answers to all exercisesmissbuckles.weebly.com/uploads/2/2/4/6/22466700/ch5...answers to all...

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ANSWERS TO ALL EXERCISES 51

CHAPTER 5 • CHAPTER CHAPTER 5 • CHAPTER 5REFRESHING YOUR SKILLS FOR CHAPTER 5

1a. between 3 and 4 (about 3.3)

1b. between 6 and 7 (about 6.9)

1c. between 7 and 8 (about 7.4)

1d. between 8 and 9 (about 8.2)a. b. c. d.

8 100 2 4 6

2a. 2 � __

6

2b. 5 � __

3

2c. 3 � __

5

2d. 2 � ___

10

2e. 10 � __

3

3a. iii, B

3b. i, C

3c. iv, D

3d. ii, A

4a. �

4b. �

4c. �

4d. �

Answers to All Exercises

DAA2TE_985_ANS_a.indd 51DAA2TE_985_ANS_a.indd 51 3/12/09 9:44:01 PM3/12/09 9:44:01 PM

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52 ANSWERS TO ALL EXERCISES

LESSON 5.1

1a. f (5) � 3.52 1b. g(14) � 19,528.321c. h(24) � 22.92 1d. j(37) � 3332.202a. 16, 12, 9; y � 16(0.75) x

2b. 24, 36, 54; y � 24(1.5) x

3a. f(0) � 125, f(1) � 75, f(2) � 45; u 0 � 125 and u n � 0.6u n�1 where n � 1

3b. f(0) � 3, f(1) � 6, f(2) � 12; u 0 � 3 and u n � 2u n�1 where n � 1

4a. 0.75; 25% decrease

4b. 1. _

3 ; 33. _

3 % increase

4c. 0.94; 6% decrease

4d. 1.0638; 6.38% increase

5a. u 0 � 1.211, u n � u n�1 � 1.015

5b.

YearEstimated

population (billions)

1995 1.211

1996 1.229

1997 1.248

1998 1.266

1999 1.285

2000 1.305

2001 1.324

2002 1.344

5c. y represents the es timated population x years after 1995; y � 1.211 (1.015) x .

5d. The equation predicts that the population of China in 2006 was 1.426 billion. This is larger than the actual value. This means that the population is growing at a slower rate than it was in 1995.

6a. Let x represent the number of the day, and let y represent the height in cm. y � 2.56(2.5) x . For the fifth day, y � 2.56(2.5) 5 � 250 cm; for the sixth day, y � 2.56 (2.5) 6 � 625 cm.

6b. y � 2.56(2.5) 3.5 � 63.256c. 728 cm

6d. 0.76 day, or 18 hours

6e. 11 days 13 hours, or 9 P.M. on day 11

7a–d.

7e. As the base increases, the graph becomes steeper. The curves all intersect the y-axis at (0, 1).

7f. The graph of y � 6 x should be the steepest of all of these. It will contain the points (0, 1) and (1, 6).

8a. y � 2 x�3

8b. y � 2 � 2 x , or y � 2 x � 2

8c. y

__ 3 � 2 x , or y � 3 � 2 x 8d. y � 2 x/3

9a–d.

9e. As the base increases, the graph flattens out. The curves all intersect the y-axis at (0, 1).

9f. The graph of y � 0.1 x should be the steepest of all of these. It will contain the points (0, 1) and (�1, 10).

10a. y � 5 � 0.5 x , or y � 5 � 0.5 x

10b. �(y � 5) � 0.5 x , or y � �0. 5 x � 5

10c. y � 2 � 0. 5 x �1 , y � �2 � 0.5 x�1

10d. y _ 3 � 0.5 x/2 , y � 3(0.5) x/2

11a. 27 ___ 30 � 0.9 11b. f (x) � 30(0.9) x 11c.

x

y

5 10

30

y � f(x)

y � g(x)

11d. g(4) � 30

11e. possible answer: g(x) � 3 0(0.9) x�4


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11f. Sample answer: You can use the x- and y-values of any point on the curve and the common ratio to write the equation.

12. Answers will vary but will be in the form y � y 1 � 1.8 x� x 1 , with � x 1 , y 1 � being any point from the table.

13a. Let x represent time in seconds, and let y represent distance in meters.

x

y

5

5

10 (0, 10)

(3.5, 3)

(7, 10)

13b. domain: 0 � x � 7; range: 3 � y � 10

13c. y � 2�x � 3.5� � 314a. f (3) � 8.5

14b. y � 8.5 � 0.5(x � 3), y � 10 � 0.5(x � 6), or y � 7 � 0.5x

15a. A � 5000(1 � 0.035) 5

15b. S � 5200(1 � 0.032) 5

15c. After 14 years, Austin will have $8,093.47, and Sami will have $8,082.

