Stewart J. College Algebra: Concepts and Contexts
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A44 Answers to Selected Exercises and Chapter Tests
_2 10 _5 0 2
0_5
0 1
0_1
43. 44. 1- 5, 221- 2, 14
45. 46. 1- q, 1435, q 2
47. 48. 3- 5, - 221- 1, q 2
0_5 _2
49. (a) (b)
50. (a) (b)
51. (a) (b) [0, 2)
52. (a) (b)
53. 54.
1- 2, 045x
˛ 0 - 2 6 x … 06
5x
˛ 0 0 … x 6 26
1- 3, 545x
˛ 0 - 3 6 x … 56
3- 3, 545x
˛ 0 - 3 … x … 56
55. 56.
57. 58.
23. 24. 25. 26. 27. w 28. 25
29. 30. 31. 32. 33. 34.
35. 36. 37. 38. 39. 40.
41. 42. 43. x 44. 45.
46. 47. 48.
Algebra Toolkit B
B.1 Exercises
■
page T31
1. like; 2. like; 3. (a), (c)
4. 5. ;
6. ; 7. (a) A number times three, minus five;
4 (b) A number minus five, times three; 8. (a) A
number minus two, divided by five; (b) A number minus
two-fifths; 9. (a) A number squared, plus one; 10 (b) A
number plus one, squared; 16 10. (a) A number plus 5, times
two; 16 (b) A number plus 2, times five; 25 11. (a) ,
, 1 (b) 15 12. (a) (b) 10 13. (a)
(b) 37 14. (a) (b) 8 15. 16.
17. 18. 19.
20. 21.
22. 23.
24. 25.
26. 27.
28. 29. 30.
31. 32. 33.
34. 35. 36.
37. 38. 39.
40. 41.
42. 43. 44.
45. 46. 47. 48.
49. 50. 51. 52.
53. 54.
55. 56.
57. 58.
59. 60.
61. 62.
63. 64.
B.2 Exercises
■
page T37
1. (a) 3 (b) (c) (d)
2. 10; 7; 2, 5; 3. ;
4. ; 5.
6. 7. 8.
9. 10. 11.
12. 13.
14. 15.
16. 17. 18.
19. 20. 21.
22. 23.
24. 25.
26. 27. 91x - 521x + 1212x - 121x + 42
15x + 321x - 2 213x - 121x - 52
12x - 721x + 1 21z - 2 21z + 8 2
1y - 521y - 3 21x - 821x - 621x - 3 21x + 5 2
1x - 1
21x - 521x - 121x + 321z + 2 21z - 3 2
1y - 621y + 9 2- 7xy
2
1x
3
- 2y - 3y
2
2
xy
˛ 12x - 6y + 322x
2
1x
2
+ 2x - 72
ab
˛ 15 - 8c23y
3
12y - 52- 2x˛ 1x
2
- 82
6x
˛ 12x
2
+ 3215x
3
12 + x 2- 3˛ 1b - 4 2
5
˛ 1a - 421x + 52
2
1A + B 2
2
12x + 5212x - 5 2
1A + B 21A - B 21x + 221x + 52
2x
3
1x
2
+ 3x + 222x
3
2x
5
, 6x
4
, 4x
3
2x
3
- 5x
2
- x + 12x
3
- 7x
2
+ 7x - 5
2x
3
+ x
2
+ 1x
3
+ 4x
2
+ 7x + 6
4x
3
- 5x
2
+ 2x - 43x
6
+ 2x - 1
6x
4
+ 3x
3
+ 3x
2
- 2x - 16x
3
- 10x
2
+ 4x + 2
x
2
+ 2xy + y
2
- z
2
4x
2
+ 4xy + y
2
- 9
- x
4
- 3x
2
- 4- x
4
+ x
2
- 2x + 1
a - b
2
x - 44u
2
- √
2
x
2
- 9y
2
4y
2
- 259x
2
- 16y
2
- 9x
2
- 25
4 + 4y
3
+ y
6
x
4
+ 2x
2
+ 1r
2
- 4rs + 4s
2
4x
2
+ 12xy + 9y
2
x
2
- 6xy + 9y
2
4u
2
+ 4u√ + √
2
1 - 4y + 4y
2
9x
2
+ 24x + 16
x
2
- 4x + 4x
2
+ 6x + 914y
2
- 13y + 3
6x
2
