16
7 23
48. }}
}
5
10
111
49. }
}
2
3
5 11
50. }}
}
3
12
411
51. }
}
5
8
1 13
52. }}
}
4
10
521
53. }
}
6
2
7 2 11
54. }
}}
8
16
9 21
55. }}
}
3
10
221
56. }
}
3
6
2
1 1}
57. }
5
4
41 1 25
58. }
}}
}
5
12
6
32 3 23
59. }
}} }
4
2
10
9 2121
60. }}
}
}
5
10
2
71 3 21
61. }
}}
}
4
16
8
8122 7
62. }
}
}}
9
3
12
4 11
1 1 }}
63. }
}
15
3
6
11211
64. }
}
}
4
2
3
15 2 7 1 1
65. }}
}}
}
16
10
2
5 211 7
66. }}
}
}}
24
6
12
321
1 1}
67. }
}
5
4
2
5232 2
68. }
}
}}
5
6
15
4132 7
69. }
}
}}
4
9
12
Skills Review Handbook
n2pe-9020.indd 979
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SKILLS REVIEW HANDBOOK
Ratios and Proportions
A ratio uses division to compare two quantities.
Three Ways to Write the Ratio of a to b
You can write a ratio of two quantities a and b,
where b is not equal to 0, in three ways.
a to b
a
b
a:b
}
You should write ratios in simplest form.
EXAMPLE
Write the ratio of 12 boys to 16 girls in three ways.
Boys
Girls
12 5 12 4 4 5 3
First write the ratio as a fraction in simplest form: }}} 5 }}
}}}
}
16
16 4 4
4
3.
c Three ways to write the ratio of boys to girls are 3 to 4, 3 : 4, and }
4
A proportion is an equation stating that
two ratios are equal.
You can use cross multiplication to solve
a proportion.
EXAMPLE
9
a
b
c
d
b.
}} 5 }
If } 5 }, where b ? 0 and d ? 0, then ad 5 bc.
Solve the proportion.
n
54
5
} 5 }}
a.
Using Cross Multiplication to Solve Proportions
5 p 54 5 9 p n
270 5 9n
30 5 n
Cross multiply.
Simplify.
Solve for n.
x
40
3
8
x p 8 5 40 p 3
8x 5 120
x 5 15
Cross multiply.
Simplify.
Solve for x.
PRACTICE
Write the ratio in simplest form. Express the answer in three ways.
1. 3 to 9
2. 16 to 24
3. 10 to 8
4. 6 to 2
5. 25 to 30
6. 60 to 10
7. 4 to 4
8. 8 to 20
9. 32 to 72
10. 42 to 15
11. 14 to 2
12. 12 to 15
x 5 12
13. }}
}}
14
24
8 5 d
14. }}
}}
24
36
15 5 3
15. }}
}
4
n
9 5 5
16. }}
}
45
h
a5 4
17. }
}}
6
12
13 5 91
18. }}
}}
t
7
75 5 r
19. }}
}
120
8
b 52
20. }}
}
90
3
4 5 n
21. }}
}}
11
110
5 5 150
22. }
}}
90
z
95x
23. }
}
8
6
72 5 24
24. }}
}}
105
m
17 5 51
25. }}
}}
33
a
20 5 24
26. }}
}}
125
n
16 5 8
27. }}
}
144
x
96 5 t
28. }}
}
6
3
Solve the proportion.
980
n2pe-9020.indd 980
Student Resources
11/21/05 10:26:50 AM
Converting Units of Measurement
SKILLS REVIEW HANDBOOK
The table of measures on page 1025 gives many statements of equivalent
measures. Using each statement, you can write two different conversion factors.
Statement of Equivalent Measures
Conversion Factors
100 cm 5 1 m
}}}} 5 1 and }}}} 5 1
100 cm
1m
1m
100 cm
To convert from one unit of measurement to another, multiply by a conversion
factor. Use the one that will eliminate the starting unit and keep the desired unit.
EXAMPLE
Copy and complete.
a. 3.5 m 5 ? cm
b. 620 cm 5 ? m
100 cm 5 (3.5 3 100) cm 5 350 cm
3.5 m 3 }}}}
1 m 5 620 m 5 6.2 m
620 cm 3 }}}}
}}
c So, 3.5 m 5 350 cm.
c So, 620 cm 5 6.2 m.
1m
100 cm
100
Sometimes you need to use more than one conversion factor.
EXAMPLE
Copy and complete: 7 days 5 ? sec
Find the appropriate statements of equivalent measures.
24 h 5 1 day, 60 min 5 1 h, and 60 sec 5 1 min
24 h , 60 min , and 60 sec
Write conversion factors: }}}
}}}}
}}}
1 day
1h
1 min
Multiply by conversion factors to eliminate days and keep seconds.
24 h 3 60 min 3 60 sec 5 (7 3 24 3 60 3 60) sec 5 604,800 sec
7 days 3 }}}
}}}}
}}}
1 day
1 min
1h
c So, 7 days 5 604,800 sec.
PRACTICE
Copy and complete.
1. 6 L 5 ? mL
2. 2 mi 5 ? ft
3. 80 oz 5 ? lb
4. 4 days 5 ? h
5. 77 mm 5 ? cm
6. 5 gal 5 ? qt
7. 48 ft 5 ? yd
8. 1500 mL 5 ? L
10. 125 lb 5 ? oz
11. 800 g 5 ? kg
12. 900 sec 5 ? min
13. 72 in. 5 ? ft
14. 2.5 ton 5 ? lb
15. 90 min 5 ? h
16. 65,000 mg 5 ? g
17. 100 yd 5 ? in.
18. 3.5 kg 5 ? g
19. 6 pt 5 ? qt
20. 1 week 5 ? min
21. 2 oz 5 ? lb
22. 1 km 5 ? mm
23. 1 mi 5 ? in.
24. 5 gal 5 ? c
25. 288 in.2 5 ? ft 2
26. 24 pt 5 ? gal
27. 4 kg 5 ? g
28. 7 hr 5 ? sec
9. 40 m 5 ? cm
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SKILLS REVIEW HANDBOOK
Scientific Notation
Scientific notation is a way to write numbers using powers of 10. A number is
written in scientific notation if it has the form c 3 10n where 1 ≤ c < 10 and n is
an integer. The table shows some powers of ten in order from least to greatest.
Power of Ten
1023
1022
1021
100
101
102
103
Value
0.001
0.01
0.1
1
10
100
1000
EXAMPLE
a. 12,800,000
Write the number in scientific notation.
Standard form
b. 0.0000039
Standard form
12,800,000
Move the decimal point
7 places to the left.
0.0000039
Move the decimal point
6 places to the right.
1.28 3 107
Use 7 as an exponent of 10.
3.9 3 1026
Use 26 as an exponent of 10.
EXAMPLE
a. 6.1 3 104
Write the number in standard form.
Scientific notation
b. 5.74 3 1025
Scientific notation
6.1 3 104
The exponent of 10 is 4.
5.74 3 1025
The exponent of 10 is 25.
61,000
Move the decimal point
4 places to the right.
0.0000574
Move the decimal point
5 places to the left.
61,000
Standard form
0.0000574
Standard form
PRACTICE
Write the number in scientific notation.
1. 0.6
2. 25,000,000
3. 0.08
4. 0.00542
5. 40.8
6. 7
7. 0.000385
8. 8,145,000
9. 41,236
10. 0.0000016
11. 486,000
12. 0.000000009
13. 0.01002
14. 1,000,000,000
15. 7050.5
16. 0.37
17. 9850
18. 0.0000206
19. 805
20. 0.0005
Write the number in standard form.
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21. 5 3 103
22. 4 3 1022
23. 8.2 3 1021
24. 6.93 3 102
25. 3.2 3 1023
26. 9.01 3 1025
27. 7.345 3 105
28. 2.38 3 1022
29. 1.814 3 100
30. 2.7 3 108
31. 1 3 106
32. 4.9 3 1024
33. 8 3 1026
34. 5.6 3 104
35. 1.87 3 109
36. 7 3 1024
37. 6.08 3 106
38. 9.009 3 1023
39. 3.401 3 107
40. 5.32 3 101
Student Resources
11/21/05 10:26:53 AM
Significant Digits
SKILLS REVIEW HANDBOOK
Significant digits indicate how precisely a number is known. Use the following
guidelines to determine the number of significant digits.
• All nonzero digits are significant.
• All zeros that appear between two nonzero digits are significant.
• For a decimal, all zeros that appear after the last nonzero digit are significant.
For a whole number, you cannot tell whether any zeros after the last nonzero
digit are significant, so you should assume that they are not significant.
Sometimes calculations involve measurements that have various numbers of
significant digits. In this case, a general rule is to carry all digits through the
calculation and then round the result to the same number of significant digits
as the measurement with the fewest significant digits. When you calculate with
units that cannot be divided into fractional parts, such as number of people,
consider only the significant digits of the other number(s).
EXAMPLE
a.
12.6
3 0.05
0.63
0.6
Perform the calculation. Write your answer with the
appropriate number of significant digits.
3 significant digits
1 significant digit
b.
840
2 significant digits
1 702
3 significant digits
The product has 2 significant
digits.
1542
The sum has 4 significant
digits.
Round to 1 significant digit.
1500
Round to 2 significant digits.
c. $61.20 restaurant bill 4 6 people
The number of people is exact, so consider only the 4 significant digits of the bill,
$61.20. The answer should have 4 significant digits.
$61.20 4 6 5 $10.20
c Each person pays $10.20.
PRACTICE
Perform the calculation. Write your answer with the appropriate number of
significant digits.
1. 600 1 30
2. 5 2 2.6
3. 12 p 6.75
4. 0.098 1 0.14 1 0.369
5. 3.6053 2 1.720
6. 40 4 3.5
7. 8.0 2 3.1
8. 31.7 p 6.8 p 0.435
9. 30.5 p 6.40
13. 4016 2 3007
10. 3.18 1 2.0005
11. 0.088 4 2.44
12. 8650 1 380 2 49
14. 1.35 1 14.8
15. 320 4 18
16. 38.1 p 3.04 4 0.024
17. $1.45 per notebook p 12 notebooks
18. 10.0 liters of water 2 4.5 liters of water
19. 260 pints of milk 4 106 students
20. 0.5 yard of fabric 1 0.87 yard of fabric
21. 27,973 books 4 11 libraries
22. 12.76 gallons of gas 1 6.08 gallons of gas
23. $6.95 per ticket p 180 tickets
24. 1540 pounds 2 160 pounds 2 85 pounds
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SKILLS REVIEW HANDBOOK
Writing Algebraic Expressions
To solve a problem using algebra, you often need to write a phrase as an
algebraic expression.
EXAMPLE
Write the phrase as an algebraic expression.
a. 6 less than a number
b. The cube of a number
c. Double a number
“Less than” indicates
subtraction.
“Cube” indicates raising
to the third power.
“Double” indicates
multiplication by 2.
cn26
c n3
c 2n
EXAMPLE
Write an algebraic expression to answer the question.
a. Rebecca walks three times as far to school as Meghan does. If Meghan
walks m blocks to school, how many blocks to school does Rebecca walk?
c 3m
b. Kate is 8 inches taller than Noah. If Noah is n inches tall, how tall is Kate?
cn18
PRACTICE
Write the phrase as an algebraic expression.
1. 8 more than a number
2. 10 times a number
3. Twice a number
4. 6 less than a number
5. One fifth of a number
6. 4 greater than a number
7. 5 times a number
8. A number squared
9. 25% of a number
10. Half a number
11. 2 less than a number
12. The square root of a number
Write an algebraic expression to answer the question.
13. Allison is 4 years younger than her sister Camille. If Camille is c years old,
how old is Allison?
14. Ryan bought a movie ticket for x dollars. He paid with a $20 bill. How much
change should Ryan get?
15. Bridget spent $5 more than Tom spent at the mall. If Tom spent x dollars,
how much did Bridget spend?
16. Marc has twice as many baseball cards as hockey cards. If Marc has h hockey
cards, how many baseball cards does he have?
17. Elizabeth’s ballet class is 45 minutes long. If Elizabeth is m minutes late for
ballet class, how many minutes will she spend in class?
18. Steve drove x miles per hour for 5 hours. How many miles did Steve drive?
19. Wendy bought 10 pens priced at x dollars each. How much did she spend?
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Student Resources
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Binomial Products
EXAMPLE
Simplify (2x 1 1)(x 1 3).
Draw a rectangle with dimensions 2x 1 1 and x 1 3. Use the
dimensions to divide the rectangle into parts. Then find the
area of each part. The binomial product (2x 1 1)(x 1 3) is
the sum of the areas of all the parts.
2
There are 2 blue parts with area x , 7 green parts with area x,
and 3 yellow parts with area 1.
x
2x 1 1
x
1
x
x2
x2
x
1
1
1
x
x
x
x
x
x
1
1
1
x13
(2x 1 1)(x 1 3) 5 2x2 1 7x 1 3
SKILLS REVIEW HANDBOOK
A monomial is a number, a variable, or the product of a number and one or more
variables. A binomial is the sum of two monomials. In other words, a binomial is
a polynomial with two terms. You can use a geometric model to find the product
of two binomials.
Another way to find the product of two binomials is to use the distributive
property systematically. Multiply the first terms, the outer terms, the inner terms,
and the last terms of the binomials. This is called FOIL for the words First, Outer,
Inner, and Last.
EXAMPLE
Simplify (x 1 2)(4x 2 5).
First
Outer Inner
Last
(x 1 2)(4x 2 5) 5 x(4x) 1 x(25) 1 2(4x) 1 2(25)
2
Use FOIL.
5 4x 2 5x 1 8x 2 10
Multiply.
5 4x2 1 3x 2 10
Combine like terms.
PRACTICE
Simplify.
1. (a 1 5)(a 1 3)
2. (m 1 4)(m 1 11)
3. (t 1 8)(t 1 7)
4. (z 1 1)(z 1 6)
5. (y 1 4)(y 1 2)
6. (x 1 9)(x 1 9)
7. (y 2 2)
2
8. (n 1 6)
2
9. (4 2 z)2
10. (a 1 10)(a 2 10)
11. (y 1 3)(y 2 7)
12. (k 1 1)2
13. (5x 2 4)(5x 1 4)
14. (3 1 n)2
15. (c 1 5)(2c 2 7)
16. (a 1 5)(a 1 5)
17. (7 2 z)(7 1 z)
18. (3x 2 8)(x 2 6)
20. (3 2 g)(2g 1 3)
21. (4 2 x)(8 1 x)
22. (3n 2 1)(n 2 4)
23. (2a 1 9)(a 2 9)
24. (8x 1 1)(x 1 1)
25. (5x 1 2)(2x 2 5)
26. (2d 2 5)(3d 2 1)
27. (24z 1 3)(6z 2 1)
19. (4a 1 3)
2
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SKILLS REVIEW HANDBOOK
LCDs of Rational Expressions
A rational expression is a fraction whose numerator and denominator are
nonzero polynomials. The least common denominator (LCD) of two rational
expressions is the least common multiple of the denominators. To find the LCD,
follow these three steps:
STEP 1
Write each denominator as the product of its factors.
