Algebra 2

#sc7o.P(37n.C -5371.)!3H !5 =559.74!!.a,G.w3  h56i 1cbh. .cise6nt htcee.rp:7Ce(2r3m, 1=u)t;3a7v!t4!eio!r,tnicfeosr:m(2u,la6)f,or Lesson 11-3  pp. 688–693

Got It? 1. Independent; the number of coins is the same

after the coin is replaced. 2. 0.20, or 20% 3a. not

(2, - 4); co-vertices: (5, 1), (- 1, 1); foci: (2, 5), (2, - 3)  mutually exclusive; 2 is a prime number and an even

62. 1x - 12 2 + 1y - 12 2 = 36 is a circle (not an ellipse) number.  b. mutually exclusive; there is no even number Selected Answers

with center (1, 1) and radius 6. 63. 41x - 12 2  less than 2 in the roll of a number cube. 4a. 0.61, or

64. - 1x + 32 2 65. 3(x - 5)(x + 5) 66. 30,240  61% b. yes; the percentage of students tells which
59   5
67. 4  68. 210 language is chosen by more students. 5a. b. 9
5
Lesson Check 1. 115, or 6.6% 2. 8207, or 33.75% 
Lesson 11-2  pp. 681–687 3. 1, or 100%  4. 87, or 87.5% 5. 85, or 62.5% 

l4Gik.oe2lti,h5Iot94?o88 ,d916o.0f0go.e4r t00ti.no0gr040a00n%1e8v 4e26n.8o09r.2oo0rddo≈ris02.t00h%0e1 s8a35ma%.e, 12i .eb..21t. he 6. Events A and B are independent if the outcomes of A

do not affect the outcomes of B. The events are mutually

5. 0.05 or 5% #exclusive if A and B cannot occur at the same time. For

independent events, P(A and B) = P (A) P(B). For
mutually exclusive events, P(A and B) = 0. For any events,
Lesson Check 1. 0.75 or 75% 2. 0.80 or 80%  P(A or B) = P (A) + P(B) - P(A and B). 7. Since these are
61  4. 1
3. 3  5. Experimental probabilities are calculated on not mutually exclusive events, P(A and B) ≠ 0. The

the basis of data from an experiment, actual or student should have multiplied to get the correct answer,

simulated. Given equally likely outcomes, the basis for 123aEwrx579hee...icer0651hq8c5. u5ii2 ssa449el0 ,1.s.1t2. h23719ne ..on3o2t9ir1n5tmh2.d e1152eu%p 9stuue3..amn3mldl.yuies1net4uextv .ac5e1llul%ny1s ie. v2xde31c e5l.u4p.s34e3i8n v.7e2d49.;e34 in.f%4tt35 h 91.e3%3178n1.. u 162m144 557b..e12r8s

calculating theoretical probability is being able to

determine the no. of ways that an event can occur

within these outcomes. Comparisons of measures such

as length and area are the basis of geometric 49. 5  51. 161 05.024. 215 85.312.e216 ≈542.0371 .5715 . 5-923., 2 56. 1
probability. 6. Answers may vary. Samples: Flip a coin; 57. 21e3 ≈ 6

generate random numbers on a calculator; roll a die with
odd numbers as true and even numbers as false. 
7. Because you are averaging over more samples, you

are getting a more accurate average. { e2 ≈ { 7.39
21
Exercises 9. the number 1: 134 ≈ 15.7%; the number 60. 1  61. 1  62. 3
16 16 16
2: 11 ≈ 16.4%; the number 3: 45 ≈ 16.8%; the
67 268
11 47
number 4: 67 ≈ 16.4%; the number 5: 268 ≈ 17.5%; the Lesson 11-4  pp. 696–702

number 6: 23 ≈ 17.2% 11. Answers may vary. Sample: Got It?  1a. ≈ 0.57355 or ≈ 57.355%  b. female; there
134 are more females enrolled. 2a. ≈ 0.026448 or ≈ 2.64%
Toss 5 coins. Keep a tally of the times three or more heads b. ≈ 0.040302 or ≈ 4.03% 3. 0.2 4. 9%

are tossed. (A head represents a correct answer.) Do this Lesson Check 1. 12 2. 113, or 7.7% 3. 0% 4. 50% 
5. The sum of the probability of an event happening and
100 times. The total number of tally marks, as a percent, the probability of an event not happening is 1. Each branch
represents either the event happening or the event not
gives the experimental probability. The simulated happening. 7. Answers may vary. Sample: You can use
probability should be about 50%. 13. 130, or 30% both to determine the number of ways a sequence of
15. 45, or 80% 17. 14285, or 38.4%  events can occur. A tree diagram can represent all
#19. 110235, or 82.4%  outcomes for a given situation. If there are many branches,
21. 17275, or 61.6% 23. 30C3 120C6 ≈ 0.17879 ≈17.9% it may be easier to use the Fundamental Counting Principle
150C 9 to find the number of outcomes.
85, 34, 116 Exercises 9. 0.6 11. ≈ 0.085 13. ≈ 0.682 
25. or 62.5%  27. or 75% 29. 147 ≈ 78.9%  15. ≈ 0.709 17. ≈ 23%

31. 43 ≈ 29.3% 33. 1 chance in 2,869,685 or
147
≈ 0.00003485% 35. if there are any restrictions on the

last digit of a ZIP code

Selected Answers i 1111

19.   M 0.33 L M ϭ male beneficial.  13. 28 trials; about 1.14 correct answers per
0.67 R F ϭ female trial  15. Answers may vary. Sample answer: Yes, the
0.17
defensive driving course appears to be very effective and
0.83 0.1 L R ϭ right-handed
F 0.9 R L ϭ left-handed should be offered again. None of the drivers who took the

Selected Answers P(L 0 F) = 10%, P(M and R) ≈ 11.4%  course were involved in a major accident in the previous
year.  17. Answers may vary. Sample answer: Use a
21. 75% 23. P(S and W) 25. 23, or 66.67% 
27. 0.08, or 8% 29. 0.84 31. 0.16 graphing calculator to generate random integers from 1

33. T 0.8 I to 5. Let the integers 1, 2, 3, and 4 represent a made field
0.2 N
0.6 goal, and let 5 represent a missed field goal. Generate

0.4 0.2 I random integers in groups of 3 to simulate the attempts
R 0.8 N
in the next game. Perform the simulation at least 20 or 25

times and find the average number of field goals made.
19. A  21. C  23. 0.35  24. 0.52  25. 0.7  26. 18

T ϭ representative that completed training seminars Lesson 11-6  pp. 711–718
R ϭ representative that didn’t complete a training seminar
I ϭ representative with increased sales Got It? 1. mean: 5.25, median: 5, mode: 5 2a. yes; it is
Nϭ representative without increased sales unlikely that the water temperature of a lake would change
by 25 degrees.  b. no; 98 would represent the busiest night
P(I 0 N) = 0.2 of the week and it may relate to a weekly event. 3. Dauphin
Island: mean: 69.083, mode: 84, range: 33, Q1 = 58,
Lesson 11-5  pp. 703–709 median: 71, Q3 = 81, interquartile range: 23; Grand Isle:
mean: 73.416, modes: 61, 70, 77, 83, 85, range: 24,
Got It?  1. Answers may vary. Sample answer: No, it is Q1 = 64.5, median: 73.5, Q3 = 83, interquartile range:
18.5; The range and the interquartile range show the
not likely that both siblings have an equal chance of temperatures varying less at Grand Isle than at Dauphin
winning the race.  2. a. 1, 6, 8, 9, 3 b. Yes, each student Island. Also, the temperatures at Grand Isle are generally
higher. 4a. Use STAT PLOT, select a box-and-whisker plot.
has an equal chance of being selected for either team.  Enter data for the three remaining Gulf Coast sites. Enter the
3. Answers may vary. Sample answer: Roll the cube until window values. Draw the box-and-whisker plots. Use TRACE
on the plot to find quartiles Q1, Q2 and Q3.
you get a 6. Keep track of the results. Repeat several
Q1 Q2 Q3
times and take the average number of rolls needed. 
4. Answers may vary. Sample answer: No, almost as many 65 70 75 80 85 90
Q1 Q2 Q3
volunteers who received the placebo reported
45 50 55 60 65 70 75 80 85 90
improvement as received the drug. Fewer than half of Q1 Q2 Q3

those who received the drug reported improvement. 55 60 65 70 75 80 85 90

Lesson Check  1. about 0.89  2. about 0.86  b. yes, a box-and-whisker plot uses minimum and
3. Answers may vary. Sample answer: Flipping a coin to
maximum values, the median, and the first and third
decide who has to wash the dishes is a fair decision. Arm quartiles to display the variability in a data set. 5a. 79 
b. 98
wrestling to see who has to wash the dishes might be Lesson Check 1. outlier: 54; outlier included: mean:

unfair if one brother is stronger than the other.  22.8, median: 19.5, mode: 18; outlier not included: mean:
4. Answers may vary. Sample answer: He only conducted 19.3, median: 19, mode: 18 2. outlier: 40; outlier
included: mean: 92.6, median: 98, mode: 90; outlier not
1 trial of the simulation, which is not enough to arrive at included: mean: 99.25, median: 99, mode: 90 3. the

an accurate prediction. He should conduct the simulation mean because the sum of the data values is affected and

at least 25 times and find the average number of boxes
needed.  5. Answers may vary. Sample answer:

A simulation is an imitation or way of acting something

out. In a mathematical simulation, a probability model is

used to act out a situation that would be difficult or

impractical to actually perform.
Exercises  7. Answers may vary. Sample answer: This will

not result in a fair decision because the first person

chooses the second person and might favor someone over
someone else.  9. 01, 05, 16, 03, 08  11. a. about 0.82
b. about 0.35 c. Sample answer: Yes, a high percentage

of students who took the class passed the board exams

on their first attempt so the class appears to be

1112

the mean depends on the sum 4. 40%: 49 and below; the data set within the range of distribution. 4. Standard

80%: 58 and below 5. the mean; when data are deviation measures how widely spread the data values

somewhat sym, the best representation is the mean  are. If the data pts. are close to the mean, the standard

6. The error is in how to calculate the median. The median deviation is small; if the data pts. are far from the mean,

is the middle value or the 11th value which is 90. the standard deviation is large. The data pts. of Set B are Selected Answers

Exercises 7. mean: 112.3, median: 95, mode: none  closer to the mean of 70 than the data pts. of Sets A and C;

9. 9.8 11. Jacksonville: mean: 67.9916, mode: none, likewise, the data pts. of Set A are closer to 70 than the

range: 29.2, Q1 = 58.15, median: 68.4, Q3 = 78.6, data pts. of Set C. 5. The effect of an outlier on the
interquartile range: 20.45; Austin: mean: 68.583, mode:
standard deviation is to increase the standard deviation.
none, range: 36, Q1 = 56.85, median: 70.5, Exercises 7. x ≈ 15.1, s2 ≈ 12.4, s ≈ 3.5 
Q3 = 80.75, interquartile range: 23.9; the range and the 9. x = 43.8, s2 = 75.76, s ≈ 8.7 11. x ≈ 12320.00,
interquartile range show the temperatures varying less at s ≈ 273.71  13. 3 standard deviations 15. x = 53.8,
s ≈ 3.4; 1s: 7; 2s: 9; 3s: 10 17. Overall farm income
Jacksonville than at Austin. 

13.   Q1 Q2 Q3 increased slightly, but there was less variability among the

states in 2002. The income in 2001 clustered more tightly
around the mean. (2001: sx ≈ 2679, 2002: sx ≈ 2758)
95 100 105 110 115 120 125 130 135 140 145 150 155 21. Your first friend; one standard deviation encompasses

15. 5; 17 17. outlier: 381; outlier included: mean: all values within one standard deviation above and below

≈ 161.214, median: 158, mode: none; outlier not the mean. The graph shows that all values are within 3

included: mean: ≈ 144.308, median: 142, mode: none standard deviations of the mean.  23. a. no change to s
5
19.   Q1 Q2 Q3 b. s increases by a factor of 10  25. 13  27. 6

28.   Q1 Q2 Q3

80 82 84 86 88 90

21. 30th 23. 89 is at the 100th percentile, since 100% 25 30 35 40 45 50 55 60
of the values are less than or equal to 89. 25. The
29.   Q1 Q2 Q3
median; a few outliers can heavily influence the mean
without drastically affecting the median. 27. 83.9  20 25 30 35 40 45 50 55
29. Answers may vary. Sample: The range for women’s
30. center (2, - 1); radius 6 31. center (1, 1); radius 2 
shot put is greater than that for men’s. The men are more 1 - 13 34. 1 - 111 36. 1 1
32. 2  33. 6  35. - 9  37. 7
consistent, as indicated by the shorter box and whiskers.
Overall the men tend to throw farther.  31. G 

33. P(H 0 I) = 0.40, P(H and I) = 0.20 Lesson 11-8  pp. 725–730
P (H and I)
P (H 0 I) = P (I)
Got It? 1a. convenience sample; yes; since the location
0.40 = 0.20
P (I) is at the food court in the mall, the sample may over-
P (I) = 0.50 represent food court or fast food supporters.  b. Answers
may vary. Sample: population data for the US census 
34. 0.20 35. 0.56 36. yes; - 9 37. yes; 17 38. no  2. Controlled study; If other factors of the volunteers are
{ 141 43. 19
39. yes; 0 40. { 16 41. { 0.09 42. { 5 random, like age, gender, and overall health, are known,

Lesson 11-7  pp. 719–724 the results can be used to make a general conclusion.
3. Answers may vary. Sample: Use a systematic sample.
Got It? 1. x = 69.83, s2 = 115.1389, s = 10.7303 
2. x = 7.26, s = 3.316 3a. within 3 standard deviations Go to every fifth house in your neighborhood. State the
of the mean  b. FEMA can expect that the no. of
hurricanes for a 15-year period will fall within 3 standard first and last names of the governor and ask a household
deviations of the mean.
Lesson Check 1. x = 10, s2 = 19.8, s = 4.45  member to identify the named person. A possible
2. within 2 standard deviations of the mean 3. Measures
of central tendency are specific data pts. which give a unbiased survey question is, “Who is this person?”.
summary of the middle of the data set, whereas the Lesson Check 1a. convenience sample  b. Yes; since
measures of variation give a summary of the variation of
the location is near the exit of a history museum, the

sample may over represent people who enjoy learning
history and the results will have a bias. 2. Yes; the

question is leading and loaded. It suggests the person
wants a particular answer. 3. All members of the set are

the population. A sample is a subset of the population.

Selected Answers i 1113

Answers may vary. Sample: population: students in a high 5. 0.2646, or 26.46% 6. Answers may vary. Sample: A

school; sample: students who like to snowboard  binomial experiment has three important features:  a. The

4. It is important to have as little error as poss. in a situation involves repeated trials; flipping a coin 10 times

sample, thus giving an unbiased sample. An unbiased has 10 trials.  b. Each trial has two possible outcomes; in

sample is more representative of an entire population.  this case, heads or tails.  c. The probability of success is

Selected Answers 5. A large sample size would give a better estimate. The constant throughout the trials; the trials of flipping a coin,

size of the sample is important to the reliability of the are independent. 7. The student wrote “5” instead of “4”.

sample. It should be: nC(5-1)an-4b4 = 7C4 j31 - k24 = 35j3k4

Exercises 7. systematic sampling; no bias 9. Survey; Exercises 9. ≈ 0.1361, or ≈ 13.61% 

the statistics can be used to make a general conclusion 11. ≈ 0.0015, or ≈ 0.15% 
13. a4 + 4a3b + 6a2b2 + 4ab3 + b4 
about the population because the sample is randomly 15. 243x5 + 810x4y + 1080x3y2 + 720x2y3 +
240xy4 + 32y5 17. 896g6h 19. e6 
generated, and the survey question does not introduce

a bias into the study. 11. Controlled experiment; the

statistics from this study can be used to make a general 21. P(0) ≈ 0.1176, P (1) ≈ 0.3025, P (2) ≈ 0.3241,

conclusion about the effectiveness of the plant food for P(3) ≈ 0.1852, P (4) ≈ 0.0595, P (5) ≈ 0.0102,

this particular plant type as compared with giving no plant P(6) ≈ 0.0007 

food at all.  13. Answers may vary. Sample: convenience 23. P(0) ≈ 0.000001, P (1) ≈ 0.000054, P (2) ≈ 0.0012,

sampling; interview students at a local high school.  P(3) ≈ 0.0146, P (4) ≈ 0.0984, P (5) ≈ 0.3543,

15. Answers may vary. Sample: self-selected sampling; a P(6) ≈ 0.5314 

newspaper article invites females over the age of 21 to 25. 0.99328 27. ≈ 0.2824 29. ≈ 0.1109 

call the paper and express their opinions.  31. ≈ 0.2461 33. ≈ 0.6230 35a. 0.0914  b. The

17. self-selected sampling; biased because only those who probability that three boxes would be underweight is

spend time online will respond. 19. a. all students at the 0.0001. You can conclude that there might be a

school b. every tenth student who enters the school malfunction in the machinery or that the company’s

building the day of the survey c. Answers may vary. claim may be false. 37. The probability of a group of

Sample: A little over half of students favor the new dress 30 students having 4 or fewer left-handed students is

code.  about 77.05%. This means that more than three quarters

21. Answers will vary. Sample: No, because you would of the classes will have enough left-handed desks; 4 is an

have to assume that all registered voters will actually vote adequate no. 

on Election Day.  23. a. convenience sample  39a. P(0) = 0.001, P (1) = 0.027,

b. observational study  c. Answers may vary. Sample: P(2) = 0.243, P (3) = 0.729

The statistics do not necessarily represent the school 100%
90%
population because a random sample was not used to 80%
70%
conduct the study.  25. Yes, the question is leading the Probability 60%
50%
respondent to a particular desired answer, and it gives 40%
30%
statistics that may elicit a strong reaction. Also, it requires 20%
10%
the respondent to answer a question about whether a 0%

person should wear a safety belt, which may not

necessarily influence whether they support the law. 
27. G  29. x ≈ 2.83, s ≈ 2.54 30. x ≈ 5.62, s ≈ 3.67
0123
12{(x5- 595)x; ;yneso  3324.. Successes
31. y = y= { 1x; no 
33. y = y= x92, x Ú 0; yes 35. 6  b. P(0) = 0.166375, P (1) = 0.408375,
P(2) = 0.334125, P (3) = 0.091125
36. 1 37. 10 38. 792
100%
Lesson 11-9  pp. 731–738 Probability90%
80%
Got It?  70%
1. P(0) = 0.07776; P (1) = 0.2592; P (2) = 0.3456; 60%
50%
P(3) = 0.2304; P (5) = 0.01024  40%
2. 81x4 + 108x3y + 54x2y2 + 12xy3 + y4  30%
3. ≈ 0.1035, or about 10.4% 20%
Lesson Check  1. ≈ 0.3110, or ≈ 31.10%  10%
2. ≈ 0.1641, or ≈ 16.41% 3. 20c3d3 4. - 10x4y  0% 0 1 2 3

Successes

1114

c. The probabilities of each graph sum to 1; 3. 47.5%  4. Normal distribution means that most of the

P(0) + P(1) + P(2) + P (3) = 1. The probabilities of part examples in a data set are close to the mean; the

(a) increase with increasing success numbers; the distribution of the data is within 1, 2, or 3 standard

maximum probability occurring at P(3). The probabilities deviations of the mean. 5. The mean and median are

of part (b) peak with a maximum at P(1) and then equivalent in a normal distribution. 6. mean increases by Selected Answers

decrease with increasing success numbers. 41. Answers 10: the bell curve is translated 10 units to the rt.; standard

may vary.  deviation and shape of the distribution do not change

43. a. The graph is sym. about the line x = 3.5. since each data value increases by the same amount.
Exercises 7. ≈ 43% 9. ≈ 43 men 
b. x y 11.   13.  

0 0.0078

1 0.0547 30 35 40 45 50 55 60 39 41 43 45 47 49 51

2 0.1641 15. 68% 17. 50% 19a. set 2 
b. and c.
3 0.2734  

4 0.2734

5 0.1641

6 0.0547

7 0.0078 4 5 6 7 8 9 1011

c. No; the bulge in the graph has shifted rt. 21. 59 min 23. 47.5% 25. 81.5% 27. 84% 

Lesson 11-10  pp. 739–745 29a. 500
450
Got It? 1a. 71%  b. 88% Tornadoes 400
350
2.   Distribution of Female 300
European Eels 250
200
Ϫ3 Ϫ1 mean ϩ1 ϩ3 150
deviations 100
deviations deviation deviation
50
Frequency of Measure Ϫ2 ϩ2 00 1 2 3 4 5 6 7 8 9 10 11 12
deviations deviations
Month

4.7 4.7 4.7 4.7 4.7 4.7 b. No; the curve is skewed to the left. c. No, mean and

7.0 11.7 16.4 21.1 25.8 30.5 35.2 standard deviation are appropriate only for measuring
Length (inches) normally distributed data. 31. Yes; Elena scored within

3a. 2.5%  b. 210 students  c. The students that received the top 10% of her group. Her score is 2.75 standard

a B had scores between 165 and 180. deviations above the mean, which places her in the top

Lesson Check 1. 94%  1%. Jake did not score in the top 10%. His score is 1.16

2.   Ϫ3 Ϫ1 mean ϩ1 ϩ3 standard deviations above the mean, or at the 88th
percentile. 33. A binomial distribution has a finite no. of
deviations deviation deviation deviations
probabilities, which sum to 1 and are a subset of a larger
Frequency of Measure Ϫ2 ϩ2 normal distribution. For example, using n = 6, p = 0.5,
deviations deviations the binomial distribution probabilities are
P(0) ≈ 0.0156, P (1) ≈ 0.0938, P(2) ≈ 0.2344,
P(3) ≈ 0.3125, P(4) ≈ 0.2344, P (5) ≈ 0.0938,
P(6) ≈ 0.0156.

0.3
Probability
0.2

34% 34% 0.1

15 15 15 15 15 15 0123456
2.35% 13.5% 13.5% 2.35% Successes

135 150 165 180 195 210 225

Selected Answers i 1115

35. I  37. For Distribution A with 50 data values, median: 6, mode: 9 34. mean: 10.6, median: 7, modes:
3 and 7 35. mean: 15, median: 15, mode: 18 36. mean:
25 values are at or below 40, which is the mean. For 9.5, median: 9.5, mode: none 37. range: 35;
Q1 = 30; Q3 = 55 38. range: 35; Q1 = 25; Q3 = 50 
Distribution B with 30 data values, 15 values are at or 39. range: 65; Q1 = 42; Q3 = 87  40. heights of 3
people  41. ages of thirty college students  42. gas
below the mean 40. So Distribution A has more values at mileage of 18 automobiles of various types 
43. x ≈ 6.64, s ≈ 5.12  44. x ≈ 17.14, s ≈ 3.52 
or below 40.  38. 0.02867 39. 0.1612 40. 0.03676 45. x = 7.5, s ≈ 2.67 46. not a random sample; they
will all begin with the letter “a” 47. not a random sample;
Selected Answers 41. 10 y c ircle; center: (0, 0), radius: 8;
the lawyers will choose jurors that are likely to support their
Ϫ10 O x lines of sym.: all lines through side 48. random sample; all students have an equal chance
10 the center; domain: - 8 … x … 8, to be chosen 49. not a random sample; the five with the
range: - 8 … y … 8  largest (or smallest) circulation size will be picked 50. People
Ϫ10
at the bus station may be less likely to own a car and therefore
42. y hyperbola; center: (0, 0), less likely to be in favor of a new garage.  51. 12 52. 13 
foci: ({ 3 12, 0); lines of 53. ≈ 0.14 54. ≈ 0.0710 55. ≈ 0.2066  56. ≈ 0.1766 
4 sym.: x = 0, y = 0; 57. 21a5b2 58. 56a3b5  59. continuous  60. discrete 
domain: x … - 3 or x Ú 3; 61. discrete 62. continuous  63. 16%; 2.5%
Ϫ8 Ϫ4 O 4 x range: all real numbers
Ϫ4 8

43. 4y e llipse; center: (0, 0), foci: ({ 4, 0);
O
x lines of sym.: x = 0, y = 0;
domain: - 5 … x … 5,
Ϫ4 range: - 3 … y … 3
Chapter 12

44. y = x - 3; 45. y = x; Get Ready! p. 761
  2y    y
( )1.
O2 x 2 6 2. 31  3. - 38 4. 7  5. - 9  6. 11 7. 3 8. 21, - 4
Ϫ2 4 x 72
( ) ( )9.
Ϫ2 O 2 - 23, - 2  10. 34, 11  11. (7, 9, - 6) 12. (0, 0, 8)
3 4
Ϫ2 13. (7, 5, 0)

46. y = x - 45; Lesson 12-1 pp. 764–770

y x Got It? - 9 23
2 4 - 15 25

O2 1a. £ - 1 1 §   b. £ - 5 9 §

-2 15 05
c. yes; it does not matter in which order you add matrices.