16a. 6x

x x2 6x

–4 –4x –24

16b. x 2 � 2x � 24

16c. yes

16d. A rectangle diagram also uses the distributive property. Each term in the first binomial is multiplied by each term in the second binomial.


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LESSON 5.2

1a. 1 ____ 125 1b. �36 1c. � 1 ___ 81

1d. 1 ____ 144 1e. 16 ___ 9 1f.

7 __ 2 2a. a 5 2b. b 4

2c. c 20 2d. e 3

3a. False. Valid reasons include: You must have the same base for the product property of exponents; 243 � 16 � 35,831,808.3b. False. You must raise to the power before multiplying.

3c. False. Valid reasons include: You must raise to the power before dividing; only one factor of 4 can be divided out; or it should be 4 x�1 .

3d. true

4a. x � �2 4b. x � �3

4c. x � �5 4d. x � 0

5a. x � 3.27 5b. x � 7845c. x � 0.16 5d. x � 0.505e. x � 1.07 5f. x � 16a. x 12 6b. 8 x 12 6c. 10 x 7

6d. 12 x 5 6e. 8 x 12 6f. 1 ___ 25 x �12 7. Sample answer: (a � b) n is not necessarily equivalent to a n � b n . For example, (2 � 3 ) 2 � 25 but 2 2 � 3 2 � 13. However, they are equivalent when n � 1, or when a � b � 0 and n is odd.

8a. 49; 79.7023; 129.6418; 210.8723; 343

8b. 30.7023; 49.9396; 81.2305; 132.1277. The sequence is not arithmetic because there is not a common difference.

8c. 1.627; 1.627; 1.627; 1.627. The ratio of consecutive terms is always the same, so the sequence is growing exponentially.

8d. Possible answer: Non-integer powers may produce non-integer values. If the exponents form an arithmetic sequence, the decimal powers form a geometric sequence.

9a–d.

9e. Sample answer: As the exponents increase, the graphs get narrower horizontally or steeper verti-cally. The even-power functions are U-shaped and

always in the first and second quadrants, whereas the odd-power functions have only the right half of the U, with the left half pointed down in the third quadrant. They all pass through (0, 0) and (1, 1).

9f. Sample answer: The graph of y � x 6 will be U-shaped, will be narrower (or steeper) than y � x 4 , and will pass through (0, 0), (1, 1), (�1, 1), (2, 64), and (�2, 64).

Sample answer: The graph of y � x 7 will fall in the first and third quadrants, will be narrower (or steeper) than y � x 3 or y � x 5 , and will pass through (0, 0), (1, 1), (�1, �1), (2, 128), and (�2, �128).

9g. Power functions go through the origin and have long-run values of infinity. Exponential functions have y-intercepts at 1 (or a) and go to 0, either as x increases or as x decreases.

10a. y � 4 � x 3 , or y � x 3 � 4

10b. y � (x � 2 ) 3

10c. 4y � x 3 , or y � 1 __ 4 x 3 10d. 8( y � 2) � x 3 , or y � 2 � 1 __ 8 x3, or y �

1 __ 8 x 3 � 211a. 47(0.9)(0.9) x�1 � 47(0.9 ) 1 (0.9 ) x�1 � 47(0.9 ) x by the product property of exponents; 42.3(0.9 ) x�1 .

11b. 38.07(0.9 ) x�2

11c. The coefficients are equal to the values of f 1 corresponding to the number subtracted from x in the exponent. If ( x 1 , y 1 ) is on the curve, then any equation y � y 1 � b (x� x 1 ) is an exponential equation for the curve.

12a. y � 30.0 r x�3 ; y � 5. 2r x�6

12b. 30.0 r x�3 � 5.2 r x�6 ; r � 0.557612c. 173 cm

13a. x � 7 13b. x � � 1 __ 2 13c. x � 014a. 0.9476

14b. y � 42(0.9476 ) x�2002

14c. y � 39.8(0.9476 ) x�2003


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14d. 42(0.9476 ) 1980�2002 � 137.2; 39.8 (0.9476 ) 1980�2003 � 137.2; both equations give approximately 137.2 rads.

14e. 42(0.9476 ) 2010�2002 � 27.3; 39.8(0.9476 ) 2010�2003 � 27.3; both equations give 27.3 rads.

14f. y � 42(0.9476 ) x�2002 � 42(0.9476)(0.9476 ) x�2002�1 � 39.8(0.9476 ) x�2003

15a. x � 7 15b. x � �4

15c. x � 4 15d. x � 4.61

16. y � 2(x � 4 ) 2 � 3

x

y

–10 5 10

–5

5

17a. Let x represent time in seconds, and let y represent distance in meters.

x

y

10 20 30 40

2

4

6

8

17b. All you need is the slope of the median-median line, which is determined by M 1 (8, 1.6) and M 3 (31, 6.2). The slope is 0.2. The speed is approximately 0.2 m/s.