STEP 2 Write the product consisting of the highest power of each factor that
appears in either denominator.
STEP 3 Simplify the product from Step 2 to write the LCD.
EXAMPLE
Find the least common denominator of the rational expressions.
3 and 1
b. }}
}}
12x
8x 2
2 and 2
a. }}
}}3
5xy
y
STEP 1
Factors:
Factors:
5xy 5 5 p x p y
8x2 5 23 p x2
3
y 5y
3
Factors:
2
STEP 2 Product: 5 p x p y 3
STEP 3 LCD: 5xy
x
21 and
c. }}}
}}}}}}
3x 1 6
x2 2 3x 2 10
3
3x 1 6 5 3 p (x 1 2)
2
12x 5 2 p 3 p x
x 2 3x 2 10 5 (x 1 2) p (x 2 5)
Product: 23 p 3 p x2
Product: 3 p (x 1 2) p (x 2 5)
LCD: 24x
2
LCD: 3(x 1 2)(x 2 5)
PRACTICE
Find the least common denominator of the rational expressions.
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1 and 4
1. }}
}}
2ab
a2
5 and 6
2. }}
}}
6k 2
7k 2
2 and 2
3. }}
}}
z3
z2
4 and 23
4. }}
}}
5x
10x
m and 1
5. }}
}}}
14
18m
19 and 3
6. }}}
}}}
20xy
16xy
1 and 1
7. }}
}}
3y
3y 2
24 and 2
8. }}}
}}}
9ab2
21a2b
n and n2
9. }}}
}}}
n12
n22
21 and 3
10. }}}
}}}
x21
x13
28 and 4
11. }}}
}}}
5n 1 5
n11
y
1
12. } and }}}
8
2y 1 8
1
2
13. }}}}
and }}}}
2m 2 6
3m 2 9
a and 2a
14. }}
}}}}
n2
n2 2 6n
1 and
1
15. }}}
}}}}
x24
(x 2 4)2
3
4
16. }}}}
and }}}}
4x 1 12
6x 1 18
29
1 and
17. }}
}}}}}
2n3
10n2 1 8n
10
17b
18. }}}}
and }}}}
15b 2 30
9b 2 18
25 and
3
19. }}}}
}}}}
(k 1 3)4
(k 1 3)2
8
1 and
20. }}}
}}}}
y25
3y 2 15
n2
n
21. }}}}}
and }}}}
10n 1 20
7n 1 14
20
1
22. }}}}
and }}}}
5z 2 40
9z 2 56
2a
2
23. }}}}}}
and }}}
a12
a2 1 4a 1 4
1
21
24. }}}
and }}}}}
2z 2 6
z2 2 z 2 6
3k and
2k
25. }}}
}}}}}}
k23
k 2 2 5k 1 6
x
2x
26. }}}
and }}}}}}
x2 2 9
x 2 1 3x 2 18
m2
25
27. }}}}}}}
and }}}}}}}
2
m 2 11m 1 28
m2 1 5m 2 45
Student Resources
11/21/05 10:26:58 AM
The Coordinate Plane
Each point in a coordinate plane is represented by an ordered
pair. The first number is the x-coordinate, and the second
number is the y-coordinate.
The ordered pair (3, 1) is graphed at the right. The x-coordinate
is 3, and the y-coordinate is 1. So, the point is right 3 units and
up 1 unit from the origin.
EXAMPLE
y-axis
Quadrant II 4
(2, 1)
3
origin 2
(0, 0)
y
Quadrant I
(1, 1)
(3, 1)
1 2 3 4 5 6x
262524232221
21
22
23
(2, 2)
Quadrant III 24
x-axis
(1, 2)
Quadrant IV
SKILLS REVIEW HANDBOOK
A coordinate plane is formed by the intersection of a horizontal
number line called the x-axis and a vertical number line called
the y-axis. The axes meet at a point called the origin and divide
the coordinate plane into four quadrants, numbered I, II, III,
and IV.
Graph the points A(2, 21) and B(24, 0) in a coordinate plane.
A(2, 21) Start at the origin.
The x-coordinate is 2, so move right 2 units.
The y-coordinate is 21, so move down 1 unit.
Draw a point at (2, 21) and label it A.
B(24, 0) Start at the origin.
The x-coordinate is 24, so move left 4 units.
The y-coordinate is 0, so move up 0 units.
Draw a point at (24, 0) and label it B.
4
3
2
1
B(24, 0)
y
3 4 5 6x
1
262524232221
21
22
23
24
A(2, 21)
PRACTICE
Graph the points in a coordinate plane.
1. A(7, 2)
2. B(6, 27)
3. C(2, 23)
4. D(28, 0)
5. E(24, 28)
6. F(1, 3)
7. G(3, 0)
8. H(1, 25)
9. I(0, 22)
10. J(26, 5)
11. K(5, 8)
12. L(8, 22)
13. M(23, 24)
14. N(27, 8)
15. P(25, 1)
16. Q(22, 26)
17. R(0, 6)
18. S(24, 21)
19. T(4, 4)
20. V(23, 7)
Give the coordinates and the quadrant or axis of the point.
21. A
24. D
22. B
25. E
23. C
5
26. F
27. G
28. H
29. J
30. K
31. L
32. M
33. N
34. O
35. P
36. Q
37. R
38. S
39. T
40. U
41. V
42. W
43. X
44. Y
T 4
F
3
N
2
1
D
y
A
M
G
S
W
U
E
L
26
24
22
O
R
V
H
3 4 5 6x
1
K
J
B23 X
Œ
24
Y
P
C
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SKILLS REVIEW HANDBOOK
Transformations
A transformation is a change made to the position or to the size
of a figure. Each point (x, y) of the figure is mapped to a new point,
and the new figure is called an image.
A translation is a transformation in which each point of a figure
moves the same distance in the same direction. A figure and its
translated image are congruent.
EXAMPLE
Translation a Units Horizontally
and b Units Vertically
(x, y) → (x 1 a, y 1 b)
Translate }
FG right 3 units and down 1 unit.
y
13
F
21
F9
To move right 3 units, use a 5 3. To move down 1 unit, use
b 5 21. So, use (x, y) → (x 1 3, y 1 (21)) with each endpoint.
1
F(2, 4) → F9(2 1 3, 4 1 (21)) 5 F9(5, 3)
G(1, 1) → G9(1 1 3, 1 1 (21)) 5 G9(4, 0)
G
G9
1
x
Graph the endpoints (5, 3) and (4, 0). Then draw the image.
A reflection is a transformation in which a figure is
reflected, or flipped, in a line, called the line of reflection.
A figure and its reflected image are congruent.
EXAMPLE
Reflection in x-axis
Reflection in y-axis
(x, y) → (x, 2y)
(x, y) → (2x, y)
Reflect n ABC in the y-axis.
y
A9
A
B9 B
Use (x, y) → (2x, y) with each vertex.
A(4, 3) → A9(24, 3)
B(1, 2) → B9(21, 2)
C(3, 1) → C9(23, 1)
1
C9
Change each
x-coordinate
to its opposite.
C
x
1
Graph the new vertices. Then draw the image.
A rotation is a transformation in which a figure is turned
about a fixed point, called the center of rotation. The
direction can be clockwise or counterclockwise. A figure
and its rotated image are congruent.
EXAMPLE
Rotation About the Origin
1808 either direction
(x, y) → (2x, 2y)
908 clockwise
(x, y) → (y, 2x)
908 counterclockwise
(x, y) → (2y, x)
Rotate RSTV 1808 about the origin.
y
R(2, 2) → R9(22, 22)
S(4, 2) → S9(24, 22)
T(4, 1) → T9(24, 21)
V(1, 0) → V9(21, 0)
Change every
coordinate
to its opposite.
R
2
Use (x, y) → (2x, 2y) with each vertex.
S
V9
S9
T
V
T9
2
x
R9
Graph the new vertices. Then draw the image.
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EXAMPLE
Dilation with Scale Factor k
with Respect to the Origin
SKILLS REVIEW HANDBOOK
A dilation is a transformation in which a figure stretches
or shrinks depending on the dilation’s scale factor. A figure
stretches if k > 1 and shrinks if 0 < k < 1. A figure and its
dilated image are similar.
(x, y) → (kx, ky)
Dilate JKLM using a scale factor of 0.5.
The scale factor is k 5 0.5, so multiply every coordinate by 0.5.
Use (x, y) → (0.5x, 0.5y) with each vertex.
J(4, 4) → J9(0.5 p 4, 0.5 p 4) 5 J9(2, 2)
K(6, 4) → K9(0.5 p 6, 0.5 p 4) 5 K9(3, 2)
L(6, 21) → L9(0.5 p 6, 0.5 p (21)) 5 L9(3, 20.5)
M(4, 21) → M9(0.5 p 4, 0.5 p (21)) 5 M9(2, 20.5)
y
J
J9
K
K9
1
M9
L9 5
M
x
L
Graph the new vertices. Then draw the image.
PRACTICE
Find the coordinates of N(23, 8) after the given transformation. For rotations,
rotate about the origin.
1. Rotate 1808.
2. Reflect in x-axis.
3. Translate up 3 units.
4. Reflect in y-axis.
5. Rotate 908 clockwise.
6. Translate left 5 units.
7. Rotate 908 counterclockwise.
8. Translate right 2 units and down 9 units.
Transform n PST. Graph the result. For rotations, rotate about
the origin.
9. Reflect in x-axis.
y
T
10. Rotate 908 counterclockwise.
11. Rotate 908 clockwise.
12. Translate down 7 units.
13. Reflect in y-axis.
14. Translate left 4 units.
15. Rotate 1808.
16. Translate right 2 units.
x
1
22
P
S
17. Translate right 1 unit and up 4 units.
18. Translate left 6 units and up 2 units.
The coordinates of the vertices of a polygon are given. Draw the polygon. Then
find the coordinates of the vertices of the image after the specified dilation, and
draw the image.
19. (1, 3), (3, 2), (2, 5); dilate using a scale factor of 3
3
20. (2, 8), (2, 4), (6, 8), (6, 4); dilate using a scale factor of }
2
1
21. (3, 3), (6, 3), (3, 23), (6, 23); dilate using a scale factor of }
3
22. (2, 2), (2, 7), (5, 7); dilate using a scale factor of 2
1
23. (2, 22), (6, 22), (4, 26), (0, 26); dilate using a scale factor of }
2
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SKILLS REVIEW HANDBOOK
Line Symmetry
A figure has line symmetry if a line, called a line of symmetry, divides the figure
into two parts that are mirror images of each other. Below are four figures with
their lines of symmetry shown in red.
Trapezoid
No lines of symmetry
EXAMPLE
Isosceles Triangle
1 line of symmetry
Rectangle
2 lines of symmetry
Regular Hexagon
6 lines of symmetry
A line of symmetry for the figure is shown in red.
Find the coordinates of point A.
Point A is the mirror image of the point (3, 26) with respect
to the line of symmetry y 5 22. The x-coordinate of A is 3,
the same as the x-coordinate of (3, 26). Because 26 is the
y-coordinate of (3, 26), and 22 2 (26) 5 4, the point (3, 26)
is down 4 units from the line of symmetry. Therefore, point A
must be up 4 units from the line of symmetry. So, the
y-coordinate of A is 22 1 4 5 2. The coordinates of point A
are (3, 2).
y
A
1
x
2
y 5 22
C
B(3, 26)
PRACTICE
Tell how many lines of symmetry the figure has.
1.
2.
3.
4.
5. A parallelogram
6. A square
7. A rhombus
8. An equilateral triangle
A line of symmetry for the figure is shown in red. Find the coordinates of
point A.
9.
4
(24, 3)
10.
y
11.
y
(0, 4)
A
1x
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y5x
1
y51
A
y
A
x52
1
1
1
x
x
(2, 22)
Student Resources
11/21/05 10:27:03 AM
Perimeter and Area
SKILLS REVIEW HANDBOOK
The perimeter P of a figure is the distance around it. To find the perimeter of a
figure, add the side lengths.
EXAMPLE
Find the perimeter of the figure.
a.
b.
13 in.
5 in.
18 m
4m
4m
18 m
12 in.
P 5 5 1 12 1 13 5 30 in.
P 5 2(4) 1 2(18) 5 8 1 36 5 44 m
The area A of a figure is the number of square units enclosed by the figure.
Area of a Triangle
Area of a Rectangle
Area of a Parallelogram
Area of a Trapezoid
b1
h
w
h
l
b
1
2
b
A 5 lw
A 5 }bh
EXAMPLE
h
b2
1
2
A 5 }(b1 1 b2)h
A 5 bh
Find the area of the figure.
a.
b.
c.
7 in.
6m
5 ft
3m
15 in.
A 5 (15)(7) 5 105 in.2
A 5 (5)(5) 5 25 ft 2
1 (6)(3) 5 9 m 2
A5}
2
PRACTICE
Find the perimeter and area of the figure.
1.
2.
8 ft
3 cm
17 ft
3.
3 in.
15 ft
4.
4 in.
5 in.
5m
12 in.
6m
2 cm
5.
6.
10 yd
8 yd
7.
8.
12 mm
17 yd
8 in.
2.7 m
3m
9 mm
9 mm
21 yd
4m
12 mm
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SKILLS REVIEW HANDBOOK
Circumference and Area of a Circle
A circle consists of all points in a plane that are the same distance from a fixed
point called the center.
The distance between the center and any point on the circle is the radius. The
distance across the circle through the center is the diameter. The diameter is
twice the radius.
circle
radius
diameter
center
The circumference of a circle is the distance around the circle. For any circle,
the ratio of the circumference to the diameter is π (pi), an irrational number
22 .
that is approximately 3.14 or }}
7
To find the circumference C of a circle with radius r, use the formula C 5 2πr.
To find the area A of a circle with radius r, use the formula A 5 πr 2.
EXAMPLE
Find the circumference and area of a circle with radius 6 cm.
Give an exact answer and an approximate answer for each.
Circumference
Area
C 5 2πr
A 5 πr 2
5 2π(6)
5 π(6)2
5 12π
5 36π
< 12(3.14)
< 36(3.14)
< 37.7
< 113
c The circumference is 12π centimeters,
or about 37.7 centimeters.
6 cm
c The area is 36π square centimeters,
or about 113 square centimeters.
PRACTICE
Find the circumference and area of the circle. Give an exact answer and an
approximate answer for each.
1.
2.
3.
4.
5 in.
10 m
2 cm
5.
6.
4 in.
7.
8.
6 ft
12 ft
16 m
9 cm
9.
10.
2 cm
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11.
14 ft
12.
22 in.
36 cm
Student Resources
11/21/05 10:27:05 AM
Surface Area and Volume
SKILLS REVIEW HANDBOOK
A solid is a three-dimensional figure that encloses part of space.
The surface area S of a solid is the area of the solid’s outer surface(s).
The volume V of a solid is the amount of space that the solid occupies.