Chapter Review  pp. 751–756 3 6 -1 0 0 d   b. c -1 10 - 5 d
3 7 §  3a. c0 0 0 2 - 3
1. sample 2. outlier 3. probability distribution 4. range 2. A = £ 1 1
1 3
of a set of data 5. 6 6. 362,880 7. 12 8. 30 9. 21  4a. x = 6, y = - 6  b. x = 4, y = - 3, z = 2

10. 10 11. 30 12. 744 13. 220; 84; 20; 1  Lesson Check
74570z e1r7o.ti0m e1s8, .o25ne
14. 3.315312 * 109 15. 216 16. 1. c 1 1 2. 1 -9 8 3. c -3 4 4. 6 10
19. Not necessarily; you may pick a -2 8d c -3 -1 8d -5 11 d c 13 -4d

time, or more than once. Each time you pick, the prob. 5. Yes; the elements in each of the corresponding
a05.7i9s 22104. .200.3.  d2e5p.en0d.7e n2t 62. 114. i2n7d.ep51e n2d8e.n18t
that it will be positions are equal.
22. 0.21 23.
6. The elements were not subtracted. The correct answer is
29. This will not necessarily result in a fair decision, 6 3 3
c5d - c7d = c -2d
because the principal may aim at a particular name, which

means that not all students have an equally likely chance Exercises -8
-1§
of being chosen.  30. Yes, this will result in a fair decision, 6 5 4 3.9 - 2.3 4
c2 -1 7 c - 0.6 9.1 1
because the probabilities of each goalie being chosen are 7. d  9. d  11. £ -1
11
the same.  31. Answers may vary. 32. 9 33. mean: 6,

1116

13. c 6 2 d  15. 2 -3 4 d   matrices A and B exists only if the number of columns of
-1 3 c5 6 7
- A is equal to the number of rows of B. Since A is a 2 * 4
matrix with 4 columns and B is a 3 * 6 matrix with 3
0 5 rows and 4 ≠ 3, the product AB does not exist. Likewise,
6 3   since 6 ≠ 2, the product BA does not exist.
17. x = - 2, y = 3, z = 1 19. £ 8 - 6 §  21. c -3 3 d
0 5
Exercises Selected Answers

952 760 9 12 c - 3 - 6 d  11. 9 2
7. £ 18 9 - 3 £2 6§
-4 1 720 832 - 6 §  9. - 10
23. c -3 1 d  25a. E 1108 U ; E 1252 U 3 0 3

- 1172 1144 13. 19 11 d  15. c 8 - 2.5 d  17. c -4 8
c -12 10 - 1.5 -1 - 22 2d
1044 1064
5 - 12 - 8 0
b. Allen: 4996; Iagorashvili: 5052  19. c9 -6 d  21. c 0 -8d  23. [34 0]

27. Matrix B would have the same dimensions as A. Its - 15 0 - 2
25 0 5
elements would be the opposites of the corresponding 25. c d  27. c d  29. yes 31. yes 33. yes

elements in A. 35a. River’s Edge: 99 pts.; West River: 97 pts.  b. West River
29. c = 52, d = 25, f = 7, g = 5, h = - 1 
cCwoynszxidde.rBayntyhtewdoef2in*iti2onmoaftrmicaetsr,ixAad=dcitacionbdadnadntdhe 9 -6 17 -24 34 -1
31.
B= 37. £ 15 - 3 §  39. £ - 33 - 7 §  41. £ 6 - 13 §

-6 -12 69 -18 -7 16

Comm. Prop. of Add. -90 0

a b w x a + w b + x 43. £ - 78 42 §  45. yes  47. yes
cc dd cy zd cc + y d + z
A + B = + = d -30 -30

= w + a x + b = w x + a b Lesson 12-3 pp. 782–790
cy + c z + dd cy zd cc dd
Got It?  1a. yes  b. yes  c. no; no matrix that is
=B+A multiplied by the zero matrix will give an identity matrix.
2a. 3  b. 0  c. - 48 3a. 12 units2  b. 28 units2
33. B 35. B 37. 68%  38. 97.5% 39. 47.5% 
621 454d1 4. 523,. 21,
40. 2, - 2 42. - -6  43. 5, 0  4a. yes; 1 - 1   b. no  c. yes; 3 - 4 d
44. 9 - 2 c -5 7
c6 c 20 d c - 3 d
35 2
5a. 88, 68, 84, 60, 12, 32, 52, 72, 28, 30, 14, 18, 2, 8,

Lesson 12-2 pp. 772–779 14, 20  b. Multiply the coded information by the inverse

Got It? of the coding matrix: 4 1 7 3 1 2 3 4
c9 8 7 6 1 3 5 7d
8 24 - 19 5 - 1 - 6 0
1. c -3 9 10 d  2. J7 0 R  3a. c - 9 11 d Lesson Check  1. 16 2. 7 3. does not exist

3 4. yes; c 3 - 2 d
-7 5
  b. c - 3 7 d   c. No; explanations may vary. Sample: For
6 8 5. The student did not subtract correctly.
2 5
the matrices in parts (a) and (b), AB = c -6 0 and det c -3 1d = (2)(1) - (- 3)(5) = 2 - ( - 15)
-9 11 d = 2 + 15 = 17
-3 7
BA = c 6 8 d , so AB ≠ BA. 4. player from 1994: 6. A 2 * 3 matrix does not have a multiplicative inverse

100 pts., player from 2006: 81 pts. 5a. no  b. yes  because it is not a square matrix. The number of rows

c. yes  d. no  e. yes must equal the number of columns for a multiplicative

Lesson Check inverse to be possible. 0 15. - 11
Exercises 7. yes 9. yes 11. no 13. 6 27. 40
6 - 2 -3 11 5 7 9 - 1 466,250 mi2
1. c4 0 d  2. c - 10 6 d  3. c2 6 d  4. c -2 2 d 17. 11 19. - 6 21. - 5 23. 106 25.
1
- 1 3 0 2
5. scalar; repeated matrix addition is repeated addition of 29. yes; c 1 2 d  31. yes; £1 1 §
- - 6
each element of the matrix, which is the same as scalar 3

multiplication of the matrix. 6. The product of two

Selected Answers i 1117

- 1 - 1 0 1 eq. should be written as c 2 3x = c 5 d .
8 2 1 3 -4 5d cyd 1
33. yes; £ 3 1 §  35. yes; £ 2 1 §
-
16 4 6 6. Use matrix multiplication to combine the coefficient

37. 2, 10, 10, 6, 9, 55, 15, 15, 9, 20 39. - 120 41. 9 matrix and the variable matrix into a product matrix. Then

Selected Answers 43. - 3 45. 1 47. Answers may vary. Sample: Form a new set the first element in the product matrix equal to the first

matrix by switching the element in row 1, column 1 with element in the constant matrix and set the second element

the element in row 2, column 2. Then replace the other two in the product matrix equal to the second element in the

elements with their opposites. Finally, divide each element constant matrix. The result will be a system of equations.;
by the determinant of the original matrix.  49. 38 units2
-2p + 3q = 2
4p + q = -5
51. yes; c - 5 7 d  53. yes; 0.5 0 Exercises
3 4 c0 0.5 d
- 29

0.4 0.4 0.2 31

55. yes; £ - 0.6 - 0.6 0.2 §   7. c - 15 - 17 d  9. E - 66 U  11. 1 1x = c 5 d ;
26 29 217 c1 -2d cyd 4
- 0.2 0.8 0.4 -
57. no inverse because the determinant equals zero  34
217
ae + bg af + bh
59. 6  61. MN = c ce + dg cf + dh d coefficient matrix: 1 1 d , variable matrix: c x d ,
c1 2 y
det MN = (ae + bg)(cf + dh) - (af + bh)(ce + dg) -

= acef + adeh + bcfg + bdgh constant matrix: c - 5 d  13. 3 5a = c 0 d ; coefficient
4 c1 1d cbd 2
- acef - adfg - bceh - bdgh
3 5 a 0
# = adeh + bcfg - adfg - bceh matrix: c1 1 d , variable matrix: c b d , constant matrix: c2d

Also, det M det N = (ad - bc)(eh - fg) 1 -1 1 r 150
# = adeh - adfg - bceh + bcfg.
15. £ 2 0 1 § £ s § = £ 425 § ; coefficient
So, det M det N = det MN 
2 5 - 10 19 0 13 t 0
63. 1   65. 15  67. c1 1d 68. c - 20 7d 69. 720 
2 1 -1 1 r

70. 362,880  71. 1.08972864 * 1010 72. 110,880 matrix: £ 2 0 1 § , variable matrix: £ s § ,
73. no solution 74. (6, 0, - 3) 75. (3, - 3, 9) 
76. (- 2, - 1, - 3) 0 13 t

150

Lesson 12-4 pp. 792–800 constant matrix: £ 425 §

Got It? 0
( )17. (2, 1) 19. 12, 20  21. (3, 2) 23. (2, -1, 3)
1a. c - 8 d   b. c - 14 - 20 d
9 19 28 25. (1, 2, - 2) 27. 2.5 lb of almonds, 3.5 lb of peanuts,
and 3 lb of raisins 29. (- 2, - 1) 31. (- 1, 0) 33. (5, 0, 1)
c. Since matrix A has no inverse, the eq. has no solution. 35. (1, 0, 3) 37. (1, 1, 1, 1) 39. (6, 2) 41. (16, - 22)

2a. 3 - 7 d c x d = 8 43. (5.4, 7.4) 45. (6, 1) 47. (2, - 1, 4) 49. length = 280 ft,
c5 1 y c -2d
10
1 3 5x 12 width = 140 ft 51. c - 3 2 d   53. £ 3 §   55. 14 
- 5 8
b. £ - 2 1 - 4 § £ y § = £ - 2 § -2

7 -2 0 z 7 57. Answers may vary. Sample: y + z = 0; y + z = 1 

c. c 2 - 8 d c x d = c - 3 d 59. B  61. A
-1 1 y - 4
Lesson 12-5 pp. 801–808
3a. (5, - 21)  b. no solution 4. run: 32 min; jog: 8 min
Lesson Check Got It?  1a. Subtract 8 from each x-coordinate and add
2 3 0x 12
- 6 3x 8 £1 5 to each y-coordinate.
4 -2d cyd 10 -2
1. c = c d  2. 0 6 1§ £y§ = £ 9§ b. 0 -1 -5 1 4 + c - 3 -3 -3 -3 - 3 d
c -5 -1 0 3 0d 2 2 2 2 2
-4 z 8

3. (5, 3) 4. (- 6, - 6) 5. The student did not separate the = c -3 -4 -8 -2 1 d ; ( - 3, - 3), (- 4, 1), ( - 8, 2),
coefficient matrix and the variable matrix. The matrix -3 1 2 5 2

1118

(- 2, 5), (1, 2) y Exercises 7. (- 2, 2), (- 2, 6), (2, 6), (2, 2);

x BЈ y CЈ
8
Ϫ8 Ϫ4 O 4 4
Ϫ4
AЈ B DЈ Selected Answers
2. Answers may vary. Samples: 24 C
Ϫ2 O x
a. 0 5 5 0 d   Ϫ2
c0 0 3 3 A 6

D

b. 2c 0 5 5 0 = 0 10 10 0 d ; 9. (- 13, 7), (- 19, 6), (9, 0);
0 0 3 3d c0 0 6 6
(0, 0), (10, 0), (10, 6), (0, 6) KЈ JЈ 8y

c. 4 KJ 4
Ϫ16 Ϫ12
3a. (0, 3), (4, 4), (1, - 1); b. (- 3, 0), (- 4, 4), (1, 1); O LЈ x
4y 4y 4 8 12

Ϫ4 L

1 x 1 x 11. (0, 0), (4, 8), (10, 10), (16, 2) 
Ϫ4 Ϫ2 O 24 Ϫ4 Ϫ2 O 24 13. (- 12, 9), (3, 6),(4.5, 0), (1.5, - 6), (- 3, 0)

15. y

4a. (1, - 1), (3, - 1), (6, - 4), (1, - 3); 4

6y

2 x Ϫ2 O x
Ϫ4 O 6 Ϫ2 4

Ϫ4

b. (1, 1), (1, 3), (4, 6), (3, 1); 17. (- 3, 3), (3, - 6), (3, - 3), (- 3, - 6)
19. (- 1, - 3), (- 2, - 2), (- 3, - 2), (- 4, - 3), (- 2.5, - 5)
6y 21. y

Ϫ4 O x 4
Ϫ4 48

( ) ( )Lesson Check  1. A′ 21, 1 1
2 , B′(1, 2), C ′ 2, - 2   O x
Ϫ4 4
2. A′(1, - 1), B′(4, - 2), C ′(- 1, - 4) 3. A′(1, 1),
B′(4, 2), C′(- 1, 4) 4. Translations, rotations and
reflections leave the size of the figure unchanged. 23. (3, - 3), (- 3, - 6), (- 3, - 3), (3, - 6)
25. (3, 1), (2, 2), (2, 3), (3, 4), (5, 2.5)
Translation moves the figure to a new location. Rotation
- 8 -5 -2 -5 - 5 -1 -3
turns the figure about a fixed pt. Reflection maps a figure 27. c - 8 -5 -8 - 11 d  29. c - 2 -1 1 d

in the coordinate plane to its mirror image using a specific

line as its mirror. 5. Answers may vary. Sample: reflection 31. 3 -1 1  33. Apply a 90° rotation matrix
c -1 -2 -4d
across y = x; translation 4 units to the right and 4 units
down
0 - 1
6. They are equal. c1 0 d three times to determine the image matrix for

c - 3 01 -2 4 = - 3 c 1 01 -2 4 each frame. The fourth application of the 90° rotation matrix
0 -3d c1 -1 2d 0 1d c1 -1 2d
would show the gymnast at the starting position of Frame 1.
1 -2 4
= - 3 c 1 -1 2d 37. f: c -5 -2 1 d , g: c -1 2 5 translation
3 0 3 1 -2 1d

Selected Answers i 1119

39. c - 1.5 0.25 - 2.5 d 49. yes; Assoc. Prop. of Add.
0 1.5 1.5
y
Selected Answers 41. Answers may vary. Sample: The reflection of a matrix vw
of pts. from a function table across the line y = x
interchanges the values of y and x in the function table. v؉w

Finding the inverse of the matrix of pts. of a function from u u؉v

a function table also results in the interchanging of the u ؉ (v ؉ w) ‫( ؍‬u ؉ v) ؉ w
values of y and x.  x

43. H  45. G  47. (3, - 2) 48. (1, - 1, 2) 49. (0, 1, - 2) O

50. c 16 d  51. [12] 52. [22] 51. a and d are parallel; a and b are perpendicular; b and
12 d are perpendicular 

Lesson 12-6 pp. 809–815

Got It?  1. u = 83, 4 9 ; v = 8 - 1, - 6 9  2a. 85, 3 9 Chapter Review pp. 817–820
b. translation, reflection and dilation 3. 174 ≈ 8.60
4a. b. 4 y 1. equal matrices 2. zero matrix 3. matrix equation
y 4. square matrix

u2 u2 - 1 9 - 8 5 -4
4 0 6 c5 0
x x 5. c d  6. d  7. [1 -8 12]
24
Ϫ2 O 2 0.5u 8. c - 3 10 d  9. x = - 2, w = 8, r = 4, t = -1
Ϫ2 ؊u Ϫ4 Ϫ2 O - 3 3

5a. not normal  b. normal 10. t = - 4, y = 121, r = 4, w = 4

Lesson Check  1. 85, 2 9  2. 89, - 11 9  3. 8 - 1, - 12 9 11. 18 3 0 24  12. undefined 13. undefined
4. 80, - 15 9  5. The magnitudes of vectors a, b, and c are c -12 9 21 33 d

the same: 5. 6. Although the x-component of 88, 3 9 is 4 -6 - -
- 28 -
times the x-component of 82, 1 9, the y-component and 14. c 10 21 41 d  15. c 14 2 d
10 28 28 43 7
the magnitude of 88, 3 9 are not 4 times those of 82, 1 9,
3 ≠ 4 * 1 and 173 ≠ 4 * 15.
Exercises 7. 84, 1 9 9. 84, - 2 9 11. 80, 2 9 13. 8 - 1, 5 9 - 11 18 1 - 1
16. c - 17 -2d  17. 24; C6 24 S  
15. 82, 0 9  17. 83, 0 9  19. 83, - 4 9  21. 84, - 1 9
0 1
4
23. 8 - 3, 8 9  25. 8 - 8, 20 9  27. 8 - 6, - 12 9  29. not
3nCRo7Ar.m=aab8lo -u31t1,3. -0n44or9mm i4a>lh1  .33389.6.,8A-0R,B3-9= 1448339,. 13895-,.4B8,RC1-8=2, - 13 9
8 - 2, 3 9 , 18. 0; does not exist
9
1 2 0
45. v - v = 80, 0 9 ; 80, 0 9 , the zero vector, is the 5 1 3 - 3
additive identity for the set of all vectors and - v is the - 42 1
19. 42; £ 42 5 §  20. 6; D - 6 1 - 1 T
additive inverse of any given vector v; so v + 80, 0 9 = v 4 3 2
- 21 21 1 1
and v + (- v) = 80, 0 9 . 0
47. yes; Distributive Prop. 3 3

y 21. 1 2 d  22. ( - 4, - 7) 23. c 2 d  24. 2 1
c -1 0 2 c3 2d

25. no unique solution 26. no unique solution

kv 27. 0 -5 - 2 d  28. c - 3 2 - 1 d  29. 1 0 5
c5 4 9 1 0 5 c3 -2 1d
ku v
k(u ؉ v) ‫ ؍‬ku ؉ kv 30. c 1.5 -1 0.5 d  31. c 6 -4 2 d  32. c 1 0 5
u x 0.5 0 2.5 2 0 10 -3 2 -1d
O
33. 8 - 1, 8 9 ; about 8.1 34. 87, - 5 9 ; about 8.6
35. 8 - 9, 12 9 ; 15 36. 8 - 2, 14 9 ; about 14.1 
37. 8 - 8, 14 9 ; about 16.1  38. 8 - 4, 21 9 ; about 21.4 
39. 0; normal 40. 0; normal

1120

Chapter 13 19. y

Get Ready! p. 825 8

1. vert. asymptote: x = 3 2. vert. asymptotes: 4 Selected Answers
1e71xx3.2=p..cli-c1+-9ixt2124o-1a r6n81a,d.21-12xx=63+= 4193146.y; , 13a+a75n.n4 =2=a12b20-a n.4=n-6.311, 5161e-85;x1 3p.5,li4.cr,ei2t1c(cu;3r+asnidve=)  61.9 1c6   x
- 3n,
O
yx
1 unit on the x-axis is 0.005 s.
21. a. y b. x  23. repeating of a pattern at regular
intervals  25. a. 1 s  b. 1.5 mV
27. 3, - 3, 4;
4 y

O6 O 48 x
Ϫ4
Ϫ4

14. 1x - 222 + 1y - 522 = 1; 29. 4, - 4, 8;
4 y
y 9 4 x
O 48
2x Ϫ4

O4 - 122 - 622 31. 2 weeks  33. 1 hr  35. a. 67  b. 70  c. 70  d. 67

15. y = 1 x2 - 3; 16. 1y 9 - 1x 16 = 1; #37. C  39. B  41. 64 s; The first two functions are at
32
1  y x   the beginning of their cycles together every 6 7 = 42
y seconds: 42, 84, 126, . ​. ​. The third function is at the
12 O 4 12 beginning of its cycle every 8 seconds, starting at (42 + 20)
4 seconds: 62, 70, 78, 86, 94, 102, 110, 118, 126, . ​. ​. ​

2 O 4 x The three functions are all at the beginning of their cycles
8 12
at 126 seconds, which is 64 seconds after the third
4
function achieves its first maximum.
17. Answers may vary. Sample: Similar data tends to recur
after a certain period has lapsed. In this case, 12 months. Geometry Review p. 835

Lesson 13-1 pp. 828–834 13.11322ini.n .7 3. .sh1o2r6tefrt le5g. :h21ypfot,telonnugseer: 6 in., longer leg:

Got It? 1a. from x = - 3 to x = 1 or from leg: 13 ft
2
x = 0 to x = 4; 4 b. from x = - 4 to x = - 1 or from
x = 0 to x = 3; 3 2a. no b. yes; 4 c. 15 cycles; Lesson 13-2 pp. 836–842
1 1
3 s; 440 s 3a. 1.5; y = - 0.5  b. 1.5; y = 0.5  Got It? 1. 225°
2a. y
4. period: 0.006; amplitude: 0.25; y = - 0.75 b. y c. y
Lesson Check 1. periodic; 5 2. no 3. Answers may 85°
x O x 180Њ
vary. Sample: hands of a clock, phases of the moon  Ϫ320° x
e22q=ua1l.f (5x.) O
4. The amplitude is not 2, but f(6) = f(11) = 2; O
for any x, f(x + 5) will always because the

period is 5.  6. - 4 3. - 315°, 45°, 405° 
Exercises  7. x = - 2 to x = 3, x = 2 to x = 7; 5 
9. x = 0 to x = 4, x = 2 to x = 6; 4  11. periodic; 12  4a. cos (- 90°) = 0, sin (- 90°) = - 1; cos (360°) = 1,
sin (360°) = 0; cos (540°) = - 1, sin (540°) = 0 
13. not periodic  15. periodic; 7  17. 3; y = - 1 0 -1
b. cos -90° = 1 = 0, sin-90° = 1 = -1

cos 360° = 1 = 1, sin 360° = 0 = 0
1 1
-1 0
cos 540° = 1 = - 1, sin 540° = 1 = 0

Selected Answers i 1121

5a. 122, - 122 b. - 123, 21 c. Yes; for example, when 55. y

u = 45°, sin u = cos u. 1 unit 225°
Lesson Check 1. 135° 2. 240°
3. y ; - 332° 4. y ; - 35° 57. 1 - 122, - 12
O x 2
Selected Answers
28Њx x
O
O 45°
325Њ y

5. Answers may vary. Sample: 45° and - 315° 6. The
measure of the coterminal angle is not 310°; the measure
of the coterminal angle is 50° - 360° = - 310°. 1020° 60° 13
Exercises 7. - 315° 9. 240° 11. - 30° 1 21, - 2
x

13. y 15. y

40Њ O x
Ox Ϫ270°
59. No; yes; if the sine and cosine are both negative, the
angle is in Quadrant III. 60° is in Quadrant I and - 120° is
in Quadrant III;
17. y
y
95° I
x
60°
O Ox

19. 215° 21. 4° 23. 150°  III Ϫ120°

25. 180° 27. - 122, 122; - 0.71, 0.71  61. H y
29. - 12, 123; - 0.50, 0.87 31. 122, - 122, 63.
0.71, - 0.71 33. - 122, 122; - 0.71, 0.71 35. 0.98,
- 0.17 37. 0.00, 1.00 39. 5  45° Ϫͱ2 x
2
41–43. Answers may vary. Samples: 
ͱ2 O
21

41. 370°, - 350° 43. 40°, - 320° 45. II 
47. negative x-axis 49. positive x-axis
T he terminal side forms an angle of 45° with the negative
51a. Quadrant II y Quadrant I
12
cos ␪ is Ϫ cos ␪ is ϩ x-axis, so: sin (- 135°) = - 2 and

sin ␪ is ϩ sin ␪ is ϩ cos ( - 135°) = - 122. Then [sin ( - 135°)]2 +
x

O [cos (- 135°)]2

Quadrant III Quadrant IV ( ) ( )=-12 2+ - 12 2=
2 2
cos ␪ is Ϫ cos ␪ is ϩ

sin ␪ is Ϫ sin ␪ is Ϫ 2 + 2 = 4 = 1.
4 4 4
b. II c. If the terminal side of an angle is in Quadrants I 64. periodic; 3 65. not periodic 66. periodic; 6

or II, then the sine of the angle is positive. If the terminal 67. (0, 2 15), (0, - 2 15); 68. (0, 5 15), (0, - 5 15);
y 9 y
side of an angle is in Quadrants I or IV, then the cosine of

the angle is positive. 

53. y 4 3 x
Ox Ϫ6 Ϫ3 O 6
60° Ϫ2 2

O 1 12, 13 Ϫ9
Ϫ300° x 2

1122

69. (185, 0), (- 185, 0); 45. If two angles measured in radians are coterminal, the

8y difference of their measures will be evenly divisible by 2p. 
3p
47. ≈ 11 radians 49. ≈ 6.3 cm  51. - 2 radians
53.
4 x u = 2p
48 s 2pr
Ϫ4 O
Ϫ4 u = 1 Selected Answers
s r

ur = s
s = ur
Ϫ8 55. G  57. For a central angle of 1 radian, the length of

70. (1145, 0), (- 1145, 0); the intercepted arc is the length of the radius.
y
58. y 59. y

6 x 15° Ox
Ϫ12 Ϫ6 O 6 12 Ox Ϫ75°

Ϫ6

60. 150° y 61. y

71. 50.24 in.2 72. 3846.5 m2 73. 200.96 mi2  x O x
74. 9.0746 ft2 Ϫ270°
O

Lesson 13-3 pp. 844–850

Got It? 1a. 90° b. 5p radians c. 360° ≈ 114.59°  62. y
4 p

d. 5p radians 2. - 123, - 1  3a. 6.3 in. b. arc length x
6 2 O
would also double. 4. ≈ 15,708 km
EL3ex. se2s0r3ocpins≈eCs2h 0e7.c9. k456 pi1n,..2 5.436p.21r a9rda.ida-niasp3n≈ , 5-5.1.62.04“5pr ae1dr1fiea.cnpt9s” , 2s0l..ic31e5s3 5°  Ϫ85°
112928,°- 11252.  -2137. 20°, 1-71. 22570. °- 11293.,12-, 13
13. 2   63. mean ≈ 12.9, s.d. ≈ 3.53 64. mean = 30,
21. s.d. ≈ 8.09 65. 2 66. all real numbers 67. 1 
68. y = 0

1  27. 10.5 m  Lesson 13-4 pp. 851–858
2
29. 25.1 in. 31. 43.2 cm 33. ≈ 31.9 ft 35. ≈ 42.2 in.  Got It? 1a. ≈ 0.1411; estimates may vary. b. - 1 
37. III 39. negative x-axis 2a. 2; 2p b. 3; 43p 3a. 3; - 3 b. 0.6; 0.6
4. y
41. y 0.71, - 0.71

2

7p x u
4 O p 3p

O Ϫ2

Q√22 , Ϫ √2 R
2
y = 1
3 sin 2 u

43. (0, 1) y 0.00, 1.00 5a. 2 y b. 2 y

O pp u u
O1 3
42
x Ϫ2
5p Ϫ2
2
6. y = sin 3p20u

Selected Answers i 1123

Lesson Check 1a. 2 b. 3; p c. y = 3 sin 2u 41. p, 25; 2 y
2. 1 y
u u

O p p Op 5p
12 4 4
4
Selected Answers 3. One cycle of a sine function is an interval on the x-axis
Ϫ2

with length equal to the period. The period is the length 43. 23p, 0.4; 0.5 y

of one cycle. 4. Answers may vary. Sample: y = 5 sin u   O
5. The amplitude is 3, but since 3
y x

a 6 0, the graph is reflected 2 p
across the x-axis. Also, the period O
is 2, not p. u Ϫ0.5
24 6 8
45. 152, 1.2; 1.2 y
Ϫ2

Exercises 7. ≈ 0.1 9. ≈ - 1 11. ≈ - 0.7 13. 12; 1, 4p u
15. 1 y O 0.6 1.8 3 4.2 5.4
u
Ϫ1.2
Op
47. 0.001 y
3
u

y = 2 sin 3u O1
17. 2 y
660
Ϫ2 O p
u 49. y = sin 60pu  51. y = sin 240,000pu
53. 2p, 1; 1 y
3p
u
y = 4 sin 1 u O 246
2
Ϫ1
19. 0 .y5 u 21. y u
2 55. C
O 0.5 0.5 1 57. C
O 59. 120°; consider the point where a 60° angle intersects
the unit circle. Reflect this point across the y-axis. The
y = sin pu image is the intersection of a 120° angle and the unit
circle. These two points have the same y-coordinate.
23. y u 25. y u Therefore sin 120° = sin 60°.
4 3p 662.306..21-r69a4p9dpiraarnadsdi aia6nn2ss,.,5--.51493p.4ra0radrdiaaidannisas n,6s-4 6.4.1-1.9552p6rpardraaidadniaiasn nss,,
0.5 2 - 7.85 radians 65. ≈ 49% 66. 1 67. 0 68. - 1 69. 0
O
Op
Ϫ2

27. 2p; y = 2 sin u 29. p; y = 5 sin 2u 31. 1; 1, 2p 
33. p; 1, 2 35. 1; 5, 2p 2

37. They are reflections of each Lesson 13-5 pp. 861–867
other across the x-axis.
Got It? 1. domain: all real numbers; period: 2p; range:
When a is replaced by its p2 ; 3p
opposite, the graph is a -1 … y … 1; amplitude: 1 sine: max at min at 2 ;
reflection of the original zeros at 0, p, 2p
graph across the x-axis.
2. 2 y

39. 0.001 y u
O 3p 6p

u Ϫ2
O 0.003

1124

( )3a. f(t) = -35 cos 42p5t   b. The function would cross of cos u, so these graphs are reflections of each other
across the x-axis.
the midline at 3 hours, 7 minutes, 30 seconds. The
y
midline represents average water level. 4a. 1.15, 1.99,
x
4.29, 5.13 b. 2.21, 4.06 c. 0 … u 6 2.21 and O3 7
4.06 6 u … 2p; 2.21 6 u 6 4.06 Ϫ1
Lesson Check Selected Answers

1. y 2. 2 y u 45. a. 2 y b. 2 y

2

Ϫ1 O p 2p O 36 u u
Ϫ2 Ϫ2 O p 2p O p 2p

Ϫ2 Ϫ2

3. y = 3 cos u 4. y = 1.5 cos 2u 5. Answers may vary. shift of p units to the right They are the same.
2
c. To write a sine function as a cosine function,
( )Sample: y = 5 cos 5 u  6a. 0 … u 6 p2 , 3p 6 u … 2p  replace sin with cos and replace u with u - p2.
2 2 47. F 49. G
( )b. 2p p
p 6u6 2p c. y= 3 sin 3 x - 2 3p
2
Exercises  7. 2p, 3; max: 0, 2p; min: p; zeros: p2 ,   Lesson 13-6 pp. 868–874

9. p, 1; max: 0, p, 2p; min: p2 , 32p; zeros: p4 , 34p, 54p, 7p Got It? 1a. not defined b. 13 c. - 1
4

11. 1 y u 13. 1 y t 2a. 1 b. 2

Op Op u u
p O2
3 Op
Ϫ2
15. 1 y u 3

O1 Ϫ1

2p 3a. ≈ 46.6 ft b. ≈ 8° 133 3. - 1 4. 0 5. 23p  6. The
3 Lesson Check 1. 1 2.