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LESSON 5.3

1. a�e�j; b�d�g; c�i; f�h

2a. Power; the base is a variable.

2b. Power; the base is a variable.

2c. Exponential; the exponent is a variable.

2d. Power; x is equivalent to x 1 .

2e. Power; a square root is equivalent to the exponent 1 _ 2 .2f. Power; t 2 � 4t � 3 is equivalent to (t � 2 ) 2 � 1.

2g. Exponential; 12 __ 3t is equivalent to 12(3 ) �t .

2h. Power; 28 ____ w � 5 is equivalent to 28(w � 5 ) �1 .2i. Power; 8 __

y 4 is equivalent to 8 y �4 .

2j. Neither; the function is not a transformation of either x a or b x .

2k. Power; the fifth root of a cube is equivalent to the exponent 3 _ 5 .2l. Exponential; the exponent contains a variable.

3a. a 1/6 3b. b 4/5 , b 8/10 , or b 0.8

3c. c �1/2 , or c �0.5 3d. d 7/5 , or d 1.4

4a. a 1/6 � 4.2; raise both sides to the power of 6: a � 4. 2 6 � 5489.031744.

4b. b 4/5 � 14.3; raise both sides to the power of 5 _ 4 : b � 14. 3 5/4 � 27.808.4c. c �1/2 � 0.55; raise both sides to the power of �2: c � 0.5 5 �2 � 3.306.4d. d 7/5 � 23; raise both sides to the power of 5 _ 7 : d � 2 3 5/7 � 9.390.5. 490 W/c m 2

6a–d.

6e. Each curve is less steep than the prior one. The graphs of y � x 1/2 and y � x 1/4 are in only the first quadrant, whereas the graphs of y � x 1/3 and y � x 1/5 are in the first and third quadrants. All of the functions go through (0, 0) and (1, 1). The graphs of y � x 1/3 and y � x 1/5 both go through (�1, �1).

6f. y � x 1/7 will be less steep than the others graphed and will be in the first and third quad rants. It will pass through (0, 0), (1, 1), and (–1, –1).

6g. The domains of y � x 1/2 and y � x 1/4 are x � 0 because you can’t take a square root or fourth root of a negative num ber. The domains of y � x 1/3 and x 1/5 are all real numbers.

7a–d.

7e. Each graph is steeper and less curved than the previous one. All of the functions go through (0, 0) and (1, 1). y � x4/4 (or y � x) is not curved at all.

7f. y � x 5�4 should be steeper and should curve upward.

8. Sample answer: Power func tions with rational exponents can have limited domain. When the exponent is between 0 and 1, the curve increases slowly with a shape similar to y � �

__ x . Exponential

curves always have a steadily increasing or decreasing slope, unlike power functions.

9a. exponential

9b. neither

9c. exponential

9d. power

10a. y � 3 � (x � 2 ) 3/4

10b. y � 1 � [�(x � 5) ] 3/4

10c. y � 4 � � x __ 4 � 3/4

10d. y

__ 4 � � x � 3 _____ 2 �

3/4 , or y � 4 � x � 3 _____ 2 �

3/4

11a. x � � 13 ___ 9 � 5 � 6.29

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11b. x � 18 0 1/4 � 3.66

11c. x � � � ___

35 ____ 4 �

3/2 � 1.80

12a. 0.723 AU

12b. 29.475 yr

12c.

Planet Mercury Venus Earth Mars

Orbital radius (AU) 0.387 0.7232 1.00 1.523

Orbital time (yr) 0.2408 0.615 1.00 1.8795

Planet Jupiter Saturn Uranus Neptune

Orbital radius (AU) 5.201 9.542 19.181 30.086

Orbital time (yr) 11.861 29.475 84.008 165.02

13a. P � k V �1 ; P � k __ V ; PV � k13b. k � (40)(12.3) � 492

13c. 8.2 L

13d. 32.8 mm Hg

14a. 27 x 9

14b. 16 x 9

14c. 0.2 x �1

14d. 108 x 8

14e. 18 x 2 y 4

15a. y � (x � 4 ) 2

15b. y � x 2 � 1

15c. y � �(x � 5 ) 2 � 2

15d. y � (x � 3 ) 2 � 4

15e. y � � _____

x � 3

15f. y � � __

x � 1

15g. y � � _____

x � 2 � 1

15h. y � � � _____

x � 1 � 1

16. about 840

17a. u 1 � 20 and u n � 1.2 u n�1 where n � 2

17b. u 9 � 86; about 86 rat sightings17c. Let x represent the year number, and let y represent the number of rats; y � 20(1.2 ) x�1 .


