Algebraic Exponents & Exponential Functions Chapter Questions

Transcription

1 Algebraic Eponents & Eponential Functions Chapter Questions 1. How can you tell the difference between a linear and an eponential relationship? 2. Eplain the difference between growth factors and growth rates. 3. Eplain the difference between decay factors and decay rates.

2 Algebraic Eponents & Eponential Functions Chapter Problems Eponential Growth Introduction Classwork 1. The drama club wants to make confetti even faster. Now they decide to take one piece of paper and cut it into fourths. Then they stack the four pieces and cut them into fourths. They repeat this process creating smaller and smaller pieces of paper. a. Create a table showing the number of cuts from 1 to 5 and the pieces of confetti. b. How many pieces of confetti after 10 cuts? c. How is this process different than when the paper was cut in halves and thirds? 2. For each epression below, identify the eponent and the base and write the answer. a. 7 5 b c. 9 4 d e Write each epression in eponential form. a = b = c. 3 3 = d = e = 4. Write each epression in standard form. a. 8 3 = b. 5 6 = c. 2 7 = d. 9 3 = e = 5. Eplain how the meanings of 2 4, 4 2 and 2 4 differ. 6. Find: a. Five squared b. Two to the fifth power c. Nine cubed 7. Given 7 = 343, what is the value of? 8. Given 6 = 1296, what is the value of? 9. Given 3 = 729, what is the value of? 10. Given 6 = 4096, what is the value of?

3 11. Given 3 = 1000, what is the value of? 12. Given 2 = 196, what is the value of? Write the product as a power * 2 * 2 * * 8 * 8 * 8 * 8 * 8 * m * m * m * m * m * m Write using eponents squared 17. The fifth power of P to the sith Evaluate each epression = = = = Homework 23. The drama club wants to make confetti even faster again. Now they decide to take one piece of paper and cut it into fifths. Then they stack the five pieces and cut them into fifths. They repeat this process creating smaller and smaller pieces of paper. a. Create a table showing the number of cuts from 1 to 5 and the pieces of confetti. b. How many pieces of confetti after 10 cuts? c. How is this process different than when the paper was cut in halve, thirds, and fourths? 24. For each epression below, identify the eponent and the base and write the answer. a. 3 5 b. 8 2 c d e Write each epression in eponential form. a = b = c = d. 8 8 = e = 26. Write each epression in standard form. a = b. 4 7 = c = d. 3 6 = e =

4 27. Eplain how the meanings of 3 5, 5 3 and 3 5 differ. 28. Find: a. Eight to the fourth power b. Four cubed c. Twelve squared 29. Given 5 = 625, what is the value of? 30. Given 9 = 81, what is the value of? 31. Given 2 = 256, what is the value of? 32. Given 7 = 10,000,000 what is the value of? 33. Given 4 = 2401, what is the value of? 34. Given 4 = 81, what is the value of? Write the product as a power * 10 * 10 * 10 * * 6 * n * n 38. y * y * y * y * y * 94 * 94 * 94 Write using eponents. 40. q squared 41. seven to the fourth 42. three to the fifth 43. five cubed 44. two to the sith Evaluate each epression = = = = = Eponential Growth v. Linear Growth Classwork Write whether the equation shows a linear growth or an eponential growth. 50. y = y = y = y = 3 2

5 54. y = 4 3 Does the graph show an eponential growth or a linear growth? Do the tables show an eponential growth or a linear growth? Simplify. 61. m 2 * m 5 = 62. c 5 * c 4 * c2 = 63. y 3 * y 3 = 64. 5z 3 * z 5 = 65. 6p 8 * 4p 3 = 66. b 2 * b = 67. d 4 * d 3 * d * * 5 * 4 = 69. (3y 2 )(8y 4 ) = 70. p 3 * q 4 * q 7 = Homework Write whether the equation shows a linear growth or an eponential growth. 71. y = y = y = y = y = 2 3 Does the graph show an eponential growth or a linear growth?