Cylinder
Rectangular Prism
h
S 5 2lw 1 2lh 1 2wh
V 5 lwh
w
l
EXAMPLE
r
2
S 5 2πr 1 2πrh
h
V 5 πr 2h
Find the surface area and volume of the rectangular prism.
Surface area
Volume
S 5 2lw 1 2lh 1 2wh
V 5 lwh
7 ft
5 2(5)(3) 1 2(5)(7) 1 2(3)(7)
5 (5)(3)(7)
5 30 1 70 1 42
5 105 ft 3
5 ft
3 ft
5 142 ft 2
EXAMPLE
Find the surface area and volume of the cylinder.
Surface area
3m
Volume
2
12 m
2
V 5 πr h
S 5 2πr 1 2πrh
2
5 2π(3) 1 2π(3)(12)
5 π(3)2 (12)
5 90π m 2
Exact answer
5 108π m3
Exact answer
< 283 m 2
Approximate answer
< 339 m3
Approximate answer
PRACTICE
Find the surface area and volume of the solid.
1.
2.
6.5 mm
3.
3 in.
3 cm
12 mm
5 in.
3 cm
3 cm
8 in.
4.
2m
4m
5.
6.
14 yd
10 ft
4 yd
10 m
15 ft
Skills Review Handbook
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SKILLS REVIEW HANDBOOK
Angle Relationships
An angle bisector is a ray that divides an angle into two congruent angles.
Two angles are complementary angles if the sum of their measures is 908.
Two angles are supplementary angles if the sum of their measures is 1808.
EXAMPLE
Find the value of x.
]›
a. BD bisects ∠ ABC
and m ∠ ABC 5 648.
C
b. ∠ GFJ and ∠ HFJ are
J
D
x8
648
]›
Because BD bisects
∠ ABC, the value of
x is half m ∠ ABC.
64 5 32
x 5 }}
2
D
4x 8
(3x 2 1)8
F
G
A
supplementary.
H
(2x 2 6)8
B
c. ∠ CBD and ∠ ABD are
complementary.
E
C
Because ∠ GFJ and ∠ HFJ
are complementary
angles, their sum is 908.
(x 2 3)8
B
A
Because ∠ CBD and ∠ ABD
are supplementary angles,
their sum is 1808.
(2x 2 6) 1 4x 5 90
(3x 2 1) 1 (x 2 3) 5 180
6x 2 6 5 90
4x 2 4 5 180
x 5 16
x 5 46
PRACTICE
]›
BD is the angle bisector of ∠ ABC. Find the value of x.
1.
2.
A
D
3.
A
D
248
788
x8
C
B
A
(11x 2 19)8
(8x 1 5)8
B
(2x 2 4)8
B
D
C
C
∠ ABD and ∠ DBC are complementary. Find the value of x.
4.
5.
A
6. B
D
A
A
(3x 2 18)8
(3x 2 4)8
(7x 1 5)8
D
(4x 1 10)8
B
D
(5x 2 20)8
C
(5x 1 1)8
C
C
B
∠ ABD and ∠ DBC are supplementary. Find the value of x.
7.
8.
D
(x 2 28)8
(4x 1 17)8 (3x 1 2)8
A
994
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B
A
C
9.
D
D
3x 8
B
(4x 1 2)8 (2x 1 4)8
C
A
B
C
Student Resources
11/21/05 10:27:08 AM
Triangle Relationships
SKILLS REVIEW HANDBOOK
The sum of the angle measures of any triangle is 1808.
EXAMPLE
Find the value of x.
60 1 35 1 x 5 180
x8
95 1 x 5 180
608
358
x 5 85
The sum of the angle measures is 1808.
Simplify.
Solve for x.
In a right triangle, the hypotenuse is the side opposite the
right angle. The legs are the sides that form the right angle.
The Pythagorean theorem states that the sum of the squares
of the lengths of the legs equals the square of the length of
the hypotenuse.
Pythagorean Theorem
a2 1 b2 5 c 2
c
a
b
EXAMPLE
Find the value of x.
a.
b.
12 cm
x
6 ft
x
13 cm
8 ft
6 2 1 82 5 x 2
2
Simplify.
100 5 x2
Simplify.
36 1 64 5 x
x 5 10 ft
x2 1 122 5 132
Pythagorean theorem
Pythagorean theorem
2
x 1 144 5 169
Simplify.
x2 5 25
Solve for x2 .
x 5 5 cm
Solve for x.
Solve for x.
PRACTICE
Find the value of x.
1.
2.
688
378
3.
x8
4.
x8
348
728
x8
x8
5.
6.
40 cm
x
568
x8
7.
8 in.
8 in.
x
x8
37 ft
35 ft
30 cm
8.
5m
6m
x
x
9. A triangle with angles that measure x8, x8, and 708
Skills Review Handbook
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SKILLS REVIEW HANDBOOK
Congruent and Similar Figures
Two figures are congruent if they have the same shape and the same size.
If two figures are congruent, then corresponding angles are congruent
and corresponding sides are congruent. The triangles at the right are
congruent. Matching arcs show congruent angles, and matching tick
marks show congruent sides.
Two figures are similar if they have the same shape but not necessarily the same
size. If two figures are similar, then corresponding angles are congruent and the
ratios of the lengths of corresponding sides are equal.
EXAMPLE
Tell whether the figures are congruent, similar, or neither.
a.
7
3
3
3
3
7
7
b.
10
11
As shown, corresponding
angles are congruent, but
corresponding sides have
different lengths. So, the
figures are not congruent,
but they may be similar.
F 3.75 E
A 3 B
D
As shown, corresponding
angles are congruent and
corresponding sides are
congruent. So, the figures
are congruent.
7
6
7.5
C
G
12.5
13.75
H
The figures are similar if the ratios of the lengths of corresponding sides are equal.
BC
FG
3
3.75
AB
EF
}} 5 }} 5 0.8
6
7.5
}} 5 }} 5 0.8
CD
GH
11
13.75
}} 5 }}} 5 0.8
AD
EH
10
12.5
}} 5 }} 5 0.8
c Because corresponding angles are congruent and the ratios of the lengths
of corresponding sides are equal, ABCD is similar to EFGH.
EXAMPLE
The two polygons are similar. Find the value of x.
a.
608
x8
The angle with measure x°
corresponds to the angle with
measure 60°, so x 5 60.
308
b.
8
12
x
9
The side with length 12 corresponds to
the side with length 8, and the side
with length 9 corresponds to the side
with length x.
12
8
9
x
}} 5 }
12x 5 72
x56
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Write a proportion.
Cross multiply.
Solve for x.
Student Resources
11/21/05 10:27:10 AM
PRACTICE
SKILLS REVIEW HANDBOOK
Tell whether the figures are congruent, similar, or neither. Explain.
1.
2.
4
2
6
14
11
3.5
2
3.
1
10
7
14
3
4.
5.
7
6.
8
12
9
9
7
7
12
8
8.
6
4.5
9
1.5
12
1.6
3.2
5
1.5
3
9.
2.4
4.8
4
3
3
8
8
5
7.
12
6
4.8
7
3
3.2
7
1.6
3
3
2.4
The two polygons are similar. Find the value of x.
10.
11.
438
7
x21
12.
12
14
20
18
8
4x 1 3
x8
13.
14.
1138
34.5
27
15.
568
568
678
(5x 2 3)8
16.
18
17.
538
568
568 1248
5x 2 7
538
18.
1198
618
1198
36
15
30
x15
(3x 1 4)8
(11x 2 5)8
1198
618
(7x 1 4)8
538
Skills Review Handbook
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SKILLS REVIEW HANDBOOK
More Problem Solving Strategies
Problem solving strategies can help you solve mathematical and real-life
problems. Lesson 1.5 shows how to apply the strategies use a formula, look for a
pattern, draw a diagram, and use a verbal model. Below are four more strategies.
Strategy
When to Use
How to Use
Make a list or table
Make a list or table when a problem
requires you to record, generate, or
organize information.
Make a table with columns, rows, and any
given information. Generate a systematic list
that can help you solve the problem.
Work backward
Work backward when a problem
gives you an end result and you
need to find beginning conditions.
Work backward from the given information
until you solve the problem. Work forward
through the problem to check your answer.
Guess, check, and revise
Guess, check, and revise when you
need a place to start or you want to
see how the problem works.
Make a reasonable guess. Check to see if
your guess solves the problem. If it does not,
revise your guess and check again.
Solve a simpler problem
Solve a simpler problem when a
problem can be made easier by
using simpler numbers.
Think of a way to make the problem simpler.
Solve the simpler problem, then use what
you learned to solve the original problem.
EXAMPLE
Lee works as a cashier. In how many different ways can Lee
make $.50 in change using quarters, dimes, and nickels?
Use the strategy make a list or table. Then count the number of different ways.
Quarters
Dimes
Nickels
2
0
0
1
2
1
1
1
3
1
0
5
0
5
0
0
4
2
0
3
4
0
2
6
0
1
8
0
0
10
Start with the greatest number of quarters.
Then list all the possibilities with 1 quarter,
starting with the greatest number of dimes.
Then list all the possibilities with 0 quarters,
starting with the greatest number of dimes.
c Lee can make $.50 in quarters, dimes, and nickels in 10 different ways.
EXAMPLE
In a cafeteria, 3 cookies cost $.50 less than a sandwich. If a
sandwich costs $4.25, how much does one cookie cost?
Use the strategy work backward.
4.25 2 0.50 5 3.75
Cost of 3 cookies
3.75 4 3 5 1.25
Cost of 1 cookie
CHECK 1.25 3 3 5 3.75
Cost of 3 cookies
3.75 1 0.50 5 4.25 Cost of sandwich
c One cookie costs $1.25.
998
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Student Resources
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Nolan’s class has 6 more boys than girls. There are
28 students altogether. How many girls are in Nolan’s class?
Use the strategy guess, check, and revise. Guess a number of girls that is less than
half of 28.
First guess:
12 girls, 12 1 6 5 18 boys, 12 1 18 5 30 students
Too high ✗
Second guess: 10 girls, 10 1 6 5 16 boys, 10 1 16 5 26 students
Too low ✗
Third guess:
Correct ✓
11 girls, 11 1 6 5 17 boys, 11 1 17 5 28 students
c There are 11 girls in Nolan’s class.
EXAMPLE
SKILLS REVIEW HANDBOOK
EXAMPLE
How many diagonals does a regular decagon have?
Use the strategy solve a simpler problem. A decagon has 10 sides, so find the
number of diagonals of polygons with fewer sides and look for a pattern.
3 sides
0 diagonals
4 sides
2 diagonals
5 sides
5 diagonals
6 sides
9 diagonals
7 sides
14 diagonals
Notice that the difference of the numbers of diagonals for consecutive figures
keeps increasing by 1:
22052
52253
92554
14 2 9 5 5
So, an 8-sided polygon has 14 1 6 5 20 diagonals, a 9-sided polygon has
20 1 7 5 27 diagonals, and a 10-sided polygon has 27 1 8 5 35 diagonals.
c A regular decagon (a 10-sided polygon) has 35 diagonals.
PRACTICE
1. Ben has a concert at 7:30 P.M. First he must do 2 hours of homework. Then,
dinner and a shower will take about 45 minutes. Ben wants to allow a half
hour to get to the concert. What time should Ben start his homework?
2. Quinn and Kyle collected 87 aluminum cans to recycle. Quinn collected
twice as many cans as Kyle. How many cans did each person collect?
3. In how many different ways can three sisters form a line at a ticket booth?
4. The 8 3 8 grid at the right has some 1 3 1 squares, some 2 3 2 squares, some
3 3 3 squares, and so on. How many total squares does the grid have?
5. If Kaleigh draws 20 different diameters in a circle, into how many parts will
the circle be divided?
6. Six friends form a tennis league. Each friend will play a match with every
other friend. How many matches will be played?
7. Susan has 13 coins in her pocket with a total value of $1.05. She has only
dimes and nickels. How many of each type of coin does Susan have?
Skills Review Handbook
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ALGEBRA 2
n2pe-0010.indd i
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About Algebra 2
The content of Algebra 2 is organized around families of functions,
including linear, quadratic, exponential, logarithmic, radical, and
rational functions. As you study each family of functions, you will learn
to represent them in multiple ways—as verbal descriptions, equations,
tables, and graphs. You will also learn to model real-world situations
using functions in order to solve problems arising from those situations.
In addition to its algebra content, Algebra 2 includes lessons on
probability and data analysis as well as numerous examples and
exercises involving geometry and trigonometry. These math topics
often appear on standardized tests, so maintaining your familiarity
with them is important. To help you prepare for standardized tests,
Algebra 2 provides instruction and practice on standardized test
questions in a variety of formats—multiple choice, short response,
extended response, and so on. Technology support for both learning
algebra and preparing for standardized tests is available at classzone.com.
n2pe-0010.indd ii
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ALGEBRA 2
Ron Larson
Laurie Boswell
Timothy D. Kanold
Lee Stiff
n2pe-0010.indd iii
10/6/05 2:32:36 PM
Copyright © 2007 McDougal Littell, a division of Houghton Mifflin Company.
All rights reserved.
Warning: No part of this work may be reproduced or transmitted in any form or
by any means, electronic or mechanical, including photocopying and recording,
or by any information storage or retrieval system without the prior written
permission of McDougal Littell unless such copying is expressly permitted by
federal copyright law. Address inquiries to Supervisor, Rights and Permissions,
McDougal Littell, P.O. Box 1667, Evanston, IL 60204.
ISBN-13: 978-0-6185-9541-9
ISBN-10: 0-618-59541-4 123456789—DWO—09 08 07 06 05
Internet Web Site: http://www.mcdougallittell.com
iv
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CHAPTER
2
Linear Functions, p. 76
P(d) 5 1 1 0.03d
Unit 1
Linear Equations,
Inequalities, Functions,
and Systems
Linear Equations and Functions
Prerequisite Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.1 Represent Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2 Find Slope and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.3 Graph Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Graphing Calculator Activity Graph Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.4 Write Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Problem Solving Workshop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Mixed Review of Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5 Model Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.6 Draw Scatter Plots and Best-Fitting Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Investigating Algebra: Fitting a Line to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.7 Use Absolute Value Functions and Transformations . . . . . . . . . . . . . . . . . . . . . . 123
Investigating Algebra: Exploring Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.8 Graph Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Mixed Review of Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
ASSESSMENT
Quizzes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96, 120, 138
Chapter Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
★ Standardized Test Preparation and Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
"MHFCSB Activities . . . . 71, 73, 86, 90, 95, 98, 102, 107, 115, 133
DMBTT[POFDPN
Chapter 2 Highlights
PROBLEM SOLVING
★ ASSESSMENT
• Mixed Review of Problem Solving,
106, 139
• Multiple Representations, 95, 104, 105,
119, 129
• Multi-Step Problems, 78, 88, 95, 103,
106, 137, 139
• Using Alternative Methods, 105
• Real-World Problem Solving Examples,
74, 76, 85, 91, 100, 108, 115, 125, 134
• Standardized Test Practice Examples,
82, 132
• Multiple Choice, 77, 86, 87, 93, 102, 109,
110, 118, 127, 128, 136
• Short Response/Extended Response,
77, 78, 79, 87, 88, 94, 95, 103, 106, 111,
119, 128, 129, 136, 137, 139, 146
• Writing/Open-Ended, 76, 86, 87, 93, 94,
101, 103, 106, 109, 110, 117, 118, 127, 128,
135, 136, 139
TECHNOLOGY
At classzone.com:
• Animated Algebra, 71, 73, 86, 90, 95, 98,
102, 107, 115, 133
• @Home Tutor, 70, 78, 87, 94, 97, 103, 110,
119, 121, 128, 137, 141
• Online Quiz, 79, 88, 96, 104, 111, 120,
129, 138
• Electronic Function Library, 140
• State Test Practice, 106, 139, 149
Contents
n2pe-0050.indd ix
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9/23/05 2:02:50 PM
Sele
elecc ted Answers
Chapter 1
a
b
c
d
a d
b c
ad
5}
bc
Definition of division
ad
5}
Commutative property of
multiplication
55. } 4 } 5 } p }
1.1 Skill Practice (pp. 6–7) 1. reciprocal
3.