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LESSON 5.4

1a. x � 5 0 1/5 � 2.187 1b. x � 29.7911c. no real solution

2a. x � 625 2b. x � 1

2c. x � 512

2d. x � 12(�1 � 1.812 5 1/7.8 ) � 0.9512e. x � � 14.2 _____ 222.1 �

1/3.5 � 0.4563a. 9 x 4 3b. 8 x 6

3c. 216 x �18

4a. 100 r 6 4b. 100 r 6 � 50

4c. r � 0.891; 89.1%

5a. She must replace y with y � 7 and y 1 with y 1 � 7; y � 7 � ( y 1 � 7) � b x� x 1 .5b. y � 7 � (105 � 7) b x�1 ; � y � 7 _____ 98 �

1/(x�1) � b

5c. Possible answers: x � 0, y � 200, b � 0.508; x � 2, y � 57, b � 0.510; x � 3, y � 31, b � 0.495

5d. Possible answer: The mean of the b-values is 0.511. y � 7 � 98(0.511 ) x�1 .

6a.

6b. Sample answer: ŷ � 0.37 x 1.5 , where x is measured in units of 100,000 km. The graph of the data and the equation appear to be a good fit.

6c. approximately 1,229,200 km

6d. 545.390 d

7a. 39 tons

7b. 54 ft

8a. 19.58 cm

8b. 23.75 m

9a. 1.9 g

9b. 12.8%

10. 0.319% per month, or 3.9% per year

11a. 0.0466, or 4.66% per year

11b. 6.6 g

11c. y � 6.6(1 � 0.0466 ) x � 6.6(0.9534 ) x

11d. 0.6 g

11e. 14.5 yr

12a.

0 10 20 30 40 6050Temperature (°F)

12b. 18.9, 29.15, 40.1, 50.35, 57.4

12c. range � 38.5, IQR � 21.2

12d. The data do not support his conjecture. There are approximately the same number of cities in each category.

13. x � �4.5, y � 2, z � 2.75


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LESSON 5.5

1. (�3, �2), (�1, 0), (2, 2), (6, 4)

2a. 9 2b. 2

2c. 15 ___ 2 , or 7.53. Graph c is the inverse because the x- and y-coordinates have been switched from the original graph so that the graphs are symmetric across the line y � x.

4. a and e are inverses; b and d are inverses; c and g are inverses; f and h are inverses.

5a. f (7) � 4; g(4) � 7

5b. They might be inverse functions.

5c. f (1) � �2; g(�2) � 5

5d. They are not inverse functions, at least not over their entire domains and ranges.

5e. f (x) for x � 3 and g(x) for x � �4 (its entire domain) are inverse functions.

6a. 4 � (x � 2 ) 3/5 � 12

(x � 2 ) 3/5 � 8

x � 2 � 8 5/3 � 32

x � 34

6b. 4 � (y � 2 ) 3/5 � x

(y � 2 ) 3/5 � x � 4

y � 2 � (x � 4 ) 5/3

f�1(x) � 2 � (x � 4 ) 5/3

6c. Sample answer: The steps are the same, but you don’t have to do the numerical calculations when you find an inverse.

7a. �1, 0, 1, 2 7b. (�1, �3), (0, �1), (1, 0), (2, 2)

7c. Yes, it is a function; it passes the vertical line test.

x

y

2

2–2–2

8a. f (x) � 2x � 3; f �1 (x) � x � 3 _____ 2

, or f �1 (x) � 1 __ 2 x � 3 __ 2

8b. f (x) � �3x � 4 ________ 2 or f �1 (x) � � 3 __ 2 x � 2;

f �1 (x) � �2x � 4 ________ 3 , or f �1 (x) � � 2 __ 3 x �

4 __ 3 8c. f (x) � � x

2 � 3 _______ 2 or f �1 (x) � � 1 __ 2 x 2 �

3 __ 2 ; y � �

________ �2x � 3 (not a function)

9a. i. f �1 (x) � x � 140 _______ 6.34 9a. ii. f ( f �1 (15.75)) � 15.759a. iii. f �1 ( f (15.75)) � 15.759a. iv. f ( f �1 (x)) � f �1 ( f(x)) � x9b. i. f �1 (x) � x � 32 ______ 1.8

9b. ii. f ( f �1 (15.75)) � 15.759b. iii. f �1 ( f(15.75)) � 15.759b. iv. f � f �1 (x)� � f �1 � f (x)� � x10a. The equation of the median-median line isf(x) � �0.006546x � 14.75.10b. f �1 (x) � x � 14.75 ___________

�0.006546x , or

f �1 (x) � �152.76x � 2252.7610c. The equation of the median-median line isg(x) � �0.003545x � 58.81.10d. g �1 (x) � x � 58.814 __________

�0.003545 , or

g �1 (x) � �282.1x � 16,59110e. Use the function in 10a to find the temperature in °C first. f (6194) � �0.006546(6194) � 14.75 � �25.80°C. Then use the function from 9b to change the °C to °F: y � �14.44°F.11a. y � 100 � C 11b. C � 100 � F � 32 _______ 1.8 12. Your friend’s score is 1. Sample answers are given for explanations of incorrect answers. Problem 1 is correct. Problem 2 is incorrect: The notation f �1 (x) indicates the inverse function related to f (x), not the exponent �1. Problem 3 is incorrect: The expression 9 �1/5 can be rewritten as 1 ___

9 1/5 . Problem 4 is incorrect:

The expression 0° is not defined.