6 Do the tables show an eponential growth or a linear growth? Simplify. 82. v 7 * v 7 = 83. t * t 8 * t 10 = 84. n 7 * -6n 8 = z 6 * 3z 4 = * 6 = 87. (-5u 2 v 3 )(4uv 2 ) = 88. (-3j 3 k)(-5j 2 k 2 ) = n * -2n 3 = 90. a 3 * c 4 * c 3 * a 7 = 91. w 8 * w 7 = Eponential Relationships in Tables, Equations and Graphs Classwork 92. In the table below: a. What is the y intercept? b. What is the growth factor? c. What equation fits this data? d. Graph the data on a coordinate plane. Y

7 In the table below: a. What is the y intercept? b. What is the growth factor? c. What equation fits this data? d. Graph the data on a coordinate plane. Y In the table below: a. What is the y intercept? b. Is there a growth factor? What type of relationship does the table show? c. What equation fits this data? d. Graph the data on a coordinate plane. How does this graph differ from the first two eamples? Y

8 95. You decide to start a garden at your house and plant Black-Eyed Susans. The net summer you notice that the flowers had reproduced significantly and you wrote the following equation n = 5(4 t ). In the equation n represents the number of flowers after t time in years. a. How many flowers did you plant the first year? b. What is the growth factor of the Black-Eyed Susans in the garden? c. How many flowers will be in the garden after 4 years? d. In how many years will there be over 5,000 plants in the garden? 96. The table below shows how a population of rabbits increases over several years. a. Is the population linear or eponential or neither? b. What is the starting population? c. Write an equation to represent the data in the table. Year Total rabbits

9 97. The table below shows how a population of whales increases over several years. a. Is the population linear or eponential or neither? b. What is the starting population? c. Write an equation to represent the data in the table. Year Total whales For each table below decide if the relationship is linear, absolute value, eponential or none of them. If it represents one of them write the equation to represent the data. Table A Table B Table C X y y Y Table D Table E Table F X y Y Y

10 Simplify. 99. a 9 a 4 = = Homework a 12 5a In the table below: a. What is the y intercept? b. What is the growth factor? c. What equation fits this data? d. Graph the data on a coordinate plane. X Y In the table below: a. What is the y intercept? b. What is the growth factor? c. What equation fits this data? d. Graph the data on a coordinate plane. X Y

11 107. In the table below: a. What is the y intercept? b. Is there a growth factor? What type of relationship does the table show? c. What equation fits this data? d. Graph the data on a coordinate plane. How does this graph differ from the first two eamples? X Y You decide to start a garden at your house and plant Black-Eyed Susans. The net summer you notice that the flowers had reproduced significantly and you wrote the following equation n = 8(4 t ). In the equation n represents the number of flowers after t time in years. a. How many flowers did you plant the first year? b. What is the growth factor of the Black-Eyed Susans in the garden? c. How many flowers will be in the garden after 4 years? d. In how many years will there be over 5,000 plants in the garden?

12 109. The table below shows how a population of rabbits increases over several years. a. Is the population linear or eponential or neither? b. What is the starting population? c. Write an equation to represent the data in the table. Year Total rabbits The table below shows how a population of whales increases over several years. a. Is the population linear or eponential or neither? b. What is the starting population? c. Write an equation to represent the data in the table. Year Total whales

13 111. For each table below decide if the relationship is linear, absolute value, eponential or none of them. If it represents one of them write the equation to represent the data. Table A Table B Table C X y y y Table D Table E Table F X y y y Simplify 112. z 14 z m 20 m s 20 s

14 Growth Factors and Growth Rates Classwork 118. What is the growth factor in each of the following eponential tables below? Table A Table B Table C X y y Y Table D Table E Table F X Y y Y

15 119. Fill in the table below with the missing growth factors and growth rates. Growth factor Growth rate % 25% 7% % 42% If you invest $750 at a yearly interest rate of 7%: a. What is the growth factor? b. How much money will you have after 6 years? 121. If you invest $50 at a yearly interest rate of 3%: a. What is the growth factor? b. How much money will you have after 10 years? 122. If you invest $1225 at a yearly interest rate of 4%: a. What is the growth factor? b. How much money will you have after 6 years? Evaluate (-10) 2 = 124. ( 2) 3 = 125. ( 2) 5 = 126. ( 3) 2 = 127. ( 3) 3 = 128. ( 1) 5 = 129. ( 10) 7 = 130. ( 4) 4 = 131. ( 7) 0 = 132. ( 8) 1 =