+ 7x - 58s
2
+ 18s - 521t
2
- 26t + 8
s
2
+ 6s - 16r
2
+ 2r - 15y
2
+ 4y - 5
x
2
+ x - 12au + a√ + aw + bu + b√ + bw
uw + au + bu + √w + a√ + b√ax - ay + bx - by
ax - ay - bx + by- bu + 2c√ + 7
- 2ax + 7ay - 6a3by + b + 5
9aw + w- 2x
2
+ 4x - 37x
2
- 3x + 6
- x - 37x + 5
x
2
, b, 5
1xy 2
2
, a
2
axy, b- 10a
2ax
13
5
1
5
- 6
25 - x
2
A
2
- B
2
4x
2
+ 12x + 9A
2
+ 2AB + B
2
x
2
+ 5x + 6
xy + 8b - 5c + 24a + b + 5
s
5>4
y
1>2
1
2
x
1>4
˛y
1>4
2x
1>6
2x
-2>3
2a
7>6
10x
7>12
b
5>4
y
3>2
2u√
2
3st
2
5
1>3
˛y
4>3
x
4
s
4
64t
r
6
16q
3
u
2
√
1
x
3
y
2
k
3
1
2
r
1>8
9a
4>3
2b
1>2
y
2
x
2
1_2 0 _1 0
06 0_4 8
0_4 4
02 6
59. (a) 9 (b) 12 (c) (d) 60. (a) 100
(b) 5 (c) 12 (d) 61. 5 62. 4 63. (a) 15
(b) 24 (c) 64. (a) (b) 19 (c) 0.8
A.3 Exercises
■
page T18
1. 2. base; exponent 3. add; 4. subtract;
5. multiply; 6. (a) (b) (c) 2 7. (a) 81 (b)
8. (a) 64 (b) 9. (a) 25 (b) 10. (a) 10,000
(b) 11. (a) (b) 12. (a)
(b) 13. (a) 1 (b) 14. (a) 1 (b) 1
15. (a) (b) 16. (a) (b) 17. 125
18. 32 19. 64 20. 1 21. 1 22. 23. 24. 9
25. 1000 26. 27 27. 28. 25 29. 1 30. 31.
32. 33. 34. 35. 36. 37. 38.
39. 40. 41. 42. 43. 44.
45. 46. 47. 48. 49. 50.
51. 52.
A.4 Exercises
■
page T23
1. 2. 3. Yes 4. ;
5. 6. 7. 8. 9.
10. 11. 12. 13. 14.
15. (a) 4 (b) 2 (c) 16. (a) 8 (b) (c) 2
17. (a) (b) 4 (c) 18. (a) 14 (b) 4 (c) 4
19. (a) 7 (b) (c) 20. (a) 2 (b) (c)
21. (a) (b) 4 (c) 22. (a) (b) (c) 4
1
81
1
4
125
512
3
2
3
2
-
1
3
1
5
- 2
1
2
2
3
- 4
1
2
x
-5>2
2
5
a
2
1
12
5
3>5
1
211
3
2
3
4
2
7
2>3
5
-1>2
-
1
3
2
3
4
3>2
= 14
3
2
1>2
= 8
4
3>2
= 14
1>2
2
3
= 8155
1>3
-
8x
6
y
9
81u
20
16√
12
1
32
1
24z
4
3x
2
4
1
64a
3
z
4
a
6
1
x
4
y
3
16w
4
- 27a
3
16a
20
a
18
64x
2
8y
6
1
x
4
x˛
10
1
2
1
100
27
8
9
16
1
2
1
9
- 9- 1
1
1,000,000
1
10
1
27
1
3
- 1- 2187
- 2187- 216- 216- 10,000
- 25
1
16
1
243
1
8
1
2
3
8
3
3
3
9
5
6
18
35
7
4
Answers to Section C.2 A45
28. 29. (a)
(b) 30. (a)
(b) 31.
32. 33.
34. 35.
36. 37.
38. 39. 40. 41.
42. 43. 44. 45.
46. 47. 48.
49. 50. 51.
52. 53.
54. 55.
56. 57. 58.
59. 60.
61. 62.
63. 64.
65. 66.
B.3 Exercises
■
page T43
1. (a), (c) 2. numerator, denominator; 3. (a) False
(b) True 4. (a) numerators, denominators;
(b) invert; 5. (a) 3 (b)
(c) 6. (a) False (b) True 7. 5 8.
9. 10. 11. 12. 13.
14. 15. 16. 17.
18. 19. 20. 21.
22. 23. 24. 25. 26.
27. 28.
29. 30. 1 31. 32.
33. 34.
35. 36. 37.
38. 39. 40.
41. 42. 43.
44. 45. 46.