13a. i, ii, iii 13b. ii, v

13c. i, iv 13d. i, ii, iii

14a. c(x) � 7.18 � 3.98x, where c is the cost in dollars and x is the number of thousands of gallons

14b. $39.02

14c. g(x) � x � 7.18 _______ 3.98 , where g is the number of thousands of gallons and x is the cost in dollars

14d. 12,000 gal

14e. g �c(x)� � g(7.18 � 3.98x) � 7.18 � 3.98x � 7.18 _________________ 3.98 �

3.98x _____ 3.98 � xc�g(x)� � c � x � 7.18 _______ 3.98 � � 7.18 � 3.98 �

x � 7.18 _______ 3.98 � � 7.18 � x � 7.18 � x

14f. about $16

14g. Answers will vary, but volume should equal

231 � 1500, or 346,500 in 3 or approximately 200 ft 3 .15. possible answers:

3 � ____

125 2 , � 3 � ____

125 � 2 , 5 2 ,

3 � ______

15,625 , 25

16. f (x) � 12.6(1.5 ) x�2 or f(x) � 42.525(1.5 ) x�5

17a. x � 9 __ 2 , or 4.517b. x � � 1 __ 2 , or �0.517c. x � 1

18. y � 3(x � 3 ) 2 � 2 and x � 1 __ 9 (y � 2 ) 2 � 319. x � �1, y � 1, z � 0


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LESSON 5.6

1a. 10 x � 1000

1b. 5 x � 625

1c. 7 1/2 � x

1d. x 3 � 8

1e. 5 �2 � x

1f. 6 x � 1

2a. x � 3

2b. x � 4

2c. x � � __

7 � 2.652d. x � 2

2e. x � 1 ___ 25 2f. x � 0

3a. x � lo g 10 0.001; x � �3

3b. x � lo g 5 100; x � 2.86143c. x � lo g 35 8; x � 0.58493d. x � lo g 0.4 5; x � �1.7565 3e. x � lo g 0.8 0.03; x � 15.7144 3f. x � lo g 17 0.5; x � �0.24474a. This is a translation of the graph of y � log x horizontally �2 units. Note that it actually continues downward indefinitely.

4b. This is a vertical dilation of the graph ofy � log x by a factor of 3.

4c. This is a reflection of the graph of y � log x across the x-axis and a translation vertically �2 units.

4d. This is a translation of the graph of y � 10 x horizontally �2 units.

4e. This is a vertical dilation of the graph of y � 10 x by a factor of 3.

4f. This is a reflection of the graph of y � 10 x across the x-axis and a translation vertically �2 units.

5a. false; x � log 6 12

5b. false; 2 x � 5

5c. false; x � log 5.5

______ log 3

5d. false; x � log 3 7

6. approximately 25 min

7a. sometime in 1977

7b. 8.3%

7c. 8.7 yr

8a. y � 100(0.999879 ) x

8b. 6025 yr ago. The technique is approximate and assumes that the carbon-14 concentration in the atmosphere has not changed over the past 6000 yr.

9a. y � 88.7(1.0077 ) x 9b. 23 or 24 clicks

10a. x � 345

10b. x � 7 1/2.4 � 2.25


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11a. Median-median line equation using years since 1900 is ŷ � �1121 � 17.1x.

Yearx

Passengers(millions)

y

1991 433.0

1992 452.1

1993 462.3

1994 503.4

1995 517.7

1996 546.6

1997 562.7

1998 573.8

1999 596.4

2000 621.7

2001 579.4

2002 571.2

2003 603.4

2004 650.4

2005 676.0

2006 676.1

x

y

1990 1995 2000Years since 1990

2005

450

500

550

600

650

700

Pas

sen

gers

11b. residuals: �1.85, 0.15, �6.75, 17.25, 14.45, 26.25, 25.25, 19.25, 24.75, 32.95, �26.45, �51.75, �36.65, �6.75, 1.75, �15.25

11c. 25.288196. Predictions based on this model will gen erally be within 25.3 million of the correct number of passengers.

11d. Based on the model, about 875.4 million passengers. A better estimate might be to say between 850 million and 900 million passengers.