17. y = p cos u 19. y = -3 cos 2u  student found 3u instead of u. Divide each side of the
2
21. 0.52, 2.62, 3.67, 5.76  equation 3u = - 1.33 by 3 to get u = - 0.44. 
7. y = tan 2u; the period of the function y = sin 4u is p2.
23. 0.55, 1.45, 2.55, 3.45, 4.55, 5.45 25. 0.00 

27. 2p, - 3 … y … 3, 3 29. 4p, - 2 … y … 2, 2  For a tangent function to have the same period, p2 , b must
31. 6p, - 3 … y … 3, 3 33. 43, - 16 … y … 16, 16  equal 2.

35. y = 70 + 13 cos p (x - 1) where x represents the Exercises 9. 0 11. undefined 13. 0 15. undefined 
6 23p; 3p 2 3p 2
17. p  19. u = - p and u = p  21. 2 u = - 4
months of the year with January as 1, February as 2, 2 3 3 ;

March as 3, etc. 37. 0.64, 2.50  and u = 3p2
4
39. 0.50, 2.50, 4.50 23. 2 25. 2

41a. 30 y

u

20 Op O pu
Ϫ2 Ϫ2
10
t 27.
O 37
Ϫ10 6 2p
Ϫ20
Ϫ30 - 100, undefined, 100

b. 4:40 p.m.; 5:00 a.m. (next day); 5:20 p.m. (next day);
5:40 a.m. (2 days after time 0) c. 6 h 10 min; 6 h 10 min
43. On the unit circle, the x-values of - u are equal to the
x-values of u, so cos(- u) = cos u. - cos u is the opposite

Selected Answers i 1125

29a. b. ≈ 14.3 ft 4a. y 5x b. y x
c. ≈ 20.2 ft
O 2 O 2 4
Ϫ2 Ϫ2

Selected Answers Ϫ4 Ϫ4

( )5a. p
31. 2p ; 1y y = cos x + p2  b. y = 2 sin x - 4  6a. 69.9°F 
5
b. the average of the highest and lowest temperatures

Ϫ35p Ϫp5 O u c. Yes; data can be extrapolated to calculate for next year.
Lesson Check 
p 3p 1. 2 y
55

33. 1.11, 4.25 35. 0.08, 1.65, 3.22, 4.79 O2 x
37. 20 y 6

Ϫ2

O4 16 x 2. phase shift: 2 units to the right; vertical shift: 9 units up 
Ϫ20
( )3.y= cos x- 2p + 3 4. Answers may vary.
150 13 in.2 ≈ 260 in.2 3
Sample: y = 4 sin 23(x - p) - 5 5. Scott is correct;
( )39. 1 + k = cos 3 x+
200 41. 135 43. 70 45. y = - tan 2 x ( )y = a cos b(x - h) p where h = - p6 .
6
47a. y b. ≈ 27.7 ft2 The phase shift is p units to the left of y = cos 3x.
≈ 166.3 ft2 6
4  c. Exercises 7. - 2; 2 units to the left 9. 3; 3 units to the
57p; 5p
u right 11. 7 units to the right

O 30 60 13. y x

≈ 6.9 ft O 2468
49. Answers may vary. Sample: Triangles OAP and OBQ
Ϫ2
both share the angle u and each triangle has a right
Tnahnougnslneec,sogisnsoauutitv=heetyaannadrue.t hsi5emr1ei.laa2rr;ebfyoornAl0Ay .…2csosixnes 6ucuti=2opnOAs,APxo=fistOBhQBe=grtaa1nphu.of 15. 2 y 17. 4 y
the tangent function on or above the x-axis.  53. H  55. I
57. 1.32, 4.97 58. 1.77, 4.51 59. 6.15 60. 0.44, 1.56, x 2
2.44, 3.56, 4.44, 5.56 61. mean ≈ 5.9, median = 6, O 2468
modes = 4 and 6 62. 83 63. - 227 64. 145 65. - 332  x
66. 2 units to the right and up 5 units 67. 5 units to the O π2 π 32π 2π
left and down 4 units 68. 2 units to the left and up 1 unit
19. 1 y 21. 1 y x
Lesson 13-7 pp. 875–882 π 2π
O x
Got It? 1a. 5; 5 units to the right b. - 3; 3 units to Ϫ1 35 Oπ
Ϫ1 2
the left

2a. y p u b. 1 y 23. 1 unit to the left and 2 units down 25. 3 units to the
right and 2 units up
O u
Op
Ϫ2 27. 2 y 29. 4 y

Ϫ1 O x
2p
c. y = sin(x - 2) d. y = sin x - 2 p 2
3a. 6 y b. 4 y
O x

246

4 2 31. y x 33. 1 y
2 O 24
O 24 x O p 2p O x
2p
Ϫ2

x Ϫ1

1126

35. 2 y 5. 29.4 ft 6. y = 5 sec u = 5 u; there is no value of u
cos found
O 5 u equal to zero. 7. The student
x that will make cos
5
the reciprocal of (1 + cos 20°) = 1 + 1 20° ≈ 0.5155.
Ϫ2 cos
1
37. y = cos x - p  39. y = cos(x - 1.5) 41. y = The answer should be: sec 20° + 1 = cos 20° + 1 ≈ Selected Answers
2
p p 1.0642 + 1 ≈ 2.0642. 8. The graphs have the same
cos(x + 3) + p 43. y = 1.5 cos 6 (x - 6) - 2 + 2  period and range. The domain of y = sec x is all real
c d nnaitssuuymmamsbbypeemtrrosspteteeosxx.tcceeeTspph. tteTnnhg2pperad(p+wohpm4hoea(frwienyhno=efirsceysanc=nxisicncsatcaennxgienbisrte)ea,oglwlberhrte)ai,acinwlhehadirceahsitaasre

( )45. p
y= - 10 cos1p0 x; y = 10 sin 10 x - p  
2

49.

( )translation of y = sec x - p of the parent function
4
y = sec x.

Exercises 9. - 1 11. -1233 13. 0 15. - 12 

51. 17. ≈ - 1.248 19. ≈ 0.675 21. ≈ - 1.6 23. undefined
25. 2

u
O π 2π

Ϫ2

53. 27. y u
π 2π
O
Ϫ2

29. 1.1547 31. - 2.9238 33. 1.0642 35. 1.7321 
( )37. ≈104 ft and ≈164 ft 39. Answers may vary. Sample:
55. C  57. B  59. p6; u = - 1p2, 1p2 60. 4p; u = - 2p, 2p  y = csc u + p  41. C
2
43. y
61. 23p; = - p3 , p  62. 6p; = - 3p, 3p 63. 0.0064 
64. u 3 u u
69.
0.3456 65. ≈ 0.136 66. ≈ 0.198 67. 13  68. - 58   O 4p
15 t 9 Ϫ2 3
2p 70. 4m  71. - 14
Ϫ4

Lesson 13-8 pp. 883–890 45. y

Got It? 1a. 2 13  b. - 1 c. - 1 d. 43; 45; 5 by the
3 3
definition of a unit circle, the length of the hypotenuse of u
O1 3 5
the right triangle is 1; by the Pythag. Thm., the length of Ϫ2
the unlabeled leg is 54. So the triangle is similar to a 3-4-5
right triangle. 2a. ≈ 2.16 b. ≈ 4.649 c. ≈ 1.035  47a. domain: all real numbers except multiples of p,
cos x range: y Ú 1 or y … - 1; period: 2p b. 1 c. - 1 
d. undefined e. use sin x 49. csc 180° is undefined because sin 180° = 0 and
acsncduco=tsuin1=u.cs oi5ns1uu.. cot 0° is undefined because sin 0° = 0
3. 2 y 53a. 2

x x
O7 Op
Ϫ2

4. ≈ 1.4142 5. about 860 ft away and 3185 ft away
Lesson Check 1. 1 2. 2 13 3 3. ≈ 1.003 4. ≈ 1.346 

Selected Answers i 1127

b. The domain of y = tan x is all real numbers except 10. y
doodmd aminulotifplyes=ocfop2t ,xwishearllereitaslansuymmpbteorsteesxoceccputrm. Tuhlteiples
of p, where its asymptotes occur. The range of both x
O 30°

functions is all real numbers. c. The graphs have the same
p
Selected Answers period and range. Their asymptotes are shifted 2 units.  240° 12. sin (315°) = - 12 ≈
3p 2
d. Answers may vary. Sample: x = p4, x = 4 11. - 0.71,
55. y p units to cos
2 the left (315°) = 12 ≈ 0.71; sin (- 315°) = 12 ≈ 0.71,
2 2

2 cos (- 315°) = 12 ≈ 0.71  13a. p3  b. 21, 13  
2 2
θ
O p 2p 14a. - p4  b. 122, - 12  15a. p b. -1, 0 
Ϫ2 2

16a. 360° b. 1, 0 17a. 150° b. - 123, 1  
2
57. y 2 units to the left and 1 unit
4 down 18a. - 135° b. - 122, - 122  
19. 26.2 ft
x 20. 2 y
O p 2p

Ϫ4

Op 3p u

2 2

59. y p units to the right and 2 units Ϫ2
6
x down 21. 2 y
O 2p

Ϫ4 u

61a. II b. I  Op
2

Ϫ2

63. y = cos 3x cycles 3 times for each cycle of y = cos x. 22. y = 4 sin 4u
Thus, for each cycle of y = sec x, y = sec 3x cycles 3 23. 2 y
times, and each cycle of y = sec 3x is 1 as wide as one
cycle of y = sec x. 3 O 4p u
Ϫ2 3
2p

Chapter Review pp. 892–896

1. period 2. unit circle 3. tangent function  24. 2 y u
4. phase shift 5. secant function 
6. periodic; from 0 to 4 or from 4 to 6; 4; 2 O 2p 5p
7. Answers may vary. Sample: Ϫ2 3 3

y 25. y = 3 cos 2u 26. 0.58, 1.00, 2.15, 2.57, 3.72, 4.14,
5.29, 5.71 27. 0.70, 1.30, 2.70, 3.30, 4.70, 5.30
2
O1 2 3 4 t 28. 2 y

8. y t
O 2p
2 Ϫ2

O1 3 5 7 x 0.41, 1
2 29. y

9. - 225° t

Op
Ϫ2

- 1, undefined

1128

30. 4 y t 44. y p p 3p 2p u
2p
Op 2 22 7p
Ϫ4 t 2
2p O
2, undefined
31. y 45. 6 Selected Answers

4 2 u
Op
10p
Ϫ2 2p 3
3
undefined, 0
32. 2 y x Chapter 14
6
O4
Ϫ2 Get Ready!  p. 901

33. y x 1. x = { 25 2. x = { 123 3. x = { 4=5x -5352 ;4d.oxm=ain{o5f 11
5. x = { 130 6. x = { 2 7. f -1(x) range of f and 2
O p p 3p 2p f and range of f -1: all real numbers,
Ϫ2 2 2

Ϫ4 domain of f -1: all real numbers; yes 8. f -1(x) = x 2 - 3;

domain of f and range of f -1: all real numbers Ú - 3,
domain of f -1: all real numbers; range of f : all real
34. 4 y
x nf uamndberarsngÚe0o;fyfe-s1 :9a.llfr-e1a(lx)n=umx2b3e+rs4;Úd34o,mdaoinmoaifn of f -1:
2 p 2p all real numbers; range of f : all real numbers Ú 0; yes 
10. f -1(x) = 5x; domain of f off -1: all real
O x numbers except 0, domain and range range of f : all real
p 2p of f -1 and
35. 1 y 0; yes 11. f -1(x) = 10
numbers except -1: all real numbers x + 1; domain of
O f and range of f
except 1, domain of
f -1 and range of f : all real numbers except 0; yes 
Ϫ3 12. f -1(x) = x 1+01;
numbers except 0,
( )36. p  37. y = cos x - 2 38. 12  domain of f and range of f -1: all real
y = sin x - 4 domain of f -1 and range of f : all real

39. - 13  40. 2 41. 13 numbers except - 1; yes 13. x = - 32 14. x = 0.002
3 15. x = 1.0646 16. x = 18257.4 17. x = 0.00003
42. y
18. x = 5 19. 0.67; 0.74; 1.11 20. - 0.26; - 0.97; 3.73
21. 0.96; 0.28; 0.29 22. - 0.87; 0.50; - 0.58 
2 u 23. Answers may vary. Sample: The eq. is true for all
Op values of u for which tan2 u and sec2 u are defined. 
5p
2 2

Ϫ4 24. Answers may vary. Sample: the lengths of the sides

43. y of rt. triangles

4u Lesson 14-1  pp. 904–910
O p 2p 3p 4p
Ϫ4 Got It?  1. all real numbers except multiples of p 
1sin1 u2
2. csc u = = cos u = cot u; all real numbers except
sec u 1 1 u2 sin u
cos
p
multiples of 2  

Selected Answers i 1129

cos u 2 4. tan u cot u - sin2 u
sin u
( )3a. 1 + cot2 u = 1 + = tan u 1 - sin2 u
tan
= 1 + cos2 u u
sin2 u
= tan u - sin2 u
= + 1 - sin2 u tan u
1 sin2 u = 1 - sin2 u
Selected Answers
= 1 + 1 - sin2 u = cos2 u 
sin2 sin2 u 5. Answers may vary. Sample: Letting a and b be the legs, and
u
c the hypotenuse of a right triangle, the Pythagorean Theorem
= 1 + csc2 u - 1 states that a 2 + b 2 = c 2. Dividing both sides by c 2,
= csc2 u
b. No; the domains of sin u and cos u are all real ( ) ( )thena2 b2 a 2+ b 2 = 1. Calling the angle
c2 + c2 = c c
numbers, but the domains of tan u, cot u, sec u, and
between a and c u, then sin u = b and cos u = ac. By
csc u have restrictions. c
4. sec2 u - sec2 u cos2 u substitution, cos2 u + sin2 u = 1. 6. wrong calculation:
2 - cos2 u = 2 - (1 - sin2 u) = 2 - 1 + sin2 u =
( ) ( ) = 1 2- 1 2cos2 u 1 + sin2 u
cos u cos u
#
= 1 - 1 cos2 u Exercises
cos2 cos2
u u 7. cos u cot u

1 cos2 u ( ) cos u
= cos2 u - cos2 u = cos u sin u

= 1 - cos2 u = 1 - sin2 u
cos2 u sin u

= sin2 u = 1 u - sin u; all real numbers except multiples of p
cos2 u sin
9. cos u tan u
= tan2 u  ( )
5. csc u = cos u sin u = sin u; all real numbers except odd
cos u

Lesson Check multiples of p
11. cos u sec u 2
#1. tan u csc u
= sin u 1 ( ) 1 = 1; all real numbers except odd
cos u sin u = cos u cos u

= 1 u multiples of p
cos 13. sin u csc u 2

= sec u  ( ) 1 sin u
2. csc2 u - cot2 u = sin u sin u = sin u = 1; all real numbers except

2- cos u 2
sin u
( ) ( ) =1 multiples of p 
sin u
15. csc u - sin u
= 1 - cos2 u 1
sin2 u sin2 u = sin - sin u
u
1 - cos2 - sin2
= sin2 u u = 1 sin u u

= sin2 u = cos2 u
sin2 u sin u
= cot u cos u; all real numbers except odd
= 1  p
#3. sin u tan u multiples of 2

= sin u sin u 17. sin2 u 19. - cot2 u 21. sin u 23. 1 25. 1 27. 1
cos u 29. sec u 31. sec2 u 33. csc u 35. sin2 u 37. 1 39. 1

= sin2 u 41. { 21 - cos2 u 43. { 21 - sin2 u
cos u 45. { 2csc2 u - 1 sin u

= 1 - cos2 u
cos u

= 1 u - cos2 u
cos cos u

= sec u - cos u 

1130

47. sin2 u tan2 u = sin2 u a sin2 ub 59. tan u = sin uu, sin2 u + cos2 u = 1, which
cos2 u can cos sin2 u = 1 - cos2 u
be rewritten as

= (1 - cos2 u) acsoins22 u b 1.2 = sin u
u cos u

= sin2 u - sin2 u cos2 u 1.22 sin2 u Selected Answers
cos2 u cos2 u
sin2 u sin2 u cos2 =
cos2 u cos2 u
= - u 1.44 1cos2 u2 = sin2 u

= tan2 u - sin2 u 1.44 1cos2 u2 = 1 - cos2 u
49. sin u cos u(tan u + cot u) 1.44 1cos2 u2 + cos2 u = 1
( )
= sin u cos u sin u + cos u 2.44 1cos2 u2 = 1
cos u sin u
1
= sin2 u cos u + cos2 u sin u cos2 u = 2.44
cos u sin u
cos2 u = 0.409836066
= sin2 u + cos2 u = 1 

51. cot sec u u cos u = { 10.409836066
u+ tan
Since u is in the first quadrant, cos u is positive;
= cos u#1 sin u cos u cos u = 0.640184400
sin u cos u
cos u + sin u tan = sin u
sin u cos u u cos u

= sin u = sin u = sin u sin u
u + sin2 1 0.640184400
cos2 u 1.2 =

53. 1 - sin2 u  sin u = 0.76822128
sin2 u

55. sin2 u + cos2 u = 1 61. sin2 u + cos2 u = 1
(0.5)2 + cos2 u = 1 (0.2)2 + cos2 u = 1
0.25 + cos2 u = 1 0.04 + cos2 u = 1
cos2 u = 1 - 0.25 cos2 u = 1 - 0.04
cos2 u = 0.75 cos2 u = 0.96

cos u = { 10.75 cos u = { 10.96
Since sin u is positive and tan u is negative, u
Since u is in the first quadrant, cos u is positive;
is in the fourth quadrant, so cos u is positive;
cos u = 0.866025404 cos u = 0.97979590

tan u = sin u 63. cos (u + p) = 0 cos u 0 , but is also in Quadrant III and is
cos u
0.5 negative, so cos (u + p) = - cos u  65. 1
= 0.866025404
67. If n2 7 n1, then u1 7 u2; if n2 6 n1, then u1 6 u2; if
= 0.577350269 n2 = n1, then u2 = u1.  69. H  71. F
73. By the Difference of Squares Property and the second
57. sin2 u + cos2 u = 1 Pythagorean Identity: (sec u + 1)(sec u - 1) = sec2 u - 1
sin2 u + ( - 0.6)2 = 1 = tan2 u
sin2 u + 0.36 = 1 74. 2 y
sin2 u = 1 - 0.36
sin2 u = 0.64 u
Op
sin u = { 10.64
Since u is in the third quadrant, sin u is Ϫ2

negative; sin u = - 0.8 75. 2 y

tan u = sin u O pu
cos u Ϫ2
= -- 00..68
= 1.333333333

Selected Answers i 1131

76. 2 y 23. 0.46, 3.61 25. no solution 27. p2, p, 32p 
p4 , 34p, 54p, 74p  31. 76p, 11p
u # # # #29. 0, p 33. 6  35. 30° +
Op
n 360° and 150° + n 360° 37. 210° + n 360° and
Ϫ2 3p
2
Selected Answers 330° + n 360° 39.  41. 3.04, 6.18 43. 0.0028 s;

77. y 0.019 s+ 425p.n0, 5+6p2+p2np, 23np, 32+p 2pn, 43p + 2pn 
+ 2pn 
4 47. p
6
2 p 5p p
u 49. 6 + 2pn, 6 + 2pn, 2 + pn 

Op 51. p + 2pn, 7p + 2pn, 11p + 2pn 
Ϫ2 2 6 6
p 5p p p2n  57. p p2n  
Ϫ4 53. 6 + 2pn, 6 + 2pn 55. 4 + 4 +

59. 2p + 2pn, 4p + 2pn 61a. Answers may vary.
3 3
Sample: cos u = - 1, 2 cos u = - 2, 3 cos u = - 3
b. Start with cos u = - 1, and then multiply both sides of
78. 35° 79. 45° 80. 135° 81. 211°  the eq. by any nonzero number. 
x + 3 
82. f -1(x) = x - 1  83. f -1(x) = 2 ( ) ( )63. u = sin-1 y y-1
84. f -1(x) = { 1x -4 3 - 2  65. u = cos-1 2   67. D 

69. C  71. D  73. cot u 74. tan2 u  75. 1 76. 1 
23cous 8p42u.
Lesson 14-2  pp. 911–918 77. sin u 78. tan u 79. y =4 4 83. 21 
80. y = 3 cos u 81. y = p cos
Got It?  1a. 120°  b. 30°  c. 45° 2a. 0.46 + 2pn and 5221 4
84.
2.69 + 2pn  b. - 0.82 + 2pn and 3.96 + 2pn 
3a. 0.41 + 2pn and 3.56 + 2pn, or just 0.41 + pn
b. - 0.63 + 2pn and 2.51 + 2pn, or just - 0.63 + pn Lesson 14-3  pp. 919–926

c. tan (u + p) = tan u 4. 11p and 7p  5. p and 3p   Got It?  1. sin u = 1123, cos u = - 153, tan u = - 152,
6 6 2 2
6. The air conditioner comes on about 7 hours after csc u = 1132, sec u = - 153, cot u =-3515,2sec
2a. 27.1 m  b. 32.3 m 3. sin =
midnight, 7 a.m., and goes off about 7 hours before E F = 5  
3
midnight, 5 p.m. 4. ≈ 76,430 ft ≈ 14.5 mi 5a. 23.58°  b. 56.25° 6a. 72 ft
b. Answers may vary. Sample: Build the ramp in
# #Lesson Check  1. -30° + 360° n and 210° + 360° n
# #2. 60° + 360° n and 120° + 360° n 3. 2.30, 3.98  3 sections, each of which is 24 ft, and with landings

4. 0 5. Answers may vary. Sample: To find the inverse of between sections.
Lesson Check  1.
y = 3x - 4, you interchange x and y and solve for y: sin 57° = bc , cos 57° = ac, tan 57° = b
x + 4. f -1(x) a
x = 3y - 4, y = 3 Replace y with to find
2ta.n1353.4° =3.ba sin 33° = ac3=6.90°.5  ,5c.oAsn3s3w°e=rsbcm=ay0.v8a,ry.
ifn-t1e(rxc)h=anxg+3e4u. To find the inverse of y = 3 sin u - 4, you = 0.6  4.
and y and solve for sin y and then solve for
( )y: Sample: Using the inverse of cosine, you can find the
u = 3 sin y - 4, sin y = u + 4 and y = sin-1 u+4 .
3 3
( )Replace y with f -1(u) to find f -1(u) = sin-1 acute angle between the shortest side and the
u+4 . ( )hypotenuse, u = cos-1
3 8.4 ≈ 49.4°. Because the
The procedure for finding the inverse is the same; for 12.9
triangle is a right triangle, the remaining acute angle is
y = 3 sin u - 4 you will also use the inverse sine function. ≈ 90° - 49.4° = 40.6°. 6. The student confuses sin-1 u
6. The student divided both sides of the equation by with 1 u. He or she should have divided by sin 0.45.
sin
sin u, which in the given interval can be equal to zero. 4
= 0.45 ≈
There is an error in that the student failed to take into x sin 9.20 cm

account the fact that division by zero is not possible. Exercises  3443, 
3553,, 4554,,
# #Exercises 7. 90° + n 360° 9. 240° + n 360° 7. sin u = cos u = - tan u = -
# #and 300° + n 360° 11. 90° + n 360° and 270° + csc u = sec u = - cot u = -
# #n 360°, or just 90° + n 180° 13. 0.79 + pn, or just
p + pn 15. - 0.89 + 2pn and 4.04 + 2pn  9. sin u = - 5 12626, cos u = 12266, tan u = - 5,
4 5p 5p
17. + + 2pn 19. p6 , 6  21. p4 , 4
2.67 2pn and 3.62 csc u = - 1526, sec u = 126, cot u = - 1  
5

1132

11. sin u = 147, cos u = - 34, tan u = - 137, ( )47.sec A = c = 1 = 1 A
b cos
csc u = 4 17 7, sec u = - 34, cot u = - 3 17 7  b
c

13a. 15 ≈ 0.88  b. 17 ≈ 2.13  c. 8 ≈ 0.53 ( ) ( ) 49. cos2 A + sin2 A = b 2+ a2
17 8 15 c c

d. 17 ≈ 2.13  e. 17 ≈ 1.13  f. 8 ≈ 0.53  15. 41.8  = b2 + a2 = 1 Selected Answers
8 15 15 53. G  55. I c2
17. 25.2 19. a ≈ 8.7, m∠A = 60.0°, m∠B = 30.0°
21. a = 9.0, m∠A ≈ 36.9°, m∠B ≈ 53.1° 23. a ≈ 8.0,
( )m∠A ≈ 61.8°, m∠B ≈ 28.2° 25a. m∠A = cos-1 1200 Lesson 14-4  pp. 928–934
d
b. 37°  c. 53° Got It?  1. 36.6 in.2 2. 31.0 yd 3a. 68.4°  b. yes;
sin T sin990°;
27. height = height ≈ 9 sin 47° ≈ 6.6 4. 104.7 ft

3V39 20 Lesson Check  1. ≈ 10.7 square units 2. ≈ 19.8 

u 3. ≈ 26.3° or ≈ 153.7° 4. AAS, ASA 5. dNeon;oFmorinssaiinnto42r52,°°
7 you find the sine of each numerator and
sin 22° and 1s7i42n.5294°°.251.°;inf.o2r 9si.n11024.259°° 21y1o.u7f.i4n d1t3h.e3s3in.5e°o  f
sin u = 3 12039, tan u = 3 1739, csc u = 20111739, quotient of the

sec u = 270, cot u = 7 139   Exercises 
117 15. 31.7° 17. 32 cm 19. 66° 21. m∠E ≈ 40.3°,
m∠F ≈ 85.7°, f ≈ 12.3 m 25. 44.4 27. 49.4
29. 29. 28.0 ft 31. 4.0 cm  33a. 56.4°, 93.6°, 26.4°  b. No;

24 25 △EFG could be congruent to △ABC instead of △ABD.

u 35. No; you need at least one side in order to set up a
proportion you can then solve.
7
37. I
sin u = 2254, cos u = 275, csc u = 2245, 39. A = 1 ab sin C
2
sec = 275, cot = 7  
u u 24 sin C = 2(A) = 2(31.5) = 0.5
ab 9(14)
31. m∠C = sin-1 0.5 = 30°

4 V41 So, the measure of the included angle for the given

sides is 30° or 150°.
u 40. y
5

sin u = 4 14141, cos u = 5 14141, tan u = 54, 2

csc u = 1441, sec u = 141 u
5 Op

Ϫ2

33.