47. 48. x
2
+ x +
6
x - 2
x - 2 -
16
x - 4
211x
+ 1y2
y113
- 1y2
3 - y
1x - 1
x - 1
2117
- 122
5
3 + 15
2
2 + 13
x
2
+ x + 1
x
3
2x + 1
x
2
1x + 12
215x - 92
12x - 32
2
3x + 2
1x + 12
2
x
2
+ 3x + 12
1x - 421x + 6 2
1
1x + 121x + 2 2
2x
1x + 121x - 1 2
3x + 7
1x - 321x + 5 2
x - 5
x + 4
31x + 22
x + 3
12x + 1212x - 1 2
1x + 52
2
1
12x - 5213x - 2 2
x + 5
1x + 4212x + 3 2
x
1
t
2
+ 9
-
x - 1
x + 1
x + 3
x - 3
x - 5
x - 4
1
41x - 22
y - 6
y + 1
y
y - 1
x + 4
x - 2
x + 2
x + 1
x - 2
x - 1
1
x + 2
x + 1
31x - 22
x + 2
21x - 12
14t - 1
7
5y
10 + y
9x
2
2
2
x
-
1
3
- x
2
- x + 1
x˛ 1x + 12
2
x˛ 1x + 12
2
31x + 22
x˛ 1x + 52
2x
1x + 121x + 3 2
x + 1
x + 3
13s + 521s
2
- 221t + 2212t
2
+ 12
1y - 121y
2
+ 121y - 321y - 221y + 22
1x + 121x
4
+ 121x + 121x
2
+ 12
13u + 1211 - 3u
2
212r + 121r
2
- 32
13x - 121x
2
+ 221x + 421x
2
+ 12x˛ 1x + 1 2
31x - 121x + 222x
2
y13x + y213x - y2
x
2
y
3
1x + y 21x - y 22x˛ 1x + 22
2
x
2
1x + 321x - 1 2x˛ 1x + 12
2
3x˛ 1x + 321x - 32
1r - 3s 2
2
12w + y 2
2
15u - 12
2
14z - 32
2
1t + 52
2
1t - 32
2
1y + 52
2
1x + 62
2
4
x
4ab14x + 5214x - 12
1x + 521x + 1 212t + 3s212t - 3s2
17 + 2y 217 - 2y 212x + 5212x - 52
13a + 4213a - 4 21y + 1021y - 102
1x + 621x - 6 212a + 2b - 121a + b + 3
2
12x - 121x + 3 213x + 4 213x + 82
1x + 221x + 6 212x + 1214x + 32
49. 50.
51. 52.
Algebra Toolkit C
C.1 Exercises
■
page T54
1. (a), (c) 3. (a) True (b) False (c) False 4. cube; 5
5. (a) Five times a number minus three is twelve. (b) No
(c) 3 6. (a) Four times a number plus nine is one. (b) No
(c) 7. (a) Four times a number plus seven equals nine
times that number minus three. (b) No (c) 2 8. (a) Two
minus five times a number equals eight plus that number.
(b) Yes (c) 9. 10. 4 11. 12. 13.
14. 15. 16. 3 17. 12 18. 19. 20. 30
21. 22. 23. 24. 25. 26. 27.
28. 29. 30. 31. No solution
32. No solution 33. 34. 35.
36. 37. 38. 39. No real solution
40. No real solution 41. 42. 43.
44. 45. 46. 47. ; 2
4
8; 1
6
15; 1
4
21.3- 2
- 35 ; 15
- 4, 0
; 15
; 2; 17
; 216;312; 7
-
4
9
-
1
3
- 39
- 20
13
6
29
2
35
24
10
3
5
18
1
17
21
11
- 70
32
9
- 3
-
3
4
5
2
- 3- 9- 1
- 2
3x + 1 +
51x - 12
2x
2
+ 5
x
4
+ 1
1
3
ax
2
+ x + 2 -
1
x + 2
b2x
2
- 1 -
2
2x + 1
48. 49. 50. 51.
52. 53. 54. No real solution 55. 8
56. 57. 58. 59.
60. 61. 62. 63.
64. 65. 66. 67. 68.
69. 70. 71. 72. 73.
74.
C.2 Exercises
■
page T61
1. (a) (b)
2. (a) (b)
(c) 3. 4.
5. 3, 4 6. 7. 8. 9.
10. 11. 12. 13. 14.
15. 16. 17.
18. 19. 20. 21.
