alg ii analyzing and transforming functions 1...

Alg II – Analyzing and Transforming Functions 1 NJCTL.org

Intercepts – Class Work Find the x and y intercepts for the following graphs. Write your answers as coordinates.

1. 2.

3. 4.

Find the x and y intercepts algebraically. Write your answers as coordinates.

5. y = 2x3 – 5x2 – 3x 6. f(x) = 3x2 – 13x + 4

7. g(x) = -4x + 3 8. y = -2x2 +3x - 1

9. y = x3 + 2x2 + 4x 10. h(x) = 4

3𝑥 −

3

2


Intercepts – Home Work Find the x and y intercepts for the following graphs. Write your answers as coordinates.

11. 12.

13. 14.

Find the x and y intercepts algebraically. Write your answers as coordinates.

15. 𝑓(𝑥) =3𝑥

2+ 4 16. y = 4x3 – 4x2 + x

17. y = x2 + x – 20 18. g(x) = 2x2 + 3x – 1

19. h(x) = 8x3 + 6x2 – 9x 20. m(x) = -3x + 5


Value, Change & Rate of Change – Class Work Use the table below from Center for Disease Control (CDC) to answer questions 21 - 25. The chart shows stature for age of males.

Age (in

months)

5th Percentile

Stature (in

centimeters)

10th Percentile

Stature (in

centimeters)

25th Percentile

Stature (in

centimeters)

50th Percentile

Stature (in

centimeters)

75th Percentile

Stature (in

centimeters)

90th Percentile

Stature (in

centimeters)

95th Percentile

Stature (in

centimeters)

24 80.72977 81.99171 84.10289 86.4522 88.80525 90.92619 92.19688

24.5 81.08868 82.36401 84.49471 86.86161 89.22805 91.35753 92.63177

25.5 81.83445 83.11387 85.25888 87.65247 90.05675 92.22966 93.53407

26.5 82.56406 83.84716 86.00517 88.42326 90.8626 93.07608 94.40885

27.5 83.27899 84.56534 86.73507 89.17549 91.64711 93.89827 95.25754

28.5 83.98045 85.26962 87.44977 89.91041 92.41159 94.69757 96.08149

21. What is the height of a boy in the 90th percentile at age 26.5 months?

22. What is the rate of change for a boy 26.5 months to 27.5 months in 25th percentile?

23. What is the rate of change for a boy 24 months to 24.5 months in 50th percentile?

24. What is the average rate of change of a boy who is always in the 75th percentile from 24 to 28.5 months?

25. What is the average rate of change of a boy who is always in the 10th percentile from 24 to 28.5 months? Use the graph of Pressure vs. Altitude to answer questions 26 - 30.

26. What is the pressure when the altitude is 2,000 ft?

27. What is altitude when the pressure is 200 hPa?

28. What is the rate of change from 2,000 ft to 4,000ft?

29. What is the rate of change from 4,000ft to 6,000ft?

30. What is the rate of change from 2,000 ft to 6,000ft? Spiral Review Factor each of the following

31. 3𝑥2 − 2𝑥 − 5 32. 5𝑥2 + 19𝑥 + 12 33. 7𝑥2 + 53𝑥 + 28


Value, Change & Rate of Change – Home Work Use the table below from Center for Disease Control (CDC) to answer questions 34 - 38. The chart shows stature for age of females.

Age (in

months)

5th Percentile

Stature (in

centimeters)

10th

Percentile

Stature (in

centimeters)

25th

Percentile

Stature (in

centimeters)

50th

Percentile

Stature (in

centimeters)

75th

Percentile

Stature (in

centimeters)

90th

Percentile

Stature (in

centimeters)

95th

Percentile

Stature (in

centimeters)