5 26 41. y

u 2 u
V651
O 1
Ϫ2
sin u = 256, cos u = 126651, tan u = 5 1651
651

sec u = 266156151, cot u = 16551  42. 1 y

35. 33.4 ft 37. 20.3 m2 39. c ≈ 12.2, m∠A ≈ 35.0°, u
m∠B ≈ 55.0° 41. a ≈ 3.9, c ≈ 6.9, m∠B = 55.8°  Op
43. a ≈ 19.8, b ≈ 2.9, m∠A = 81.7°  45. Using inverse
sine, you can find that u = 30°. Since sine is positive in the 4
first and second quadrants, another solution is 150°. All the
Ϫ1
# #solutions would be 30° + n 360° and 150° + n 360°.
43. 8 - 1, 7 9  44. 85, 3 9  45. 8 - 3,- 1 9  46. 8 - 6, 4 9  
47. 53.1° 48. 24.6° 49. 38.7° 50. 54.7° i 1133

Selected Answers

Lesson 14-5  pp. 936–942 Lesson Check
( ) ( )1. sin
Got It?  1a. 6.4  p + u + sin p - u
2 2
( ) ( ) p p
b. 1 mi to 6 mi; = sin 2 - (- u) + sin 2 - u
a2 = 2.52 + 3.52 - 2(2.5)(3.5) cos A
a2 = 18.5 - 17.5 cos A; = cos (- u) + cos u
if A = 0°, then cos A = 1 and a = 1; if A = 180°, = cos u + cos u
Selected Answers then cos A = - 1 and a = 6. 2. 75.3° 3. 30.7° = 2 cos u 
Lesson Check  1. ≈ 11.85 in. 2. ≈ 52.4° 3. ≈ 34.1°  5p 122  4. 12 +4an1d63 2p ,
2. p4 , 4  3. -

4. ≈ 60.3° 5. Use the Law of Sines when you have two 5. There are 2 solutions, p between 0 and 2p
because: - cos u = cos u 2
sides and a non-included angle or two angles and a side;
2 cos =0
use the Law of Cosines when you have two sides and an cos u s=in0p;2uc=osp2u, 32p 
u - cos
included angle or three sides. 6. The denominator should -u =
( )6. sin
be negative. The answer should be: p p sin u
2 2
C = 152 - 112 - 172 ≈ = (1) cos u - (0) sin u
cos - 2 (11)(17) 0.495 = cos u

C = cos-1(0.495) ≈ 60.3° Exercises
Exercises  7. 37.1 9. 13.7 11. 27.0 13. 33.7° 15. 47.2° 1
17. 50.8 19. 27.0° 21. b2 = a2 + c2 - 2ac cos B ( ) ( )7. cscu - p =
2 - p
sin B sin C sin C sin A sin u 2
23. b = c  25. c = a  27. ≈  59.1 nautical
1
miles 29. b ≈ 34.7, m∠A ≈ 26.7°, m∠C ≈ 33.3°  ( ( )) =
31. m∠A ≈ 56.1°, m∠B ≈ 70.0°, m∠C ≈ 53.9°  sin - p -
aABn,ywtwhiochsicdaenlebnegftohusnadasnindcbe,AthaenrdatBioabareisgievqeuna. l 2 u

33. For to the = 1
( )
ratio sin - sin p - u
sin 2
35a. ≈  45.4 mi  b. 14.4° left; 4.4° west of north 
37. 11.0 cm 39. 27.0° 41. 13.0 cm 43. 21.5° = 1
- cos u

45. 8.3 ft 47. 79.6° 49. 18 cm  = - sec u 
( ) (( ))9. cot
51. a. 2.1 m p - = cos p - u
2 2
b. 9.8 m2 u
p -
53. a. cos A 7 0 if b2 + c2 7 a2; sin 2 u
cos A = 0 if b2 + c2 = a2;
cos A 6 0 if b2 + c2 6 a2 = sin u
cos u
= tan u
b. acute △ if cos A 7 0; right △ if cos A = 0; ( ) ( ( ))11. tan
obtuse △ if cos A 6 0 u - p = tan - p - u
2 2
( )=
- tan p - u
2
Lesson 14-6  pp. 943–950 = - cot u

Got It? 13. tan (90° - A) = cot A 15. cot (90° - A) = tan A 
( ) ( ( ))1. cos p p 32p  19. 3p 12
u - 2 = cos - 2 - u 17. p2 , p 21. p2 , 2  23. 2  25. 0 

( ) p - 12 + 16
= cos 2 u 27. - 13 - 2 29. - 12 435. 21 
31. -2 + 13 33.
= sin u  
1 1
2. sec (90° - A) = cos (90° - A) = sin u = csc u 37. sin (A - B) = cos 3p2 - (A - B)4

sec (90° - A) = csc u  ( )= cos c p - + Bd
3a. 0, p  b.  yes; pn  2 A

4. 16 - 12   ( ) ( )= cosp - p
4 2 A cos B - sin 2 - A sin B

5. sin (A + B) = sin (A - (- B)) = sin A cos B - cos A sin B 
= sin A cos (- B) - cos A sin (- B)
= sin A cos B - cos A (- sin B)
= sin A cos B + cos A sin B
6. - 2 - 13

1134

39. tan (A + B) = sin (A + B) 2. 123 3. 2 cos 2u = 2 (2 cos2 u - 1) = 4 cos2 u - 2
cos (A + B)
12   13
= sin A cos B + cos A sin B 4a. b. - 3  
cos A cos B - sin A sin B
3 - 34  c.
sin A cos B + cos A sin B 5a. - 5   b. If 270° 6 u 6 360°,

= cos A cos B w13il5l °re6m2uain6 180° and u is also in Quadrant II. The answers Selected Answers
cos A cos B - sin A sin B the same. 2
cos A cos B
13
sin A cos B + cos A sin B Lesson Check  1. 2  2. 1 3a. - 153  b. 13 + 2 113
cos A cos B - cos A cos B 5 26
= cos A cos B sin A sin B - 513 - 2 113
cos A cos B cos A cos B c. 26  

= tan A + tan B   4. The student did not correctly
1 - tan A tan B
wd9e0ilt°leb6rme2unine6egi1ant3iwv5e°h., ict5hh.eqnsuin2uad4isrAainn  tQ2uuwadilrlabnet. If 180° 6 u 6 270°,
41. (5 cos u - 5 13 sin u, 5 sin u + 5 13 cos u)  II and the tangent
43. sin 5u 45. cos 5u 47. tan 2u 
49. a. even: cosine, secant; odd: sine, cosecant, 6. sin 5A = - 51 - cos 5A if 360° 6 5A 6 450°,
2 2
tangent, cotangent 5A
180° 6 2 6 225° and the sine is negative in Quadrant III.
b. No; answers may vary. Sample:
Exercises
x = p4, p - p
- cos 4 4
( ) ( )y = sin x - cos x. For f (x) = sin cos = 0, 7. sin 2u = sin (u + u)
=-
and f(- x) = sin - p - p 12. Because = sin u cos u + cos u sin u
4 4
f(x) ≠ f(- x), and - f(x) ≠ f(- x), the function is neither
even nor odd. = 2 sin u cos u  
+
51. sin (p - u) = sin p cos u - cos p sin u 9. - 13  11. - 13 13. - 21 15. - 1  17. 22 2 13  
2 2

= 0 - (- 1) sin u = sin u 19. 22 - 13 21. 22 + 12 23. 0 25. 3 110  27. 3
2 2 10
53. cos (p + u) = cos p cos u - sin p sin u
29. 4 117  31. -4
= (- 1) cos u - 0 = - cos u 17
B
( )55. cos u + 3p = cos u cos 3p - sin u sin 3p 33. cos B = 2 cos2 2 - 1
2 2 2
B
= cos u(0) - sin u(- 1) = sin u 2 cos2 2 = cos B + 1

57. I cos2 B = cos B + 1
2 2
59. sin (165°) = sin (15°)
= sin (45° - 30°)
Since cos B = a ,
# # = sin 45° cos 30° - cos 45° sin 30° c

= 12 13 - 12 1 cos2 B = a + 1 = a+ c.
2 2 2 2 2 c 2c

16 12 16 - 12 2
4 4 4
= - = 35. cos 2R = cos2 R - sin2 R
( ) ( )
60. ≈ 16.34 ft  61. ≈ 10.0 cm 62. 4p and 1.40  = s 2- r2
63. 0.87  64. - 9 t t

- 5p and - p and - 0.26  = s2 - r2
18 12 t2 t2
7p 19p
65. 18 and 1.22 66. 18 and 3.32 = s2 - r2  
t2
67. cos A cos B - sin A sin B 
tan A + tan B ( )37. S S 2
68. sin A cos B + cos A sin B 69. 1 - tan A tan B sin2 2 = sin 2

( )= - cos S
Lesson 14-7  pp. 951–957 { 1 2 2
5

Got It? 1 - cos S 1 - r
1. cos 2u = cos2 u - sin2 u = 2 = t

= cos2 u - (1 - cos2 u) 1 r t - r 2
2 2t 2t
= cos2 u - 1 + cos2 u = - =

= 2 cos2 u - 1  

Selected Answers i 1135

( )39. S S 2 24. 0, p2, p 25. p3, 53p 26. sin u = 45, cos u = - 35,
tan2 2 = tan 2

( )= - 2 1- cos S 1 - r tan u = - 34, csc u = 54, sec u = - 53, cot u = - 3  
{ 1 + cos S = 1+ cos S = t 4
5 1 cos S r
1 + t 27. sin u = - 1175, cos u = - 187, tan u = 185, csc u = - 1157,

Selected Answers = t - r sec u = - 187, cot u = 8   28. 53, 0.6 29. 45, 0.8 
t + r 15

41. - 2254 43. 274 45. 155 47. - 21  30. 43, 0.75 31. 35, 1.6 32. g ≈ 9.5, ∠F ≈ 18.4,
49. cos u (8 sin u - 3) = 0; p2, 32p, 0.384, 2.757  ∠H ≈ 71.6  33. h = 16, ∠F ≈ 36.9, ∠H ≈ 53.1 
34. f ≈ 37.7, ∠F ≈ 43.3, ∠H ≈ 46.7 
51. cos u (2 sin2 u - 1) = 0; p2, 32p, p4, 34p, 54p, 74p  35. g ≈ 6.4, ∠F ≈ 51.3, ∠H ≈ 38.7 36. 13.7 
53. 1 55. cos u - sin u 37. 29.4 38. 13.14 m2  39. 92.12 ft2 40. 7.1 in. 
41. 43.9° 42. 52.2° 
59. 4 sin u cos u (cos2 u - sin2 u) ( )43. cos
- tan2 u) u + p = cos u cos p - sin u sin p
61. 4 tan u (1 tan2 u + 1 2 2 2
tan4 u -6 = cos u * 0 - sin u * 1 = - sin u 


1 11 1 ( ) ( )44. sin2- p = - p 2
63. { 62 { 252 + 2 cos u u 2 c sin u 2
d

A 1 - cos A = csin cos p - cos sin p 2
2 51 + cos A 2 2
# 65. a. tan = { u u d

= { 1 - cos A 1 + cos A = [sin u * 0 - cos u * 1]2
51 + cos A 51 + cos A = ( - cos u)2 = cos2 u 

= { A 1 - cos2 A 45. 2 - 13 46. - 13  47. 12 - 16 48. -2 - 13
+ 2 4
11 cos A22

= { A11 sin2 A 49. 13  50. 13  51. - 13 52. - 13
2 2 2
+ cos A22

= 1 sin A A
+ cos

Since tan At2ananA2d sin A have the same sign Skills Handbook
wherever is defined, only the positive
occurs. sign

p. 972  1. 46%  3. 0.7%  5. 1.035  7. 25% 

67. G  69. I  71. 12   72. 13  73. 13 74. about 9. 66.6% 11. 115% 13. 12.5 15. 75 17. 20%
2 2 41252   5 5 19 1
4.1; 12 75. 2; 4 76. 45.2 77. 26.6 78. 57.6 p. 973  1. 3. 6 6   5. 21   7. 2 20   9. 8  11. 1 2  
13. 2 15.
14
Chapter Review  pp. 959–962 p. 974  1. 3 to 4  3. 19 g in 2 oz  5. 5   7. 8  9. 1.8 
11. 1.95  13. 45.5 15. {6
1. Law of Sines 2. trig. ratios 3. Law of Cosines 4. trig.
sin p. 975  1. 1 3. - 38 5. - 17 7. 4 9. 28 11. - 12 
ident. 5. Law of Sines 6. sin u tan u = sin u cos u = 13. - 90 15. 12 17. 19 19. 9 21. - 10
- cos u; domain u
sin2 u 1 - cos2 1 976  1. 14 m2 3. 30 cm2 5. 1 ft3 7. in.3 
cos u = cos u u = cos of validity: p. 91 8 100p
9.
u p 110.5 in.2 11. 121 1 ft2
2 2
all real numbers except odd multiples of   2pp11535.. ...99ab77a125455287 ,  1-115..1.I2x ba3344  .31I.V7 a.54cb.1 2II5I  .179c1.4. 54u7 19.2.yv7x065z  3 12911... 15 13. 15, - 3
2 2  
7. cos2 cot2 = cos2 cos2 u = cos2 u (1 - sin2 u) =
u u u sin2 u sin2 u

cos2 u - cos2 u sin2 u = cot2 u - cos2 u; domain of d8 11. c6 
sin2 u
validity: all real numbers except 0 and multiples of p  a3 23. 1  
mg3
8. cos2 u 9. 1 10. 1 11. - sin2 u 12. cos2 u 13. 1 
14. - 60° 15. 60° 16. - 30° 17. 30° 18. 0.34 
p3 , 53p  23. p6 , 7p p. 979  1. x2 + 10x - 5 3. 12x4 - 20x3 + 36x2 
19. - 1.11 20. 2.27 21. 0.20 22. 6  

1136

5. x2 - 2x - 15 7. (a - 6)(a - 2) 9. (x + 4)(x + 1)  p. 983  1. 3.7; 5; 5  3. 3.96; 2.4; 2.4  5. 1.5; 1.5;
11. (y + 8)(y - 3) 13. 2x(x2 + 2x - 4) no mode 
p. 980  1. 1.34 * 106 3. 7.75 * 10-4  5. 111,300 
7. 1.895 * 103 9. 1.234 * 105 11. 6.4 * 105  7.
13. 8.52 * 102  15. 17.5  17. 8.95 * 10-12 
19. 3.77 * 1010 21. 1.8 * 10-6 5
p. 981  1. 10  3. 15  5. 54.7  7. 5  9. 13  11. 22.7 
13. 2.8  15. 9  17. 7
p. 982  1.

NASA Space Shuttle Expenses, 2000
Millions of Dollars 4 Selected Answers
Frequency
3

2

1

04 567 8
Number

p. 984  1. a  3. 1  5. 4x   7. 2   9. x   11. 74x  
3b2 2 h 10 35

1200 13. 4x2   15. 16   17. 16   19. 2x
1000 5 x 5

800 Propulsion Mission, Flight Ground
600 launch operations operations
400 operations
200 Operation

0 Orbiter,
integration

NASA Space Shuttle Expenses, 2000

Ground
operations

16% Orbiter,
integration
Flight 8% 22%
Operations

Mission, Propulsion
launch 31%
operations
23%



Selected Answers i 1137

Index Index agriculture, 222, 746 battery life, 735
board games, 597
A algebraic expressions, 18–24 boat building, 365
defined, 5 bonus, 588
absolute value dividing, 21 break-even point, 147
of algebraic expressions, 41, 52 evaluating, 19, 52 bridges, 204, 323
of complex numbers, 249 modeling words with, 18–19 bus travel, 30
defined, 41 representing patterns using, 5, 51 calendar, 833
equations, 41–48 simplifying, 21, 52 camp, 708
inequalities, 41–48, 117 subtracting, 21 camping, 752
of real numbers, 41 career, 712
symbols, 36, 44 The Ambiguous Case, 935 carpentry, 294
car racing, 45
absolute value function(s) amplitude, 830, 853, 894 car rental, 66
defined, 107 ceramics, 873
family of, 108, 126 angle(s) class project, 23
graphing, 107–113, 126 acute, 922 climate, 47
piecewise function, 90 central, 844, 895 clubs, 511
reflection, 125 in a circle, 844 coins, 223
simplest form of, 107 in a coordinate plane, 836 collecting, 9
writing, 110 cosine of, 838, 839, 845, 894 college enrollment, 153
coterminal, 837, 838 comets, 651
Absolute Value Parent Function, 107 identities of, 943–956 communications, 212, 600
initial side of, 836, 894 community service, 708
act it out, 675 of inverse trigonometric functions, 912 competition, 228
measuring, 836 construction, 39, 466, 512, 530, 873,
Activity radian measure, 833–850, 895
Graphing Inverses, 413 sine of, 838, 839, 845, 894 923
Graphs in Three Dimensions, in standard position, 836, 837, 894 consumer issues, 679
164–165 tangent of, 868, 896 cooking, 47, 118, 161, 245, 343, 448
Linear Programming, 163 terminal side of, 836, 894 coordinate geometry, 799
Measuring Radians, 843 of triangles, 928–932, 935, 936–939 crafts, 461
Probability Distributions, 694–695 in unit circle, 836–842, 894 critical thinking, 40
Quadratic Inequalities, 256–257 customer satisfaction, 729
Systems with Rational Equations, 549 angle difference identities, 946, 947, data analysis, 97, 680
Using Logarithms for Exponential 962 data collection, 716, 747
Models, 477 demographics, 136, 440
Writing Equations From Roots, 232 angle sum identities, 947, 948, 951, depreciation, 461
962 digital media, 803
addition. See also summation discounts, 425
of complex numbers, 250 applications distance, 71, 888
of function, 398 academics, 364 elections, 729
inverse operation to, 27 acoustics, 475 electricity, 365, 425, 540, 917
of matrices, 764–770, 802, 818 activities, 732 electronics, 419
properties of, 767 advertising, 717, 752 endangered species, 438
of radical expressions, 374–380 aerodynamics, 643 education, 696, 699
of rational expressions, 534, 535, 556 aircraft, 544 energy, 723, 728
of vectors, 811 airline ticket pricing, 757 energy conservation, 915, 916
air quality, 160, 246 entertainment, 30, 35, 116, 148, 570,
Addition Property air traffic control, 652
of Equality, 27, 143 animation, 807 729
of Inequality, 34 aquarium, 6 environment, 160, 246, 438, 440,
athletics, 585
additive identity, 13, 14 auto maintenance, 756 628, 915, 916
automobiles, 45, 66, 213, 513, 538, estimation, 857
additive identity matrix, 767 547, 675, 678, 735 exercise, 569
aviation, 30, 814, 922 financial planning, 19, 23, 53, 187,
Additive Identity Property, 767 awards, 678
bacteria growth, 484 448, 461, 491, 576, 579, 599
additive inverse baking, 245, 448 firefighter, 229
of complex number, 250 banking, 19, 53, 585 fitness, 31, 237, 269, 569, 798
defined, 14 barometric pressure, 485 flight, 544, 652
of a matrix, 767 basketball, 210, 737
opposites, 14

Additive Inverse Property, 767

additive trigonometric identities, 943

1138

food, 16, 95, 180, 592 rocket, 31 geometric vs., 583 Index
food production, 96, 323, 333 rocket science, 924 identifying and generating, 573, 604
forestry, 933 sailing, 934
fuel economy, 513, 538, 547 sales, 187, 402, 403, 791 arithmetic series, 587–593
fundraising, 151, 245, 503, 504 satellite dish, 649, 650 defined, 587, 605
games, 686 satellites, 372, 482 sum of, 588, 589
gardening, 271, 317, 728 savings, 19, 53
gears, 949 scheduling, 678 assessment
genetics, 737 science, 484, 924 Apply What You've Learned, 17, 24,
geometry, 17, 30, 31, 39, 52, 66, 73, security, 679 32, 80, 88, 141, 148, 173, 208,
shopping, 708, 821 214, 231, 287, 295, 317, 388, 397,
88, 154, 173, 180, 213, 222, 238, sky diving, 61 420, 441, 476, 514, 541, 548, 586,
246, 294, 301, 308, 309, 329, 366, snacks, 180 601, 636, 644, 652, 687, 702, 779,
379, 410, 411, 426, 449, 570, 577, solar reflector, 625 800, 815, 834, 867, 890, 926, 942,
684, 686, 789, 799, 807, 808, 874, sound, 619, 628, 858 957
889, 925, 933, 941 sound waves, 831 Chapter Review, 50–52, 122–126,
golf, 238, 585, 642 space, 480, 481, 482, 849 183–185, 267–272, 347–352, 422–
grades, 39, 365, 522, 547, 591, 717 sports, 20, 31, 72, 139, 172, 238, 426, 487–490, 553–556, 603–606,
grocery, 592 663–666, 751–756, 817–820, 893–
harmony, 540 246, 269, 439, 440, 511, 548, 579, 897
health, 833 585, 642, 676, 678, 686, 700, 737, Chapter Test, 53, 127, 187, 273, 353,
hydraulics, 395 744, 754, 765, 769, 775, 778, 940 427, 491, 557, 607, 667, 757, 821,
income, 576, 723 sports arena, 574 907, 963
indirect measurement, 889, 924, 925, statistics, 692, 737 Common Core Performance Task, 3,
927 storage, 546 49, 59, 121, 133, 182, 193, 266,
investing/investment, 23, 187, 448, summer job, 23 279, 346, 359, 421, 433, 486, 497,
461, 491, 579 surveying, 931, 932 552, 563, 602, 613, 662, 673, 750,
landscaping, 207, 230, 418, 449 swimming, 31 763, 816, 827, 891, 903, 958
languages, 690 teamwork, 547 End-of-Course, 964–971
lottery, 686 technology, 200, 292, 482 Got It? 4, 5, 6, 12, 13, 14, 18, 19, 20,
machinery, 635 television, 212 21, 27, 28, 29, 30, 34, 35, 36, 37,
manufacturing, 96, 172, 199, 440, temperature, 66, 84, 410, 713, 881 42, 43, 44, 45, 61, 62, 63, 64, 69,
532, 729, 733, 745 temperature cycles, 879 70, 75, 76, 77, 82, 83, 84, 85, 93,
marketing, 736 testing, 682, 708, 715 94, 95, 100, 101, 102, 103, 108,
market researcher, 698 ticket price, 64 109, 110, 115, 116, 117, 135, 136,
measurement, 294, 427, 933 tides, 866, 918 137, 143, 144, 145, 150, 151, 152,
media postage, 91 time, 47, 841 158, 159, 167, 168, 169, 170, 175,
metalwork, 294 track and field, 269, 676, 718 176, 177, 178, 195, 196, 197, 198,
money, 19, 23, 53, 146, 187, 448, traffic signs, 395 203, 204, 205, 209, 210, 211, 217,
461, 491, 576, 579, 585, 588, 599 transportation, 126, 146, 546, 576, 218, 219, 220, 227, 228, 233, 234,
motor vehicles, 675 701, 849 235, 236, 237, 241, 242, 243, 244,
movie rentals, 30, 35 travel, 30, 544, 781 249, 250, 251, 252, 253, 259, 260,
musical, 53 utilities, 93 261, 281, 283, 284, 288, 289, 290,
nature, 198, 578 volume, 343, 395 291, 292, 297, 298, 299, 304, 305,
navigation, 544, 657, 658, 814, 941 weather, 148, 388, 485, 701, 716, 306, 307, 313, 314, 315, 320, 321,
nutrition, 97, 798, 799 736, 744 327, 328, 332, 333, 334, 340, 341,
optics, 540 weather satellite, 847 362, 363, 367, 368, 369, 370, 375,
optimal height, 386 wind generated power, 342 376, 377, 382, 383, 384, 385, 391,
paint/painting, 180, 503 woodworking, 547 392, 393, 394, 399, 400, 401, 406,
Pascal’s Triangle, 737 407, 408, 409, 415, 416, 417, 426,
picnic, 689 arc, intercepted, 844, 845, 895 435, 436, 437, 438, 443, 444, 445,
planetary motion, 383 446, 447, 452, 453, 454, 455, 463,
population, 136, 440, 692 archaeology, 448, 481 464, 465, 469, 470, 471, 472, 479,
postage/postal rates, 91, 213 480, 499, 500, 501, 502, 508, 509,
potential energy, 503 architecture, 234, 375, 592, 871 510, 511, 517, 518, 519, 520, 528,
product testing, 754 529, 530, 535, 536, 537, 538, 543,
projectile motion, 244 area 544, 545, 565, 566, 567, 568, 573,
public opinion, 726 of circles, 310, 407, 414 574, 580, 581, 582, 583, 588, 589,
quality control, 736 Law of Sines and, 928–934 590, 596, 597, 598, 615, 616, 617,
reading, 691 of polygons, 784, 785 624, 625, 626, 631, 632, 633, 639,
recreation, 127 of triangles, 784, 928–929 640, 641, 647, 648, 649, 654, 655,
restaurant, 16 656, 657, 675, 676, 677, 681, 682,
road safety, 213 arithmetic mean, 574, 604 683, 684, 688, 689, 690, 697, 698,