12a. C 1 � 32.7, C 2 � 65.4, C 3 � 130.8, C 6 � 1046.4, C 7 � 2092.8, C 8 � 4185.6

12b. y � 16.35(2 ) x , where x represents C-note number and y represents frequency in cycles per second

13a. y � 1 � x � 3, or y � x � 4

13b. y � 4 � (x � 5 ) 2 , or y � (x � 5 ) 2 � 4

13c. y � 2 � �x � 6�, or y � �x � 6� � 213d. y � 7 � �

_____ x � 2 , or y � �

_____ x � 2 � 7

14a. 2l � 2w � 155

l � 2w � 7

14b. l � 54, w � 23.5; length: 54 in., width: 23.5 in.

15a.

x

y

–6 6

–6

6

�1

�2

They are parallel.

15b. possible answer: A(0, �3); P(1, 1); Q(4, 3)

15c. Possible answer: Translate horizontally 1 unit and vertically 4 units. 2(x � 1) � 3(y � 4) � 9.

15d. Possible answer: Translate horizontally 4 units and vertically 6 units. 2(x � 4) � 3(y � 6) � 9.

15e. Possible answer:

2(x � 1) � 3(y � 4) � 9 → 2x � 2 � 3y � 12 � 9 → 2x � 3y � �1, which is l 2 2(x � 4) � 3(y � 6) � 9 → 2x � 8 � 3y � 18 � 9 → 2x � 3y � �1, which is also l 2


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LESSON 5.7

1a. log 55

1b. log 8

1c. log 4

1d. log 1 ___ 36 1e. log 63

2a. log 2 � log 11

2b. many possible answers, such as log 26 � log 2

2c. log 3 � log 13

2d. many possible answers, such as log 14 � log 2

3a. x log 5

3b. 2 log x

3c. 1 __ 2 log 33d. 2x log 7

4a. true

4b. false; possible answer: log 5 � log 3 � log 15

4c. true

4d. true

4e. false; possible answer: log 9 � log 3 � log 3

4f. false; possible answer: log � __

7 � 1 __ 2 log 7

4g. false; possible answer: log 35 � log 5 � log 7

4h. true

4i. false; possible answer: log 3 � log 4 � log 3 __ 4 4j. true

5a. g h � g k ; product property of exponents5b. log st; product property of logarithms

5c. f w�v ; quotient property of exponents

5d. log h � log k; quotient property of logarithms

5e. j st ; power property of exponents

5f. g log b; power property of logarithms

5g. k m/n ; definition of rational exponents

5h. log u t; change-of-base property

5i. w t�s ; product property of exponents

5j. 1 __ p h

; definition of negative exponents

6a. y � 100(0.999879 ) x 6b. 11,460 yr

6c. x � 3891.968; about 1981 � 3892 � 1910 B.C.E.6d. y � 100(0.999879 ) x ; y � 100(0.999879 ) 100,000,000 ; y � 0. There is virtually nothing left to measure, so you could only use carbon-14 for dating coal if you had very sensitive instruments to detect the radioactivity.

7a. Let x represent the note’s number of steps above middle C, and let y represent the note’s frequency in hertz. y � 261.6 � 2 x/12 � because the starting value is 261.6 and there are 12 intermediate frequencies to get to the last C note, which has double that frequency.

7b.

Note Frequency (Hz)

Do C4 261.6

C# 277.2

Re D 293.6

D# 311.1

Mi E 329.6

Fa F 349.2

F# 370.0

Sol G 392.0

G# 415.3

La A 440.0

A# 466.1

Ti B 493.8

Do C5 523.2

8a. x � 3.38168b. x � 11.4958c. x � 11.1748d. x � 42.7399a. y � 14.7(0.8022078 ) x 9b.

Altitude (mi)

(0, 14.7)

(2, 9.467)

Air

pre

ssu

re (

lb/i

n.2

)

Air pressure (lb/in2)

(14.7, 0)

(9.467, 2)

Alt

itu

de

(mi)

9c. y � 8.91 lb/i n 2

9d. x � 6.32 mi

10a. 96.5%

10b. y � 100(0.965 ) x , with x in minutes

10c. 19.456 min

10d. In one day, the carbon-11 is virtually gone, so you could never date an archaeological find.


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11. Graphs will vary. If a horizontal line intersects the graph in more than one point, its inverse is not a function.

12a. y � 5 � 3x 12b. y � 2 � 3x

13a. The graph has been vertically dilated by a factor of 3, then translated horizontally 1 unit and vertically �4 units.

x

y

–5 5

–5

5

13b. The graph has been horizontally dilated by a factor of 3, reflected across the x-axis, and translated vertically 2 units.

x

y

5

5

14a. False. If everyone got a grade of 86% or better, one would have to have gotten a much higher grade to be in the 86th percentile.

14b. False. Consider the data set {5, 6, 9, 10, 11}. The mean is 8.2; the median is 9.

14c. False. Consider the data set {0, 2, 28}. The range is 28; the difference between the mean, 10, and the maximum, 28, is 18.