24 79.25982 80.52476 82.63524 84.97556 87.31121 89.40951 90.66355

24.5 79.64777 80.91946 83.04213 85.39732 87.74918 89.86316 91.12707

25.5 80.44226 81.73541 83.8943 86.29026 88.68344 90.83505 92.12168

26.5 81.22666 82.53699 84.72592 87.15714 89.58751 91.77421 93.08254

27.5 81.9954 83.31968 85.53389 87.99602 90.46018 92.67969 94.00873

28.5 82.74411 84.07998 86.31589 88.80551 91.30065 93.55097 94.89974

34. What is the height of a girl in the 90th percentile at age 26.5 months?

35. What is the rate of change for a girl 26.5 months to 27.5 months in 25th percentile?

36. What is the rate of change for a girl 24 months to 24.5 months in 50th percentile?

37. What is the average rate of change of a girl who is always in the 75th percentile from 24 to 28.5 months?

38. What is the average rate of change of a girl who is always in the 10th percentile from 24 to 28.5 months? In Questions 39 - 43, refer to the graph of the participation of the players on the field of a soccer team. 𝑃(𝑡) is

amount of participation at any given 𝑡, time in minutes.

39. What was the amount of participation at 𝑡 = 7?

40. What was the rate of change in participation from 𝑡 = 2 to 𝑡 = 3?

41. What was the rate of change in participation from 𝑡 = 2 to 𝑡 = 5?

42. At what time was there a participation of 4 players?

43. What was the rate of change in participation from 𝑡 = 6 to 𝑡 = 7? Spiral Review

Factor each of the following

44. 4𝑥2 − 35𝑥 + 49 45. 6𝑥2 + 7𝑥 − 49 46. 15𝑥2 − 27𝑥 − 6


Maxima and Minima – Class Work 47. A box manufacturer wants to make a box with a square base that holds 10,000 in3 and has a height of more

than 1 inch. To minimize materials, what dimensions should the box have? 48. A farmer has 300’ of fence and wants to maximize his materials. He wants to make two equal size pens that

share a side. What are the dimensions of one pen? 49. An 8 in by 10 in sheet of paper is to have squares removed from its corners so that the remaining edges can

be folded into a lid-less box. What is the greatest volume possible?

50. An isosceles triangle is to have an area of 30 𝑐𝑚2. What side lengths will minimize the perimeter? 51. A 10 in by 20 in sheet of material will have squares cut out of the corners

and two squares cut out from the middle of each long side so that when folded the net forms a box. Calculate the size of the squares that will maximize the volume.

Maxima and Minima – Home Work 52. A box manufacturer wants to make a box with a square base that holds 20,000 𝑖𝑛3 and has a height more

than 1 inch. To minimize materials, what dimensions should the box have? 53. A farmer has 450’ of fence and wants to maximize his materials. He wants to make two equal size pens that

share a side. What are the dimensions of one pen? 54. An 8 in by 12 in sheet of papers is to have squares taken out of its corners so that the remaining edges can

be folded into a lid-less box. What is the greatest possible volume?

55. An isosceles triangle is to have an area of 100 𝑐𝑚2. What side lengths minimize the perimeter? 56. A 10 in by 8 in sheet of material will have squares cut out of the corners

and two squares cut out from middle of each long side so that when folded, the net forms box. Calculate the size of the squares that will maximize the volume.


Increasing and Decreasing Functions– Class Work Use the graph of f(x) to answer the following questions. 57. Interval(s) on which 𝑓(𝑥) is

increasing 58. Interval(s) on which 𝑓(𝑥) is

decreasing. 59. 𝑥-value of any local maxima

60. 𝑥-value of any local minima 61. 𝑥-value of any absolute maximum 62. 𝑥-value of any absolute minimum

Use the table to answer the following questions. The table represents the scores one student received on

practice math exams leading up to the SAT’s.

Week 1 2 3 4 5 6 7 8 9

Score 510 520 550 560 530 550 560 580 590

63. During what interval(s) were the scores

increasing?

64. State any relative minimum scores.

65. What was the greatest rate of change and when did it occur?


Increasing and Decreasing Functions – Home Work Use the graph of f(x) to answer the following questions.

66. Interval(s) on which 𝑓(𝑥) is increasing

67. Interval(s) on which 𝑓(𝑥) is decreasing.

68. 𝑥-value of any local maxima

69. 𝑥-value of any local minima 70. 𝑥-value of any absolute maximum 71. 𝑥-value of any absolute minimum

Use the table to answer the following questions. The table represents the number of assignments one student received in math class for a marking period

Week 1 2 3 4 5 6 7 8 9

Assignments 80 120 130 140 145 135 120 130 135

72. During what interval(s) was the number of

assignments increasing?