22. 23. 24. 25. 2, 5- 6, 1- 3, 5-
3
2
;
1205
10
5
2
;
313
2
1 ;
2
13
- 2 ;
A
7
2
-
7
2
,
1
2
3 ; 2152 ; 12- 1 ; 16
1
9
; -
1
3
1
16
; -
1
4
1
4
;
1
2
49
4
;
7
2
-
1
2
,
3
2
-
1
2
, - 3-
3
2
,
5
2
-
1
3
, 2- 6, - 2
- 4, 1- 4, 3x =
- 1- 42; 21- 42
2
- 41121- 52
2112
1x - 22
2
= 91x - 521x + 12= 0
1
2
, - 1, - 4; - 2, 4
- b ; 2b
2
- 4ac
2a
A
3
5
4x
2
1
315a
2
2L
5
d
s
a
8
5A
b
10>7
a
1
2A
b
5>6
a
7A
8
b
5>9
a
s
1.2
b
5>2
Fr
2
GM
PV
RTA
3
3V
4p
P - 2l
2
5
s - 1
3y
y + 2
5l
21l + 22
3y
5y - 4A
3
2t
5
;
135x
7
1
1
5
4
- 2
- 5, 1; 1
6
15
a
2
5
b
1>4
a
5
1.8
b
8>7
12.6 2
2>3
1
3
5
A46 Answers to Selected Exercises and Chapter Tests
26. 27. 28.
29. 30. 31. 32.
33. No real solution 34. 35. No real solution
36. No real solution
C.3 Exercises
■
page T66
1. (a) (b) (c) (d) 2. (a) True (b) False
3. 4 4. 5. 6. 1, , 2, 4 7. 4
8. , 2, 4 9. 10.
11. 1- q,
7
2
4
- 1, 0,
1
2
, 1- 2, - 1, 2, 412
1212, 2, 4- 2, - 1, 0,
1
2
7……6
-
5
2
;
113
2
1
4
;
15
4
3
4
3 ; 212- 1 ;
216
3
2 ; 12
-
3
2
;
15
2
- 10, - 20
6. A (5, 1), B (1, 2), C , D , E ,
F , G , H 12, - 221- 1, - 321- 2, 02
1- 4, - 121- 6, 221- 2, 62
y
0x
1
1
(2, 3)
(_2, 3)
(4, 5)
(4, _5)
(_4, 5)
(_4, _5)
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32. 1- 1, 22´ 15, q 2
1- q, - 24´ 31, 34
1- q, - 34´ 33, q 2
1- 2, 22
1- 1, 22
1- q, - 14´ 3
1
2
, q 2
1- q, - 32´ 1- 2, q 2
3- 3, 64
1- q, - 44´ 35, q 2
1- 2, 32
1- 1, 44
12, 6 2
33, 64
3- 3, - 12
1- q, - 14
31, q 2
37, q 2
1- q, 2 4
1- q, - 22
14, q 2
1- q, -
5
2
4
33.
34.
Algebra Toolkit D
D.1 Exercises
■
page T70
1. 2. II 3. ; 10
4. ; (4, 6)
5.
a
a + c
2
,
b + d
2
b
21a - c2
2
+ 1b - d2
2
13, - 52
1- 1, q 2
1- q, - 22´ 1- 2, 42
7
2
0
2
0
_
5
40
0_2
20
0 7
0 1
0_1
_1 0_3
360
260
04_1
03
_2
50_4
06_3
0_3 _2
0
1
2
_1
02_1
02_2
0_3 3
01 3_2
052_1
04_2
_1 0
y
0x
1
1
y
0x
1
1
7. 8.
y
0x
1
1
y
0x
1
1
9. 10.
11. (a) (b)
(1.5, 1) 12. (a) 5 (b) 13. (a) 10
(b) (1, 0) 14. (a) (b) 11, - 22140
10,
1
2
2113
Answers to Section D.2 A47
(b) 10 (c) (0, 0) (b) (c) (2.5, 3)161
y
0x
1
3
(_3, _6)
(4, 18)
y
0
x
2
2
(9, 9)
(_1, _1)
y
0
x
2
2
(6, _2)
(_1, 3)
y
0
x
1
2
(_1, 6)
(_1, _3)
y
0
x
2
2
(11, 6)
(7, 3)
(b) 25 (c) (b) (c) (4, 4)12001
1
2
, 62
(b) (c) (2.5, 0.5) (b) 9 (c)
21. 22.