23
21
1
3
5
22
21
0
1
2
5.
cb
Definition of multiplication
of fractions
11. Associative property of addition 13. Commutative
property of multiplication 15. Distributive property
a d
5}
p}
1
17. 6 p (a 4 3) 5 6 p a p }
3
1
56p }pa
3
Definition of multiplication
of fractions
a b
5}
4}
Definition of division
1
1
2
2
1
5 6p}
pa
3
1
2
Definition of division
Commutative property
of multiplication
Associative property
of multiplication
5 2a
1
23. Sample answer: a 5 22, b 5 }
25. $8.50 per h
4
2
27. $36.25 29. 195 mi 31. 116 } yd 33. 2200 g
3
35. 1.75 gal 37. 0.00175 ton 39. The unit multiplier
0.82 euro
0.82 euro
should be }
; 25 dollars p }
5 20.5 euros.
1 dollar
1 dollar
41. 29.3 ft/sec 43. 31.1 mi/h 45. 0.04 oz/sec
47. 1800 mi/h 49. Always; this represents the
associative property of addition, which is true for all
real numbers. 51. Sometimes; it is true for b ≤ 0 and
c ≤ 0. 53. Always; this represents the distributive
property, which is true for all real numbers.
c
b
d
1.1 Problem Solving (pp. 8–9) 57. a. Lance: 6, Darcy: 2,
Javier: 3, Sandra: 22 b. Sandra, Darcy, Javier, Lance
59. a. Pluto, Neptune, Uranus, Saturn, Jupiter, Mars,
Earth, Mercury, Venus b. Mercury, Venus, Earth, Mars,
Jupiter, Saturn, Uranus, Neptune, Pluto c. Sample
answer: The planets are in opposite orders in parts
(a) and (b) with the exception of Mercury and
Venus. d. Mercury or Venus 61. a. cheetah: 102.67;
three-toed sloth: 0.15; squirrel: 17.6; grizzly bear: 30
b. Sample answer: The cheetah is about 467 times
faster than the three-toed sloth.
1.2 Skill Practice (pp. 13–15) 1. base: 12, exponent: 7
3. The negative sign should be applied after
evaluating the power, 234 5 281. 5. 81 7. 49 9. 232
11. 210,000 13. 264 15. 64 17. 25 19. 2100 21. 75
23. 6 25. 5x 1 5 27. 13z 2 2 2z 1 10 29. 11m 2 1
SELECTED ANSWERS
Multiplication
Definition of
subtraction
5 c 1 ((23) 1 3) Associative
property of
addition
5c10
Inverse property
of addition
5c
Identity property
of addition
21. 7a 1 (4 1 5a) 5 7a 1 (5a 1 4) Commutative
property of
addition
5 (7a 1 5a) 1 4 Associative
property of
addition
5 12a 1 4
Combine like
terms.
19. (c 2 3) 1 3 5 (c 1 (23)) 1 3
c
1
31. 25p 2 1 21 35. 10n 1 24; 44 37. 26 39. 49 41. }
9
43. 27d 1 11c 45. 2m 2 1 n 2 2 8m 47. 13m 2 2 5
49. 28s 1 8t 51. Sample answer: 3k 1 4k 1 (28) 2 2j ;
7k 2 8 2 2j 53. (4 1 3) p (5 2 2) 5 21
55. (3 p 4)2 2 (23 1 3)2 5 23
1.2 Problem Solving (pp. 15–16) 57. 0, 10, 20, 30;
$1.89, $3.20, $4.51, $5.82 59. 270 2 4.5x; no; when
x > 60 there will be a negative balance on the card,
which means you will have spent more than what
you had on the card. 63. 26.5x 2 6y 1 200; $88
1.3 Skill Practice (pp. 21–23) 1. solution 3. 3 5. 12
2
7. 6 9. 2 }
11. 4 13. 21 15. 18 17. 29 21. 1 23. 4
9
1
2
25. 27 27. 21 }
29. 4 31. 22 }
33. 4 35. 27 37. 22
3
3
39. 28 41. Both sides of the equations should be
3
3
divided by }
instead of subtracting }
from each side;
7
7
3
3
} x 5 15, x 5 15 4 }, x 5 35. 43. 12 45. 60 47. 223
7
7
Selected Answers
n2pe-9090.indd SA1
SA1
12/1/05 12:19:41 PM
2
49. 1}
51. 6; 15, 8 53. 2; 6, 6, 3 55. 4 57. 2 59. 4
3
29.
3x 1 18 5 72,
18 in., 24 in., 30 in.
61. 2.9 63. no solution 65. all real numbers
d2b
a2c
67. x 5 }; a 5 c and b ? d or a ? c and b 5 d;
x 1 12
a 5 c and b 5 d
x16
1.3 Problem Solving (pp. 23–24) 69. 3 h 71. 9 h
5 1
1 1
73. a. 3c 1 2g 5 8 b. 2 }
; }; 2 }
; } 75. 18 min
4 2 12 4
A
1.4 Skill Practice (pp. 30–31) 1. formula 3. l 5 };
w
2A
5 mm 5. h 5 }
; 6 cm 7. y 5 26 2 3x; 5
b1 1 b2
6
31
3
21
9. y 5 2 } x 1 }; 11 11. y 5 } x 2 }
; 23
5
5
2
2
7
11
13. y 5 } x 2 }
; 6 17. The variable y should only
4
4
appear on one side of the equation, not both;
SELECTED ANSWERS
S
9
4y 2 xy 5 9, y(4 2 x) 5 9, y 5 }
. 19. h 5 }
2 k;
42x
pr
40 1 3x
16x 1 28
about 4.96 cm 21. y 5 }
; 11 23. y 5 }
;
x
3x
15
5
2
7}
25. y 5 }; 5 27. Method 1: y 5 } x 2 3,
3
1 2 2x
3
5
1
y5}
p 2 2 3, y 5 }
; Method 2: 15 p 2 2 9y 5 27,
3
3
1
30 2 9y 5 27, 29y 5 23, y 5 }
; Sample answer:
3
Method 1 is more efficient because it is already
x1y
xy 2 1
xy
xy 2 y 2 x
solved for y. 29. z 5 } 31. z 5 }
C
1.4 Problem Solving (pp. 31–32) 33. d 5 }; about 36 in.
p
5
35. C 5 } (F 2 32); 108C 37. R 5 80c 1 150d;
9
R 2 80c
d5}
; 80 designer tuxedos; 160 designer
150
tuxedos; 240 designer tuxedos b. Sample answer:
r 5 1.5, R 5 2.25; r 5 1.15, R 5 3.83; r 5 0.849,
2
2
4p
4p
lw
w l
R 5 7.03 39. V 5 }
;V5}
x
31. 40x 1 7(20 2 x ) 5 404; 8 boxes of books,
12 boxes of clothes 33. about 4.07 in.
1.6 Skill Practice (pp. 44–45) 1. solution
3.
0
22
2
4
6
5.
26
24
0
22
2
11. 23 ≤ x ≤ 1 13. x < 22 or x > 4
17.
24
0
22
2
4
19.
22
23. x ≤ 22
25. x < 4
0
2
4
23
22
0
22
6
0
21
2
1
4
6
35. The inequality symbol should not be reversed
when subtracting; 10 > 2x, 5 > x.
37. 26 < x < 3
39. 1 < x ≤ 7
28
24
0
4
2
4
6
8
0
43. x < 24 or x > 2
1
45. x ≤ 2 }
or x ≥ 1
2
24
22
0
22
21
8
0
2
1
4
2
49. no solution 51. no solution
1.6 Problem Solving (pp. 46–47) 53. 45x 1 35 ≤ 250,
7
x ≤ 4}
days 55. a. 0 ≤ e < 500 b. 1400 ≤ e < 2429
9
pattern shows the output is decreased by 10 each
time; an equation that represents the table is
y 5 75 2 10x. 23. y 5 7x 2 16
c. 0 ≤ e < 500 or 1400 ≤ e < 2429 57. 50 ≤ F ≤ 95;
10 ≤ C ≤ 35 59. a. Amy: 0.65(84) 1 0.15(80) 1 0.2w ≥ 85,
Brian: 0.65(80) 1 0.15(100) 1 0.2x ≥ 85, Clara:
0.65(75) 1 0.15(95) 1 0.2y ≥ 85, Dan: 0.65(80) 1
0.15(90) 1 0.2z ≥ 85 b. w ≥ 92; x ≥ 90; y ≥ 110; z ≥ 97.5
c. Amy, Brian, and Dan. Sample answer: It is
impossible to score over 100 points on a test, so Clara
will not be able to achieve a grade of 85 or better.
1.5 Problem Solving (pp. 38–39) 25. 3.75 km/min
1.6 Problem Solving Workshop (p. 49)
27. y 5 1.5x 1 15; no; the bamboo shoot will
1. y 5 235x 1 200; x < 20 3. x ≥ $7000
1.5 Skill Practice (pp. 37–38) 1. verbal model
3. 0.5 h 5. 90 mi 7. 54 ft 9. 20 m 11. y 5 4x 1 11
13. y 5 46 2 10x 17. 4x 1 9 5 12, 0.75 ft 19. The
eventually slow it’s growth rate and stop growing.
1.7 Skill Practice (pp. 55–56) 1. An apparent solution
that must be rejected because it does not satisfy
the original equation. 3. solution 5. not a solution
7. solution
SA2
n2pe-9090.indd SA2
Selected Answers
12/1/05 12:19:44 PM
9. 29, 9
11. 0
212
0
26
6
5. domain: 22, 1, 6, range: 23, 21, 5, 8
12
y
24
0
22
2
4
6
7
4
21. 24, 9 23. }, 2 25. 27, 4 27. 27, 2 29. 1 }
;3
9
Input
Output
22
23
21
5
8
1
2
31. 220, 4 33. No; the equation has no solutions
6
x
22
because an absolute value will never be negative.
3
4
1
1
1
35. 23 37. 21 }
, 2}
, 39. 2 }
, 23 } 41. When writing
2
2
6
the second equation, the right side of the equation
should be 2x 2 3; 5x 2 9 5 2x 2 3, 6x 2 9 5 2 3,
6x 5 6, x 5 1, the solutions are 3 and 1.
25
5
43. 25 ≤ x ≤ 5
26
45. 25 < m < 9
2
22
26
6
2
22
10
6
10
65. c > 0, c 5 0, c < 0 67. no solution
69. x < 9 or x > 9
0
3
6
9
11. Yes; each input has exactly one output. 13. Yes;
each input has exactly one output. 15. x is the input
and y is the output, so there should be one value of y
for each value of x; the relation given by the table is
not a function because the inputs 1 and 0 each have
more than one output. 17. No; the input 22 has more
than one output. 19. No; the input 21 has more than
one output. 21. function 23. not a function
25.
Chapter Review (pp. 61–64) 1. exponent, base
3. extraneous solution 5. Sample answer: 3(x 2 4)
and 3x 2 12 7. Inverse property of multiplication
9. Distributive property 11. 3x 2 6y 13. 18b 233
1
15. 22t 4 1 5t 2 17. 2 }
19. 9 21. 21 23. $74.99
6
2.1 Problem Solving (pp. 78–79) 43. Yes; each input has
exactly one output. 45. About 905; V(6) represents the
volume of a sphere with radius 6.
47. a. h(l)
domain:
74
15 ≤ l ≤ 24,
72
70
range:
68
57.95 ≤ h(l) ≤ 75.5
25. y 5 210x 1 7; 223 27. y 5 }; 15 29. y 5 } x 2 5;
S 2 2p r
220 31. h 5 }
; about 7.73 cm 33. 602 mi
2p r
0
1
37. x ≤ 2 }
2
39. 23 ≤ x ≤ 3
2
4
6
8
22
21
0
1
24
0
22
0
24
0 15
17
19
21
Length (inches)
23 l
11,350,000, 12,280,000, 12,420,000, 15,980,000,
18,980,000, 20,850,000, 33,870,000, range: 20, 21, 27,
31, 34, 55 b. Yes; each input p has exactly one output.
c. No; the input 21 has more than one output.
4
2
41. 23, 1 }
43. no solution
3
45. y < 21 or y > 6
66
64
62
60
58
0
b. 59 in. or 4 ft 11 in. c. 21.7 in. 49. a. domain:
2
2
x
35. not linear; 10 37. linear; 6 39. linear; 23
5
2
215
x26
21
SELECTED ANSWERS
81. e 2 6008 ≤ 5992, m 2 46,000 ≤ 45,000
1
x
21
Height (inches)
75. p 2 6.5 ≤ 1 77. b 2 21 > 1 79. x 2 45 ≤ 15
35. x ≤ 6
y
1
12
2c 2 b
c2b
b2c
c1b
71. x ≤ } or x ≥ } 73. x < } or x > }
a
a
a
a
2
27.
y
4
8
12
47. v 2 26 ≤ 0.5, 25.5 in. ≤ v ≤ 26.5 in.
Extension (p. 81)
1.
discrete; 21, 1, 3, 5, 7
y
Chapter 2
2.1 Skill Practice (pp. 76–78) 1. independent, dependent
3. domain: 24, 22, 1, 3, range: 23, 21, 2, 3
y
1
21
x
Input
Output
24
22
1
3
23
21
2
3
1
21
x
Selected Answers
n2pe-9090.indd SA3
SA3
12/1/05 12:19:46 PM
3.
continuous; y > 26
y
1
2.3 Skill Practice (pp. 93–94) 1. slope-intercept
3.
x
21
Both graphs have a
y-intercept of 0, but the
graph of y 5 3x has a slope
of 3 instead of 1.
y
1
x
21
5.
5. d (x ) 5 3.5x
Both graphs have a slope
of 1, but the graph of
y 5 x 1 5 has a y-intercept
of 5 instead of 0.
y
Distance (miles)
d(x)
14
10
1
6
2
0
x
21
9.