16 Homework 133. What is the growth factor in each of the following eponential tables below? Table A Table C Table E X Y y Y Table B Table D Table F X Y X y Y Fill in the table below with the missing growth factors and growth rates. Growth factor Growth rate % % 88% % 30%

17 135. If you invest $900 at a yearly interest rate of 4%: a. What is the growth factor? b. How much money will you have after 7 years? 136. If you invest $2000 at a yearly interest rate of 7%: a. What is the growth factor? b. How much money will you have after 4 years? 137. If you invest $505 at a yearly interest rate of 5%: a. What is the growth factor? b. How much money will you have after 10 years? Evaluate ( 6) 2 = 139. ( 6) 3 = 140. ( 3) 4 = 141. ( 3) 5 = 142. ( 5) 5 = 143. ( 10) 9 = 144. ( 7) 0 = 145. ( 10) 8 = 146. ( 6) 1 = 147. ( 4) 3 = Eponential Decay Classwork 148. You have a bag of 500 pieces of candy and you eat a third of the pieces a day. a. Create a table showing the candy remaining for days 0-8. b. What is the y-intercept? c. What is the decay factor? d. What equation matches this situation? 149. You have another bag of candy with 700 pieces and you eat a fourth of the pieces a day. a. Create a table showing the candy remaining for days 0-8. b. What is the y-intercept? c. What is the decay factor? d. What equation matches this situation?

18 150. Fill in the table below with the missing decay factors and decay rates. Decay factor Decay rate % % 45% % 8% Simplify ( 4 ) 3 = 152. (y 2 ) 5 = 153. (c 3 ) 2 = 154. (2 2 ) 8 = 155. (3 6 ) 7 = 156. (56 9 ) 20 = 157. (43 18 )3= 158. (w 8 ) 60 = 159. (12 3 ) 9 = 160. (z 5 ) 6 = Homework 161. You have a bag of 850 pieces of candy and you eat half of the pieces a day. a. Create a table showing the candy remaining for days 0-8. b. What is the y-intercept? c. What is the decay factor? d. What equation matches this situation?

19 162. You have another bag of candy with 1,000 pieces and you eat a fifth of the pieces a day. a. Create a table showing the candy remaining for days 0-8. b. What is the y-intercept? c. What is the decay factor? d. What equation matches this situation? 163. Fill in the table below with the missing decay factors and decay rates. Decay factor Decay rate % % 72% % 40% Simplify (16 3 ) 9 = 165. (12 6 ) 3 = 166. ( 10 ) 7 = 167. (10 4 ) 8 = 168. (a 6 ) 7 = 169. (t 9 ) 8 = 170. (r 7 ) 7 = 171. (3 3 ) 2 = 172. (q 8 ) 3 = 173. (54 12 ) 3 =

20 Rules for Eponents Classwork 174. Complete each equation for the missing value: a. (5 2 )(5 5 ) = 5? b. (12 7 )(12 3 ) = 12? c. (3-2 )(3 5 ) = 3? d. (4 9 )(4-3 ) = 4? e. (5 4 )(5? ) = 5 12 f. (10 7 )(10? )(10-6 ) = 10 3 g = 3? 9 5 h. 6 = 5? i. 8 = 9? 9 j = 12? k ? = 10 3? 2 l. 3 = Simplify. a. (4m 5 ) 2 = b. (-4z 2 ) 3 = c. (a 2 b 3 ) 4 = d. (-2 2 y 3 ) 5 e. (2 3 ) 4 = f. (-3h 3 ) 5 = g. (-4b 3 ) 3 = h. (5a 3 ) 5 = i. (3 2 y 4 ) 4 = j. (-j 7 ) 2 = k. 3-4 = l. (-4) -3 = m. 7-1 = n. (-3) -2 = o. 11m -5 = p. 2a -3 = q. 7s -4 = r. 7s -4 t2= s. n -5 v 2 = t. 5-2 p -1 = Homework 176. Complete each equation for the missing value: a. (12 2 )(12 7 ) = 12? b. (2 5 )(2 2 ) = 2? c. (5-3 )(5 5 ) = 5? d. (15 8 )(15-5 ) = 4? e. (6 7 )(6? ) = 6 15 f. (11-6 )(11? )(11 8 ) = 10 5 g = 7? h. 6 = 11? 11 i = 3?