arithmetic sequences, 572–577 i 1139
defined, 572
formulas for, 572

Index

Index 699, 704, 706, 707, 712, 714, 726, of Multiplication, 773, 776 transformations, 816, 817
727, 733, 734, 740, 741, 742, 765, of Real Numbers, 14, 51 variable, 49, 50, 346, 602, 603
766, 767, 773, 774, 775, 776, 783,
784, 785, 786, 787, 793, 794, 795, astronomy, 467, 636, 644, 857 bimodal, defined, 711
797, 802, 803, 805, 810, 811, 812,
829, 830, 831, 837, 838, 839, 845, asymptotes binomial(s)
846, 847, 852, 853, 854, 855, 862, defined, 435, 487 defined, 281
863, 864, 869, 871, 876, 877, 878, oblique, 524–525 distributions, 731–738
879, 884, 885, 886, 887, 905, 906, of rational functions, 517–518 expanding, 326–330, 733–734
907, 913, 914, 915, 920, 921, 922, experiment, 731, 734, 755
923, 929, 930, 931, 937, 938, 939, average. See mean multiplying, 376
944, 945, 947, 952, 953, 954 probability, 732, 734
Mid-Chapter Quiz, 25, 89, 156, 224, axis radical expressions, 374–380, 424
311, 389, 461, 526, 579, 637, 710, of ellipse, 639, 665
791, 859, 927 of hyperbola, 645, 646 Binomial Theorem, 326–330, 351, 733,
Mixed Review, 10, 40, 48, 67, 73, 98, imaginary, 249, 250 734, 755
106, 113, 120, 155, 162, 181, 201, major, 639, 665
223, 239, 247, 255, 264, 302, 310, minor, 639, 665 biology, 83, 84, 135, 482, 743, 866
324, 330, 338, 345, 366, 373, 380, real, 249, 250
404, 412, 450, 458, 468, 483, 505, of symmetry, 107, 194, 622, 646 botany, 448
523, 533, 571, 577, 593, 620, 629, of systems of inequalities, 194
660, 680, 693, 718, 724, 730, 738, transverse, 645 boundary line, 107, 114, 115, 126
745, 770, 790, 808, 842, 850, 858, x and y, 101, 102, 164, 195, 197, 226,
874, 882, 910, 918, 934, 950 256, 435 box-and-whisker plots, 714, 754
Open-Ended exercises, 8, 9, 24, 40,
47, 72, 105, 126, 127, 138, 140, B branches
147, 154, 160, 173, 179, 181, 187, defined, 508
200, 213, 222, 224, 230, 254, 262, base e, exponential function, 442–444, of hyperbola, 616, 646, 649
273, 286, 294, 302, 311, 317, 323, 446 of reciprocal function graph, 508, 554
365, 372, 379, 385, 387, 396, 403,
427, 440, 449, 457, 475, 491, 505, base ten logarithms, 453, 463 business, 159, 170, 180, 245, 263, 522,
512, 513, 525, 526, 531, 539, 545, 777
548, 557, 570, 578, 579, 585, 592, Basic Identities, 904
600, 607, 619, 635, 643, 651, 660, C
667, 678, 691, 700, 701, 707, 710, best fit line, 94, 95, 125
730, 736, 737, 757, 791, 799, 821, CALC feature, 163, 202, 227
833, 840, 841, 842, 849, 855, 857, bias, 751, 755
859, 864, 881, 898, 909, 927, 933, calculator. See also graphing calculator
941, 956, 963 Big Ideas, 3, 59, 133, 193, 279, 359, Change of Base Formula, 464
Quick Review, 51–52, 121–126, 182– 433, 497, 563, 613, 673, 763, 827, degree mode, 920
185, 266–272, 346–352, 421–426, 903 EDIT option, 771
486–490, 552–556, 602–606, 662– exercises, 96, 180
666, 752–756, 818–820, 894–897, Connecting, 50, 122, 183, 267, 347, inverses of trigonometric functions,
960–962 422, 487, 553, 603, 663, 751, 817, 913
Standardized Test Prep, 10, 17, 40, 48, 893, 959 linear regression modeling, 136
54–56, 67, 73, 98, 106, 113, 120, linear systems, 178
128–129, 141, 148, 155, 162, 181, coordinate geometry, 662, 663 log key, 464
188–190, 201, 223, 231, 239, 247, data collection and analysis, 751 matrix equation, 793
255, 264, 274–276, 302, 310, 324, data representation, 750, 816, 817 matrix feature, 771
330, 338, 345, 354–356, 366, 373, equations, 182 quadratic equations, 227
380, 388, 397, 404, 412, 428–430, equivalence, 121, 122, 133, 183, 266, reciprocal trigonometric functions, 885
441, 450, 458, 468, 476, 483, 492– right-triangle trigonometry, 920
494, 505, 523, 533, 541, 548, 558– 267, 346, 347, 422, 487, 552, 553, RREF function, 177, 178
560, 571, 577, 586, 593, 608–610, 602, 603, 662, 663, 958, 959 standard deviation using, 720
620, 629, 660, 668–670, 680, 693, functions, 122, 133, 182, 183, 266, STAT CALC feature, 720
718, 724, 730, 738, 745, 758–760, 267, 346, 347, 422, 486, 487, 552, STAT EDIT feature, 720
770, 790, 800, 808, 822–824, 834, 553, 892, 950, 959 TABLE feature, 227
842, 850, 858, 867, 874, 882, 898– geometry, 958, 959
900, 910, 918, 926, 934, 950 modeling, 121, 122, 421, 486, 487, The Celebrated Jumping Frog of
602, 603, 662, 663, 816, 817, 892, Calaveras County (Twain), 228
Associative Property 893
of Addition, 767 probability, 750, 751 center
properties, 49, 50, 346 of a circle, 630, 633, 665
proportionality, 552, 553 of an ellipse, 639
Pull It All Together, 49, 121, 182, 266, of hyperbola, 646, 666
346, 421, 486, 552, 602, 662, 750, of rotation, 804
816, 892, 958
related expressions, functions, and central angle, 844, 895
equations, 346
solving equations, 266, 267, 347, 422 central tendency, measures of, 711,
solving equations and inequalities, 49, 712, 754
50, 133, 182, 183, 421

1140

Challenge exercises, 9–10, 17, 24, 32, combinatorics, 683 compression Index
40, 48, 66–67, 71, 73, 80, 88, 98, absolute value functions, 108, 109
106, 112, 120, 140, 148, 154, 162, combined defined, 102, 125
173, 181, 200–201, 207, 214, 223– transformations, 103 exponential functions, 434, 444
224, 230–231, 239, 247, 255, 256, translations, 109, 509 logarithmic functions, 455
263–264, 287, 296, 302, 310, 317, variation, 500, 501, 502 quadratic functions, 195
324, 330, 337–338, 345, 366, 372,
380, 397, 404, 412, 420, 440, 449, common Concept Byte
458, 467, 475–476, 482–483, 505, difference, 572, 589, 604 The Ambiguous Case, 935
513, 523–524, 532, 540–541, 546– factors, 218 Drawing Conclusions from Samples,
548, 570–571, 576, 585, 592–593, logarithm, 453, 489 748–749
601, 619, 629, 636, 644, 651–652, ratio, 580, 605 Exponential and Logarithmic
660, 680, 687, 693, 701, 717–718, Inequalities, 484–485
724, 730, 737, 745, 770, 779, 780, Common Core Performance Task, 3, The Fibonacci Sequence, 578
799, 809, 814–815, 833, 842, 850, 49, 59, 121, 133, 182, 193, 266, Fitting Curves to Data, 459–460
857–858, 866, 874, 882, 890, 909, 279, 346, 359, 421, 433, 486, 497, Geometry and Infinite Series, 594
917–918, 925–926, 934, 941–942, 552, 563, 602, 613, 662, 673, 750, Graphing Conic Sections, 621
950, 946 763, 816, 827, 891, 903, 958 Graphing Inverses, 413
Graphing Polynomials Using Zeros, 325
Change of Base Formula, 464 communication. See Reasoning; Writing Graphing Rational Functions, 506
Graphing Trigonometric Functions, 860
Chapter Review. See assessment, Commutative Property Graphs in Three Dimensions, 164–165
Chapter Review of Addition, 767 Identifying Quadratic Data, 215
of Real Numbers, 14, 51 Linear Programming, 163
Chapter Test. See assessment, Chapter Margin of Error, 746–747
Test Compare and Contrast exercises, 7, 15, Measuring Radians, 843
22, 37, 45, 96, 103, 111, 118, 145, Networks, 780–781
check for reasonableness, 236, 624, 152, 160, 198, 206, 212, 221, 229, Oblique Asymptotes, 524–525
785 261, 329, 342, 378, 394, 456, 461, Piecewise Function, 90–91
503, 557, 583, 598, 617, 650, 700, Powers of Complex Numbers, 265
check your answers, 608 722, 729, 743, 887, 889 Probability Distributions, 694–695
Properties of Exponents, 360
chemistry, 39, 67, 147, 379, 457, 504 complement of an event, 702 Quadratic Inequalities, 256–257
Rational Inequalities, 550–551
circle, 630–636 completing the square Special Right Triangles, 835
angles in, 844, 895 defined, 235 Systems with Rational Equations, 549
area of, 310, 407, 414 solving equations by, 233–239, 240, Using Logarithms for Exponential
center of, 630, 633, 665 271 Models, 477
circumference of, 15, 17, 839 Using Polynomial Identities, 318
conic sections, 614 complex conjugates, 251, 314 Working with Matrices, 771
defined, 630, 665 Writing Equations From Roots, 232
equations of, 630, 631–632, 665 complex fractions
graphing, 615, 632, 633 defined, 536 conditional probability, 696–702
intercepted arc, 844, 845, 895 simplifying, 537 defined, 696
Moiré pattern, 617 finding using tables, 696–697
radius of, 414, 633, 665 complex number(s), 249–256 finding using tree diagrams, 699
transformations, 632 absolute value of, 249 formula for, 697, 698, 753
translations, 631 adding, 250 in statistics, 697
additive inverse of, 250
circumference, 839 conjugates, 251, 314 confidence interval, 746
defined, 249, 272
classification of events, 688 dividing, 251 confidence level, 746
factoring polynomials using, 319–322
Closure Property, 54 identifying, 249 conic sections, 614–620
of Addition, 767 multiplying, 251 circle, 614, 630–636
of Matrix Multiplication, 776 operations with, 250 defined, 614, 664
of Real Numbers, 14 powers of, 265 ellipse, 638–644
of Scalar Multiplication, 773 simplifying, 249 family of, 614
simplifying quotients of, 251 graphing, 616, 621
coefficient, 20, 94 solutions to quadratic equations, 248– hyperbola, 614, 645–652
255 identifying, 616
coefficient matrix, 175, 794, 819 subsets of, 248 modeling with, 617, 657
subtracting, 250 Moiré pattern, 617
cofunction identities, 944, 945, 962 writing using imaginary numbers,272 parabola, 614, 622–629
translations, 653–660, 666
combinations, 674–680 complex number plane, 249, 250, 272
defined, 676, 696
finding the number in n items, 676, composition
677 of functions, 399, 400–401
importance of order in, 677 of inverse functions, 408, 409
numbers of, 676
compound interest, 446, 488

Index i 1141

Index types of, 664 families of, 877 range, 713, 754
graphing, 861–863 sample, 725–730, 751, 755
conjecture, 295, 317, 420, 578, 594, modeling with a, 863 standard deviation, 711, 719–724
651, 890 properties of, 862, 895 variance, 719, 720, 754
translations, 875–882, 896, 944 z-score, 748
conjugate axis, 640, 646
cotangent data collection and analysis, as Big
Conjugate Root Theorem, 314, 350 curve, 944 Idea, 751
function, 883, 897
conjugates identity, 904, 960 data modeling, 209–213, 269, 331–334,
complex, 251, 314 352
constructing polynomials using, 314 coterminal angles, 837, 838
defined, 377 data representation, as Big Idea, 750,
multiplying, 377 counting, fundamental principle of, 816, 817
radical, 377 674, 675
data sets
Connecting Big Ideas, 50, 122, 183, co-vertices of ellipse, 639, 665 bimodal, 712
267, 347, 422, 487, 553, 603, 663, comparing, 713
751, 817, 893, 959 cricket chirps per minute, 84 range of, 713
of scatter plots, 92, 93, 125
conservation, 71 cube root function, 417
decay factor, 436, 488
consistent system, 127, 137, 184 cubes, sum or difference of, 297
decoding matrix, 787
constant, 5 cubic functions, 283–284, 339–340
matrix, 793, 819 degrees
ratio, 74 cubic polynomial, 281, 331 converting to and from radians, 844,
term, 20 845
CUBICREG feature, 331, 332, 334 of polynomials, 280–284
constant of proportionality, 341
cumulative DTbl feature, 470
constant of variation, 68, 554 frequency, 695
probability, 695 demography, 474
constraint(s), 157, 185
curves denominator
contingency table, 696 conic sections. See conic sections least common (LCD), 535, 537
cosine, 944 rationalizing the, 369, 370, 377, 423
continuous function, 516 cotangent, 944
fitting to data, 459–460 dependent
continuously compounded interest, sine, 852, 895, 944 events, 688, 698, 753
446, 488 tangent, 944 system, 137, 184
variable, 63
continuous probability distribution, cycles
739, 756 identifying, 828 Descartes’ Rule of Signs, 315, 350
of periodic functions, 828–829
controlled experiment, 727 tangent function, 869 design, 871

convenience sample, 725 cylinder, 839 determinants, 782–790
defined, 783
convergence of a series, 598, 606 D of matrices, 784, 819

Coordinate Geometry, as Big Idea, data, describing, 729 diagrams
662, 663 drawing and using for problem
data analysis, 711–718 solving, 39, 53, 54, 55, 56, 60, 62,
coordinate plane binomial distributions, 731–738 65, 238, 294, 302, 316, 355, 375,
angles in, 836 binomial experiment, 731, 734, 755 378, 379, 395, 421, 449, 453, 492,
boundary line, 114, 126 box-and-whisker plots, 714, 754 532, 570, 594, 617, 628, 629, 631,
movement in, 74 exercises, 769 657, 814, 841, 921, 933, 938, 940
points on, 164 interquartile range, 713 mapping, 60
reflection matrices for, 804 mean, 711, 754 tree, 699, 732
rotation matrices for, 804 measures of central tendency, 711,
719, 754 difference
coordinate space, 164 measures of variation, 719, 754 common, 572, 589, 604
median, 711, 754 of cubes, factoring, 297
coordinate system mode, 711, 754 opposite of a, 21
three-dimensional, 164–165 normal distribution, 739–745, 756 of squares, factoring, 220, 297
two-dimensional, 164 outliers, 712, 754 using to determine degree, 284
percentiles, 715
correlation, 92, 125 probability distributions, 694, 734, dilation, 801, 802, 803, 820
739, 756
correlation coefficient, 94 quartile, 713, 754 dimensional analysis, 845
random sample, 725, 755
cosecant, 887 dimensions, determining, 234

cosecant function, 883, 897 directrix of parabola, 622, 664

cosine direct variation, 68–73, 123, 498–499
of an angle, 838, 839, 845, 894
curve, 944
equation, 864–865

cosine function, 851–867

1142

discontinuity DrawInv feature, 413 endpoints of, 639, 665 Index
non-removable, 516 equation of, 639–640, 653–654
point of, 516–517, 555 Dynamic Activities foci of, 638, 640–641, 665
removable, 516 Absolute Value, 107 graphing, 615
Box-and-Whisker Plots, 712 horizontal, 639, 653–654
discontinuous function, 516 Circles, 630 major and minor axis, 639, 665
Cosine Function, 861 reflection property, 640
discrete Dilations, 801 translations, 653–654
function, 437 Ellipse, 639 vertical, 639, 653–654
probability distribution, 739, 756 Factored-form, 226 vertices of, 639, 665
Geometric Probability, 682
discriminant, 242, 243, 244, 271 Growth/Decay, 435 ellipsoid, 641
Hyperbola, 645
distribution Independent and Dependent Events, encoding matrix, 787
binomial, 731–738 689
continuous probability, 739, 756 Inequality Systems, 149 end behavior, 282–283
discrete probability, 739, 756 Linear Factors, 289
normal, 739–745, 756 Linear Programming, 158 endpoints, 91, 639, 665
probability, 694 Log Functions, 451, 478
skewed, 740 Matrix Addition, 764 equality, properties of, 26–27, 34, 143,
uniform, 694 Parabolas, 622 946
Probability Binomial, 731
Distributive Property Quadratic Functions, 203 equally likely sample space, 682, 752
of Matrix Multiplication, 776 Radical Functions, 415
of Real Numbers, 14, 51 Rational Functions, 515 equal matrices, 767, 818
of Scalar Multiplication, 773 Real Number Line, 11
for Subtraction, 21 Roots, 240 equation(s). See also function(s)
Sequences, 572 absolute value, 41–48
divergence of a series, 598, 606 Simplifying, 361 as Big Idea, 182
Sine Function, 851 of a circle, 630, 631–632
division Special Solutions, 143 cosine, 864–865
of algebraic expressions, 21 Sum and Difference Identities for Sine defined, 26
of complex numbers, 251 and Cosine, 943 direct variation, 68–73
definition of, 21 Synthetic Division, 303 of ellipse, 639–640, 653–654
of functions, 398, 399 Systems of Linear Equations, 135 exponential. See exponential equations
inverse operation to, 27 Tangent Function, 868 of hyperbola, 647, 648, 655–656
of logarithms, 462 Translation, 99, 507 as an identity, 28, 29, 52
of polynomials, 302–310, 349 Trigonometric Functions, 875 for the inverse, 406
of radical expressions, 367–373, 423 Trigonometric Identities, 904 of a line. See linear equations
of rational expressions, 529 Variation, 499 literal, 29
of square roots, 225 Vertex Form, 194 logarithmic, 469–476
synthetic, 303, 306–307, 349 matrix, 765, 792–793, 818
E with no solution, 28, 29, 52
Division Property of parabola, 209, 623–624, 627
of Equality, 27 e parametric, 413
of Inequality, 34 base of exponential function, 442– polynomial. See polynomial equations
of Square Roots, 225 444, 446 quadratic. See quadratic equations
natural logarithmic function, 478 radical, 390–397, 416, 425
domain rational, 542–548, 549, 556
extrapolating, 334 earthquakes, 456, 474, 543 solving, as Big Idea, 266, 267, 347,
of functions, 62, 398, 399, 408, 414, 422
425, 434, 435, 515, 516 earth science, 392, 628 solving by completing the square,
predicting, 334 233–239, 240, 271
of rational expressions, 527 eccentricity, 643 solving by finding square roots, 233,
of relations, 61, 62, 123 235, 271
of validity, 904, 905 economics, 140, 206, 402 solving multi-step, 27
solving one-step, 27
dot product, 812, 820 EDIT option, 771 solving using inverses, 911–918
solving using properties, 26–27, 52
double-angle identities, 951–947, 952 elimination, 144, 148, 166–173, 186 solving with logarithmic properties,
472
drawing ellipse, 638–644 square root, 390–397, 425
a diagram for problem solving, 814, center of, 639 systems of. See systems of equations
842, 921, 933, 938, 940 conic sections, 614 of transformations, 510
a graph for problem solving, 36, 857, co-vertices of, 639, 665 of a trend line, 94
862, 864, 865, 869, 873, 878, 881 defined, 638, 665 trigonometric, 879, 911, 918, 935,
sketch angles in standard position, 837 eccentricity of, 643 960
sketch sine curves, 854 writing from roots, 232
sketch solutions in a plane, 164, 184
i 1143
Index

Index writing in slope-intercept form, 74–80, exponential F
83, 124 growth and decay, 434–441, 488
inequalities, 484–485 factor(s)
writing to describe a translation, 879 models, 434–441, 445, 477, 488 common, 218
greatest common (GCF), 218, 297
Equations and Inequalities, solving as exponential equation(s) growth and decay, 435, 436, 488
Big Idea, 49, 50, 133, 182, 183, base e, 446
421, 422 common base, 469 factorial notation, 675
defined, 469, 490
Equivalence, as Big Idea, 121, 122, different base, 470 factoring
133, 183, 266, 267, 346, 347, 422, exponential growth, 488 defined, 216
487, 552, 553, 602, 603, 662, 663, general form of, 438 difference of cubes, 297
958, 959 modeling, 471 difference of squares, 220, 297
solving by graphing, 470 to find discontinuities, 516–517
equivalent solving using logarithms, 479 the GCF out, 297
compound inequalities, 36 solving using tables, 470 by grouping, 297
equations, 142, 144, 168 writing, 452 perfect square trinomials, 219, 220,
expressions, 381 234–235, 297
exponential function(s) polynomials, 288–295, 297, 349
Error Analysis exercises, 7, 16, 22, 24, base e, 442–444, 446 quadratic expressions, 216–223, 270
30, 31, 37, 39, 47, 64, 72, 78, 79, defined, 434 quadratic trinomial, 297
86, 87, 97, 105, 112, 140, 147, discrete, 437 rational expressions, 527
152, 161, 171, 180, 206, 207, 212, families of, 444 solving polynomial equations by, 296–
222, 230, 237, 244, 253, 263, 285, general form of, 488 302
293, 301, 309, 315, 316, 323, 329, graphing, 434–435, 437 solving quadratic equations by, 218–
337, 342, 364, 370, 372, 379, 385, graphing base e, 442–450 219, 226–231, 270, 271
389, 396, 401, 409, 418, 439, 448, inverses of, 454, 489 sum or difference of cubes, 297
461, 466, 473, 480, 482, 503, 522, natural base, 446
530, 539, 545, 547, 568, 575, 583, properties of, 442–450, 488 Factor Theorem, 290
591, 598, 600, 627, 634, 642, 650, transformations, 443–444
651, 658, 691, 710, 715, 717, 723, Family(ies)
735, 744, 768, 777, 787, 788, 797, exponents of absolute value functions, 108, 126
813, 831, 840, 849, 855, 871, 880, properties of, 360, 462 of conic sections, 614
887, 908, 915, 923, 932, 939, 948, rational, 381–388, 424 of cosine function, 877
950, 955 of exponential functions, 444
ExpReg feature, 445, 460 of functions, 99–106, 125
Essential Questions, 3, 59, 133, 193, of logarithmic functions, 455
279, 359, 433, 497, 563, 613, 673, expression(s) of radical functions, 415
763, 827, 902 algebraic. See algebraic expressions of reciprocal functions, 507–514,
as Big Idea, 346 554
answering Essential Questions, 50, equivalent, 381 of sine function, 877
122, 267, 347, 422, 553, 603, 663, greatest common factor (GCF), 218
751, 817, 958 logarithmic, 451, 478–479 feasible region, 157, 185
numerical, 5
estimation using line of best fit, 94, quadratic. See quadratic expressions Fibonacci Sequence, 578
95, 125 radical. See radical expressions
rational. See rational expressions finance, 51, 94, 172, 568, 571
evaluating representing patterns using, 6
algebraic expressions, 19, 20, 52 writing, 20 find needed information, 128, 274,
ex, 446 558, 608, 668
logarithmic expressions, 451–458 Extended Response exercises
sequences, 573, 581 end of chapter problems, 130, 190, finite geometric series, 595, 596
series, 598 276, 356, 430, 494, 560, 610, 670,
760, 824, 900, 966 finite sequences, 587
events end of lesson problems, 40, 120, 141,
dependent, 688, 753 214, 223, 239, 295, 310, 330, 366, finite series, 587, 595, 596
independent, 688, 689, 753 412, 441, 548, 577, 629, 738, 834,
multiple, 688 918 flips. See reflections
mutually exclusive, 689, 753
theoretical probability of, 683 Extension FLOAT feature, 568
Exponential and Logarithmic
expanding Inequalities, 484–485 flower carpets, 28
binomial(s), 326–330, 733–734 The Fibonacci Sequence, 578
logarithms, 463 Linear and Exponential, 477 focus (foci)
of ellipse, 638, 640–641, 665
experimental probability, 681–682, extraneous solutions, 42, 43, 393 of hyperbola, 645, 666
744 of parabola, 622, 664
extrapolation, 334
experiments FOIL, 376
binomial, 731, 734, 755
controlled, 726, 727 formulas
for area, 310, 407, 414, 558, 928
Explicit Formulas, 565, 567, 604, 605

1144

arithmetic sequences, 572, 604 surface area, 66, 276, 531 power, 341 Index
atmospheric pressure, 475, 513 surface area of a cylinder, 53 quadratic. See quadratic functions
for binomial probability, 732, 734 tapered cylinders, 547 quartic, 341
brightness of stars, 467 temperature conversions, 29, 410 radical, 414–420
for centripetal force, 372 for time as a function of acceleration, rational. See rational functions
Change of Base, 464 reciprocal. See reciprocal functions
circumference of a circle, 15, 17 371 reciprocal trigonometric, 883–890,
for conditional probability, 697, 753 variance, 719
converting between radians and for velocity, 372, 404, 480 896
for volume, 67, 292, 309, 365, 392, rewriting by completing the square,
degrees, 844
for density, 372 410, 421, 426, 531 233–239
direct variation function, 68 wind chill, 387 secant, 883
for distance, 408 wind power, 327 simplest form of, 99, 125
earthquake intensity, 453 sine, 851–858, 875–882, 894
electric current, 16 fractions, complex, 536, 537 square root, 414–420, 426
error of margin for drilling a hole, 48 step, 90
for expanding a binomial, 327 frequency, cumulative, 695 subtracting, 398, 399
experimental probability of an event, transformations. See transformations
frog-jumping competition, 228 translations. See translations
681 trigonometric, 860, 886
explicit, 565, 567, 604, 605 function(s). See also equation(s) vertical-line test, 62–63
generating mathematical patterns, absolute value, 90, 107–113, 125, zeros of. See zero(s)
126
564, 565, 567 adding, 398, 399 function notation, 62, 63, 407
for geometric sequences, 565, 566, as Big Idea, 122, 133, 182, 183, 266,
267, 346, 347, 422, 486, 487, 552, function rule, 63
567 553, 892, 958, 959
for geometric series, 597, 606 composition of, 399, 400–401, 408, Fundamental Counting Principle, 674,
harmonic mean, 540 409 675
for height as a function of time, 344 continuous, 516
hourly wage, 548 cosecant, 883 Fundamental Theorem of Algebra,
Ideal Gas Law, 504 cosine, 875–882, 895 319–324, 351
independent events, 753 cotangent, 883
interest, 446, 488 cube root, 417 G
inverses of, 408 cubic, 283–284, 339–340
for kinetic energy, 343, 344 defined, 62 Galton box, 731
mass energy equivalence, 493 dependent variable in, 63
maximum flow of water in a pipe, 395 discontinuous, 516 GCF (greatest common factor), 218,
mean, 711, 754 discrete, 437 297
measuring bacteria populations, 482 dividing, 398, 399
median, 711 domain of, 62, 398, 399, 408, 414, geography, 849
mode, 711 425, 434, 435, 515, 516
mutually exclusive events, 689, 753 exponential. See exponential functions geometric
number of combinations, 676 family of, 99–106, 125 mean, 583, 605
number of permutations, 676 graphing quadratic, 194–201 probability, 684
perimeter of a rectangle, 28 graphing trigonometric, 860, 886 series, 595–601, 606
for planetary motion, 383 greatest integer, 90 transformations, 801–808, 820
point-slope form, 81 identifying, 60–65
potential energy, 502 independent variables in, 63 geometric sequences, 580–586
probability of A and B, 688, 753 input of the, 63 analyzing, 581
probability of A or B, 689, 753 inverse. See inverse functions arithmetic vs., 583
quadratic, 240–248 linear. See linear functions defined, 580
radioactive to nonradioactive carbon, logarithmic. See logarithmic functions formula for, 565, 566, 567, 605
maximum and minimum values, 158, geometric series, 595–601, 606
387 185, 195, 291, 292 identifying, 580–581
radius of a circle, 414 monomial, 352
radius of a sphere, 365 multiplying, 398, 399 Geometry as Big Idea, 958, 959
recursive, 565, 566, 604, 605 natural logarithmic, 478–483
resistance, 540 objective, 157, 185 Geometry exercises, 17, 30, 31, 39, 52,
slope, 81, 84 one-to-one, 408 66, 73, 88, 154, 173, 180, 213,
solving for one variable, 142, 184 operations with, 398–404, 425 222, 238, 246, 294, 301, 308, 309,
sound, measuring, 466, 475 parent. See Parent Function 329, 366, 379, 410, 411, 426, 449,
standard deviation, 711, 719 periodic. See periodic function 570, 577, 684, 686, 789, 799, 807,
sum of arithmetic series, 587, 605 piecewise, 90–91 808, 874, 889, 925, 933, 941
sum of geometric series, 595, 606 polynomial. See polynomial functions
Geometry Review
Special Right Triangles, 825