0
1
2
3
Hours
11.
y
1
x
4
y
x
21
domain: x ≥ 0, range: d (x ) ≥ 0; continuous
7. m (x ) 5 3x m(x)
Gallons of milk
SELECTED ANSWERS
30
27
24
21
18
1
15
12
9
6
3
0
x
21
21. The slope and y-intercept
were switched around.
y
1
0 1 2 3 4 5 6 7 8 9 10 x
Weeks
1
x
domain: whole numbers, range: multiples of 3;
discrete
25. x-intercept: 215, y-intercept: 23 27. x-intercept: 5,
y-intercept: 210 29. x-intercept: 6, y-intercept: 24.5
3
2.2 Skill Practice (pp. 86–87) 1. slope 3. }; rises
2
5
7
5. 2 }; falls 7. 24; falls 9. }; rises 11. undefined;
4
3
31.
is vertical 13. 0; is horizontal 15. The x and y
coordinates were not subtracted in the correct
21 2 (23)
2 2 (24)
1
order; } 5 }
. 19. neither 21. perpendicular
33.
y
1
1
(8, 0)
x
22
43.
3
2
6
21
(4, 0)
x
21
45.
y
y
1
x
21
1
23. parallel 25. 13 mi/gal 27. 2 m/sec 29. 2 31. }
3
2
y
(0, 2)
x
33. 2 } 35. No; no. Sample answer: The slope of
120
221
1
PQ 5 }
5 2}
. The slope of QR 5 }
5
23 2 (21)
2
21 2 (22)
325
21 2 1
1
1
2}
, the slope of ST 5 } 5 2 }
.
2
2
7
2.2 Problem Solving (pp. 87–88) 41. } 43. 6.5%
12
3
1
47. a. } b. yes c. }
8
8
SA4
n2pe-9090.indd SA4
A
55. Sample answer: x 5 3, y 5 22 57. slope: 2 }
,
B
C
y-intercept: }
B
Selected Answers
12/1/05 12:19:48 PM
19
4
2.3 Problem Solving (pp. 94–96)
1
27. y 5 22x 1 6 29. y 5 2 }
x 1 } 31. y 5 23x 1 11
59.
2
33. y 5 2 }
x 1 7 35. y 5 5x 1 23 37. y 5 23x 1 17.5
$480
y
600
Cost
4
3
525
450
375
41. 24x 1 y 5 23 43. 4x 2 5y 5 27 45. 4x 1 3y 5 32
1
47. Sample answer: y 5 2 }
x18
2
300
225
150
75
0
2.4 Problem Solving (pp. 103–104) 51. n 5 15t 1 50
53. 15x 1 9y 5 4500
0 1 2 3 4 5 6 7 8 x
y
500
Months
Cost
61.
400
$1.50; $3
C(g)
36
300
32
28
24
200
20
16
100
12
8
4
0
0
0 1 2 3 4 5 6 7 8 g
Number of games
63. 30; fall; the value of the card will decrease after
Walking time (hours)
b.
8
0
4
8
12
16
c. Sample answer:
r
x
12
0
2
200
16
1
1
150
w
4
0
100
Find the point on the line where x is 200 then the
corresponding y-coordinate is how many student
tickets were sold. 55. y 5 1.661x 1 21.62; $48.20
57. a. 2l 1 2w 5 24
2
0
50
SELECTED ANSWERS
you buy each smoothie, so the line will fall from left
to right.
65. w
Sample answer:
r 5 0 and w 5 4,
4
r 5 1.75 and w 5 1,
3
r 5 0.875 and w 5 2.5
0
l
l
w
6
6
7
5
8
4
9
3
10
2
Running time (hours)
t
(minutes)
h
(feet)
0
200
1
350
2
500
3
650
4
800
5
950
b.
h
900
800
Height (feet)
67. a.
700
600
500
400
300
200
100
0
0
1
2
3
4
m
Minutes
c. h (t ) 5 150t 1 200
2.4 Skill Practice (pp. 101–103) 1. standard 3. y 5 2
5
5. y 5 6x 7. y 5 2 } x 1 7 9. y 5 4x 2 2 11. y 5 2x 1 11
4
4
1
13. y 5 29x 1 85 15. y 5 2 }
x 1 1 17. y 5 2 }
x22
7
3
2.4 Problem Solving Workshop (p. 105) 1. y 5 4x 1 7
1
3. y 5 2 }
x 1 16 5. y 5 32.14x 1 1764.36;
2
about 2825 years old
2.5 Skill Practice (pp. 109–110) 1. Sample answer:
If y 5 ax, then a is the constant of variation. a is a
constant ratio of y to x for all ordered pairs (x, y ).
y
3. y 5 3x
1
19. The x- and y-coordinates were transposed;
y 2 1 5 22(x 2 5), y 2 1 5 22x 1 10, y 5 22x 1 11.
21
x
1
1
21. y 5 2x 1 8 23. y 5 23x 1 13 25. y 5 2 }
x2}
4
4
Selected Answers
n2pe-9090.indd SA5
SA5
12/1/05 12:19:49 PM
5. y 5 23.5x
19. y 5 0.05x 1 1.14
y
3
y
6
4
x
21
2
1
11. y 5 2x; 24 13. y 5 20.2x; 22.4 15. y 5 }
x; 4
0
3
20
40
60
x
80
19. not direct variation 21. direct variation; 2.5
21. a. Sample answer: Measuring the depth of water
1
4
4
23. direct variation; }
25. y 5 2 }
x; 3 27. y 5 27x; }
7
6
3
5
1
29. y 5 27.2x; } 31. direct variation; y 5 2 } x
9
3
at different times while filling a swimming pool.
The number of gallons of milk you buy and the
total cost. b. Sample answer: The age of a car and its
current value. The number of miles you have driven
since you last put gas in the tank and the amount of
gas left in the tank. c. Sample answer: The height of
a person and the month they were born. The age
of a person and the number of vehicles they own.
to be compared to each other, not the products;
24
1
8
3
12
2
6
4
} 5 24, } 5 6, } ø 2.7, } 5 1.5, because the ratios
are not equal, the data does not show direct variation.
2.5 Problem Solving (pp. 110–111) 39. w 5 3600d;
6300 lb 41. direct variation; t 5 5.1s 43. a. direct
variation; P 5 4s b. Not a direct variation; the ratios
of A to s are not equal. c. Not a direct variation; the
ratios of A to P are not equal.
2.6 Skill Practice (pp. 117–118) 1. best-fitting line
2.6 Problem Solving (pp. 119–120) 25. Sample answer:
y 5 101.3x 1 2236.6 27. a. (0, 37), (4, 49), (8, 57),
(12, 64), (14, 67), (18, 72), (22, 77)
y
b.
c. Sample answer:
80
y 5 1.8x 1 40.7;
102 countries
60
Countries
33. direct variation; y 5 24x 35. The quotients need
SELECTED ANSWERS
0
3. negative correlation 5. no correlation 7. 0 9. 21
11. a. y
b. Sample answer:
120
20
y 5 220x 1 141
c. about 2259
96
40
0
0
6
12
18
x
24
Years since 1980
72
2.7 Skill Practice (pp. 127–128) 1. vertex
48
3.
translated down 7 units
y
2
24
x
22
0
13. a.
0
1
2
3
5 x
4
b. Sample answer:
y
120
y 5 6.7x 1 1
c. about 135
90
5.
translated left 4 units and
down 2 units
y
1
x
21
60
30
0
0
4
8
12
16
1
1
15. y 5 23x 17. y 5 }
x 19. y 5 }
x 1 2 2 1
3
x
21.
17. The line should go through the middle of the
data points. Sample answer:
(24, 0)
y
60
1
1
x
y
1
2
(22, 1 )
2
(21, 21)
(0, 23)
40
20
0
2
23.
y
x
21
29. The graph should be
0
2
4
6
8
x
(1, 1)
(2, 0)
y
translated left 3 units.
1
21
x
33. No. Sample answer: It does not pass the vertical
line test.
SA6
n2pe-9090.indd SA6
Selected Answers
12/1/05 12:19:51 PM
23.
2.7 Problem Solving (pp. 128–129)
37.
y
1
15,000 pairs of shoes
s
50
Sales (thousands)
25.
y
1
40
30
1
20
x
21
10
0
0
10
20
30
40
29. solution, not a solution 31. solution, not a solution
y
y
35.
33.
t
Weeks
1
140
39. y 5 2 }x 2 69 1 140
69
41. a.
b.
1
0
0.5
1
1.5
2
2.5
3
d
90
60
30
0
30
60
90
2
3
3
5
39. Sample answer: y > x 1 3 41. y > 2 } x 1 3; pick
60
40
20
1
2
3
4
t
5
Extension (p. 131) 1. 21 3. }
2
2.8 Problem Solving (pp. 137–138) 43. 0.03x 1 0.06y ≤ 20
7.
y
45. 1.5x 1 2.5y ≤ 75
y
1
1
x
21
y
30
Linen
5.
20
10
x
21
0
9.
y
0
10
20
30
40
50 x
Cotton
y ≤ 15.6 yd
47. a. 11x 1 26y ≤ 120
1
x
y
10
Bike
22
2.8 Skill Practice (pp. 135–136) 1. half-plane 3. solution,
not a solution 5. solution, solution
7.
9.
y
1
21
8
6
4
2
0
0 2 4 6 8 10 12 14 16 18 x
Canoe
b. Sample answer: 2 days canoeing and 5 days biking,
y
1
3 days canoeing and 2 days biking, 2 days canoeing
and 2 days biking
y
c. 11x 1 26y ≤ 96
5
Sample answer: 1 day
4
canoeing and 3 days biking, 3
4 days canoeing and 2 days 2
1
biking, 2 days canoeing and 0
0 1 2 3 4 5 6 7 8 9 x
2 days biking
x
21
x
19. The boundary line should
be a dashed line.
y
1
21
x
Selected Answers
n2pe-9090.indd SA7
SELECTED ANSWERS
two points on the boundary line to find the slope
and then use the point-slope form of an equation to
find the equation. The boundary line is dotted, so the
inequality dos not include points on the boundary.
Then choose a point to determine which inequality
sign to use. Sample answer: You and your sister want
to spend at least $15 on your little brother’s birthday.
You want to buy him some racecars that cost $3 each
and some building block sets that cost $5 each.
1
3
} ≤ t ≤ 2}
0
x
21
c. d 5 60t 2 1.5;
80
x
22
t
d
0
x
SA7
12/1/05 12:19:52 PM
Chapter Review (pp. 141–144) 1. standard 3. direct
variation 5. domain: 22, 21, 2, 3, range: 22, 0, 6, 8;
function 7. linear function; 51 9. undefined 11. 0
13.
15.
y
1
x
21
y
1
x
21
3
4
17. y 5 2 } x 1 2 19. y 5 28x; 224 21. y 5 20.8x; 22.4
23. Sample answer: y 5 2x 1 2.3
y
25.
shrunk vertically by
1
1
1
2
21. (28, 6) 23. no solution 25. (7, 3)
27. Failed to multiply the constant by 22.
26x 2 4y 5 214
5x 1 4y 5 15
2x 5
1
x 5 21
29. (25, 26) 31. infinitely many solutions 33. (28, 0)
3
2
1
2
1
3
4 2
2
45. about (2.90, 22.16) 47. (21, 2) 49. (7, 1)
1
51. 2 }
, 6 53. (5, 4)
1
x
21
2
2
1
35. (7, 26) 37. 2 }, 4 39. 2 }, }
41. (2, 3) 43. (3, 2)
3
a factor of }
4
1
3.2 Skill Practice (pp. 164–165) 1. substitution 3. (6, 21)
4
5. no solution 7. }
, 2 9. (0, 3) 11. (23, 8)
3
1
1 1
13. (44, 217) 15. 7, }
17. (26, 22) 19. 2 }
,}
2
2 6
2
9
3.2 Problem Solving (pp. 165–166) 55. 5 acoustic,
27.
$1.75, $1.25
y
4 electric 57. The company can fill its orders by
operating Factory A for 5 weeks and Factory B for
3 weeks. 59. 12 doubles games, 14 singles games
61. 80 pounds of peanuts, 20 pounds of cashews
1
x
SELECTED ANSWERS
21
29. solution
31.
3.3 Skill Practice (pp. 171–172) 1. The ordered pair
33.
y
1
21
x
must satisfy each inequality of the system.
y
5.
7. no solution
y
1
21
x
1
x
21
9.
17.
y
y
Chapter 3
3.1 Skill Practice (pp. 156–157) 1. independent
1
3. (1, 21) 5. (4, 21) 7. (5, 0) 9. (22, 4) 11. (6, 0)
1
81 51
23 23
x
21
1
2
13. }, } 17. (2, 21); consistent and independent
19. no solution; inconsistent 21. infinitely many
solutions; consistent and dependent 23. (2, 0);
consistent and independent 25. (3, 21); consistent
and independent 27. infinitely many solutions;
consistent and dependent 31. no solution
33. (24, 2) and (24, 2)
3.1 Problem Solving (pp. 157–158) 35. lifeguard: 6 h,
cashier: 8 h 37. 11 days; the number of days will
decrease; the number of days will be divided by a
larger number, which will decrease the quotient,
which is the number of days. 39. a. m 5 20.096x 1
50.8 b. w 5 20.12x 1 57.1 c. in the year 2192 d. No.
Sample answer: It is not likely that women’s times
will ever catch up to men’s times or that the times
will continue to decrease infinitely.
SA8
n2pe-9090.indd SA8
x
21
19.
y
1
x
23
3
27. Sample answer: y < x 2 1, y < 2 }x 1 4
4
y
29.
1
22
x
Selected Answers
12/1/05 12:19:54 PM
Chapter 8
Lesson 7.6 (pp. 519–522)
15.
5x
5 33
Lesson 8.1 (pp. 554–557)
5x
5 log11 33
15. y 5 } → 2 5 }
→ 14 5 a
7
11
log11 11
a
x
log 33
log 11
5x 5 log11 33 5 } → x ø 0.2916
35. 5.2 log4 2x 5 16
14
log4 2x ø 3.0769
21. x p y: 12(132) 5 1584
y/x: 132/12 5 11
18(198) 5 3564
198/18 5 11
23(253) 5 5819
253/23 5 11
29(319) 5 9251
319/29 5 11
34(374) 5 12,716
374/34 5 11
4(log4 2x) ø 43.0769
WORKED-OUT SOLUTIONS
14
y5}
→y5}
x
3
Check:
2x ø 71.2020
a
Intersection
X=35.601007 Y=16
x ø 35.6010
57. R 5 100e20.00043t → 5 5 100e20.00043t
x and y show direct variation because the
ratios y/x are equal.
0.05 5 e20.00043t
ln 0.05 5 ln e20.00043t
a
A
22.9957 ø 20.00043t
172
An equation is P 5 }
.