21 2 2 6 j = 2? k.? = l. 5? 5 6 = Simplify. a. (-3r) 3 = b. (3y 2 ) 5 = c. (-2r 5 s 2 ) 6 = d. (y 2 ) 5 = e. (-a 2 b 3 ) 3 = f. (-2r 2 s) 4 = g. (4uv 5 ) 3 = h. (c 3 ) 2 = i. (-d 5 ) 2 = j. (4 6 y 2 ) 5 = k. 4-2 = l. (-) -2 = m. (-5) -2 = n = o. 5-4 = p. -7 = q. 7ab -2 = r. r -4 s 2 = s. 3s -2 = t. -5 y 7 = Unit Review Multiple Choice Choose the correct answer for each question. 1. Find three cubed. a. 3 2 = 6 b. 3 3 = 9 c = 27 d. 3 (3 3 3) = Epressed in simplest form, (3 3 )(3y 2 ) 2 4 (4 ) is equivalent to: a y 4 c. 108 b y 4 12 y 4 d y 4 3. Given 6 = 216, find. a. 36 b. 3 c. 216 d In the equation y = 5(2 ), what value does the 2 represent? a. y-intercept b. starting value c. growth factor d. growth rate

22 5. The epression is equivalent to: a b c. 3-8 d Given the equation y = 4(2 ) if = 4, what is the value of y? a. 12 b. 32 c. 64 d Which of the following equations shows a linear relationship? a. y = 2 b. y = c. y = 3 2 d. y = What is the y intercept of the following relationship? y a. 3 b. 15 c. 0 d What is the y intercept of the following relationship? y a. 0 b. 1 c. 10 d What is the growth factor in the following relationship? y a. 3 b. 3.2 c. 3.4 d If the growth rate is 10%, what is the growth factor? a. 10 b. 1.1 c d If the growth factor is 1.6, what is the growth rate? a. 6% b. 60% c. 16% d. 160%

23 13. What is the equation for this table? y a. y = 1.3 b. y = 1.3(20 ) c. y = 20(1.3 ) d. = 1.3 y 14. Identify the decay factor in the following table y a. 900 b. 3 c. -3 d. 1/3 15. In a science e p e r i m e n t, the a m o u n t o f b a c t e r i a d e c r e a s e d e a c h d a y. The table below shows the am ount of bacteria that remained at the start of work on successive days. Day Fractional Part of the bacteria remaining Which fractional part of the rock will remain at noon on day 6? 1 2 a c b d Short Constructed Response Write the correct answer for each question. 16. Eplain how the meanings of 4 2, 2 4 and 4 2 differ. 17. Create an eponential relationship and eplain what the growth factor is for the relationship.

24 18. Suppose there is an initial rabbit population in the forest of 5,000 deer. The growth factor for the population is 1.2 per year. How large will the deer population be in year 4? 19. What will y equal if equals 6 in the equation y = 15(2.1 )? 20. What kind of growth does this table show? X y Etended Constructed Response - Solve the problem, showing all work. 21. Given the following table: a. Eplain why the relationship is eponential. b. Identify the: i. Growth factor ii. Growth rate iii. y intercept c. Graph the relationship. d. Write the equation. X y

25 Algebraic Eponents & Eponential Functions Chapter Problems Answer Key Classwork 4. 1 a a b c d e = =16, 4 2 =4 4 =16, 2 4 =8 6. a. 25 b c. number of cuts fourths halves thirds b. 32 c a. eponent 5, base 7, b. eponent 3, base 11, 1331 c. eponent 4, base 9, 6561 d. eponent 5, base 10, e. eponent 7, base 4, a. 4 4 b. 7 6 c. 3 2 d m P e. 12 4