Index i 1145

Index Get Ready! of real numbers, 11, 13 DOT plotting mode, 506
for chapters, 1, 57, 131, 191, 277, 431, of reciprocal functions, 507–511, 554 DrawInv feature, 413
495, 561, 611, 671, 761, 825, 901 of reciprocal trigonometric functions, encoding and decoding with matrices,
for lessons, 10, 17, 24, 32, 40, 48, 67,
73, 80, 88, 98, 106, 113, 120, 141, 886 787
148, 155, 162, 165, 173, 181, 201, of relations, 60–61, 406–407 exercises, 87, 88, 90, 97, 120, 121,
208, 214, 223, 231, 239, 247, 255, scatter plot, 92–93, 125
264, 287, 295, 302, 310, 317, 324, of sine function, 851, 842–854 138, 139, 140, 151, 180, 213, 229,
330, 338, 345, 366, 373, 380, 388, solving exponential equations using, 265, 300, 337, 413, 418, 419, 420,
397, 404, 412, 420, 441, 450, 458, 440, 448, 449, 458, 460, 473, 483,
468, 476, 483, 505, 514, 523, 533, 470 485, 512, 513, 514, 522, 545, 546,
541, 548, 571, 577, 586, 593, 601, solving linear-quadratic system using, 600, 619, 634, 651, 685, 722, 737,
620, 629, 636, 644, 652, 660, 680, 771, 788, 797, 798, 799, 841, 847,
687, 693, 702, 718, 724, 730, 738, 259 864, 865, 866, 867, 867, 872, 873,
745, 770, 779, 780, 800, 808, 815, solving linear systems using, 134–141, 882, 888, 889, 917
834, 842, 850, 858, 867, 874, 882, exponential and logarithmic
890, 910, 918, 926, 934, 942, 950 166, 168 inequalities, 484–485
solving polynomial equations using, exponential equations, 470
Glossary, 994 exponential functions, 437, 444, 445,
296–302 446, 477
Got It? See assessment, Got It? solving quadratic equations using, ExpReg feature, 445, 460
extraneous solutions, 393
graph(s) 226–231 FLOAT feature, 568
of absolute value functions, 107–113 solving radical equations using, 416 hyperbola, 647, 648
of absolute value inequalities, 117 solving systems of inequalities using, INTERSECT feature, 136, 137, 163,
boundary lines, 107, 114, 115, 126 260, 299, 416, 511, 550, 864
of circles, 615, 632, 633 115, 117, 150–151, 185 INTERSECT option, 163
complex number plane, 250 of square root function, 416 inverse of a matrix, 786
of conic sections, 616, 621 of tangent function, 870 linear regression, 136
of cosine function, 861–863 in three-dimensions, 164–165, 166 linear systems, 178
of cube root function, 417 transformations. See transformations line of best fit, 94, 95
of cubic functions, 283–284 translations. See translations LINREG function, 94, 95, 136, 137,
of direct variation equations, 70 of trigonometric functions, 852 331, 334
drawing for problem solving, 857, turning point, 291 LIST feature, 590
862, 864, 865, 869, 873, 878, 881 using for problem solving, 6, 9, 13, 16, logarithmic equations, 472
of ellipse, 615 matrices, 771
end behavior of, 282–283 60, 61, 65, 70, 77, 78, 79, 83, 84, MAXIMUM feature, 292
of exponential functions, 434–435, 87, 93, 94, 100, 105, 106, 109, MODE feature, 506, 568
437 110, 111, 115, 116, 117, 118, 119, natural logarithmic function, 478
of exponential functions with base e, 121, 127, 129, 134, 141, 150, 152, oblique asymptotes, 524–525
442–450 153, 154, 155, 156, 158, 159, 161, parametric mode, 413
feasible region, 157, 185 162, 189, 190, 195, 196, 197, 199, piecewise function, 90
of hyperbola, 616, 647 200, 205, 210, 227, 229, 230, 246, polynomial equations, 298
of inequalities, 33, 35, 114–120, 126 250, 254, 257, 268, 275, 276, 286, polynomial functions, 282, 290, 332
of inverse functions, 413, 508, 912 289, 291, 319, 333, 337, 338, 340, polynomial inequalities, 317
of linear equations, 77, 83, 84, 124 341, 344, 345, 346, 352, 355, 356, polynomial models, 331, 333–334
of linear inequalities, 115, 117, 150– 373, 406, 414, 415, 416, 418, 419, quadratic equations, 227, 228
151 429, 430, 434, 437, 440, 441, 442, quadratic formula, 241
line of best fit, 94, 95, 125 443, 444, 449, 450, 454, 455, 468, quadratic functions, 195, 202
of logarithmic functions, 454–455 484, 493, 494, 499, 508, 509, 510, quadratic regression, 211
multiple zeros effect on, 291 514, 515, 519, 533, 541, 554, 559, quadratic systems, 260, 661
numbers on a number line, 13 593, 608, 609, 615, 616, 618, 623, QuadReg feature, 211, 331
ordered pairs, 60, 110, 121–122, 123, 624, 632, 633, 635, 640, 641, 642, rational functions, 506, 521
164 644, 647, 649, 655, 656, 657, 659, rational inequalities, 549, 550
of parabola, 194, 197, 268 664, 668, 669, 670, 716, 721, 723, real roots, 299
piecewise function, 90–91 740, 741, 742, 744, 789, 807, 811, reciprocal function, 509, 511
points on a coordinate plane, 164 833, 852, 871, 912 relative maximums or minimums, 292
of polynomial functions, 283–284, roots of polynomial equations, 315
291, 339–345 graphing calculator. See also calculator RREF function, 177–178, 332
of quadratic functions, 194–201 activity, 163 sequences, 568
of quadratic functions in standard area of a polygon, 785 sine curves, 852, 853
form, 202–208 box-and-whisker plots, 714 STAT EDIT feature, 714
of radical functions, 414–420, 426 CALC feature, 163, 202, 227 STAT feature, 95, 459, 460, 477
of rational functions, 506, 515–523 conic sections, 621
CONNECTED mode, 506
cosine equation, 864–865
CubicReg feature, 331, 332, 334
DTbl feature, 470
determinant of a matrix, 784
direct and inverse variation, 498–499,
500

1146

STAT PLOT feature, 459, 460, 477, 714 relating radians and degrees, 845 cofunction, 944, 945, 962 Index
sum of a series, 590 Remainder Theorem, 307 cotangent, 904, 960
systems of equations, 795 sum and differences of cubes, 298 domain of validity, 904
TABLE feature, 140, 202, 227, 437, vertical in-line test, 62 equations as, 28, 29, 52
multiplicative, 782, 819
871 Homework Online, 7–10, 15–17, Pythagorean, 906, 960
tangent function, 871 22–24, 30–32, 38–40, 46–48, reciprocal, 904, 960
TBLSET feature, 470, 472 65–67, 71–73, 78–80, 86–88, tangent, 904, 947, 960
Tmax feature, 413 96–98, 104–106, 111–113, 118– trigonometric, 904–910, 960
Tmin feature, 413 120, 138–140, 146–148, 153–154, verifying, 905, 906, 952–953
TRACE feature, 506 160–162, 171–173, 179–181, 199– writing without parentheses, 906
transformations, 110 201, 206–207, 212–214, 217–220,
trigonometric functions, 860, 886 221–223, 229–231, 237–239, 245– image, 801, 820
VALUE option, 163 247, 253–255, 262–264, 293–295,
VARS feature, 413, 460 301–302, 308–309, 316–317, 322– imaginary
vertex form, 236 324, 329–330, 335–337, 343–345, axis, 249, 250
Xmax feature, 413 364–366, 371–373, 378–380, 386– numbers, 248, 249, 272
Xmin feature, 413 388, 395–397, 402–404, 410–412, solutions to quadratic equations, 252,
YLIST feature, 477 418–420, 439–440, 447–449, 456– 272
ZERO option, 163, 227, 299 458, 466–467, 473–476, 481–483, unit, 248, 249
zeros of polynomial function, 321 503–505, 512–513, 521–523, 531–
ZOOM feature, 460 532, 539–540, 546–548, 569–571, inconsistent system, 137
575–576, 584–585, 591–593, 599–
Graphing Conic Sections, 621 601, 618–619, 627–629, 634–636, independent
642–644, 658–660, 678, 685–686, events, 688, 689, 753
Graphing Trigonometric functions, 860 691–692, 700–701, 715–716, 722, system, 137, 184
728, 735, 743, 768, 777, 778, 788, variable, 63
Greatest Common Factor (GCF), 218, 789, 797, 798, 806, 813, 832, 840,
297 841, 848, 849, 856, 864, 865, 872, index of the radical expression, 362,
880, 881, 888, 908, 916, 924, 932, 423
Greatest Integer Function, 90 939, 940, 948, 949, 955
industrial design, 531
Gridded Response horizontal
exercises, 32, 48, 56, 73, 130, 148, asymptotes, 518, 554 industry, 548
173, 190, 208, 247, 276, 317, 338, ellipse, 639
356, 380, 388, 430, 476, 483, 494, line, 75, 414 inequality(ies)
523, 560, 571, 601, 610, 636, 670, translation, 100–101, 102, 108, 415 absolute-value, 41–48, 117, 152
681, 693, 724, 811, 815, 929, 942 compound, 36–37, 43, 52
problems, 27, 175, 242, 307, 400, Horse Hollow Wind Energy Center, exponential, 484–485
447, 520, 574, 760, 790, 824, 900 342 graphing, 33, 35, 114–120, 126
linear, 114, 115, 117, 149–155, 166,
growth, exponential, 434–441, 488 hyperbola, 645–652 185
analyzing, 648 logarithmic, 484–485
growth factor, 435, 436, 488 axes of, 646 polynomial. See polynomial equations
branches of, 616, 646, 649 properties of, 34
guess and check, 286, 312 center of, 646, 666 quadratic, 256–257, 261, 272
conic sections, 614 rational, 550–551
H defined, 645, 666 real numbers as a solution, 36, 52
equation of, 647, 648, 655–656 solving, 33–40
half-angle identities, 951–957, 962 foci of, 645, 666 solving, as Big Idea, 49, 50, 133, 182,
graphing, 616, 647 183, 421, 422
half-plane, 114 horizontal, 655 systems of, 157, 166, 184, 185
reflection property, 649 writing, 33, 34, 117
Here’s Why It Works, 674, 928 translations, 655
angle difference identities, 946 vertical, 655 infinite
area of a triangle, 928 vertices of, 645, 646 geometric series, 594, 598
Change of Base Formula, 464 series, 587
cofunction identities, 944 I
double-angle identities, 951 initial
Fundamental Counting Principle, 674 i, imaginary number, 248, 249, 272 point, 809, 820
half-angle identities, 953 side of angles, 836, 894
inequalities, 34 Iberian lynx, 438
Law of Cosines, 937 integers, 12
length of an intercepted arc, 846 identity(ies)
logarithms, 462 additive, 14, 767 intercepted arc, 844, 845, 895
negative angle identities, 943 of angles, 943–956, 962
parabolas, 623 interdisciplinary
point-slope form, 81 agriculture, 222, 746
quadratic function, 203 archaeology, 448, 481
architecture, 234, 375, 592, 871
astronomy, 467, 636, 644, 857

Index i 1147

Index biology, 83, 84, 135, 482, 743, 866 J How Multiple Zeros Affect a Graph,
botany, 448 291
business, 159, 170, 180, 245, 263, joint variation, 501, 554
Identity and Multiplicative Inverse
522, 777 justifying steps, 39 Matrices, 782
chemistry, 39, 67, 147, 379, 457, 504
conservation, 71 justify your reasoning, 54 Infinite geometric series, 598
demography, 474 Inverse of a 2 3 2 Matrix, 785
design, 871 K Inverses of Three Trigonometric
earth science, 392, 628
economics, 140, 206, 402 Kepler’s Third Law of Orbital Motion, Functions, 911
finance, 51, 94, 172, 568, 571 383 Length of an Intercepted Arc, 846
geography, 849 Logarithm, 451
industrial design, 531 Key Concepts Matrix Addition and Subtraction,
industry, 548 Absolute Value, 41
language arts, 833 Absolute Value Functions, 107 764
meteorology, 475, 717, 720, 721 Area of a Triangle, 784 Matrix Multiplication, 774
music, 143, 476, 849, 857 Arithmetic Sequence, 572 Measures of Central Tendency, 711
oceanography, 440, 863 Basic Identities, 904 The (n 1 1) Point Principle, 332
packaging design, 286 Binomial Experiment, 731 Natural Logarithmic Function, 478
physics, 22, 212, 231, 245, 269, 273, Binomial Probability, 732 Normal Distribution, 739
Binomial Theorem, 733 The nth Root, 361
344, 345, 372, 387, 396, 419, 449, Box-and-whisker plots, 714 Number of Combinations, 676
483, 502, 504, 581, 600, 815, 989, Combined Variations, 501 Number of Permutations, 676
942 Completing The Square, 235 Parabolas, 622
psychology, 449 Complex Numbers, 249 Parallel Lines, 85
science, 466, 584 Composition of Functions, 400 Perpendicular Lines, 85
seismology, 474 Composition of Inverse Functions, 408 Point of Discontinuity, 516
social studies, 98 Compression, 195 Point-slope Form, 81
sociology, 736 Conditional Probability, 697 Power Functions, 341
zoology, 440, 740, 741 Conic Sections, 614 Probability of A and B, 688
Continuously Compounded Interest, Probability of A or B, 689
interest, continuously compounded, 446 Properties of Ellipses with Center
446, 488 Converting Between Radians and
Degrees, 845 (0, 0), 639
interpolation, 334 Cosecant, Secant, and Cotangent Properties of Hyperbolas with Centers
Functions, 883
interquartile range, 713 Cosine and Sine of an Angle, 838 (0, 0), 645
Determinants of 2 3 2 and 3 3 3 Properties of Rational Exponents, 383
INTERSECT feature, 136, 137, 163, 260, Matrices, 784 Pythagorean Identities, 906
299, 416, 511, 550, 864 Discriminant, 242 The Quadratic Formula, 241
Division Algorithm for Polynomials, Rational Exponent, 382
inverse 304 Reciprocal Family Function, 509
additive, 14 Ellipse, 638 Reflection, 195
of matrices, 782–790, 792–800, 819 Experimental Probability, 681 Roots, Zeros, and x-intercepts, 289
multiplicative, 13, 14 Exponential Growth and Decay, 436 Row Operations, 176
operations, 27 Factoring a Difference of Two Squares, Sampling Types and Methods, 725
relations, 405–413, 426 220 Scalar Multiplication, 772
solving trigonometric equations, 911– Factoring Perfect Square Trinomials, Slope, 74
918, 960 219 Slope-intercept Form, 76
variation, 498–505, 554 Families of Radical Functions, 415 Solutions of a Linear-quadratic System,
Family of Absolute Value Functions,
inverse functions, 405–413 108 258
composition of, 408, 409 Four Ways To Represent Relations, 60 Solving an Equation by Completing the
domain and range, 426 Function Operations, 398
exponential, 454 Fundamental Counting Principle, 674 Square, 235
graphing, 413, 508, 912 General Form of Absolute Value Square Root of a Negative Real
logarithmic, 451–458, 489 Function, 110
representative part, 911 General Form of the Reciprocal Family Number, 248
sine, 913, 931 Function, 507 Standard Form of a Linear Equation,
tangent, 913 Geometric Sequences, 580
trigonometric, 911–918 Horizontal Asymptote of a Rational 82
Function, 518 Standard Form of an Equation of a
Inverse Property of Real Numbers, 14
Circle, 630
inverse variation Function, 508 Standard Form of a Polynomial

irrational Function, 281
numbers, 12 Stretch, 195
roots, 314, 350 Study Methods, 726
Summation Notation and Linear

Functions, 589

1148

Sum of a Finite Arithmetic Series, 595 intercept of, 76 independent, 137, 184 Index
Tangent of an Angle, 868 parallel, 81, 85, 124 of inequalities, 157, 184
Theoretical Probability, 683 perpendicular, 81, 85, 124 linear term, 281
Transformations of a Parabola, 626 slope of, 74, 75, 124 with no unique solution, 145, 166,
Transforming a Circle, 632 vertical, 75, 107, 414
Translation of the Parabola, 197 writing equations of, 83 184
Trigonometric Ratios for a Circle, 919 x-intercept, 76, 289 solving algebraically, 142–155, 184
Trigonometric Ratios for a Right y-intercept, 76 solving by elimination, 144, 184
solving by graphing, 134–141, 150,
Triangle, 920 linear data, 215
Variables and Expressions, 5 166, 168
Variance and Standard Deviation, 719 linear equation(s) solving by substitution, 142–143, 169,
Vectors in Two Dimensions, 809 defined, 75
Vertex Principle of Linear direct variation, 68–73, 123 184
graphing, 77, 83, 84, 124 solving with calculators, 178
Programming, 158 line of best fit, 94–95 solving with matrix equations, 174–
Vertical Asymptotes of Rational matrix representation of, 174–178
modeling, 92–98 178
Functions, 517 for parallel lines, 85 solving with tables, 134–141, 149–150
Writing and Graphing Inequalities, 33 for perpendicular lines, 85
Piecewise Function, 90–91 line of best fit, 94, 95, 125
Know/Need/Plan problems, 20, 69, 77, point-slope form, 81–82, 83, 124
82, 94, 117, 136, 151, 205, 284, slope formula, 74 LIST feature, 590
289, 313, 320, 391, 400, 445, 471, slope-intercept form, 77, 83, 124
502, 511, 597, 615, 626, 631, 640, solving using calculators, 178 literal equation, 29
690, 713, 721, 732, 766, 803, 839, solving using matrices, 174–178, 186
847, 854, 863, 922, 954 standard form, 82–83, 124 logarithm(s)
systems of. See linear systems base ten, 453, 463
L for a trend line, 94 Change of Base Formula, 464
writing, 76 common, 453, 489
language arts, 833 defined, 451
linear factors, 288, 289 evaluating, 452
Law expanding, 463
of Cosines, 936–942, 961 linear functions, 74–80 natural, 478–483, 490
of Sines, 928–934, 961 defined, 75 properties of, 462–468, 472, 489
graphs of, 124 simplifying, 463
Least Common Denominator (LCD), slope-intercept form of, 76 using for exponential models, 477
535, 537
linear inequality(ies), 149–155 logarithmic
Least Common Multiple (LCM), 534– defined, 114 expressions, 451, 478–479
535 solving by graphing, 115, 117, 150– inequalities, 484–485
151, 185 scale, 453, 465
length solving using tables, 149–150
of an intercepted arc, 845, 846 systems of, 166, 185 logarithmic equation(s)
sides of triangles, 928–932, 935, 938, defined, 490
939 linear polynomial, 281 natural, 479
solving, 469–476
Lesson Check, 7, 15, 22, 30, 37, 45, 64, linear programming, 157–162
71, 78, 86, 96, 103, 111, 118, 138, activity, 163 logarithmic function(s)
145, 152, 160, 171, 179, 198, 206, constraints in, 157, 185 define, 454
212, 221, 229, 237, 244, 253, 261, defined, 157 families of, 455
285, 293, 300, 308, 315, 322, 328, vertex principle of, 158 graphing, 454–455
335, 342, 364, 370, 378, 385, 394, as inverses, 451–458, 489
401, 409, 418, 439, 447, 456, 465, linear-quadratic system, 258–264 natural, 478, 490
473, 480, 503, 512, 521, 530, 539, solving by graphing, 259 translations, 455
545, 568, 575, 583, 591, 598, 617, solving by substitution, 259
627, 634, 642, 650, 658, 678, 691, long division of polynomials, 302–304
700, 722, 728, 735, 743, 768, 777, linear regression, 94, 136, 352
787, 798, 806, 813, 831, 840, 846, look for a pattern, 522
855, 864, 861, 880, 887, 908, 915, linear regression (LinReg), 94
923, 932, 939, 948, 955 look for key words, 34, 72
linear system(s)
like radicals, 374, 424 absolute-value, 152 Looking Ahead Vocabulary, 1, 57, 131,
classifying, 137 191, 277, 357, 431, 495, 561, 611,
like terms, 21, 52 consistent, 137, 184 671, 761, 825, 901
defined, 134, 184
limits, 589, 605 dependent, 137, 184 M
equivalent, 142, 144, 168
line(s). See also linear equations inconsistent, 137, 184 magnitude, 809, 820
boundary, 107, 114, 115, 126
horizontal, 75, 414 major axis of an ellipse, 639, 665

make a drawing, 188

Index i 1149

Index manipulatives matrix 494, 608, 668–670, 758–760, 822–
number cubes, 694–695 element, 174, 175, 767 824, 899–900
protractor, 839 equation, 765, 792–783, 818 problems, 12, 18, 109, 112, 119, 146,
158, 169, 197, 222, 259, 314, 323,
mapping diagram, 60 MAXIMUM feature, 292 340, 370, 387, 396, 416, 452, 469,
501, 510, 513, 583, 589, 627, 632,
Math Behind the Scenes, 3, 59, 133, maximums or minimums 656, 690, 697, 712, 796, 838, 855,
193, 279, 359, 433, 497, 563, 613, amplitude of periodic functions, 830 912, 935, 937
763, 903 defined, 195
of functions, 158, 185, 195, 268, 291, multiple events, 688–693
mathematical modeling 292
exponential equations, 472 linear programming for finding, 157, multiple representations
exponential functions, 445, 477 158 with diagrams, 249, 683, 684, 690,
exponential growth and decay, 434– relative, 291 694, 695, 699, 712, 732, 741, 780,
441, 488 of tolerance, 44 835, 843, 847, 866, 920, 921, 922,
infinite series, 594 923, 928, 929, 930, 931, 935, 936,
inverse variation, 499 max-zero-min-zero-max pattern, 863 937, 938, 943, 944, 945, 946
linear equations, 92–98 with examples, 5, 26, 34, 41, 74, 176,
logarithmic equations, 472 mean 242, 248, 249, 341, 383
polynomial functions, 331–334, 352 arithmetic, 574, 604 exercises, 888, 925
quadratic functions, 209–213, 269 average, 574 with graphs, 33, 44, 60, 85, 107, 194,
trend line, 94 defined, 711 195, 197, 258, 291, 434, 478, 507,
formula for finding, 711, 754 622, 632, 638, 639, 645, 646, 714,
mathematical patterns, 564–571, 604. geometric, 583, 605 721, 734, 739, 740, 741, 742, 801,
See also patterns 802, 805, 809, 810, 811, 828, 829,
measurement 830, 837, 838, 851, 852, 854, 855,
mathematical quantity, 5 of angles, 836, 844–850 861, 862, 863, 864, 865, 875, 876,
in radians, 843, 844–850, 895 877, 878, 886, 905, 912, 913, 914,
MathXL for School, 25, 53, 89, 127, 915, 919, 920, 943, 944, 946
156, 187, 224, 273, 311, 353, 389, measures with photographs, 84, 785, 903
427, 461, 491, 526, 557, 570, 579, of central tendency, 711, 754 with tables, 60, 107, 694, 695, 696,
607, 637, 667, 710, 757, 791, 859, of variation, 719, 754 697, 698, 712, 713, 720, 721, 765,
927, 963 775, 796, 816, 879
median, 711, 754
matrices multiplication
adding, 764–770, 802, 818 Mental Math, 232, 396, 474, 619, 949 of binomials, 376
additive identity, 767 of complex numbers, 251
additive inverse, 767 Meteor Crater, 392 of conjugates, 377
coefficients, 175, 819 counting using, 674
constant, 176, 793, 819 meteorology, 475, 717, 720, 721 of exponents, 360
defined, 134, 174 of functions, 398, 399
determinants, 784, 819 Mid-Chapter Quiz. See assessment, inverse operation to, 27
dilation, 781, 802, 803 Mid-Chapter Quiz of logarithms, 462
dimensions of, 764 of matrices, 772–779, 818, 820
elements of, 174, 175 midline, 830 by minus one, 21
encoding with, 787 properties of exponents, 360
equal, 767, 818 minor axis of an ellipse, 639, 665 of radical expressions, 367–373, 423
inverse of, 766, 782, 790, 792, 800, of rational expressions, 528
819 Mixed Review. See assessment, Mixed scalar, 772–773, 802, 811, 812
multiplicative identity, 782, 819 Review of square roots, 225
multiplying, 772–779, 818, 820 by zero, 21
properties of, 776 mode, 711, 754
reflection, 801, 804, 805 Multiplication Property, 21
rotation, 801, 804 MODE feature, 506, 568 of Equality, 27
row operations, 176 of Inequality, 34
scalar multiplication, 772–773, 802, modeling of Square Roots, 225
818 algebraic expressions with words,
solving systems of equations using, 18–19 multiplicative identity, 14
174–178, 186, 795, 796 as Big Idea, 121, 122, 421, 486, 487, matrix, 782, 819
solving using scalars, 773–774 602, 603, 662, 663, 816, 817, 892, property, 773
square, 776, 782, 819 893
subtracting, 764–770, 818 with a cosine function, 863 multiplicative inverse
translations of geometric, 801–808, light waves with sine function, 855 of any nonzero number, 14
820 mathematical. See Mathematical defined, 13, 14
variable, 793, 819 modeling matrix, 782
writing vectors as, 810 trigonometric equations, 879 zero and the, 13
writing vertices as, 801 word problems, 274