A
t ø 6967 years
172
Boots: P 5 }
→ P ø 2.87 lb/in.2
60
Lesson 7.7 (pp. 533–536)
5.66 2 2.89
521
ln y
11. m 5 } ø 0.69
(5, 5.66)
(3, 4.28)
(4, 5.00)
(2, 3.58)
(1, 2.89)
ln y 2 2.89 5 0.69(x 2 1)
ln y 5 0.69x 1 2.2
Lesson 8.2 (pp. 561–563)
5.
1
The graph of y 5 }
x
5
2
(22, )
y 5 2x
lies farther from the
axes than the graph
x
y 5 e 2.2(e 0.69)x ø 9(2) x
1
(25, 1)
1
1
ln x
0
0.693
1.099
1.386
1.609
ln y
20.511
1.411
2.518
3.296
3.902
y5
(5, 21)
1
x
5
2
(2, 2 )
(1, 25)
3.902 2 (20.511)
m 5 }}
ø 2.743
1.609 2 0
ln y
21.
ln y 2 (20.511) 5 2.743(ln x 2 0)
ln y 5 ln x2.743 2 0.511
y 5 eln x
25
y
(21, 5)
5
1
y 5 e 0.69x 1 2.2
23.
a
400
39. Snow shoes: P 5 } → 0.43 5 } → 172 5 a
1
(1, 0)
2.743
x
5
1 ln x
2.743 2 0.511
y 5 e20.511 p eln x
y (3, 2)
1
x
of y 5 }x , and it lies
in Quadrants II
and IV instead of
Quadrants I and III.
The domain is all real
numbers except 4, and
the range is all real
numbers except 21.
(7, 22)
ø 0.6x2.743
x
(5, 24)
y5
23
x24
21
33. a. A model is y 5 0.48(2.08) .
b. Linear if a graph of (x, y) appears linear;
exponential if a graph of (x, ln y) appears
linear; power if a graph of (ln x, ln y)
appears linear. The graph of (x, y) appears
linear, so a model is y 5 33.8x 1 28.
1000
0.6T 1 331
1000
0.6(25) 1 331
39. a. t 5 } 5 }} ø 2.89
2.89 seconds to travel 1 kilometer;
2.89(5) 5 14.45 seconds to travel 5
kilometers
WS14 Worked-Out Solutions
n2pe-9080.indd WS14
10/13/05 11:40:57 AM
Time (seconds)
39. b.
From the graph,
you can estimate
the temperature to
be 3.98C.
t
3.02
3.01
3.00
2.99
2.98
2.97
0
0 1 2 3 4 5 6 T
Air temperature (8C)
Lesson 8.5 (pp. 586–588)
2x
9 2 2x
9
5. }
2}
5}
x11
x11
x11
5
32
15x
Pi
Pi
43. a. M 5 }}
5 }}
12t
1
1
12 }
11i
1
(1 1 i )
5
(x 1 1)(x 2 1)
(2
2x
x2 2 1
1 4
,
2 3
(2, )
4
3
b. P 5 15,500; i 5 0.06; t 5 4
(3, )
(0, 0)
(23, 2 )
(22, 2 )
3
4
(1 1 0.06)
(
1
,
2
4
23
3x
21
4
9 0}
Check: }
(6) 1 2
3(6)
4
9 0}
}
8
18
1
1
}5}✓
2
2
4(3x) 5 9(x 1 2)
)
4
3
x56
Depth
Temp. b.
1000
4.7634
1050
4.5796
1100
4.4094
1150
4.2515
1200
4.1044
1250
3.9672
1300
3.8389
Temperature (8C)
33. a.
4
x12
9
}5}
5.
x
2
22
15,500(0.06)(1 1 0.06)48
m 5 }}
ø $990.41
48
Lesson 8.6 (pp. 593–595)
3
4
)
21
(1 1 i )
WORKED-OUT SOLUTIONS
y5
Pi(1 1 i )12t
(1 1 i ) 2 1
}}
12t
No x-intercept; x 5 21 and x 5 1 are vertical
asymptotes.
y
(1 1 i )
5}
5 }}
12t
12t
7. y 5 }
→ y 5 }}
2
15.
2
12}
12t
Pi
Lesson 8.3 (pp. 568–571)
5
x 21
32 2 15x
8
17. }2 2 }
5 }2 2 }2 5 }
4x
12x
12x
12x 2
3x
15.
T
4.8
4.6
4.4
3x
2
6x }
1
3x
1
4.2
4.0
3.8
3.6
1
4
6
3x
1
4
} 5 6x }
6
3x
1
2
}1}5}
2
1
4
2
}1}0}
1 2
12
2(2) 1 1(x) 5 4(2) → x 5 4
0
0
1000 1100 1200 1300
Depth (meters)
d
4
2
0}
Check: }
1}
6
3(4)
3(4)
6
12
4
4
}5}✓
12
12
635t 2 2 7350t 1 27,200
n 5 }}
2
35.
t 2 11.5t 1 39.4
635t 2 2 7350t 1 27,200
The mean temperature is
48C at about 1238 meters.
720 5 }}
2
t 2 11.5t 1 39.4
2
720t 2 8280t 1 28,368 5 635t 2 2 7350t 1 27,200
Lesson 8.4 (pp. 577–580)
(x 2 5)(x 1 4)
7. }} Cannot be simplified
(x 1 5)(x 2 3)
48x7y4
6x y
6 p 8 p x3 p x4 p y4
8x4
y
}
25. }
3 6 5 }}
3
4
2 5
2
6px py py
2407t 1 7220
5.92t 2 131t 1 1000
S 4 A 5 }}
4 }}
2
2
6.02t 2 125t 1 1000
26420t 1 292,000
}}
930 6 Ï(930)2 2 4(1168)(85)
t 5 }}}
ø 1.45, 9.45
2(85)
Because 9.45 is not in the domain (0 ≤ t ≤ 9),
t ø 1.45 → 1995.
49.
26420t 1 292,000
85t 2 2 930 1 1168 5 0
2
5.92t 2 131t 1 1000
5 }}
p }}
2
2407t 1 7220
6.02t 2 125t 1 1000
247,060 373.08
For 1999, t 5 7: S 4 A 5 }
p}
ø $50.21
419.98
4371
Chapter 9
Lesson 9.1 (pp. 617–619)
}}}
7. d 5
}
Ï(6 2 2)2 1 (25 2 (21))2 5 4Ï2
12 1 6
21 1 (25)
2
2
Midpoint 5 }
, } 5 (4, 23)
2
Worked-Out Solutions
n2pe-9080.indd WS15
WS15
10/13/05 11:40:59 AM
27. A(24, 1), B(22, 6), C(0, 21)
Lesson 9.3 (pp. 629–632)
}}}
2
}
}}}
2
}
}}}
}
AB 5 Ï(22 2 (24)) 1 (6 2 1) 5 Ï 29
17. 15x 2 1 15y 2 5 60
BC 5 Ï (0 2 (22)) 1 (21 2 6) 5 Ï 53
2
2
2
x 1y 54
}
AC 5 Ï(0 2 (24)) 1 (21 2 1) 5 Ï20 5 2Ï 5
2
2
y 2)
x 2 1 y 2 5 4 (0,
2
1
}
(22, 0)
r 5 Ï4 5 2
1
2 1
}}
2
}
x 2 1 y 2 5 (Ï260 )2 → x 2 1 y 2 5 260
}
b. VS 5 Ï (26 2 0) 1 (5 2 0) 5 Ï 61
Î1
2
65.
WORKED-OUT SOLUTIONS
}}}
SM 5
}
2
Ï117
3
2
}
2}
2
(26)
1
(8
2
5)
5
2
2
2
y
x 2 1 y 2 5 225 x 2 1 y 2 5 25
}
Ï117
}
VS 1 SM 5 Ï 61 1 }
2
20 x
A
0.1 mi
ø 13.2 units p } 5 1.32 mi
1 unit
}
Ï117
c. MP 5 }
2
}}
}
Ï117
E F
x 2 1 y 2 5 100
A: x 2 1 (24)2 5 225 → x ø 614.5 → (214.5, 24)
B: x 2 1 (24)2 5 100 → x ø 69.2 → (29.2, 24)
}
C: x 2 1 (24)2 5 25 → x 5 63 → (23, 24)
MP 1 PV 5 }
1 Ï130
2
0.1 mi
1 unit
ø 16.8 units p } 5 1.68 mi
Lesson 9.2 (pp. 623–625)
D: x 2 1 (24)2 5 25 → x 5 63 → (3, 24)
E: x 2 1 (24)2 5 100 → x ø 69.2 → (9.2, 24)
F: x 2 1 (24)2 5 225 → x ø 614.5 → (14.5, 24)
15. 5x2 5 215y → x2 5 23y
a. AF ø ⏐14.5 2 (214.5)⏐ 5 29 mi
y
2
3
4p 5 23 → p 5 2}
4
y5
2
Focus: 1 0, 2}
42
3
3
4
x
b. BE ø ⏐9.2 2 (29.2)⏐ 5 18.4 mi
c. CD ø ⏐3 2 (23)⏐ 5 6 mi
(0, 2 )
3
4
3
4
C D
220
}
2
B
y 5 24
PV 5 Ï (0 2 3) 1 (0 2 11) 5 Ï 130
2
Directrix: y 5 }
}
39. r 5 Ï (28 2 0)2 1 (14 2 0)2 5 Ï260
}}
2
x
(0, 22)
AB Þ BC Þ AC, so nABC is scalene.
26 1 3 5 1 11
3
53. a. M 5 }, } 5 2}, 8
2
2
2
(2, 0)
1
Lesson 9.4 (pp. 637–639)
Axis of symmetry: x 5 0
11. 16x 2 1 9y 2 5 144
(0, 4) y
2
27. Focus: (25, 0) → p 5 25 → y 5 220x
57. a.
y2
x2
} 1 } 5 1; a 5 4, b 5 3
16
9
y
y
48 in.
(0, 248)
Vertices: (0, 64);
48 in.
146 in.
x
146 in.
(248, 0)
b. x 2 5 4(48)y → x 2 5 192y
x
y 2 5 4(248)x → y 2 5 2192x
2
c. Using x 5 192y and x 5 73, y ø 27.8
Using y 2 5 2192x and y 5 73, x ø 227.8.
The dish is about 27.8 inches deep.
(0,
1
(23, 0)
(3, 0)
x
1
Co-vertices: (63, 0);
}
(0, 24)
Foci: (0, 6Ï 7 )
(0, 2
7)
}
29. b 5 Ï7 ; c 5 3; a2 5 b2 1 c 2 → a 5 4
2
y
y2
x2
x2
}2 1 }
} 2 5 1, or } 1 } 5 1
(Ï 7 )
4
16
7
49. Largest field:
2a 5 185 → a 5 92.5; 2b 5 155 → b 5 77.5
2
y2
2
92.5
6006.25
y2
8556.25
x
x
}2 1 }2 5 1, or } 1 } 5 1
77.5
A 5 π(92.5)(77.5) ø 22,521 square meters
WS16 Worked-Out Solutions
7)
Smallest field:
1 6 23 1 1
Center: (h, k) 5 1 6}
, }2
2a 5 135 → a 5 67.5; 2b 5 110 → b 5 55
Distance between vertex (6, 23) and (h, k):
2
y2
y2
x2
x2
}2 1 }2 5 1, or } 1 } 5 1
4556.25
3025
67.5
55
2
a 5 23 2 k 5 23 2 (21) 5 2
A 5 π(67.5)(55) ø 11,663 square meters
Distance between focus (6, 26) and (h, k):
11,663 ≤ A ≤ 22,521
c 5 26 2 k 5 26 2 (21) 5 5
}
b2 5 c 2 2 a2 5 25 2 4 5 21 → b 5 Ï 21
Lesson 9.5 (pp. 645–648)
(0, 9)
y
49. x 2 2 10x 1 4y 5 0; A 5 1, B 5 0, C 5 0
2
y
x2
}2 }51
81
16
B2 2 4AC 5 0 2 4(1)(0) 5 0 → Parabola
4
}
a 5 4; b 5 9; c 5 Ï 97 (2
(
97 ,0)
(24, 0)
2
97 , 0)
(4, 0)
Vertices: (64, 0)
x 2 2 10x 1 4y 5 0
x
(x 2 2 10x 1 25) 5 24y 1 25
(0, 29)
}
Foci: (6Ï 97 , 0)
25
(x 2 5)2 5 24 1 y 2 }
42
9
Asymptotes: y 5 6}
x
4
25
25
(h, k) 5 1 5, }
5 6.25 feet
2; height 5 }
}
23. c 5 4Ï 5 ; a 5 4; b2 5 c 2 2 a2 2 a → b 5 8
y
2
2
x2
8
y
16
4
41. a. A(30.5, 0); B(85, 240)
Lesson 9.7 (pp. 661–664)
b. Vertices: (630.5, 0); horizontal trans. axis
y2
y2
x2
x2
}2 2 }2 5 1 → }2 2 }2 5 1
b
b
a
30.5
2
2
(240)
85
2
b
y2
2
x
So, an equation is }
2}
5 1.
236.5
930.25
2
5.
Intersection
X=.47247477 Y=-2.582576
5 1 → b ø 236.5
}2 2 }
2
30.5
4
When y 5 0, the x-intercepts are 0 and 10, so
the distance of the jump is 10 feet.
x2
64
}2 2 }2 5 1, or } 2 } 5 1
4
(x 2 6)2
WORKED-OUT SOLUTIONS
13. 81x 2 2 16y 2 5 1296
(y 1 1)2
4
An equation is } 2 }
5 1.
21
Intersection
X=3.5275252 Y=6.5825757
The solutions are approximately (0.5, 22.6)
and (3.5, 6.6).
y2
42
c. x 5 42: }
2}
5 1 → y ø 14.6
236.5
930.25
15. 4x 2 2 5y 2 5 276
2x 1 y 5 26 → y 5 22x 2 6
h 5 40 1 14.6 5 54.6 feet
Substitute 22x 2 6 for y in Equation 1.
Lesson 9.6 (pp. 655–657)
3. (x 1 4)2 5 28(y 2 2)
4x 2 2 5(22x 2 6)2 5 276
y
y54
(24, 2)
Parabola; vertical axis;
vertex at (h, k) 5 (24, 2).
4x 2 2 20x2 2 120x 2 180 5 276
2
2 x
(24, 0)
4p 5 28 → p 5 22
216x 2 2 120x 2 104 5 0
2x 2 1 15x 1 13 5 0
Focus: (h, k 1 p) 5 (24, 0)
13
13
(2x 1 2) 1 x 1 }
5 0 → x 5 21, x 5 2}
22
2
Directrix: y 5 k 2 p → y 5 4
19. Vertices: (6, 23), (6, 1); Focus: (6, 26), (6, 4)
(y 2 k)2
(x 2 h)2
a
b
2}
51
Vertical transverse axis; }
2
2
When x 5 21: y 5 22(21) 2 6 5 24
13
13
: y 5 22 1 2}
2657
When x 5 2}
2
22
13
The solutions are (21, 24) and 1 2}
, 7 2.