73. State any relative minimum assignment weeks.

74. What was the greatest rate of change and when did it occur?


-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

Odd and Even Functions – Class Work Is the equation given an odd function, an even function, or neither? Show work.

75. 𝑓(𝑥) = 3𝑥5 + 2𝑥3 + 6𝑥 76. 𝑔(𝑥) = −5𝑥4 − 3𝑥2 + 2 77. ℎ(𝑥) = 2𝑥 + 1

78. 𝑓(𝑥) = 3𝑥4 79. 𝑔(𝑥) = 5𝑥3 − 1

Is the graphed function odd, even, or neither? Explain why.

80. 81. 82.

Spiral Review

Simplify each of the following.

83. √512𝑏2 84. √80𝑝3 85. √28𝑥3𝑦3

Simplify and add each of the following

86. −2√3 + 3√27 87. 2√6 − 2√24 88. 2√6 + 3√54


-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

-8 -6 -4 -2 2 4 6 8

-3

-2

-1

1

2

3

x

y

Odd and Even Functions – Home Work Is the equation given an odd function, an even function, or neither? Show work.

89. 𝑓(𝑥) = 𝑥7 + 2𝑥3 + 6 90. 𝑔(𝑥) = −𝑥6 − 𝑥2 + 2 91. ℎ(𝑥) = 2𝑥4 + 1

92. 𝑓(𝑥) = 6𝑥4 + 𝑥 93. 𝑔(𝑥) = −7𝑥3 − 𝑥

Is the graphed function odd even or neither? Explain why.

94. 95. 96.

Spiral Review

Simplify each of the following

97. √147𝑚3𝑛3 98. √200𝑚4𝑛 99. √384𝑥4𝑦3

Simplify and add each of the following.

100. −√12 + 3√3 101. 3√3 − √27 102. −3√20 − √5


-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

Symmetry and Periodicity – Classwork State whether the function is has symmetry over the x-axis, y-axis, diagonal (y = x), origin, or none. Explain why.

103. 𝑓(𝑥) = 3𝑥5 + 2𝑥3 + 6𝑥 104. 𝑔(𝑥) = −5𝑥4 − 3𝑥2 + 2 105. ℎ(𝑥) = 2𝑥 + 1

106. 𝑓(𝑥) = √3𝑥 + 15 107. 𝑔(𝑥) = 5𝑥3 − 𝑥 108. ℎ(𝑥) = −1

3𝑥

109. 110.

Find the period of the function.

111. 112.


Symmetry and Periodicity – Homework State whether the function is has symmetry over the x-axis, y-axis, diagonal (y = x), origin, or none.

113. 𝑓(𝑥) = 𝑥7 + 2𝑥3 + 6 114. 𝑔(𝑥) = −𝑥6 − 𝑥2 + 2 115. ℎ(𝑥) =1

2𝑥4+1

116. 𝑓(𝑥) = 6𝑥4 + 𝑥 117. 𝑔(𝑥) = −7𝑥3 − 𝑥 118. ℎ(𝑥) = √6𝑥4 − 3𝑥

119. 120.

Find the period of the function.

121. 122.


End Behavior – Class Work Find the end behavior of the following functions. Use appropriate notation.

123. 124.

125. 126.

127. 128.


End Behavior – Home Work Find the end behavior of the following functions. Use appropriate notation.

129. 130.

131. 132.

133. 134. Spiral Review: Simplify the following:

135. √24𝑥6𝑦8 136. (−3𝑚3𝑛5)4 137. (

3𝑎−5𝑏−8

−9𝑎5𝑏−4)−2


Analyzing Functions – Class Work Analyze the following functions. Use appropriate notation.

138.

Domain:

Range:

Minimum (Min):

Maximum (Max):

 x-intercepts:

 y-intercepts:

Increasing:

Decreasing:

Odd or Even Function:

End Behavior:

f(x) = -x4 – 2x3

139.