1- 1, 1.52174
23. 24.
y
0
x
2
2
(2, 13)
(7, 1)
y
0x
1
1
(3, 4)
(_3, _4)
y
0
x
1
1
(0, 6)
(5, 0)
D.2 Exercises
■
page T77
1. 3, 2; yes 2. 1, 2; no 3. (a) y, x; (b) x, y;
4. (1, 2); 3 5. (a) Subtracting y from x gives 7. (b) No
6. (a) Adding x to gives 6. (b) No 7. (a) Subtracting 2x
from gives 5. (b) Ye s 8. (a) Adding 3x to , then
subtracting 2y, gives 0. (b) Yes
9.
y
2
y
3
y
2
1
2
- 1
y
0
x
1
2
x y
0 3
1 1
2
- 1
3
- 3
4
- 5
5
- 7
6
- 9
0
2
2
y
x
(0, 8)
(6, 16)
0
1
2
y
x
(_2, 5)
(10, 0)
(b) 10 (c) (3, 12) (b) 13 (c) (4, 2.5)
17. 18.
19. 20.
15. 16.
(b) 5 (c) (9, 4.5) (b) 13 (c) (4.5, 7)
x y
0
- 4
1 1
2 6
3 11
4 16
5 21
6 26
10.
11.
y
0
x
1
5
y
0
x
1
2
x y
- 3
9
- 2
4
- 1
1
0 0
1 1
2 4
3 9
A48 Answers to Selected Exercises and Chapter Tests
13. 17. 18.
x y
- 3 - 27
- 2
- 8
- 1
- 1
0 0
1 1
2 8
3 27
x y
- 3 - 30
- 2 - 11
- 1
- 4
0
- 3
1
- 2
2 5
3 24
x y
- 3
12
- 2
7
- 1
4
0 3
1 4
2 7
3 12
x y
- 3
6
- 2
5
- 1
4
0 3
1 4
2 5
3 6
x y
0 3
1 2
2 1
3 0
4 1
5 2
6 3
12.
y
0
x
1
10
y
x
1
10
14.
15.
16.
y
0
x
1
2
y
0
x
1
1
y
0
x
1
1
y
0
x
1
1
y
x
1
2
y
0
x
1
5
19. 20.
y
0
x
1
_2
21. 22.
23. 24.
25. (a)
4 (b) 1, 3 (c) 0, 4; 0
26. (a) (b) (c) 3 (d)
27. (a), (b) x-intercepts 3; y-intercepts - 2, 2- 3,
- 2, 8; - 4, 4- 1, 74, - 4
y
0
x
1
2
y
0
x
1
2
y
0
x
1
2
y
0
x
1
2
Answers to Section D.3 A49
37. (3, 0); 4 38. (0, 2); 2
y
0x
1
1
y
0
x
1
1
y
x
2
2
y
0x
2
2
−10
400
−2
2
39. 40. 1- 1, - 22; 61- 3, 42; 5
41.
42.
43.
44.
45.
46.
47.
48.
D.3 Exercises
■
page T84
1. viewing rectangle 2. zoom in 3. (c)
4. (c) 5. (c) 6. (d)
x
2
+ 1y + 32
2
= 7; 10, - 32; 17
1x - 22
2
+ 1y + 52
2
= 16; 12, - 52; 4
1x - 12
2
+ 1y - 12
2
= 4; 11, 12; 2
1x - 12
2
+ 1y + 22
2
= 4; 11, - 22; 2
1x + 12
2
+ 1y - 12
2
= 9
1x + 22
2
+ 1y - 22
2
= 4
1x + 12
2
+ 1y + 42
2
= 64
1x - 22
2
+ 1y + 12
2
= 9
9. 10.
11. 12.
13. 14.
−1000
100
−55
−10
20
−4
10
−10
20
−10
5
−1
5
−20
20
20
0
10
−2000
2000
−50
150
−250
150
−10
10
y
0x
1
1
y
0x
1
1
15. 16.
17.
No 18. No 19. Yes; 2
20. Yes; 1
21. 22.
−5
5
−5
5
−7
7
−5
8
−4
4
−6
6
−1
3
−3
3
28. (a), (b) x-intercepts ; y-intercepts
29. 30. 2, 3; 6 31.
32. ; 1 33. 34.
35. (0, 0); 3 36. (0, 0); 15
- 1; 1- 2, 2; - 2, 2
1
2
- 3, 3; - 93; - 3
- 4, 4- 2, 2 7. 8.
A50 Answers to Selected Exercises and Chapter Tests
D.4 Exercises
■
page T89
1. x 2. above 3. (a) (b)
4. (a) 1, 4 (b) (1, 4) 5. 6. 7. 8.
9. No real solution 10. No real solution 11. 12. 2.61
13. 3.00, 4.00 14. 0.25, 0.50 15. 1.00, 2.00, 3.00
16. 17. 1.62 18. 0, 2.31 19. 0.75
20. 21. No solution 22. No solution
23. 24. 3- 2.00, 0.2543- 2.00, 5.004
- 0.31, 0.81
- 1.00, - 0.25, 0.25
;
5
2
1
3
16132- 6- 4
3- 1, 04´ 31, 34- 1, 0, 1, 3
25.