14d. true

15a. Let h represent the length of time in hours, and let c represent the driver’s cost in dollars. c � 14h � 20. The domain is the set of possible values of the number of hours, h � 0. The range is the set of possible values of the cost paid to the driver, c � 20.

15b. Let c represent the driver’s cost in dollars, and let a re present the agency’s charge in dollars. a � 1.15c � 25. The domain is the money paid to the driver if she had been booked directly, c � 20. The range is the amount charged by the agency, a � 48.

15c. a � 1.15(14h � 20) � 25, or a � 16.1h � 48


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LESSON 5.8

1a. 2.90309

1b. 11

1c. �4

1d. 1.4123

1e. 2.9303

1f. 5.3246

2a. log� 10 n�p � � log� 10 n � 10 p �(n � p)log 10 � log 10 n � log 10 p

(n � p)log 10 � n log 10 � p log 10

(n � p)log 10 � (n � p)log 10

Because the logarithm of the left side equals the logarithm of the right, the left and right sides are equal. Or, because log� 10 n�p � � log � 10 n � 10 p �, 10 n � p � � 10 n ��1 0 p �.2b. log � 10 d ___ 10 e � � log� 10 d�e �log 10 d � log 10 e � log� 10 d�e �

d log 10 � e log 10 � (d � e)log 10

(d � e)log 10 � (d � e)log 10

Because the logarithm of the left side equals the logarithm of the right, the left and right sides are equal. Or, because log � 10 d ___ 10 e � � log� 10 d�e �,

10 d ___ 10 e � 10 d�e .

3. t � log 3

________ log 1.005625 � 195.9; about 195.9 mo, or about 16 yr 4 mo

4a. h � 146(0.9331226 ) T�4

4b. about 24.1 h at 30°C; about 63.6 h at 16°C

4c. 147 � 146(0.9331226 ) T�4 ; 1.00685 �

0.9331226 T�4 ; T � 4 � log 1.00685

____________ log 0.9331225

;

T � �0.0986 � 4 � 3.9°C

4d.

4e. A realistic domain is 0° to 100°C; these are the freezing and boiling points of water.

5a. f (20) � 133.28. After 20 days, 133 games have been sold.

5b. f (80) � 7969.17. After 80 days, 7969 games have been sold.

5c. x � 72.09. After 72 days, 6000 games have been sold.

5d. 12000 ______________ 1 � 499(1.09 ) �x

� 6000; 2 � 1 � 499(1.09 ) �x ;

1 � 499(1.09 ) �x ; 0.002 � (1.09 ) �x ; log 0.002 � log(1.09 ) �x ; log 0.002 � �x log 1.09;

x � � log 0.002

________ log 1.09

� 72.1

5e.

Sample answer: The number of games sold starts out increasing slowly, then speeds up, and then slows down as everyone who wants the game has purchased one.

6a. D � 10 log 10 �13 _____

10 �16 � 30 dB

6b. D � 10 log 3.16 � 10 �10 __________

10 �16 � 65 dB

6c. I � 1 0 10.7 � 10 �16 � 10 �5.3 � 5.01 � 10 �6 W/ cm 2 6d. about 3.16 times as loud

7a.

7b. (log x, y) is a linear graph.

7c. y � 6 � 20x; ŷ � 6 � 20 log x7d.

Sample answer: Yes; the graph shows that the equation is a good model for the data.


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8a. Let x represent time in min, and let y represent temperature in °F.

8b. Because the curve is both ref lected and translated, first graph points in the form (x, �y). Then translate the points up so that the data approach a long-run value of 0. Estimating that the new points approach the long-run value –74, graph points in the form (x, log(�y � 74)), which appears to be linear. The median-median line for these altered data is log(�y � 74) � 1.823 � 0.0298x. Solving for y gives the equation ŷ � 74 � 1 0 1.823�0.0298x , or ŷ � 74 � 66.52(0.9338 ) x .9a. The data are the most linear when viewed as (log(height), log(distance)).

9b. The median-median line equation for these altered data is ŷ � 0.555 � 0.49909x. Or, in terms of the original data, log(distance) � 0.555 � 0.49909 log(height). Solving for distance gives the equation ŷ � 10 0.555�0.49909 log x , ŷ � 10 0.555 � 10 0.49909 log x , ŷ � 3.590� 10 log x � 0.49909 , or ŷ � 3.590 x 0.49909 .

10a. 14.6 qt after 1 day; 13.41 qt after 2 days; u 0 � 16, u n � u n�1 (1 � 0.15) � 1, n � 1

10b.