2
Worked-Out Solutions
n2pe-9080.indd WS17
WS17
10/13/05 11:41:04 AM
1
29. (2s4 1 5)5 5 5C 0(2s4)550 1 5C1(2s4)451
41. a. Oak Lane: m 5 2}
, (x1, y1) 5 (22, 1)
7
1
1
1 5C2(2s4)352 1 5C3(2s4)253 1 5C4(2s4)154
5
y 2 1 5 2}
(x 1 2) → y 5 2}
x1}
7
7
7
1 5C5(2s4)055 5 1(32s20) 1 5(16s16)(5)
Circle: x 2 1 y 2 5 1
1 10(8s12)(25) 1 10(4s8)(125) 1 5(2s4)(625)
5 2
1
x 1 2}
x1}
51
7
7
1
2
b.
2
10
1
1 1(1)(3125) 5 32s 20 1 400s16 1 2000s12
1 5000s8 1 6250s4 1 3125
25
x2 1 }
x2 2 }
x1}
51
49
49
49
49. You can choose 3 of the 18 types of flowers.
WORKED-OUT SOLUTIONS
49x 2 1 x 2 2 10x 1 25 5 49
18 3
4
5
3
5
1 4
1 3
4
} 1 } 5 }; y 5 2} 2} 1 } 5 }
y 5 2}
7 5
7
5
7 5
7
5
1 2
3 4
4 3
The solutions are }
, } and 2}
,} .
5 5
5 5
1
Î1 2
1
2
18 p 17 p 16 p 15!
15! p 3!
3
(5x 2 4)(10x 1 6) 5 0 → x 5 }
, x 5 2}
5
5
1 2
18!
15!3!
C 5 } 5 }} 5 816
50x 2 2 10x 2 24 5 0
2
Lesson 10.3 (pp. 702–704)
7. Factors of 150 from 1 to 50: 1, 2, 3, 5, 6, 10, 15,
25, 30, 50
Factors of 150
10
1
P 5 }} 5 }
5}
5
50
Integers from 1 to 50
}}
c. d 5
3
5
}
3 2
4 2
4
1 }
2}
5 Ï2 ø 1.4 mi
5
5
5
}2}
2
1
2
Chapter 10
numbers. Only 1 is the correct combination.
1
Lesson 10.1 (pp. 686–689)
13. a. 26 p 26 p 26 p 26 p 10 p 10 5 45,697,600
b. 26 p 25 p 24 p 23 p 10 p 9 5 32,292,000
362,880
9!
9!
35. 9P 2 5 } 5 } 5 } 5 72
5040
(9 2 2)!
7!
65. Permutations of 9 objects taken 3 at a time:
9!
(9 2 3)!
17. There are 48C 6 different combinations of 6
362,880
9!
6!
P 5}5}5}
5 504 ways
9 3
720
Area of smallest circle
Area of entire target
and 4 of the 48 that are not queens.
4!
48!
C p 48C4 5 } p } 5 778,320
4 1
3!1!
44!4!
No queen: Choose 5 cards from the 48 in a
deck that are not queens.
48!
43!5!
C 5 } 5 1,712,304
48 5
The total number of possible hands is
778,320 1 1,712,304 5 2,490,624.
π p 82
π p 40
1
25
39. P 5 }} 5 }2 5 } 5 0.04
Lesson 10.4 (pp. 710–713)
11. P(A or B) 5 P(A) 1 P(B) 2 P(A and B)
0.71 5 0.28 1 0.64 2 P(A and B)
20.21 5 2P(A and B) → P(A and B) 5 0.21
21. P(K or ♦) 5 P(K) 1 P(♦) 2 P(K and ♦)
13
4
1
4
5 1}
1 1}
2 1}
5}
52 2
52 2
52 2
13
Lesson 10.2 (pp. 694–697)
17. Exactly one queen: Choose 1 of the 4 queens
1
P(correct numbers) 5 } 5 }
12,271,512
C
48 6
45. The number of combinations of 6 food items
is 106. The number of combinations of 6
different food items is 10 p 9 p 8 p 7 p 6 p 5. So,
the probability that at least 2 bring the same
item is P 5 1 2 P(none are the same) 5
10 p 9 p 8 p 7 p 6 p 5
10
1 2 }}
5 0.8488.
6
Lesson 10.5 (pp. 721–723)
13. P 5 P(blue) p P(green) p P(red)
3
60
5
4
p } p } 5}
ø 0.015
5 1}
16 2 1 16 2 1 16 2
4096
WS18 Worked-Out Solutions
n2pe-9080.indd WS18
10/13/05 11:41:06 AM
d. p 5 P(Rh1) 5 P(O1) 1 P(A1) 1 P(B1)
25. Primes from 1 to 20: 2, 3, 5, 7, 11, 13, 17, 19.
P(number of odd primes)
Event C: player wins
55
39.
0.
5
0.
Event A:
player
0
wins toss .45
Total #
of
matches 0.5
Event D: player loses
Event C: player wins
0.
47
Event D: player loses
P(C) 5 P(A and C) 1 P(B and C)
P(k 5 5) 5 10C5(0.85) 5(1 2 0.85)10 2 5 ø 0.008
P(k 5 6) 5 10C6 (0.85) 6 (1 2 0.85)10 2 6 ø 0.040
P(k 5 7) 5 10C7(0.85)7(1 2 0.85)10 2 7 ø 0.130
P(k 5 8) 5 10C 8 (0.85) 8 (1 2 0.85)10 2 8 ø 0.276
P(k 5 9) 5 10C9 (0.85) 9 (1 2 0.85)10 2 9 ø 0.347
P(k 5 10) 5 10C10 (0.85)10 (1 2 0.85)10 2 10 ø 0.197
P(k ≥ 5) 5 P(k 5 5) 1 P(k 5 6) 1 P(k 5 7)
1 P(k 5 8) 1 P(k 5 9) 1 P(k 5 10) ø 0.998
5 P(A) p P(CA) 1 P(B) p P(CB)
5 (0.50) p (0, 55) 1 (0, 50)(0.47) 5 0.51
Lesson 10.6 (pp. 727–730)
N
Outcomes
P(N)
1
10
}5}
10
100
Lesson 11.1 (pp. 747–749)
5. Mean:
1
8
10
1
100
90
1000
9
100
900
1000
9
10
2
90
} 5}
3
900
} 5}
Probability
5.
Chapter 11
69 1 70 1 75 1 84 1 73 1 78 1 74 1 73 1 78 1 71
}}}}}
10
6
10
WORKED-OUT SOLUTIONS
Event B:
player
loses toss 0.5
3
1 P(AB1) 5 0.85
7
P(oddprime) 5 }}} 5 }
8
P(number of primes)
5 74.5
4
10
Median: 69, 70, 71, 73, 73, 74, 75, 78, 78, 84
2
10
73 1 74
2
} 5 73.5
0
1
2
3
Number of digits
21. Each question has 4 possible answers, so
the probability of guessing a correct answer
is p 5 0.25. There are 30 questions, so
n 5 30. The probability of randomly
guessing 11 correct answers is P(k 5 11) 5
C (0.25)11(1 2 0.25)30 2 11 ø 0.055.
30 11
45. a. p 5 0.34;
Mode: 73 and 78
15. Range: 158 2 135 5 23
135 1 142 1 148 1 136 1 152 1 140 1 158 1 154
} 5 }}}}
x
8
5 145.625
s5
Î
}}}}}
P(k 5 5) 5 10C5(0.34) 5(1 2 0.34)10 2 5 ø 0.14
b. p 5 P (Rh2) 5 P (O2) 1 P (A2) 1 P (B2)1 P (AB2)
5 0.15
2
10 2 2
P(k 5 2) 5 10C2(0.15) (1 2 0.15)
c. p 5 P(O) 5 P (O
ø 0.28
) 1 P(O ) 5 0.43
1
2
P(k 5 0) 5 10C 0 (0.43) 0 (1 2 0.43)10 2 0 ø 0.004
P(k 5 1) 5 10C1(0.43)1(1 2 0.43)10 2 1 ø 0.027
(135 2 145.625)2 1 (142 2 145.625)2 1 . . . 1 (154 2 145.625)2
}}}}}
8
Î
}
519.875
5 }
ø 8.1
8
29. a. The outlier is 5.
b. With outlier:
20 1 23 1 . . . 1 23
Mean: }
x 5 }}
5 20.2
10
P(k 5 2) 5 10C2(0.43)2(1 2 0.43)10 2 2 ø 0.093
Median: 22 Mode: 23 Range: 25 2 5 5 20
P(k ≤ 2) 5 P(k 5 0) 1 P(k 5 1) 1 P(k 5 2)
Std. Dev.:
ø 0.124
(20 2 20.2)2 1 (23 2 20.2)2 1 . . . 1 (23 2 20.2)2
s 5 }}}}
10
ø 5.4
Worked-Out Solutions
n2pe-9080.indd WS19
WS19
10/13/05 11:41:08 AM
Without outlier:
19.4 2 20
0.25
20.4 2 20
20.4 ounces: z 5 }
5 1.6
0.25
33. a. 19.4 ounces: z 5 } 5 22.4
20 1 23 1 . . . 1 23
Mean: }
x 5 }}
ø 21.9
9
b. The table shows that P(x ≤ 22.4) 5 0.0082.
Median: 23 Mode: 23 Range: 25 2 19 5 6
So, the probability is 0.0082.
Std. Dev.:
Î
c. P(x ≤ 20.4) 5 0.9452; P(x ≤ 19.4) 5 0.0082
}}}}
s5
(20 2 20.2)2 1 (23 2 20.2)2 1 . . . 1 (23 2 20.2)2
}}}}
9
P(x ≤ 20.4) 2 P(x ≤ 19.4) 5 0.937
ø 2.1
WORKED-OUT SOLUTIONS
c. The outlier causes the mean and median
to decrease, and the range and standard
deviation to increase. The mode stays the
same.
Original
data set
1
Ïn
Adding 17 to
data values
60.056 5 6 }
}
Mean
78
78 1 17 5 95
77
77 1 17 5 94
Mode
77
77 1 17 5 94
Range
9
9
Standard
deviation
2.8
2.8
Mean
Original
data set
Multiplying
data values by 4
61.9
61.9(4) 5 247.6
Median
62
62(4) 5 248
Mode
58
58(4) 5 232
Range
9
9(4) 5 36
Standard
deviation
3.4
3.4(4) 5 13.6
19.
The margin of error is about 63.2%.
19. Margin of error 5 6 }
}
Ïn
Median
11.
1
1
7. Margin of error 5 6}
} 5 6}
} ø 60.032
Ïn
Ï1000
1
Lesson 11.2 (pp. 753–755)
5.
Lesson 11.4 (pp. 769–771)
Heights
without stilts
Heights
with stilts
Mean
70.8
70.8 1 28 5 98.8
Median
72
72 1 28 5 100
Mode
72
72 1 28 5 100
Range
8
8
Standard
deviation
2.4
2.4
1
0.003136 5 }
→ n ø 319
n
29. Sample Answer: It is reasonable to assume
that Kosta is going to win the election,
because the margin of error is 65%. If the
margin of error works in favor of Murdock
and against Kosta, Kosta will have 49%
(54% 2 5%) and Murdock will have 51%
(46% 1 5%).
Lesson 11.5 (pp. 778–780)
3. Model: y 5 20.38x 2 1 1.1x 1 16
11. A model for the data is y 5 0.55x 1 1.1
Chapter 12
Lesson 12.1 (pp. 798–800)
2
2
2
2
19. }
,}
,}
,}
,. . .
3p1 3p2 3p3 3p4
Lesson 11.3 (pp. 760–762)
2
11. 29 and 37 are one standard deviation on
either side of the mean, which accounts for
68% of the data. So, the probability is 0.68.
2
2
Next term: }
5}
; A rule is an 5 }
.
15
3n
3. P(x ≤ }
x 2 s) 5 0.0015 1 0.0235 1 0.135 5 0.16
3p5
4
47.
∑ n3 5 03 1 13 1 23 1 33 1 43
n50
5 0 1 1 1 8 1 27 1 64 5 100
WS20 Worked-Out Solutions
n2pe-9080.indd WS20
10/13/05 11:41:10 AM
65.
n
1
2
3
4
5
an
1
3
7
15
31
Lesson 12.4 (pp. 823–825)
`
13.
k51
a6 5 26 2 1 5 63 moves for 6 rings
1
a1 5 7; r 5 2}
;s5 }
5}
5}
9
12r
17
8
}
a
0.625
625
WORKED-OUT SOLUTIONS
n 5 345 1 345(0.783) 1 345(0.783)2
1 345(0.783)3 1 . . .
A rule for the nth term is
a
a20 5 2(20) 2 5 5 35
345
1
5}
5}
ø 1590 million
12r
1 2 0.783
an 5 a1 1 (n 2 1)d 5 23 1 (n 2 1)2 5 2n 2 5
Lesson 12.5 (pp. 830–833)
8
∑ (23 2 2i)
15. Geometric sequence; a1 5 4; r 5 23
i51
an 5 r p an 2 1 5 23an 2 1
a1 5 23 2 2(1) 5 25; a8 5 23 2 2(8) 5 219
25 1 (219)
2
A recursive rule is a1 5 4, an 5 23an 2 1.
2
s8 5 8 } 5 296
1
2
27. f (x) 5 } x 2 3, x0 5 2
65. a. a1 5 1; d 5 8
x1 5 f (x0)
an 5 a1 1 (n 2 1)d 5 1 1 (n 2 1)8 5 27 1 8n
3
2
}
3
19. Geometric sequence; a1 5 2; r 5 5 }
4
2
3
an 5 a1r n 2 1 5 2 1 }
2
1
1
5}
(2) 2 3
2
5}
(22) 2 3
2
5}
(24) 2 3
2
5 22
5 24
5 25
a1 5 2000, an 5 1.014an 2 1 2 100.
Because a24 5 62.14, the balance at the
beginning of the 24th month is $62.14. So,
she will be able to pay off the balance at the
end of the 24th month.
4
729
5}
5}
4096
2048
8
∑ 6(4)i 2 1
Chapter 13
i51
a1 5 6(4)1 2 1 5 6; r 5 4
Lesson 13.1 (pp. 856–858)
2 r8
2 48
s8 5 a1 1}
5 6 1 1}
2 5 131,070
1 12r2
5 f (24)
45. Recursive rule:
n21
1458
x3 5 f (x2)
5 f (22)
1
Lesson 12.3 (pp. 814–817)
}
x2 5 f (x1)
5 f (2)
b. a12 5 27 1 8(12) 5 89 blocks are visible
721
625(0.001)
39. B;
a1 5 23; d 5 21 2 (23) 5 2
4
63
1
5}
5}
5}
5}
12r
1 2 0.001
0.999
999
15. Arithmetic sequence
3
a7 5 2 1 }
2
7
27. 625(0.001) 1 625(0.001)2 1 625(0.001) 3 1 . . .
Lesson 12.2 (pp. 806–809)
1
a
8
1 2 1 29 2
a8 5 28 2 1 5 255 moves for 8 rings
49.
k21
A formula for the sequence is an 5 2n 2 1.
a7 5 27 2 1 5 127 moves for 7 rings
41.