26 23. c a. d e = = 243, 5 3 = = 125, 3 5 = 15 b c. Number of Cuts Fifths Halves Thirds Fourths n 2 a b. 64 c a. eponent 5, base 3, 243 b. eponent 2, base 8, 64 c. eponent 4, base, d. eponent 3, base 12, 1728 e. eponent 7, base 6, a. 5 3 b. 2 6 c d. 8 2 e a b y q linear 51. linear 52. linear

27 53. eponential 54. eponential 55. linear 56. eponential 57. eponential 58. linear 59. eponential 60. linear 61. m c y z u 3 v j 5 k n a 10 c w a. 1 b. 3 c. y = 3(1 ) 64. 5z p b d y p 3 q linear 72. eponential 73. linear 74. linear 75. eponential d a. 1 b. 5 c. y = 5(1 ) 76. eponential 77. linear 78. eponential 79. linear 80. eponential 81. eponential 82. v t d n 15

28 a a. 1 b. no, linear c. y = d a. 5 b. 4 c d. 5 years a. eponential b. 12 c. 12(3 ) a. linear b. 50 c. y = A. eponential, 60(4 ) B. eponential, 12(3 ) C. absolute value, -4 D. linear, y = E. neither F. eponential, 64(8 ) a a. 1 b. 4 c. 4(1 ) d a. 1 b. 7 c. 7(1 ) d a. 9

29 z m 2 b. none, linear c. y = d a. 32 b. 8 c d. 5 a. eponential b. 50 c. 50(3 ) a. linear b. 250 c. y= A-linear, y=-3+12 B-eponential, 55(5 ) C-neither D-absolute value, -7 E-eponential, 10(3 ) F-eponential, 864(6 ) 114. s A-2.4 B-1.8 C-3.5 D-4.7 E-1.2 F , 1.07, 42%, 150%, 20%, 1.5, 1.42, 43% 120. a b a b a b

30 A- 1.7 B- 2.5 C- 4.2 D- 3.4 E- 2.3 F %, 7%, 1.04, 1.88, 23%, 40%, 1.28, 1.3 a b a b. 500 c..66 d. 500(2/3) b a. a b b. 700 c..75 d. 700(3/4) %, 75%,.70,.55, 10%, 95%,.1,.92

31 152. y c w z b c..8 d. 1000(4/5) %, 35%,.72,.28, 68%, 52%,.47, b. 850 c..5 d. 850(1/2) 168. a t r q a. 5 7 b c. 3 3 d. 4 6 e. 5 8

32 f g. 3 2 h. 5 3 i. 9-3 j k l. 2 7 d e. 6 8 f g. 7 4 h i. 3-2 j k a. 4m 10 l. 5 9 b. -4z c. a 8 b 12 d y 15 e f h 15 g b 9 h. 5 5 a 15 i y 16 j. j 14 k. 1/3 4 l. 1/(-4) 3 m. 1/7 n. 1/(-3) 2 o. 11/m 5 p. 2/a 3 q. 7/s 4 r. 7t 2 /s 4 s. v 2 /n 5 t. 1/5 2 p a r 3 b. 3 5 y 10 c r 30 s 12 d. 5 y 10 e. a 6 b 9 f r 8 s 4 g. 4 3 u 3 v 15 h. c 6 i. d 10 j y 10 k. 1/4 2 l. 1/(-) 2 m. 1/(-5) 2 n. 1/78 o. 5/ 4 p. 1/ 7 q. 7a/b 2 r. s 2 /r s. 3/s 2 a t. y 7 / 5 b c. 5-15

33 Unit Review 1. C 2. B 3. B 4. C 5. C 6. C 21. a. The relationship is eponential because the previous number is multiplied by 3.4 to receive the net number b. i. 3.4 ii iii B 8. A 9. B 10. C 11. B 12. B 13. C 14. D 15. B =16, =16, 4 2=8 17. There is a petri dish full of bacteria. Every day the bacteria quadruple. In this problem the growth factor is 4 because the previous population is multiplied by four every day c d y= eponential