0 1 2 3 4 5 6

16 14.6 13.41 12.40 11.54 10.81 10.19

7 8 9 10 11 12 13

9.66 9.21 8.83 8.50 8.23 7.99 7.80

14 15 16 17 18 19 20

7.63 7.48 7.36 7.26 7.17 7.09 7.03

x

y

Days after first treatment

Ch

lori

ne

(qt)

10

20

5 10 15 20

y � 6. _ 6 � 9.33(0.85 ) x

11a. y � 18( � __

2 ) x�4 , y � 144( � __

2 ) x�10 , or y � 4.5( �

__ 2 ) x

11b. y � log x � log 18

__________ log �

__ 2 � 4, y �

log x � log 144 __________

log � __

2 � 10, or

y � log x � log 4.5

__________ log �

__ 2

12a. cost: y � 1.75x � 19,000; income: y � 1.92x

12b.

x

y

0 1,000,000

250,000

500,000

750,000

1,000,000

Fish sticks (lb)

Cos

t/in

com

e ($

)

12c. 111,765 lb

12d. $66,000

13.

x

y

–5 –1–2–3–4

10

(–5, 8)(–5, 6)

(–5, 10)

(–5 , 8)1_4 (–4 , 8)3_4

14a. x � 341 8 1/5 � 5.0914b. x � 25 6 1/4 � 5.1 � 9.1 or x � 1.1

14c. x � � 55 ___ 7.3 � 1/6 � 1.40


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CHAPTER 5 REVIEW

1a. 1 ___ 16 1b. � 1 __ 3 1c. 125

1d. 7 1e. 1 __ 4 1f. 27 ___ 64

1g. �1 1h. 12 1i. 0.6

2a. lo g 3 7 � x or x log 3 � log 7

2b. log 4 5 � x or x log 4 � log 5

2c. log 75 � x

3a. 1 0 1.72 � x 3b. 1 0 2.4 � x

3c. 5 �1.47 � x 3d. 2 5 � x

4a. log xy 4b. log z � log v 4c. 2.1 x 6.8

4d. k log w 4e. x 1/5 4f. log t

_____ log 5

5a. x � log 28

______ log 4.7

� 2.153

5b. x � � ________

log 2209

_______ log 4.7

� 2.231

5c. x � 2. 9 1/1.25 � 2. 9 0.8 � 2.3445d. x � 3. 1 47 � 1.242 � 1 0 23

5e. x � � 101 ____ 7 � 1/2.4

� 3.041

5f. x � log 18

________ log 1.065

� 45.897

6a. x � 0.5�243 2 1/8 � 1� � 0.8256b. x � 11 4 1/2.7 � 5.779

6c. x � log 734 ____ 11.2 _______ log 1.56

� 9.406

6d. x � 20.2

6e. x � � 147 ____ 12.1 � 1/2.3

� 1 � 1.962

6f. x � 5.7 5 2 � 3 � 36.063

7. x � 16 log 8 ___ 45 ________

log 0.5 � 39.9; about 39.9 h

8a. 1 8b. (x � 1 ) 3 � 2

___________ 4 8c. 1 __ 2 8d. 129. y � 5 � 32 ___ 5 �

(x�1)/6

10.

x

y

–5 5

–5

5

11a. a � 0.50

11b. b � 2.94 ______ log 15

� 2.4998

11c. log x � �0.50 _____ 2.4998 � �0.2; x � 1 0 �0.2 � 0.63. The real-world meaning of the x-intercept is that the first 0.63 min of calling is free.

11d. $4.19

11e. about 4 min

12a.

12b. domain: 0 � x � 120; range: 20 � y � 100

12c. Vertically dilate by a factor of 80; reflect across the x-axis; vertically shift by 100.

12d. 55%

12e. about 4 yr old

13a. approximately 37 sessions

13b. approximately 48 wpm

13c. Sample answer: It takes much longer to improve your typing speed as you reach hig her levels. 60 wpm is a good typing speed, and very few people type more than 90 wpm, so 0 � x � 90 is a reasonable domain.

14a. u 0 � 1, u n � � u n�1 � � 2, n � 114b. y � 2x

14c.

14d . Answers will vary but can include curving upward, increasing, increasing at an increasing rate, discrete.

14e. after 20 cell divisions

14f. after 29 divisions


Discovering Advanced Algebra Teaching ResourcesAssessment ResourcesCalculator Notes for the Texas Instruments TI-83 Plus and TI-84 PlusCalculator Notes for the Texas Instruments TI-Nspire and TI-Nspire CASCondensed Lessons: A Tool for Parents and TutorsCondensed Lessons in SpanishInvestigation WorksheetsMore Practice Your Skills with AnswersSolutions ManualTeaching and Worksheet MastersTechnology DemonstrationsAnswers to All ExercisesChapter 0Chapter 1Chapter 2Chapter 3Chapter 4Chapter 5Refreshing Your Skills for Chapter 5Lesson 5.1Lesson 5.2Lesson 5.3Lesson 5.4Lesson 5.5Lesson 5.6Lesson 5.7Lesson 5.8Chapter 5 Review

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