Moiré pattern, 617

monomial, 280, 281

monomial function, 352

Multiple Choice
exercises, 54–56, 128–130, 188–190,
274–276, 354–356, 428–430, 492–

1150

Multiplicative Property of Zero, 773, roots of. See roots of numbers P Index
776 simplifying, 249
whole, 12 packaging design, 286
multiplicity of a zero, 291
number cubes, 694–695 parabola(s), 622–629
multi-step equations, 27 axis of symmetry, 194, 624
number line, 13 conic sections and, 614
music, 143, 476, 849, 857 defined, 194, 622, 664
numerical expression, 5 directrix, 622, 664
mutually exclusive events, 689, 690, equation of, 209, 623–624, 627
753 O focal length, 622, 664
graphing, 194, 197, 268
My Math Video, 3, 59, 133, 196, 279, objective function, 157, 185 maximums and minimums, 195
359, 433, 497, 563, 613, 673, 763, points on, 622–623
827, 903 oblique asymptotes, 524–525 properties of, 194, 268
transformations, 195, 626
N observational study, 727 translations, 197
vertex form, 194, 196, 198
The (n 1 1) Point Principle, 332, 333, oceanography, 440, 863
352 parallel lines, 81, 85, 124
one-step equations, 27
natural parameters, 266
logarithmic equation, 478–479 one-to-one function, 408
logarithms, 478–483, 490 parametric equations, 413
numbers, 12 online activities. See Dynamic Activities;
Homework Online; MathXL for Parent Function
natural base exponential functions, School; My Math Video; absolute value, 107, 108
446 PowerAlgebra.com; Problems defined, 99, 125
Online; Solve It!; vocabulary, exponential functions, 444
natural logarithmic function, 478 Vocabulary Audio Online logarithmic functions, 455
quadratic, 194
negative Open-Ended exercises, 8, 9, 24, 40, 47, radical functions, 415
angle identities, 943, 962 72, 105, 126, 127, 138, 140, 147, reciprocal functions, 509
numbers, 21 154, 160, 173, 179, 181, 187, 200,
square root, 248 213, 222, 224, 230, 254, 262, 273, Pascal, Blaise, 327
286, 294, 302, 311, 317, 323, 365,
Networks, 780–781 372, 379, 385, 387, 396, 403, 427, Pascal’s Triangle, 326–330
440, 449, 457, 475, 491, 505, 512,
Newton’s Law of Cooling, 483 513, 526, 531, 539, 545, 548, 557, patterns, 4–9. See also sequences
570, 579, 585, 592, 600, 607, 619, expressing with algebra, 5–6, 51
n factorial (n!), 675, 752 635, 643, 651, 660, 667, 678, 691, formulas to generate, 564, 565, 567
700, 701, 707, 710, 730, 736, 737, identifying, 4, 51
non-mutually exclusive events, 690 757, 791, 799, 821, 833, 840, 841, mathematical, 564–571, 604
842, 849, 855, 857, 859, 864, 881, Moiré, 617
non-removable discontinuity, 516 898, 909, 927, 933, 941, 956, 963
percentile
normal operations defined, 715
curve, 741 with complex numbers, 250 finding, 715
vectors, 812 with functions, 398–404, 425
inverse, 27 perfect square trinomial
normal distribution, 739–745 with rational expressions, 528, 534, defined, 220
analyzing, 742 535, 536 factoring, 219, 220, 234–235, 297
defined, 744, 756 row, 176, 185
sketching a normal curve, 741 period
opposites defined, 828
notation additive inverse, 14 of functions, 828, 893
alternative for radicals, 381 defined, 14 of sine curve, 852
factorial, 675 of a difference, 21 of tangent function, 869
function, 62, 63, 407 of an opposite, 21
summation, 589, 590, 605 of a product, 21 periodic
of real numbers, 14 cycle, 828–829
nth root, 361–366, 423 simplifying algebraic expressions, 21 data, 828–834, 894
of a sum, 21
nth term of a sequence, 565 periodic function(s)
ordered amplitude of, 830
number(s) pairs, 60, 110, 121–122, 123, 164 cycles of, 828–829
of combinations, 676 triples, 164, 167 defined, 828, 893
complex. See complex number(s) identifying, 829
imaginary, 249, 272 outliers, 712, 754
integers, 12 permutations, 674–680
irrational, 12 defined, 675
natural, 12
of permutations, 676
rational, 12, 377
real. See real numbers

Index i 1151

Index finding the number in n items, 675, PowerAlgebra.com, 2, 25, 53, 58, 89, 404, 410–412, 418–420, 439–440,
676 127, 132, 135, 156, 187, 192, 224, 447–449, 456–458, 466–467, 473–
273, 278, 311, 353, 358, 389, 427, 476, 481–483, 503–505, 512–513,
importance of order in, 677 432, 461, 491, 495, 526, 557, 562, 521–523, 531–532, 539–540, 546–
n factorial (n!), 675, 752 570, 579, 607, 612, 637, 667, 671, 548, 569–571, 575–576, 584–585,
710, 757, 762, 791, 826, 859, 902, 591–593, 599–601, 618–619, 627–
perpendicular lines, 81, 85, 124, 812 927, 963 629, 634–636, 642–644, 658–660
Solve It! 4, 11, 18, 26, 33, 41, 60, 68,
phase shift, 876 power function, 341, 352, 415 74, 81, 92, 99, 107, 114, 134, 142,
149, 157, 166, 174, 194, 202, 209,
physics, 22, 212, 231, 245, 269, 273, Power Property of Logarithms, 462 216, 226, 234, 241, 249, 259, 280,
344, 345, 372, 387, 396, 419, 449, 288, 296, 303, 312, 319, 326, 331,
483, 502, 504, 581, 600, 815, 898, powers of complex numbers, 265 339, 361, 367, 374, 381, 390, 398,
942 405, 414, 434, 442, 451, 462, 469,
prediction, 334 478, 498, 507, 515, 527, 534, 542,
Piecewise Function, 90–91 564, 572, 580, 587, 595, 614, 622,
preimage, 801, 820 630, 638, 645, 653, 674, 681, 688,
plane 696, 711, 719, 725, 731, 739, 764,
complex number, 249, 250, 272 principal (positive) root, 361, 369, 381, 772, 782, 801, 809, 828, 836, 844,
coordinate, 74, 114, 126, 164 423 868, 875, 883, 904, 919, 928, 936,
sketching solutions in a, 164, 185 943, 951
solutions as the intersection of, 166 Principles Think About a Plan, 9, 16, 17, 23, 31,
Fundamental Counting Principle, 674, 39, 46, 66, 72, 79, 87, 105, 112,
point(s) 675 119, 139, 147, 153, 161, 172, 180,
on a coordinate plane, 164 The (n 1 1) Point Principle, 332, 333, 199, 207, 213, 222, 230, 238, 245,
on parabolas, 622–623 352 254, 262, 286, 294, 301, 309, 316,
terminal, 809, 820 Vertex principle of linear programming, 323, 329, 336, 343, 365, 371, 379,
test, 115 158, 185 387, 395, 403, 411, 419, 440, 448,
turning, 282, 348 457, 466, 474, 481, 504, 512, 522,
probability, 681–688 532, 540, 546, 569, 575, 585, 592,
point of discontinuity, 516–517, 555, of A and B, 688, 753 599, 618, 628, 635, 643, 650, 659,
868 of A or B, 689, 753 709
as Big Idea, 750, 751
point-slope form, 81–82, 83, 124 binomial, 734, 735, 737 problem solving strategies
conditional, 696–702, 753 act it out, 675
polygons, area of, 784, 785 cumulative, 695 check for reasonableness, 236, 624,
of dependent events, 698, 753 785
polynomial(s), 312–317 distributions, 694–695 draw a diagram, 492, 657, 814, 841,
classifying, 281 experimental, 681–682, 752 921, 933, 938, 940
constructing using conjugates, 314 geometric, 684 draw a graph, 36, 857, 862, 864, 855,
defined, 280 of independent events, 753 869, 873, 878, 881
degree of, 280–284 of multiple events, 688–693 guess and check, 286, 312
dividing, 302–310, 349 of mutually exclusive events, 753 look for a pattern, 522
equivalent, 289 model, 705 look for key words, 34, 72
expansion, 326 normal distribution, 739–745, 753 make a conjecture, 295, 317, 420,
factoring, 288, 297 sample space, 752 578, 594, 651, 890
identities, 318 theoretical, 682, 683, 752 solve a simpler problem, 12, 18, 19,
standard form of, 281, 348 using combinatorics, 683 21, 23, 24, 25, 26, 28, 35, 36, 42,
49, 51, 52, 53, 63, 64, 69, 70, 76,
polynomial equation(s) probability distribution(s) 82, 84, 101, 124, 128, 167, 169,
with complex solutions, 319–322, 351 for binomial experiments, 734 170, 176, 177, 182, 204, 225, 236,
roots of. See roots of polynomial continuous, 739, 756 240, 241, 242, 244, 249, 250, 251,
equations defined, 694 252, 254, 256, 271, 272, 286, 290,
solving by factoring, 296–302, 349 discrete, 739, 756 297, 300, 304, 321, 328, 330, 342,
solving by graphing, 296–302 349, 354, 360, 365, 366, 367, 368,
writing, 232 probability model, 705 369, 370, 371, 372, 375, 376, 377,
378, 379, 382, 384, 385, 386, 387,
polynomial function(s), 280–287 problem solving 388, 389, 391, 392, 393, 394, 403,
behaviors of, 280 exercises, 7–10, 15–17, 22–24, 30–32, 404, 417, 422, 423, 424, 425, 426,
defined, 280 38–40, 46–48, 65–67, 71–73, 430, 436, 441, 447, 453, 463, 465,
degree of, 280–284 78–80, 86–88, 96–98, 104–106, 466, 469, 472, 478, 480, 481, 490,
end behavior, 282–283 111–113, 118–120, 138–140, 146– 493, 494, 501, 502, 505, 510, 516,
graphing, 283–284, 291, 339–345 148, 153–154, 160–162, 171–173, 517, 527, 528, 531, 532, 534, 536,
modeling, 331–334, 352 179–181, 199–201, 206–207, 212–
relative maximums or minimums, 291, 214, 217–220, 221–223, 229–231,
292, 348 237–239, 245–247, 253–255, 262–
standard form of, 281, 348 264, 293–295, 301–302, 308–309,
transformations, 339–345, 352 316–317, 322–324, 329–330, 335–
zeros of, 288, 289, 348 337, 343–345, 364–366, 371–373,
378–380, 386–388, 395–397, 402–
positive (principal) root, 361, 369, 381,
423

1152

537, 538, 539, 540, 542, 543, 555, 390–394, 399–401, 405–409, 415– of symmetry, 26, 34 Index
556, 565, 573, 574, 577, 581, 588, 417, 434–438, 443–447, 452–455, of tangent function, 870, 895
596, 597, 631, 633, 640, 647, 648, 463–466, 469–472, 478–479, 498– Zero-Product, 226, 270, 289, 349
649, 652, 656, 657, 807, 908 502, 508–511, 516–520, 527–530, of zeros, 13, 226, 270, 289, 349, 773
use a diagram, 39, 53, 54, 55, 56, 60, 534–538, 543–545, 565–568, 573–
62, 65, 238, 294, 302, 316, 355, 574, 580–583, 588–590, 596–598, proportionality
375, 378, 379, 395, 421, 449, 453, 615–617, 623–625, 631–633, 639– as Big Idea, 552, 553
532, 570, 594, 617, 628, 629, 631 641, 647–649, 654–657, 675–677, constant of, 341
use a graph, 6, 9, 13, 16, 60, 61, 65, 681–685, 688–691, 696–699, 712–
70, 77, 78, 79, 83, 84, 87, 93, 94, 715, 720–721, 727, 732–734, 740– psychology, 449
100, 105, 106, 109, 110, 111, 115, 742, 783–787, 792–797, 810–812,
116, 117, 118, 119, 121, 127, 129, 828–831, 837–839, 845–847, 852– Pull It All Together, 49, 121, 182, 266,
134, 141, 150, 152, 153, 154, 155, 855, 862–864, 869–871, 876–879, 346, 421, 486, 552, 602, 662, 750,
156, 158, 159, 161, 162, 189, 190, 884–887, 905–907, 913–915, 920– 816, 892, 902, 958
195, 196, 197, 199, 200, 205, 210, 923, 929–931, 944–948, 952–954
227, 229, 230, 246, 250, 254, 257, Pythagorean identities, 906, 950
268, 275, 276, 286, 289, 291, 319, product matrix, 774, 776
333, 337, 338, 340, 341, 344, 345, Pythagorean Theorem, 640, 666, 809,
346, 352, 355, 356, 373, 406, 414, Product Property 906
415, 416, 418, 419, 429, 430, 434, of Exponents, 462
437, 440, 441, 442, 443, 444, 449, of Logarithms, 462 Q
450, 454, 455, 468, 484, 493, 494,
499, 508, 509, 510, 514, 515, 519, proof, 32, 629 quadratic
533, 541, 554, 559, 593, 608, 609, data, 215
615, 616, 618, 623, 624, 632, 633, properties formula, 240–248, 271
635, 640, 641, 642, 644, 647, 649, of addition, 27, 34, 143, 374, 767 inequalities, 256–257
655, 656, 657, 659, 664, 668, 669, additive identity, 13, 767 polynomial, 281, 314
670, 716, 721, 723, 740, 741, 742, additive inverse, 767 regression, 211
744, 789, 807, 811, 822, 842, 871, angles, 943, 946, 947, 951 systems, 258–264, 271
912 asymptotes, 518 trinomials, 196–197, 198, 218–219,
use a matrix, 174, 175, 176, 177, 179, as Big Idea, 49, 50, 346 297
180, 181, 186 change of base formula, 464
use a spreadsheet, 200, 504, 524 cofunction identities, 944 quadratic equation(s)
use a table, 5, 6, 8, 9, 51, 54, 60, 61, combining radicals, 367, 369, 374 complex number solutions, 248–255,
65, 67, 68, 69, 72, 92, 93, 94, 95, of cosine function, 862, 895 271
96, 97, 98, 100, 102, 104, 106, of division, 27, 34, 225, 369 discriminant, 242, 243, 244, 271
121, 123, 125, 127, 130, 135, 136, of ellipse, 639 imaginary solutions, 272
138, 149, 150, 153, 159, 160, 211, of equality, 26–27, 34, 143, 946 parabola, 623
212, 213, 215, 229, 230, 232, 256, of exponential functions, 442–450, perfect square trinomials, 219, 220,
257, 259, 261, 262, 263, 266, 269, 488 234–235, 271
270, 283, 286, 287, 335, 336, 337, of exponents, 360 solving by completing the square,
338, 352, 405, 406, 410, 434, 437, factoring differences of squares, 220, 233–239, 271
442, 443, 444, 448, 454, 455, 458, 297 solving by factoring, 218–219, 226–
460, 473, 474, 477, 485, 493, 499, of hyperbola, 645 231, 270, 271
503, 508, 526, 533, 544, 551, 554, of inequalities, 34 solving by graphing, 226–231
565, 570, 584, 585, 609, 615, 616, of inequality, 34 solving using imaginary numbers, 252,
636, 644, 660, 669, 720, 723, 769, of logarithms, 462–468, 472, 489 272
778 of matrices, 767, 773, 776, 804 solving using tables, 227
use substitution, 142, 143, 144, 231, of multiplication, 13, 27, 34, 225, 367, solving using the quadratic formula,
259 773 240–248, 271
work backwards, 362 multiplicative identity, 773 standard form of, 270
nth Roots of nth Powers, 363 Zero-Product Property, 226, 270, 288,
Problems Online, 4–7, 12–14, 18–22, operations with vectors, 811 289, 349
27–30, 34–37, 42–46, 61–64, of parabolas, 194, 268
69–70, 71, 75–77, 82–85, 93–95, of a product matrix, 776 quadratic expressions
100–103, 108–110, 115–117, 135– of quadratic functions, 203 factoring, 216–223, 270
137, 142–145, 150–152, 158–159, of rational exponents, 383 perfect square trinomials, 219, 220
167–170, 175–178, 195–198, 204– of real numbers, 11–17, 51
205, 209–211, 217–220, 226–228, reflexive, 26 quadratic function(s)
233–236, 241–244, 249–252, 259– relating radians and degrees, 844 defined, 194
261, 285–287, 293–295, 301–307, for simplifying, 21 graphing, 194–201
313–315, 320–321, 332–334, 339– of sine function, 853, 894 graphing standard form of, 202–208
342, 367–370, 375–377, 381–385, square roots, 225 maximum and minimum values, 268
of subtraction, 21, 27, 34, 369 modeling, 209–213, 269
of summation, 587, 588 standard form of, 202–208, 268, 269
transformations, 194–201
Index
i 1153

Index vertex form, 194, 196–197, 198, 203, radical symbol, 225 53, 62, 64, 66, 70, 71, 72, 73, 75,
204, 268 76, 79, 82, 87, 89, 93, 97, 102,
radicand, 362, 423 103, 105, 108, 112, 116, 117, 118,
Quadratic Parent Function, 194 119, 135, 138, 139, 140, 145, 152,
radius 157, 158, 162, 167, 168, 169, 171,
quadratic-quadratic systems, 260 circle, 630, 633, 665 176, 177, 195, 200, 205, 210, 212,
unit circle, 838, 894 214, 215, 219, 221, 223, 224, 228,
QuadReg feature, 211, 331 229, 230, 235, 242, 244, 246, 249,
random sample, 725, 755 255, 261, 263, 284, 285, 286, 287,
quantity, 5 290, 299, 300, 305, 308, 309, 310,
range 311, 315, 316, 321, 323, 324, 328,
quartic polynomial, 281, 341 of a data set, 713, 754 329, 334, 335, 337, 341, 344, 345,
interquartile, 713 353, 360, 362, 364, 365, 366, 369,
quartic regression, 352 of relations, 61, 62, 123, 124 370, 372, 375, 377, 379, 382, 385,
387, 389, 393, 396, 397, 401, 403,
quartile, 713, 754 rate of change, 437 406, 409, 411, 417, 435, 437, 438,
439, 440, 445, 447, 454, 456, 459,
Quick Review, 51–52, 121–126, 182– ratio(s) 465, 466, 470, 473, 475, 480, 502,
185, 266–272, 346–352, 421–426, common, 581, 605 505, 506, 508, 513, 520, 521, 522,
486–490, 552–556, 602–606, 662– constant, 74 525, 526, 529, 530, 532, 535, 540,
666, 752–756, 818–820, 894–897, of outcomes, 682 544, 545, 549, 557, 567, 571, 574,
960–962 trigonometric, 868, 919, 926, 961 576, 582, 588, 591, 592, 598, 600,
607, 615, 621, 624, 629, 632, 634,
quintic polynomial, 281 rational 637, 640, 642, 647, 651, 656, 659,
inequalities, 550–551 660, 667, 676, 678, 679, 683, 685,
Quotient Property of Logarithms, 462 numbers, 12, 377 686, 690, 693, 694, 695, 697, 700,
701, 710, 714, 717, 721, 722, 724,
quotients, 251, 369, 462 rational equation(s) 726, 728, 736, 742, 743, 744, 745,
solving, 542–548, 556 754, 765, 769, 774, 783, 787, 790,
R systems with, 549 791, 797, 799, 802, 803, 806, 810,
829, 831, 833, 839, 891, 892, 843,
radian(s) rational exponent(s) 847, 848, 849, 859, 864, 856, 871,
converting to and from degrees, 844, converting to radical form, 381–382, 874, 879, 881, 884, 885, 887, 889,
845 424 890, 897, 905, 913, 917, 923, 925,
cosine, 845 simplifying expressions with, 381–388 927, 933, 937, 941, 942, 945, 948,
defined, 844 950, 954, 956
measuring in, 844–850, 895 rational expression(s) Here’s Why It Works, 34, 62, 81, 203,
reciprocal trigonometric functions, adding, 534, 535, 556 298, 307, 464, 623, 674, 845, 846,
883–885 complex fractions, 537 928, 937, 943, 944, 946, 951, 953
defined, 527 Think / Write problems, 6, 35, 101,
radical conjugates, 377 dividing, 529 108, 144, 168, 176, 198, 220, 234,
Least Common Denominator, 535, 537 243, 252, 259, 300, 328, 364, 368,
radical equations Least Common Multiple, 534–535 376, 384, 417, 437, 465, 519, 529,
defined, 390 multiplying, 528 538, 543, 566, 582, 588, 648, 654,
solving, 390–397, 416, 425 operations with, 528, 534, 535, 536 677, 682, 698, 786, 795, 830, 870,
simplifying, 527–533, 535, 536, 555 879, 884, 907, 914, 930, 938, 948
radical expression(s) subtracting, 536, 556
binomial, adding and subtracting, reciprocal, 14
374–380 rational function(s) identities, 904, 960
binomial, multiplying and dividing, asymptotes, 517–518
376 defined, 515 reciprocal function(s)
combining, 374 graphing, 506, 515–523 defined, 507
conjugates, 377 point of discontinuity, 516–517, 555 family of, 507–514, 554
dividing, 367–373, 423 graphing, 507–511, 554, 886
equivalent form of, 381 rationalizing the denominator, 369, transformations, 508
index of, 362, 423 370, 377, 423 translations, 509, 510
like radicals, 374, 424 trigonometric, 883–890, 897
multiplying, 367–373, 423 Rational Root Theorem, 312, 313, 350
principal root, 361, 423 recursive formula, 565, 566, 604, 605
radicand, 362, 423 real-number roots, 362–363
rational exponents and, 381–388 reduced row echelon form, 177
rationalizing the denominator, 369, real number(s)
370, 377, 423 absolute value of, 41 Reference
simplifying, 363, 368–370, 376, 423 graphing, 11, 13 Formulas of Geometry, 993
square roots, 361–368, 377 negative, 248 Glossary, 994
ordering, 13
radical function(s), 414–420, 426 properties of, 11–17, 51
square root of a negative, 248, 249
radical(s) subsets of, 12
combining, 367, 369, 374, 384
like, 374, 424 Reasoning
simplest form of, 368 exercises, 6, 9, 13, 14, 15, 16, 19, 23,
25, 29, 30, 36, 37, 40, 44, 45, 48,

1154

Math Symbols, 986 of matrices, 801, 804, 820 808, 842, 850, 858, 867, 874, 882, Index
Measures, 985 measures of angles, 836 910, 926, 934, 950, 957
Properties and Formulas, 987–992 of vectors, 810
show your work, 54
reflection matrix, 804 row operation, 176, 185
shrink. See compression
reflection(s) S
absolute value functions, 108 sigma
defined, 101, 125 sample(s), 725–730 standard deviation, 719
of ellipse, 640 bias in, 751, 755 summation, 589
exponential functions, 434, 444 convenience, 725
geometric, 801, 805, 820 defined, 725, 755 simplest form
of hyperbola, 649 random, 725, 755 absolute value functions, 107
logarithmic functions, 455 self-selected, 725 defined, 527
quadratic functions, 195 systematic, 725 of functions, 99, 125
reciprocal function, 509 of radical expressions, 368
sample proportion, 747 of rational expressions, 527, 535, 536,
Reflexive Property of Equality, 26 555
sample space writing expressions in, 385
regression defined, 682
cubic, 331–334 equally likely, 682, 752 simplifying
exponential, 459 expressions, 21, 52
linear, 94–95, 333 scalar multiplication, 772–773, 802, imaginary numbers, 249
logarithmic, 459 811, 812, 818 logarithms, 463
polynomial, 331–334 numbers, 249
quadratic, 211, 331 scalars, 772, 773–774 problem solving strategy, 12, 18, 19,
21, 23, 24, 25, 26, 28, 35, 36, 42,
relation(s), 60–65 scatter plots, 92–93, 125 49, 51, 52, 53, 63, 64, 69, 70, 76,
defined, 60, 123 82, 84, 101, 124, 128, 167, 169,
domain of, 61, 62, 123 science, 466, 584 170, 176, 177, 182, 204, 225, 236,
graphing, 60–61, 406–407 240, 241, 242, 244, 249, 250, 251,
inverse, 405–413 secant function, 883, 897 252, 254, 256, 271, 272, 286, 290,
mapping diagram, 60–61 297, 300, 304, 321, 328, 330, 342,
range of, 61, 62, 123 seismology, 474 349, 354, 360, 365, 366, 367, 368,
table of values, 60–61 369, 370, 371, 372, 373, 375, 376,
self-selected sample, 725 377, 378, 379, 382, 384, 385, 386,
Remainder Theorem, 307, 349 387, 388, 389, 391, 392, 393, 394,
sequences 403, 404, 417, 422, 423, 424, 425,
removable discontinuity, 516 arithmetic, 572–577, 604 426, 430, 436, 441, 447, 453, 463,
defined, 564, 604 465, 466, 469, 472, 478, 480, 481,
Review explicit formulas, 565, 567, 604, 605 490, 493, 494, 501, 502, 505, 510,
Square Roots and Radicals, 225 Fibonacci Sequence, 578 516, 517, 527, 528, 531, 532, 534,
Properties of Exponents, 360 finite, 587 536, 537, 538, 539, 540, 542, 543,
Special Right Triangles, 835 geometric, 580–586, 605 555, 556, 565, 573, 574, 577, 581,
infinite, 587 588, 596, 597, 631, 633, 640, 647,
right triangles nth term, 565 648, 649, 652, 656, 657, 907, 908
side lengths of, 920, 922 recursive formula, 565, 566, 604, 605 radical expressions, 363, 368–370,
special, 835 terms of, 564, 568 376, 423
trigonometric ratios for, 919–926 rational exponents, 381–388
series rational expressions, 528
roots of numbers analyzing, 598
cube, 361, 362, 417 arithmetic, 587–595, 605 simulations, 682, 731, 752
nth, 361–366, 423 defined, 587, 604, 605
principal, 361, 381, 423 finite, 587, 595, 596 sine, of angles, 838, 839, 845, 894
real, 362–363 geometric, 594, 595–601, 606
square. See Square root(s) infinite, 587, 594, 598 sine curve, 852, 895, 944
writing equations from, 232 sum of a, 589, 590
sine function, 851–858
roots of polynomial equations Short Response exercises, families of, 877
Conjugate Root Theorem, 350 end of chapter problems, 56, 130, graphing, 851, 852–854
Descartes’ Rule of Signs, 315, 350 190, 276, 356, 430, 494, 560, 610, inverse, 913
determining the number of, 319–322 670, 760, 824, 900, 966 modeling light waves, 855
finding by graphing, 299 end of lesson problems, 10, 17, 24, properties of, 853, 894
identifying, 314 67, 80, 88, 98, 106, 113, 155, 162, radian measures, 851
irrational, 314, 350 181, 201, 231, 255, 264, 287, 302, translations, 875, 882, 896, 944
theorems of, 312, 313, 350 324, 345, 373, 397, 404, 420, 450, unit circles and, 851
writing equations from, 232 458, 468, 505, 514, 533, 541, 586,
593, 620, 644, 652, 660, 680, 687, Skills Handbook
rotation 702, 718, 730, 745, 770, 779, 800, Area and Volume, 972
center of, 804
Index i 1155