∑ 71 2}89 2
5.
124
x
1
59. a. a1 5 1024 4 2 5 512; r 5 }
2
1
an 5 a1(r) n 2 1 5 512 1 }
2
u
Using the Pythagorean theorem:
11
}
}
x 5 Ï 112 2 82 5 Ï 57
8
n21
8
sin u 5 }
2
11
b. n 5 10; after 10 passes, the number of
10 2 1
1
items remaining is a10 5 512 1 }
2
2
5 1.
11
csc u 5 }
8
}
Ï 57
cos u 5 }
11
}
11Ï57
sec u 5 }
57
}
8Ï 57
tan u 5 }
57
}
Ï 57
cot u 5 }
8
Worked-Out Solutions
n2pe-9080.indd WS21
WS21
10/13/05 11:41:12 AM
opp
adj
7
3
37.
11. tan u 5 } 5 }
}
x
7
}
x 5 Ï 72 1 32 5 Ï 58
Stop
u
}
}
7Ï 58
sin u 5 }
58
}
WORKED-OUT SOLUTIONS
33.
250 ft
3
Start
3
cot u 5 }
7
d
tan 158 5 }
1500
15⬚
10 ft
Ground
u9 5 2708 2 2558 5 158
h
sin 158 5 }
75
d ø 402
1500 ft
19.4 ø h
The total depth is 402 1 250 5 652 feet.
When the ride stops, you are about 10 1 75
1 19.4 5 104.4 feet above the ground. If the
radius is doubled, your height above the
ground is doubled only if your starting height
above the ground is also doubled.
As the angle of the dive increases, the depth
increases.
Lesson 13.2 (pp. 862–865)
5 p radians
1808
5p
11. }
5}
} 5 508
18
1 π radians 2
18
x
75 ft
}
d 75⬚
2558
158
7
tan u 5 }
3
Ï58
sec u 5 }
3
15⬚
75 ft
h
3Ï 58
cos u 5 }
58
Ï58
csc u 5 }
7
y
Lesson 13.4 (pp. 878–880)
y
p
p
7. When 2} ≤ u ≤ }, or 2908 ≤ u ≤ 908, the
2
2
508
}
}
p
Ï3
Ï3
angle whose sine is }
is u 5 sin21 }
5}
,
3
x
2
2
}
Ï3
or u 5 sin21 }
5 608.
2
p radians
2p
23. 408 5 408 } 5 } radians
1
2
1808
9
23.
p
15 rev 1 min 2p rad
51. a. } }
} 5 } rad/sec
2
1 min 60 sec 1 rev
21
1
2
tan u 5 3.2; 1808 < u < 2708
y
u
tan21(3.2) ø 72.68, which is in
Quadrant I. To find the angle
in Quadrant III (1808 < u <
2708): u ø 180 1 72.6 5 252.68
72.68
x
p
b. Arc length: s 5 r u 5 29 1 } 2 ø 45.6 feet
2
Lesson 13.3 (pp. 870–872)
}
37.
}}
11
}
5. r 5 Ï x 2 1 y 2 5 Ï (27)2 1 (224)2 5 Ï 625 5 25
y
224
sin u 5 }r 5 }
25
y
224
r
225
24
tan u 5 }x 5 }
5}
27
7
sec u 5 }x 5 }
7
17.
27
r
225
x
7
csc u 5 }y 5 }
24
cot u 5 }y 5 }
24
u9 5 1808 2 1508 5 308
y
u 9 5 308
x
f
cos u 5 } 5 }
25
u 5 1508
tan u 5 }
17
u
11 ft
11
u 5 tan211 }
2 ø 338
17 ft
17
Because the angle of repose remains the
same, you can use u 5 338 to find the radius
of the 15 foot high pile:
15
338
15 ft
r
15
tan 338
tan 338 5 }
→r5}
r
ø 23
The diameter is about d 5 2r 5 2(23) 5 46 ft.
x
WS22 Worked-Out Solutions
n2pe-9080.indd WS22
10/13/05 11:41:14 AM
1
2
Lesson 13.5 (pp. 886–888)
45. nADB: s 5 }(743 1 1210 1 1480) 5 1716.5
sin 1048
B
13. sin
}5 }
Area 5
25
16
}}}}}
Ï1716.5(1716.5 2 743)(1716.5 2 1210)(1716.5 2 1480
16 sin 1048
sin B 5 }
ø 0.6210 → B ø 38.48
25
ø 447,399
A ø 1808 2 1048 2 38.48 5 37.68
25
sin 1048
1
25 sin 37.68
sin 1048
a
} 5 } → a 5 } ø 15.7
sin 37.68
nCDB: s 5 }
(1000 1 858 1 1480) 5 1669
2
Area 5
}}}}}
45. a.
Ï1669(1669 2 1000)(1669 2 858)(1669 2 1480)
B
ø 413,697
C
1 acre
Area 5 (447,399 1 413,697) ft 2 }
2
62 ft
1 43,560 ft 2
A
sin 588
sin C
b. } 5 }
62
ø 20 acres
54
If you first found the length of }
AC , you could
repeat the same process using nABC and
nADC.
54 sin 588
ø 0.7386 → C ø 47.68
sin C 5 }
62
A ø 1808 2 588 2 47.68 5 74.48
62
sin 588
62 sin 74.48
sin 588
a
} 5 } → a 5 } ø 70.4
sin 74.48
1
2
Chapter 14
WORKED-OUT SOLUTIONS
588
54 ft
Lesson 14.1 (pp. 912–914)
1
2
c. Area 5 } bc sin A 5 }(62)(54)(sin 70.48)
5. Amplitude: 1; period: 2
ø 1577
17. f (x) 5 4 tan x
1577 ft 2 4 200 ft 2/bag ø 7.9 bags
y
Period: π; intercept: (0, 0)
You will need 8 bags of fertilizer.
4
Lesson 13.6 (pp. 892–894)
x
␲
4
p
Asymptotes: x 5 6}
2
p
p
Halfway points: 1 }
, 4 2 1 2}
, 24 2
4
4
17. a2 5 b2 1 c 2 2 2bc cos A
31. a. Equation has the form y 5 a cos bt.
102 5 32 1 122 2 2(3)(12)cos A
1
a5}
(3.5) 5 1.75
2
53
72
} 5 cos A → A ø 438
2p
b
p
Equation is y 5 1.75 cos }
t.
3
B ø 128; C ø 180 2 438 2 128 5 1258
b. Choose y 5 a cos bt because at t 5 0, the
buoy is at its highest point.
In nABC, A ø 438, B ø 128, and C ø 1258.
1
2
1
2
Lesson 14.2 (pp. 919–922)
25. s 5 }(a 1 b 1 c) 5 }(5 1 11 1 10) 5 13
11. y 5 2 cos x 1 1
}}
Area 5 Ïs(s 2 a)(sb)(s 2 c)
}}}
}
5 Ï 13(13 2 5)(13 2 11)(13 2 10) 5 Ï 624
ø 25 square units
p
Period is 6, so 6 5 } → b 5 }
.
3
3(sin 438)
3
10
} 5 } → } 5 sin B
sin
B
10
sin 438
Amplitude: a 5 2
2p
Period: }
5 2p
b
y (0, 3)
(2p, 3)
␲
2
( , 1) (
2
␲
4
3␲
2,
)
1
x
(p , 21)
Horizontal shift: h 5 0; Vertical shift: k 5 1
Worked-Out Solutions
n2pe-9080.indd WS23
WS23
10/13/05 11:41:16 AM
1
23. y 5 2sin }
x13
y
2
Closest distance:
(3p, 4)
(0, 3) (2p, 3)
Amplitude: a 5 1
0.5 a.u. p }}
5 46.5 million mi
1 a.u.
(p, 2)
2p
2p
Period: }
5}
5 4π
1
b
93,000,000 mi
(4 p, 3)
2
Farthest distance:
x
␲
2
}
2
93,000,000 mi
35.6 a.u. p }}
ø 3.31 billion mi
1 a.u.
h 5 0; k 5 3; a < 0, so graph is reflected.
200 2 d
300
53. a. } 5 tan u
WORKED-OUT SOLUTIONS
d 5 2300 tan u 1 200
u
300 ft
d
b.
␲
,
4
(2
Lesson 14.4 (pp. 935–937)
d Your friend
200 ft
You
p
5. 12 sin 2 1 } 2 2 3 0 0
6
2
1
12 1 }
2 2300→32350✓
)
500
2
}
Ï3
3
13. 4 cos2 x 2 3 5 0 → cos2 x 5 }
→ cos x 5 6}
4
(0, 200)
2200
(
␲
,
4
2
u
3␲
8
)
5p
p
c. 100 5 2300 tan u 1 200 → u ø 18.48
x5}
1 2nπ or x 5 }
1 2nπ
6
6
1
5
5. cos u 5 }, 3π < u < 2π
6
1
b.
5 2
sin 2 u 1 1 }
51
62
}
Negative because u is
in Quadrant III
}
sin u
Ï
1
sin u
26
Ï 11
cos u
sin u
11
tan u 5 }
5 2}
cos u
5
5
Ï 11
X
119
120
121
122
123
124
X=122
1
6
sec u 5 }
5}
cos u
5
2sin u
sin(2u)
11. } 5 }
5 tan u
2cos u
cos(2u)
1.069
1.069
41. a. r 5 }}
5 }}
π
1 2 0.97 cos u
1 2 0.97 sin1 }
2 u2
2
sin u
2
A value of u ø 54.78
minimizes the surface
area.
c.
Minimum
X=54.735619 Y=7.9432427
Lesson 14.5 (pp. 944–947)
5. M 5 6, m 5 2
M1m
b.
2
When u ø 1228,
S 5 9 in.2
Y1
8.8886
8.9246
8.9619
9.0005
9.0405
9.0819
cot u 5 } 5 2}
}
csc u 5 } 5 }
}
}
3 2 cos u
5 6.75 1 0.84375 Ï
}
sin 2 u 1 cos2 u 5 1
Ï 11
}
3
Ï3 2 cos u
43. a. S 5 6(1.5)(0.75) 1 }(0.75)2 }
2
sin u
Lesson 14.3 (pp. 927–930)
sin u 5 2}
←
6
5p
p
In 0 ≤ x < 2π, x 5 }
and x 5 }
.
6
6
2100
612
Vertical shift: k 5 }
5}
54
2
2
The graph is a cosine curve with h 5 0.
2p
b
p
Period 5 4 5 } → b 5 }
2
c.
u
0
}
p
4
}
p
2
}
3p
4
π
r
35.6
3.4
1.1
0.6
0.5
u
}
5p
4
}
3p
2
}
7p
4
2π
r
0.6
1.1
3.4
35.6
M2m
622
5}
52
a 5 }
2
2
The graph is a reflection, so a 5 22.
p
x 1 4.
The function is y 5 22 cos }
2
WS24 Worked-Out Solutions
n2pe-9080.indd WS24
10/13/05 11:41:17 AM
9. M 5 6, m 5 26
43. a.
6 1 (26)
M1m
WQ
NA
Vertical shift: k 5 }
5}
50
2
2
f tan(u 2 t) 1 f tan t
h tan u
} 5 }}
f
h
The graph is a sine curve with h 5 0.
1
5 } (tan(u 2 t) 1 tan t)1 }
2
2p
b
1
Period 5 2(3π 2 π) 5 4π 5 } → b 5 }
2
tan u
f tan u 2 tan t
tan t(1 1 tan u tan t)
1
5 } }}
1 }}
1}
2
1 1 tan u tan t)
1
f
u 1 tan u tan2 t
1
5 } tan
}} 1 } 2
h
The graph is not a reflection, so a 5 6.
1
f tan u(1 1 tan2 t)
h 1 1 tan u tan t tan u
1
2
25. When t 5 0, m 5 4; when t 5 1, M 5 9.
2
f
h 1 1 tan u tan t
sec t
5 } }}
1
M1m
914
13
k5}
5}
5}
2
2
2
2p
b
t51
M2m
924
5
5}
5}
a 5 }
2
2
2
t50
5 ft
5
The graph is a reflection, so a 5 2}
.
2
Ground
p
1 p
7. cos } 5 cos } 1 } 2 5
8
2 4
Î
3
2
Î
p
1 1 cos }
4
}
2
Ï2
}
Î
3p
4
}
2
a
2
a
2
} < } < π → } is in Quadrant II.
2
Î
}
1
}
}
12}
Ï3
a
2 cos a
1
sin }
5 1}
5 }3 5 }
5}
3
2
2
2
3
Î
}
}
Î
Î
}
1 2
1
}
}
11}
Ï6
a
1 cos a
2
cos }
5 2 1}
5 2 }3 5 2 }
5 2}
3
2
2
2
3
Î
}
}
Î
}
3p
3p
3p
23. sin x 2 } 5 sin x cos } 2 cos x sin }
2
f
h
1 3p
13. cos a 5 }, } < a < 2π
Ï6 2 Ï2
5}
4
1
2
}
}
11}
Î2 1 Ï}2
2
1 Ï2
5 }
5 2}
5}
2
4
2
Ï3 Ï2
Ï2
1
5}
2}
1}
}
2
2
2 2
}
1
}
}
3p
3p
p
p
5 cos }
cos }
1 sin }
sin }
4
4
3
3
2
f
h 110
2
}
4 ft
Lesson 14.6 (pp. 952–954)
}
1
Lesson 14.7 (pp. 959–962)
5
13
A model is y 5 2}
cos π x 1 }
.
2
2
5p
3p
p
9. cos 2} 5 cos } 2 }
4
12
3
f
sec2(0)
h 1 1 tan u tan(0)
WQ
NA
1
} 5 } }} 5 } } 5 }
Period 5 2 5 } → b 5 π
1
2
b. When t 5 0:
The graph is a cosine curve with h 5 0.
1
WORKED-OUT SOLUTIONS
2
2
2 tan u
1 1 tan u tan t
1
5 } }} 1 }
2
1
The function is y 5 6 sin }
x.
1
2 tan u
h 1 1 tan u tan t
6 2 (26)
M2m
5}
56
a 5 }
2
2
2
2
5 (sin x)(0) 2 (cos x)(21) 5 cos x
Ï3
a
}
}
}
sin }
}
Ï3 23
2Ï3
3
2
2Ï 2
a
tan } 5 }a 5 }
} 5 } p }
} 5 }
} 5 }
3
2
2
Ï6
Ï6
Ï6
cos }
2}
2
3
1
1
u
53. When M 5 2.5: sin } 5 } 5 }
2.5
M
2
Using the Pythagorean Theorem:
}
Ï5.25
cos }u 5 }
2.5
2
}
Ï5.25
1
sin u 5 2 sin }u cos }u 5 2 1 }
ø 0.7332
2}
2
2
2.5
1
2.5
2
u ø 478
Worked-Out Solutions
n2pe-9080.indd WS25
WS25
10/13/05 11:41:19 AM
n2pe-9080.indd WS26
10/13/05 11:41:22 AM