Domain:

Range:

Minimum (Min):

Maximum (Max):

 x-intercepts:

 y-intercepts:

Increasing:

Decreasing:


End Behavior:

𝒚 = |𝒙 + 𝟏| + 𝟐


Analyzing Functions – Home Work Analyze the following functions. Use appropriate notation.

140.

Domain:

Range:

Minimum (Min):

Maximum (Max):

 x-intercepts:

 y-intercepts:

Increasing:

Decreasing:


End Behavior:

f(x) = -x3 – 1

141.

Domain:

Range:

Minimum (Min):

Maximum (Max):

 x-intercepts:

 y-intercepts:

Increasing:

Decreasing:


End Behavior:

𝒚 = (𝒙 − 𝟐)𝟐 + 𝟏

Spiral Review: Simplify the following:

142. (𝑚3𝑛4𝑝−2)−3 143. (−4𝑥−4𝑦−7)(−3𝑥−3𝑦−2) 144. √36𝑟6𝑡8


Vertical Shifts

Class Work

Part 1: In each question below, g(x) is shown. Re-graph g(x) using the transformation indicated.

145. a) move up 3 146. a) move down 3

b) move down 2 b) move up 1

Part 2: In each exercise the function h(x) is given. (i) Identify its parent function, (ii) describe the

transformation(s) needed to go from the parent function to h(x), (iii) draw the graph of both on one graph.

147. ℎ(𝑥) = 𝑥2 − 2

148. ℎ(𝑥) =1

𝑥+ 3

149. ℎ(𝑥) = √𝑥 + 1

150. ℎ(𝑥) = |𝑥| − 1

151. ℎ(𝑥) = 𝑒𝑥 − 4

Vertical Shifts

Homework


152. a) move up 3 153. a) move down 5

b) move down 1 b) move up 2



154. ℎ(𝑥) = 𝑥 + 3

155. ℎ(𝑥) =1

𝑥− 2

156. ℎ(𝑥) = cos(𝑥) + 2

157. ℎ(𝑥) = |x| + 3

158. ℎ(𝑥) = log (𝑥) − 4

Spiral Review

Factor: Simplify: Multiply: Work out:

159. 2x2 + x – 3 160. −2𝑥4𝑦7

8𝑥6𝑦4 161. (x – 1)(x2 – 2x + 3) 162. (2x + 3)2


Horizontal Shifts

Class Work


163. a) move right 2 164. a) move right 2

b) move left 5 b) move left 4



165. ℎ(𝑥) = (𝑥 − 2)2

166. ℎ(𝑥) =1

(𝑥+3)

167. ℎ(𝑥) = √𝑥 + 1

168. ℎ(𝑥) = |𝑥 − 1|

169. ℎ(𝑥) = 𝑒𝑥−4

Horizontal Shifts

Homework


170. a) move right 3 171. a) move left 2

b) move left 2 b) move right 4



172. ℎ(𝑥) = (𝑥 − 3)3

173. ℎ(𝑥) =1

𝑥−2

174. ℎ(𝑥) = |𝑥 + 4|

175. ℎ(𝑥) = √𝑥 − 5

176. ℎ(𝑥) = log (𝑥 − 2)

Spiral Review

Simplify: Factor: Simplify: Simplify:

177. (x – 1)3 178. 6x2 – 19x + 10 179. 𝑎5𝑏4𝑐8

𝑎4𝑏8𝑐2 180.

1

𝑥22

𝑥3


Reflections

Class Work


181. a) reflect over x-axis 182. a) reflect over x-axis

b) reflect over y-axis b) reflect over y-axis



183. ℎ(𝑥) = −(𝑥)2

184. ℎ(𝑥) =1

(−𝑥)

185. ℎ(𝑥) = √−𝑥

186. ℎ(𝑥) = −|𝑥|

187. ℎ(𝑥) = 𝑒−𝑥

Reflections

Homework


188. a) reflect over x-axis 189. a) reflect over x-axis

b) reflect over y-axis b) reflect over y-axis



190. ℎ(𝑥) = (−𝑥)2

191. ℎ(𝑥) = −𝑥3

192. ℎ(𝑥) = −cos (𝑥)