Index Bar and Circle Graphs, 973 630, 638, 645, 653, 674, 681, 688, 395, 396, 419, 438, 440, 445, 448,
Descriptive Statistics and Histograms, 696, 711, 719, 725, 739, 764, 772, 449, 456, 457, 465, 466, 467, 471,
782, 792, 801, 809, 828, 836, 844, 474, 475, 476, 480, 481, 482, 483,
974 851, 861, 868, 875, 883, 904, 911, 484, 502, 504, 512, 520, 521, 530,
Factoring and Operations With 919, 928, 936, 943, 951 531, 532, 540, 547, 548, 582, 584,
592, 600, 619, 625, 628, 635, 636,
Polynomials, 965 Solving Equations, as Big Idea, 266, 641, 643, 649, 650, 651, 660, 717,
Operations With Exponents, 976 267, 346, 347, 421, 422 720, 721, 736, 737, 740, 741, 744,
Operations With Fractions, 977 795, 807, 833, 849, 851, 860, 865,
Operations With Rational Expressions, Solving Equations and Inequalities, as 868, 873, 881, 883, 912, 919, 920,
Big Idea, 49, 50, 133, 182, 183, 925, 926, 944, 951
978 421, 422
Percent and Percent Applications, step function, 90
space, graphing points in, 164
979 stretch
Ratios and Proportions, 980 special right triangles, 835 absolute value functions, 108, 109
Scientific Notation and Significant exponential functions, 434, 444
spreadsheets, using for problem logarithmic functions, 455
Digits, 981 solving, 200, 504, 524 quadratic functions, 195
Simplifying Expressions With Integers, reciprocal function, 509
square(s) vertical defined, 102, 125
982 completing the, 233–239
The Coordinate Plane, Slope, and difference of, factoring, 220, 297 study methods
perfect square trinomial, 219, 220, controlled experiment, 726
Midpoint, 983 297 observational study, 726
The Pythagorean Theorem and the survey, 726
square matrix, 776, 782, 819
Distance Formula, 984 substitution
square root(s) solving linear-quadratic system using,
slope of a negative real number, 248, 249 259
defined, 74, 124 principal root, 361 solving linear systems using, 144
formula for, 74, 124 properties of, 225 solving systems of equations using,
horizontal lines, 75 radical expressions, 361–368, 377 142–143, 166–173
negative, 75 solving equations using, 233, 271, solving three-variable systems, 186
of parallel lines, 85 390–397, 425
of perpendicular lines, 85 Substitution Property of Equality, 26
positive, 75 square root function, 414–420, 426
undefined, 75 subtraction
vertical lines, 75 standard deviation, 719–724 of algebraic expressions, 21
zero, 75 defined, 719 of complex numbers, 250
formula for, 711, 719 definition of, 21
slope-intercept form, 74–80 using to describe data, 721 of functions, 398, 399
defined, 76 z-score, 748 inverse operation to, 27
writing equations in, 74–80, 83, 124 of matrices, 764–770, 818
standard form of radical expressions, 374–380
social studies, 98 defined, 202 of rational expressions, 536, 556
of equation of circle, 630, 665 of vectors, 811
sociology, 736 of equation of ellipse, 639
of equation of hyperbola, 647 Subtraction Property, 21
The Soldier’s Reasonable Request, of equation of parabola, 209 of Equality, 27
596, 597 of linear equations, 82–83, 124 of Inequality, 34
of polynomial functions, 281, 348
solution(s) of quadratic equation, 270 Sumatra, earthquake in, 453
of absolute value statements, 44 of quadratic functions, 202–208, 268,
complex, 248–255, 319–322, 351 269 summation. See also addition
extraneous, 42, 43, 393 of cubes, 297
imaginary, 252, 272 Standardized Test Prep. See assessment, notation, 589, 590, 605
as the intersection of planes, 166 Standardized Test Prep opposite of a, 21
to polynomial equations. See of a series, 588, 589, 590, 596, 598,
polynomial equations standard position of angle, 836, 837, 605, 606
to quadratic equations. See quadratic 894
equations Sum of a Finite Arithmetic Series, 587
real numbers as, 36, 52 STAT CALC feature, 720
to systems of equations. See systems surveys, 725–730
of equations STAT EDIT feature, 714, 720 analyzing questions in, 727
defined, 727
Solve It! 4, 11, 18, 26, 33, 41, 60, 68, STAT feature, 95, 459, 460, 477 designing, 727
74, 81, 92, 99, 107, 114, 134, 142,
149, 157, 166, 174, 194, 202, 209, STAT PLOT feature, 459, 460, 477, 714 symbols
216, 226, 233, 240, 248, 258, 280, absolute value, 36, 44
288, 296, 303, 312, 319, 326, 331, STEM, 16, 22, 31, 39, 66, 67, 84, 135, inequality, 13
339, 361, 367, 374, 381, 390, 398, 147, 148, 172, 199, 205, 207, 210,
405, 414, 434, 442, 451, 462, 469, 212, 229, 231, 234, 244, 245, 246,
478, 498, 507, 515, 527, 534, 542, 294, 323, 333, 342, 344, 345, 365,
564, 572, 580, 587, 595, 614, 622, 372, 375, 379, 383, 387, 388, 392,

1156

radical, 225 Take Note Find Needed Information, 128, 274, Index
sigma, for standard deviation, 719 Concept Summary, 21, 44, 83, 102, 558, 608, 668
sigma, for summation, 589 297, 322, 435, 444, 455, 653–654,
square root, 225 655, 877 Justify Your Reasoning, 54
summation, 505, 589, 590 Family(ies). See Family(ies) Make a Drawing, 188
Key Concepts. See Key Concepts Model Word Problems, 274
Symmetric Property of Equality, 26 Polynomial Factoring Techniques, 297 Show Your Work, 54
Properties. See Properties Simplify, 354
symmetry, axis of, 107, 194, 622, 646 Solutions of Absolute Value tips for success, 54, 128, 188, 274,
Statements, 44
synthetic division, 303, 306–307, 349 Theorems. See Theorems 354, 428, 492, 558, 608, 668, 758,
Transformations of f(x), 102 822, 899
systematic sample, 725 Translating Horizontal and Vertical Use a Formula, 558
Ellipses, 653–654 Use What You Know, 274, 558
systems of equations Translating Horizontal and Vertical
defined, 134, 184 Hyperbolas, 655 Test-Taking Tips, 54, 128, 188, 274,
linear. See linear systems Writing Equations of Lines, 83 354, 428, 492, 558, 608, 668, 758,
linear-quadratic, 258–264 822, 899
with no unique solution, 145, 166, tangent
184, 795 of angle, 868, 896 Theorems
quadratic, 258–264, 272 curve, 944 Binomial Theorem, 326–330, 351
solving by elimination, 144, 166–173, identity, 904, 947, 960 Conjugate Root Theorem, 314, 350
184, 186 Descartes’ Rule of Signs, 315, 350
solving by substitution, 166–173, 186 tangent function, 868–874 Factor Theorem, 290
solving using matrices, 174–178, 186, graphing, 870 Fundamental Theorem of Algebra,
795, 796 inverse, 913 319–324, 351
in three-variables, 166–173, 186 properties of, 870, 894 Law of Cosines, 936–942, 961
writing as a matrix equation, 793, 794 translations, 944 Law of Sines, 929, 961
Pythagorean Theorem, 640, 666, 809,
systems of linear inequalities, 149– TBLSET feature, 470, 472 906
155, 157, 166, 184, 185 Rational Root Theorem, 312, 313,
technical writing, 525 350
systems with rational equations, 549 Remainder Theorem, 307, 349
Technology
T Fitting Curves to Data, 459–460 theoretical probability, 682, 683, 752
Graphic Conic Sections, 621
table(s) Graphing Inverses, 413 Think About a Plan exercises, 9, 16,
finding conditional probability using, Graphing Rational Functions, 506 17, 23, 31, 39, 46, 66, 72, 79, 87,
696–697 Graphing Trigonometric Functions, 105, 112, 119, 139, 147, 153, 161,
identifying direct variation from, 68–69 860 172, 180, 199, 207, 213, 222, 230,
for problem solving, 5, 6, 8, 9, 51, 54, Linear Programming, 163 238, 245, 254, 262, 286, 294, 301,
60, 61, 65, 67, 68, 69, 72, 92, 93, Oblique Asymptotes, 524–525 309, 316, 323, 329, 336, 343, 365,
94, 95, 96, 97, 98, 100, 102, 104, Rational Inequalities, 550–551 371, 379, 387, 395, 403, 411, 419,
106, 121, 123, 125, 127, 130, 135, Using Logarithms for Exponential 440, 448, 457, 466, 474, 481, 504,
136, 138, 149, 150, 153, 159, 160, Models, 477 512, 522, 532, 540, 546, 570, 575,
211, 212, 213, 215, 229, 230, 232, 585, 592, 599, 619, 628, 635, 643,
256, 257, 259, 261, 262, 263, 266, term(s) 650, 659, 679, 686, 692, 701, 709,
269, 270, 283, 286, 287, 335, 336, in algebraic expressions, 20 716, 723, 729, 736, 744, 769, 778,
337, 338, 352, 405, 406, 410, 434, in arithmetic sequences, 604 789, 798, 807, 814, 841, 849, 857,
437, 442, 443, 444, 448, 454, 455, constant, 20 865, 873, 881, 888, 908, 917, 925,
458, 460, 473, 474, 477, 485, 493, defined, 20, 564 933, 940, 949, 955
499, 503, 508, 526, 533, 544, 551, degree of, 281
554, 565, 570, 584, 585, 609, 615, in geometric sequences, 580 Think/Write problems, 6, 35, 101, 108,
616, 636, 644, 660, 669, 720, 723, like, 21, 52 144, 168, 176, 198, 220, 234, 243,
769, 778 in polynomials, 281 252, 259, 300, 328, 364, 368, 376,
representing relations, 60 in sequences, 568, 604 384, 417, 437, 465, 519, 529, 538,
solving exponential equations using, 543, 566, 582, 588, 648, 654, 677,
470 terminal 682, 698, 786, 795, 830, 870, 879,
solving quadratic equations with, 227 point, 809, 820 884, 907, 914, 930, 938, 948
solving systems of equations using, side of angles, 836, 894
134–141, 149–150 three-dimensional coordinate space,
using to find patterns, 6 test point, 115 164–165

TABLE feature, 140, 202, 227, 437, Test-Taking Strategies three-variable system of equations,
871 Answering Word Problems, 274 166–173, 186
Check Your Answers, 608
Table of Values, 60 Draw a Diagram, 492 Tips for Success, 54, 127, 188, 274,
354, 428, 492, 558, 608, 668, 758,
822, 899

Index i 1157

Index tip-to-tail method, 811 tangent function, 944 cosine of angle, 861
types of, 108 defined, 838
Tmax feature, 413 using matrix addition, 802 inverse trigonometric functions,
vertical, 100, 102, 108, 109, 125, 415,
Tmin feature, 413 912–913
639 sine of angle, 851
tolerance, 44 writing, 877, 879 tangent of angle, 860
to verify identity, 906
TRACE feature, 506 transverse axis, 645, 646
V
transformation(s) tree diagrams, 699, 732
as Big Idea, 816, 817 variable(s)
of circles, 632 trend line, 93–94, 125 as Big Idea, 49, 50, 346, 602,
combining, 103 603
compressions, 102, 108, 109, 125, triangles clarifying, 12
195, 434, 444, 455 angles of, 928–932, 935, 936–939 classifying, 12
defined, 99 area of, 784, 928, 929 dependent, 63
dilation, 801, 802, 803, 820 congruent, 935 independent, 63
of exponential functions, 443–444 finding height of, 921
of functions, 99–106 finding sides of, 928–932, 935, 936– variable
geometric, 801–808, 820 939 matrix, 793, 819
identifying, 110 Law of Cosines, 936–942 quantity, 5
image, 801 Law of Sines, 928–934
of parabolas, 195, 626 right, 919–926 variance, 719, 720, 754
of polynomial functions, 339–341, 352 special right, 835
preimage, 801, 820 variation
of quadratic functions, 194–201 trigonometric combined, 500, 501, 502
of reciprocal functions, 508 expressions, 904, 907 constant of, 68, 123, 554
reflection, 101, 108, 125, 195, 434, ratios, 919–926, 961 direct, 68–73, 123, 498–499
444, 455, 509, 801, 805, 820, 944 inverse, 498–505, 554
rotation, 801, 804, 810, 820, 836 trigonometric equations joint, 501, 554
stretches, 102, 108, 109, 125, 195, modeling with, 879 measure of, 719, 754
434, 444, 455, 509 solving by factoring, 914
translations. See translations solving using angle identities, 935 VARS feature, 413, 460
types of, 125, 801 solving using inverses, 911–919, 960
trigonometric identities vs., 914 vectors, 809–815
Transitive Property adding, 811
of Equality, 26, 946 trigonometric functions defined, 809
of Inequality, 34 graphing, 860, 886 direction of, 809, 820
inverse, 911–918 dot product, 812
translations. See also transformations non-additive, 947 initial point, 809, 820
of circles, 631 reciprocal, 883–890, 896 magnitude of, 809, 820
compressions, 102, 108 normal, 812
of conic sections, 653–660, 666 trigonometric identities, 904–910 operations with, 811
cosine function, 875–882, 896, 944 additive, 943 representing, 810
defined, 99 angles of, 943 rotating, 810
dilations, 801, 802, 803, 820 cotangent, 904 scalar multiplication of, 811
of ellipses, 653–654 defined, 904, 960 size and direction, 809
equations of, 631 domain of validity, 904 subtracting, 811
of exponential functions, 444 reciprocal, 904 terminal point, 809, 820
geometric, 801–808, 820 tangent, 904 in two dimensions, 809
graphing, 108, 125, 126, 196, 510, trigonometric equations vs., 914 writing in matrix form, 810
861, 876–878 verifying, 907
horizontal, 100–101, 102, 108, 125, vertex(ices)
415 trinomial of absolute value functions, 107,
of hyperbola, 655 defined, 281 126
image, 820 perfect square, 219, 220, 234–235, defined, 107
of logarithmic functions, 455 271, 297 of ellipse, 639, 665
of parabola, 197 quadratic, 196–197, 198, 218–219, of feasible region, 158, 185
phase shifts, 876 297 of hyperbola, 645, 646
of quadratic functions, 195, 196 matrix representation of, 801
of reciprocal functions, 509, 510 turning point, 282, 348 of parabolas, 194, 196
rotation, 801, 804, 810, 820, 836
sine and cosine graphs, 861 Twain, Mark, 228 vertex form of quadratic functions,
sine function, 875–882, 896, 944 194, 195, 196–197, 198, 203, 204,
square root function, 415 U 236, 268

uniform distribution, 694

unit circle
angles in, 836–842, 894

1158

Vertex Principle of Linear W x-coordinate, 61, 62, 74, 123, 124, 203, Index
Programming, 158, 185 416
Whispering Gallery, 641
vertical x-intercept, 76, 243, 289, 291, 319, 348
asymptotes, 518, 554 whole numbers, 12 Xmax feature, 413
ellipse, 639 Xmin feature, 413
line, 75, 107, 414 words modeling algebraic xy-coordinate plane, 164
translation, 102, 108, 109, 125, 195, expressions, 18–19
415 Y
work backwards, 362
Vertical-line test, 62–63, 414 y-axis, 101, 102, 164
writing y-coefficient, 175
vocabulary equations, 74–80, 83, 124, 232, 452 y-coordinate, 61, 62, 74, 123, 124, 203
Chapter Vocabulary, 50, 122, 183, exercises, 16, 17, 24, 25, 31, 32, 39, y-intercept, 76
267, 347, 422, 487, 553, 751, 47, 48, 53, 72, 89, 96, 97, 105, YLIST feature, 477
817, 893 112, 120, 127, 140, 145, 147, 154,
exercises, 15, 22, 30, 45, 64, 71, 78, 171, 179, 180, 187, 207, 213, 222, Z
86, 138, 145, 152, 160, 171, 179, 224, 245, 273, 294, 302, 310, 311,
198, 221, 229, 235, 237, 285, 317, 329, 336, 345, 353, 372, 379, z-axis, 164
293, 300, 315, 322, 328, 335, 411, 413, 418, 427, 460, 461, 466, zero(s)
342, 370, 378, 394, 409, 439, 474, 477, 482, 484, 491, 503, 505,
456, 461, 465, 512, 530, 557, 512, 513, 522, 540, 547, 557, 576, additive identity, 13
568, 591, 617, 627, 658, 678, 585, 598, 600, 607, 621, 629, 636, multiplication by, 21
685, 691, 707, 710, 715, 722, 644, 661, 667, 679, 680, 685, 686, multiplicative inverse and, 13
728, 735, 743, 768, 777, 813, 701, 709, 717, 723, 728, 736, 744, multiplicative property of, 773, 776
848, 855, 871, 880, 908, 932 752, 757, 769, 778, 781, 789, 790, multiplicity of, 291
Lesson Vocabulary, 4, 11, 18, 26, 33, 791, 806, 807, 814, 821, 831, 833, of polynomial functions, 288, 289,
41, 60, 68, 74, 81, 91, 99, 107, 841, 849, 859, 860, 865, 871, 874,
114, 125, 134, 142, 157, 166, 174, 889, 898, 915, 917, 923, 927, 934, 348
194, 202, 209, 216, 226, 234, 241, 939, 941, 950, 956, 963 properties of, 13, 226, 270, 289, 773
249, 259, 280, 288, 296, 303, 312, expressions, 20, 385 of quadratic equation, 226, 270
319, 326, 331, 339, 361, 367, 374, functions, 110, 451–458 of quadratic function, 226
390, 398, 405, 414, 434, 442, 451, inequalities, 33, 34, 117 of quartic functions, 341
462, 469, 478, 498, 515, 527, 534, technical writing, 525 as a real number, 13
542, 564, 572, 580, 587, 595, 614, Think/Write problems, 6, 35, 101, 108, of transformed cubic function, 339–
622, 630, 638, 645, 674, 681, 688, 144, 168, 176, 198, 220, 234, 243,
696, 711, 719, 725, 739, 764, 772, 252, 259, 300, 328, 364, 368, 376, 340
782, 792, 801, 809, 829, 836, 844, 384, 417, 437, 465, 519, 529, 538, ZERO option, 163, 227, 299
851, 861, 868, 875, 883, 904, 919, 543, 566, 582, 588, 648, 654, 677, Zero-Product Property, 226, 270, 288,
928, 936 682, 698, 786, 795, 830, 870, 879,
Looking Ahead Vocabulary, 57, 131, 884, 907, 914, 930, 938, 948 289, 349
191, 277, 431, 495, 561, 611, 671, translations, 877 zoology, 440, 740, 741
761, 825, 901 ZOOM feature, 460
Vocabulary Audio Online, 2, 58, 132, X z-score, 748
192, 278, 358, 432, 495, 562, 612,
672, 762, 826, 902 x-axis, 101, 102, 164, 195, 197, 226,
Vocabulary Builder, 54, 128, 188, 274, 256, 435
758, 822, 899, 959
x-coefficient, 175

Index i 1159

Acknowledgments

Acknowledgments Staff Credits Technical Illustration

The people who made up the High School Mathematics team— Aptara, Inc.; GGS Book Services
representing composition services, core design digital and
multimedia production services, digital product development, Photographs
editorial, editorial services, manufacturing, marketing, and
production management—are listed below. Every effort has been made to secure permission and provide
appropriate credit for photographic material. The publisher
Emily Allman, Dan Anderson, Scott Andrews, Christopher Anton, deeply regrets any omission and pledges to correct errors called
Carolyn Artin, Michael Avidon, Margaret Banker, Charlie Bink, to its attention in subsequent editions.
Niki Birbilis, Suzanne Biron, Beth Blumberg, Tim Breeze-
Thorndike, Kyla Brown, Rebekah Brown, Judith Buice, Sylvia Unless otherwise acknowledged, all photographs are the
Bullock, Stacie Cartwright, Carolyn Chappo, Christia Clarke, property of Pearson Education, Inc.
Mary Ellen Cole, Tom Columbus, Andrew Coppola, AnnMarie
Coyne, Bob Craton, Nicholas Cronin, Patrick Culleton, Damaris Photo locators denoted as follows: Top (T), Center (C), Bottom
Curran, Steven Cushing, Sheila DeFazio, Cathie Dillender, Emily (B), Left (L), Right (R), Background (Bkgd)
Dumas, Patty Fagan, Frederick Fellows, Jorgensen Fernandez, Cover
Mandy Figueroa, Suzanne Finn, Sara Freund, Matt Frueh, Jon JL Klein & ML Hubert/Biosphoto
Fuhrer, Andy Gaus, Mark Geyer, Mircea Goia, Andrew Gorlin, Front Matter
Shelby Gragg, Ellen Granter, Jay Grasso, Lisa Gustafson, Toni ix, x Peter Mason/Getty Images; xii Jeff Greenberg/PhotoEdit, Inc.
Haluga, Greg Ham, Marc Hamilton, Chris Handorf, Angie Hanks, 28 (T) BL Images Ltd/Alamy Images; 39 (TR) Richard Wahlstrom/
Scott Harris, Cynthia Harvey, Phil Hazur, Thane Heninger, Aun Workbook/Jupiter Images/Getty Images; 45 (TR) UPI Photo/Roger
Holland, Amanda House, Chuck Jann, Linda Johnson, Blair Jones, Williams/NewsCom; 61 (TR, TCR, CR, CC) Jupiter Images/Brand
Marian Jones, Tim Jones, Gillian Kahn, Matthew Keefer, Brian X/Alamy, (TC) Roberto Mettifogo/Getty Images; 84 (BCR) William
Keegan, Jim Kelly, Jonathan Kier, Jennifer King, Tamara King, Harader, (BR) Zen Shui/SuperStock; 135 (CR) Doug Perrine/Peter
Elizabeth Krieble, Meytal Kotik, Brian Kubota, Roshni Kutty, Mary Arnold Inc./PhotoLibrary Group, Inc./Getty Images, (TCR) Paul
Landry, Christopher Langley, Christine Lee, Sara Levendusky, Lisa Nicklen/National Geographic Image Collection; 159 (TCR) Andy
Lin, Wendy Marberry, Dominique Mariano, Clay Martin, Rich Crawford/©DK Images, (TCR) Getty Images, (TC) Steve
McMahon, Eve Melnechuk, Cynthia Metallides, Hope Morley, Gorton/©DK Images; 164 (BR) Image Source/Getty Images;
Christine Nevola, Michael O'Donnell, Michael Oster, Ameer 198 (T) Design Pics Inc/Alamy; 205 (TR) Jeff Greenberg/PhotoEdit,
Padshah, Stephen Patrias, Jeffrey Paulhus, Jonathan Penyack, Inc.; 228 (TR) ©Harvey Lloyd/Getty Images, (TCR) Andy Harmer/
Valerie Perkins, Brian Reardon, Wendy Rock, Marcy Rose, Carol Photo Researchers, Inc.; 234 (TR) Rolf Hicker Photography/Alamy
Roy, Irene Rubin, Hugh Rutledge, Vicky Shen, Jewel Simmons, Images; 306 (BR) James Baigrie/Botanica/Jupiter Images/Getty
Ted Smykal, Emily Soltanoff, William Speiser, Jayne Stevenson, Images; 333 (T) Andre Gallant/Getty Images, (TC) D. Hurst/Alamy,
Richard Sullivan, Dan Tanguay, Dennis Tarwood, Susan Tauer, (TCR) Mark Sytes/Alamy, (TR) Peter Cade/Getty Images; 342 AF/
Tiffany Taylor-Sullivan, Catherine Terwilliger, Mark Tricca, Maria Fotolia; 375 Wave Royalty Free/Alamy; 383 (TR) Detlev van
Torti, Leonid Tunik, Ilana Van Veen, Lauren Van Wart, John Ravenswaay/Photo Researchers, Inc.; 392 (BC) Bob Llewellyn
Vaughan, Laura Vivenzio, Samuel Voigt, Kathy Warfel, Don 408 (T) Bob Krist/Corbis; 438 (TR) John Cancalosi/Alamy Images;
Weide, Laura Wheel, Eric Whitfield, Sequoia Wild, Joseph Will, 453 (CR) Earth Imaging/Getty Images; 467 (BR) Jerry Lodriguss/
Kristin Winters, Allison Wyss, Dina Zolotusky Science Photo Library/Photo Researchers, Inc.; 476 (TR) Dave
King/©DK Images; 480 (TR) JPL/NASA; 502 (TR) Chase Jarvis (R)
Additional Credits: Michele Cardin, Robert Carlson, Kate Corbis Super RF/Alamy; 513 (TCR) NASA; 544 (TR) JPL/NASA;
Dalton-Hoffman, Dana Guterman, Narae Maybeth, Carolyn 578 (CR) ©DK Images, (L) Bob Gibbons/Alamy Images, (R) John
McGuire, Manjula Nair, Rachel Terino, Steve Thomas Glover/Alamy Images, (CL) Peter Anderson/©DK Images;
582 (T) Thomas J. Peterson/Alamy Images; 625 (B) Hank
Illustration Morgan/Photo Researchers, Inc.; 632 (B) Guido Alberto Rossi/Tips
Images/Tips Italia Srl a socio unico/Alamy; 641 (C) Museum of
Stephen Durke: 574; Phil Guzy: 596, 597; Rob Schuster: 4, 5, Science and Industry; 675 (T) Ron Chapple Stock; 684 (C) Koji
11, 18, 26, 33, 39, 41, 48, 60, 68, 74, 81, 84, 99, 107, 114, Aoki/Aflo/Getty Images; 775 (TR, TL) iStockphoto; 803 (TC)
116, 134, 142, 143, 149, 157, 165, 166, 168, 171, 174, 194, Wolfgang Spunbarg/PhotoEdit, Inc.; 847 (CR) European Space
202, 207, 216, 226, 233, 240, 258, 280, 288, 294, 296, 303, Agency/Science Photo Library/Photo Researchers, Inc., (TCR)
308, 312, 326, 331, 367, 374, 375, 381, 390, 395, 398, 405, Steve Gorton/©DK Images; 848 (BL) David Zimmerman/Getty
414, 429, 434, 449, 451, 469, 498, 515, 522, 527, 534, 542, Images, (BR) Fnalphotos/Dreamstime LLC; 887 (TR) ART on FILE/
547, 564, 566, 567, 570, 571, 572, 580, 587, 595, 609, 614, Corbis; 889 (CR) Demetrio Carrasco/©DK Images; NewsCom;
617, 619, 630, 638, 641, 653, 865; Ted Smykel: 209; Pearson 921 (TR) age fotostock/SuperStock; 931 (C) mediacolor's/Alamy
Education: 596, 597; Judi Pinkham: 230; Pronk&Associates: Images.
12, 362, 399, 462, 589; XNR Productions: 788

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