26.
27.
28.
29.
30.
31.
32. 1- q, - 2.002´ 1- 2.00, 4.002
1- 1.00, 2.002´ 15.00, q 2
1- q, - 3.002´ 1- 2.00, q 2
3- 3.00, 6.004
1- q, - 0.5354´ 30.535, q 2
1- 1.00, 02´ 11.00, q 2
1- 1.00, - 0.252´ 1- 0.25, q 2
1- q, 1.00 4´ 32.00, 3.004
Absolute value, T11
distance between points on real line
and, T12
Absolute value function, 70
Addition
of algebraic expressions, T26–T27
graphical, of functions, 485–86, 505
of matrices, 605–6, 611–12
of polynomials, T30
of rational expressions, T41–T42
Aerodynamic lift, 503
Age and height, 5–6
Air pressure and elevation, 122–23
Alcohol, surge function and, 563, 565–66
Alcohol consumption, P1–P4, 11, 25,
52, 88
absorption, P1, P3
collecting data on, P1
metabolism, P1, P3
modeling absorption and metabolism,
P3–P4, 261, 272, 295, 376
Algebraic expression(s)
adding or subtracting, T26–T27
average rate of change of function
defined by, 147–48
combining, T25–T32
defined, T25
factoring, T33–T38
form and value of, T25–T26
multiplying, T27–T29
polynomials, combining, T29–T31
Algebraic method of solving equation,
T85, T86
Ancient objects, finding the age of,
371–72
Anglerfish, depth and pressure
experienced by, 35
Annual percentage yield (APY), 266
Archimedes, 209, 312, 568
crown problem, 568, 573
Arctic sea ice, extent of, 197–98
Area(s)
proportionality to square of “length”
measurement, 557
of similar objects, 555–56
INDEX
Arithmetic expression, evaluating, T2
Arrow notation, 509, 528
Asbestos and cancer, links between,
194, 195
Aspirin metabolism, 308
Assets, fair division of, 242–45
Associative properties, T2
Astronaut, net change in weight of,
55–56
Asymptote(s)
horizontal, 287, 288, 528, 540, 541
of rational functions, 540, 541
of reciprocal functions, 528
vertical, 329, 528, 540, 541
Atlanta, population of, 150–51
Attenuated growth, 398
Augmented matrix of linear system,
590–91, 593, 594
Average age of preschoolers, 3
Average income, 3–4
Average (mean), 2–3
definition of, 2
formula for, 104–5
Average rate of change, 142–53, 272–75
average speed, 145–46, 148
definition of, 143
for function defined by algebraic
expression, 147–48
of linear function, 155–56. See also
Rate of change
recognizing changes in, 229–33
Average speed of moving object,
145–46, 148
Bach, Johann Sebastian, 411
Back-substitution, solving triangular
system using, 581, 582
Bacteria count, in colony-forming units
per milliliter, 356
Bacterial growth, 248–49
comparing growth rates, 264
models for different time periods, 263
Bankruptcy, fair division of assets in,
242–45
Barometer, invention of, 476
Barton, Otis, 30
Base
exponential function with base a, 286,
288–90
exponential function with base e, 351
exponential notation, T14
of logarithm, 324, 326–27, 337–39
Bat cave, species-area relation in, 499,
517–18
Bathysphere, 30
Beebe, William, 30
Beer-Lambert Law, 363–64
Belgium, population of, 302–3
Bias in presenting data, 128–33, 130–33
Bicycle race, average speed in, 145–46
Billionaire ruler, 403
Binomials, T26
multiplying, using FOIL, T28
Biodiesel fuels from seeds of Jatropha
curcsa, 391–92
Biodiversity in Pasoh Forest Reserve of
Malaysia, 526
Bird flight, weight and wingspan
and, 525
Blood alcohol concentration (BAC), P1,
P3, 11, 52, 88, 189, 376
using surge functions to model data
on, 563, 565–66
Body Mass Index (BMI), 51, 189–90
Boiling point, elevation and, 33–34
Bonds, coupon, 309
Boyle’s Law, 111, 529, 534
Braking distance, finding maximum
carrying capacity of road with,
562–63
Bridge science, 237–39
Bungee jumping, 148
California, population of, 282
Camera lenses, focusing distance for,
536, 537–39
Canceling, simplifying rational
expressions by, T39
Cancer and asbestos, links between,
194, 195
I1
I2 INDEX
Carbon dioxide levels in atmosphere, 197
Carrying capacity
in logistic growth model, 277, 298
of road, 80–81, 560–63
of road, safe following distance and,
560, 561–62
of road, using braking distance to find
maximum, 562–63
Car sales, hybrid, 22, 302
Cartesian plane. See Coordinate plane
Catching up with leader on bike ride,
213–14
Categorical data, 602–11
collecting, 635
getting information from, 615–16
organizing, in matrix, 603–5, 636–37
proportions and, 604–5
tabulating, 604
Catfish, stocking pond with, 298
Causation
correlation and, 239–42
direction of, 242
Cell phone plan, 57
Cell phone usage in India, 363
Central tendency, 2
measures of, 2, 3, 4
Change, describing. See Function(s)
Change of base formula, 337–39
China, Internet usage in, 363
Chirping rate of crickets, temperature
and, 198, 208
Chocolate-powered car, 34
Chord, 412
Christmas bird count, 22
Circle(s), T76–T77
equation of, identifying, T77
equation of, standard form of, T76–T77
graphing, T76
graphing, on graphing calculator,
T83–T84
Closed interval, T9
Coefficient(s), 177–82
constant, 177–80
correlation, 195
heat transfer, 370
of x, 177, 180–82
Coefficient matrix, 621
Colony-forming units, 356
Columbia-Ecuador earthquake
(1906), 346
Columns (matrix), 590
Combining “like terms,” T26, T30
Combining logarithmic expressions, 336
Common factors
canceling, T39
factoring out, T33–T34
Common logarithms, 324–26
Common redpoll, wintering habits of, 25
Commutative properties, T2
Completing the square, T57–T59
defined, T58
to express quadratic function in
standard form, 431
for general quadratic function, 440
solving quadratic equations by,
T58–T59
Composite functions. See Composition
of functions
Composition of functions, 377–79
Compound fraction form of rational
function, 539
Compound interest, 265–66
calculating annual percentage
yield, 266
comparing yields for different
compounding periods, 266
continuously compounded, 354–55
formula, 265
investment growth and, 369
Concentration, mixtures and, 204–5
Conjugate radical, T42
Constant
of proportionality, 183, 496, 529, 557
spring, 188–89
universal gravitational, 104
Constant coefficient, 177
varying, 177–80
Constant function, 65
Constant rate of change, 153–65
linear model of, 156–57
Continuously compounded interest,
354–55
Cooling, Newton’s Law of, 370–71, 375,
399, 427
Cooling coffee, 41
Coordinate, T7
Coordinate line. See Real number line
(real line)
Coordinate plane, T67–T68, T71
distance between points on, T68–T69
graphing points and sets in, T68
graphing two-variable data in, 14–16
midpoint of line segment on, T69–T70
Correlation, causation and, 239–42
Correlation coefficient, 195
Cost
fixed, 26
model for, 89–90
unit, 26, 27
Cost comparison of gas-powered and
hybrid-electric cars, 212–13
Coughing, mathematical model of, 88
Coupon bonds, 309
Crime scene investigation, 364
Newton’s Law of Cooling and, 371
Crop yield
density (plants/acre) and, 465
rainfall and, 465
Cross-tab matrix, 603, 636
Crude oil imports in U.S., 493
Cube root function, 498
Data, 2–11. See also Modeling
analyzing, 136–37, 239
bias in presenting, 128–33
categorical. See Categorical data
collecting, 134–36, 238, 476–77
describing relationships in, 25–35
exponential, recognizing, 316–17
fitting exponential curves to, 274–75,
295–303
fitting polynomial curves to, 520–22
fitting power curves to, 517–18, 519–20
linear, recognizing, 233–37
linearizing, 408–9
log-log plot of, 518, 519
numerical, 602
one-variable, 2–5, 136–37
quadratic, recognizing, 478–79
scatter plot of, 14, 518, 519
semi-log plot of, 407–9, 518, 519
two-variable, 5–7, 13, 14–16,
37–38, 137
visualizing relationships in, 12–25
Data mining, 233
Dating, radiocarbon, 371–72
Dead Sea Scrolls, 376
Death in United States, leading causes of,
130–32
Decay factor, 253, 264
Decay rate, 253
instantaneous, 355–64
Decibel scale, 344–45
Decreasing functions, 78–79
Deforestation, paper usage and, 90–91
Degree of polynomial, 504
Demand, linear model for 170–71
Demand equation, 171, 214, 218
Demographics, life expectancy and, 23
Denominator
least common (LCD), T41
rationalizing the, T42
Dependent system, 571, 582–84, 596–98
Dependent variable, 37–38, 54
net change in, 38
Depreciation, straight-line, 176
Depth and pressure, 6–7, 14–15, 26
model for, 29–30
INDEX I3
Difference(s)
first, 28–30
of functions, 484–87
second, 478–79, 480, 481
of squares, factoring, T35–T36
Dimension of matrix, 590
Dimensions of lot, quadratic function
modeling, 453–54
Direct proportionality, 182–84
definition of, 183
to power function, 496–97
Discriminant, 452–53
Dissonance, 411–12
Distance, T11
maximum distance seen from a
height, 61
between points on real line, T12
between two points in coordinate
plane, T68–T69
Distance Formula, T68–T69
Distance function, inverse function
of, 498
Distance-speed-time problem, 574–75
Distributive property, T3–T5
expanding using, T3–T4
factoring using, T4–T5
multiplying expressions using,
T27–T28
multiplying polynomials using, T30
solving equations with, T48
Division. See also Quotient(s)
of assets, fair, 242–45
of exponents, T16
graphical, 536
long, T42–T43
of rational expressions, T40–T41
Domain
of function, 56–57, 77–78
of relation, 13, 14
Doppler effect, 111, 545
Drip irrigation system for garden, 91–92
Eames, Ray and Charles, 402
Earthquakes
intensity of, 346–47
magnitude of, 346, 349
Einstein, Albert, 101
Electrical capacity of solar panels,
182–84
Electrical resistance, 111
Elementary row operations, 591–92
Elements of set, T8
Elevation
air pressure and, 122–23
boiling point and, 33–34
temperature and, 28–29
Elimination method, 570–71
Gaussian, 581–82
Empty set, T8
End behavior
defined, 509
of polynomial function, 508–10
of rational function, 540
Energy, world consumption of, 48–49
Energy and mass formula,
Einstein’s, 101
Entries of matrix, 590
Environmental management, species
survival and, 639
Enzymes in expectant mothers, levels of,
15–16
Equation(s), 25–35. See also System of
equations
defined, T47
demand, 171, 214, 218
exponential, 365–66, 401–2
false, 582
in function form, 39
graphing, 27
of horizontal line, 171–72
inequality compared to, T62
of lines, 165–77. See also Linear
equation(s)
logarithmic, 367–68
logistic growth, 277
matrix, 621–25
operations on, T47
power, T51–T53
quadratic. See Quadratic equation(s)
reading, T48
satisfying, T72
solutions (or roots) of, T47
solving the. See Solving the equation
that represent functions, 38–40, 42
two-variable, T71–T80
of vertical line, 171–72
Equilibrium point, 214
Equivalent system, operations leading
to, 581
e (the number), 351
exponential models in terms of,
357–58
Ethiopia, population of, 374–75
Expanding expression
distributive property used in, T3–T4
logarithmic expressions, 335–36
Expectant mothers, levels of enzymes in,
15–16
Exponent(s), T14
integer, T14–T19
negative, T15–T16
rational, T20–T24
rules for working with, T16–T18,
T21, T22
zero, T15–T16
Exponential decay, 252–55, 356
of “iceman,” radiocarbon dating
and, 372
modeling 252–55, 252–55, 264–65,
290, 356–57
Exponential equation(s), 365–66
logarithms used to solve, 401–2
steps in solving, 365
Exponential form, 327
Exponential function(s), 247–61
with base a, 286
with base a, effect of varying a or C,
288–90
with base e, 351
comparing linear functions and,
275–76
comparing power functions and,
494–95
for compound interest, 265–66
definition of, 249
finding, from a graph, 290–91
fitting exponential curves to data,
295–303
graphs of, 286–95, 406–9
inverse functions of, 385–86
modeling with, 249–55, 261–66,
274–75, 290, 295–98, 356–58
modeling with, appropriateness of,
518, 519
musical scale and, 409–12
natural, 351–52
rates of change of, 273–74
semi-log graph of, 407–9
transformations of, 420–21
Exponential growth, 248–85, 356
example of, 249–50
of investment, 265–66
linear growth versus, 275–76
modeling, 249–52, 249–52, 263,
265–66, 356, 357–58
of savings, 315–16
Exponential model(s), 276
appropriateness of, 296–97
changing time period in, 262–65, 266
for data, finding, 295–96
of decay, 252–55, 264–65, 290,
356–57
fitting model to data, 274–75, 295–97
getting information from, 368–71
of growth, 249–52, 263, 265–66, 356,
357–58
of investment growth, 265–66, 369
in terms of e, 357–58