193. ℎ(𝑥) = −1

𝑥

194. ℎ(𝑥) = −log (x)

Spiral Review

Simplify: Work out: Multiply: Simplify:

195. 1

32

5

196. (2x + 1)3 197. (3x + 2)(x2 + 3x – 2) 198. −24𝑥3𝑦−4𝑧

−8𝑥−4𝑦𝑧−4


Vertical Stretches and Shrinks

Class Work


199. a) Vertical Stretch of 3 200. a) Vertical Shrink of 1

3

b) Vertical Shrink of 1

2 b) Vertical Stretch of 2



201. ℎ(𝑥) = 2(𝑥)2

202. ℎ(𝑥) =3

(𝑥)

203. ℎ(𝑥) =1

2√𝑥

204. ℎ(𝑥) = 2

3|𝑥|

205. ℎ(𝑥) = 3𝑒𝑥

Vertical Stretches and Shrinks

Homework


206. a) Vertical Shrink of 1

3 207. a) Vertical Stretch of 4

b) Vertical Stretch of 3 b) Vertical Shrink of 1

2



208. ℎ(𝑥) = 4𝑥3

209. ℎ(𝑥) =0.25

𝑥

210. ℎ(𝑥) =1

2cos (𝑥)

211. ℎ(𝑥) = 3√𝑥

212. ℎ(𝑥) = log (1

3𝑥)

Spiral Review

Simplify: Work out: Multiply: Simplify:

213. 1

𝑥2

𝑥2

214. 3 - 4(25 ÷ 5 · 2) 215. (3x + 2)3 216. −12𝑥−3𝑦−4𝑧

6𝑥−4𝑦


Horizontal Stretches and Shrinks

Class Work


217. a) Horizontal Shrink of 3 218. a) Horizontal Stretch of 1

3

b) Horizontal Stretch of 1

2 b) Horizontal Shrink of 2



219. ℎ(𝑥) = (2𝑥)2

220. ℎ(𝑥) =1

(3𝑥)

221. ℎ(𝑥) = √1

2𝑥

222. ℎ(𝑥) = |2

3𝑥|

223. ℎ(𝑥) = 𝑒3𝑥

Horizontal Stretches and Shrinks

Homework


224. a) Horizontal Shrink of 3 225. a) Horizontal Stretch of 1

2

b) Horizontal Stretch of 1

3 b) Horizontal Shrink of 2



226. ℎ(𝑥) = (4𝑥)3

227. ℎ(𝑥) =1

.25𝑥

228. ℎ(𝑥) = (1

4𝑥)

2

229. ℎ(𝑥) = √2x

230. ℎ(𝑥) = log (1

2𝑥)

Spiral Review

Multiply: Factor: Factor: Multiply:

231. (x + 5)2 232. 9x2 – 25 233. 4x2 + 25 234. -3x2(4x5 + 5x3)


Combining Transformations

Class Work


235. a) f(x) = -2g(x) + 2 236. a) f(x) = -g(x – 3) +1

b) f(x) = 1

3𝑔(𝑥 − 5) − 3 b) f(x) = 3g(x + 2) – 2



237. ℎ(𝑥) = 4(−𝑥 + 3)2 + 6

238. ℎ(𝑥) =3

(2𝑥+1)− 2

239. ℎ(𝑥) = 2√3 − 𝑥 + 4

240. ℎ(𝑥) = 2|3𝑥 − 6| − 1

241. ℎ(𝑥) = 3𝑒−2𝑥 + 4

Combining Transformations

Homework


238. a) 𝑓(𝑥) = −1

3𝑔(𝑥 − 3) 239. A) f(x) = -2g(x) – 2

b) f(x) = g(x + 3) + 2 b) f(x) = g(-2x)



240. ℎ(𝑥) = 4(2𝑥 − 3)3 + 2

241. ℎ(𝑥) =3

𝑥−2− 4

242. ℎ(𝑥) = − |1

3𝑥 − 2|

243. ℎ(𝑥) = √−x − 3

244. ℎ(𝑥) = log(2 − 𝑥) + 4

Spiral Review

Multiply: Simplify: Factor: Factor:

245. (3x – 4)2 246. 15𝑚−4𝑛−5

−25𝑚−5𝑛−3 247. 25x2 – 1 248. x2 + 36


Piecewise Functions

Class Work

249. 𝑓(𝑥) = {√𝑥2 + 3 + 2 𝑖𝑓 𝑥 < 0

3𝑥2 + 7𝑥 − 2 𝑖𝑓 𝑥 ≥ 0

Find: a. f(-2) b. f(1) c. f(4) d. state the domain and range of f(x) e. graph f(x)

251. 𝑓(𝑥) = {−3𝑥2 + 2 𝑖𝑓 𝑥 ≤ 2

1

2𝑥 + 2 𝑖𝑓 𝑥 > 2


250. 𝑓(𝑥) = {

−2|𝑥 + 3| + 3 𝑖𝑓 𝑥 < −1

𝑥2 − 4 𝑖𝑓 − 1 ≤ 𝑥 < 3

−4𝑥 + 10 𝑖𝑓 𝑥 ≥ 3

Find: a. f(-5) b. f(0) c. f(4) d. state the domain and range of f(x) e. graph f(x) 252. Find b so that f(x) is continuous:

𝑓(𝑥) = {2𝑥 + 𝑏 𝑖𝑓 𝑥 < 2

−3𝑥 + 7 𝑖𝑓 𝑥 ≥ 2

Piecewise Functions

Homework

253. 𝑓(𝑥) = {−4𝑥 + 5 𝑖𝑓 𝑥 < −2

−𝑥2 + 6 𝑖𝑓 𝑥 ≥ −2


255. 𝑓(𝑥) = {−

1

3𝑥3 𝑖𝑓 𝑥 ≤ 4

2𝑥 − 6 𝑖𝑓 𝑥 > 4


254. 𝑓(𝑥) = {

−2 𝑖𝑓 𝑥 < −3𝑥 − 4 𝑖𝑓 − 3 ≤ 𝑥 < 3

−(𝑥 − 4)2 + 1 𝑖𝑓 𝑥 ≥ 3


256. 𝑓(𝑥) = {

3𝑥 + 𝑏 𝑖𝑓 𝑥 < 4−𝑎𝑥 − 𝑏 𝑖𝑓 4 ≤ 𝑥 ≤ 6

2𝑥 + 𝑎 𝑖𝑓 𝑥 > 6 ; find a and b

so that f(x) is continuous.

Spiral Review

257. Find: f ◦ g 258. Factor: 259. Simplify 260. Describe the transformation:

If g(x) = x2 81x2 –16y2 (-2x4y)(4x3y5) 𝑦 = −|𝑥 − 2| − 5

and f(x) = 3x – 1

Alg II – Analyzing Functions ~23~ NJCTL.org

Unit Review Questions Analyzing and Working with Functions Multiple Choice – Determine the best answer for each question. 1. What is the domain of the graph to the right?

a. −10 ≤ 𝑥 ≤ 10

b. −10 < 𝑥 < 10

c. −6 ≤ 𝑥 ≤ −2 or 0 ≤ 𝑥 ≤ −6

d. −10 ≤ 𝑥 ≤ −4 or −2 ≤ 𝑥 ≤ 4 or 6 ≤ 𝑥 ≤ 10

In Questions 2 – 4, refer to the graph on the right: 2. There is a local minimum at:

a. 𝑥 = −3.5 b. 𝑥 = 0

c. 𝑥 = 1 d. There is no local minimum

3. The rate of change from 𝑥 = −2 to 𝑥 = −1.5 is the same as the

rate of change from

a. 𝑥 = −1 to 𝑥 = 0. b. 𝑥 = 2 to 𝑥 = 3

c. 𝑥 = 1 to 𝑥 = 3 d. 𝑥 = 0.5 to 𝑥 = 1

4. There is an absolute maximum at:

a. ~(−1.7, 2.3) b. ~(2.1, 8.1)

c. (0, 0) d. There is no absolute

maximum 5. In the table to the right, the rate of change between 𝑥 = 4 and 𝑥 = 6 is

a. 2 b. 1 c. 0.5 d. -1

6. A rancher has 10,000’ of fence and wants to use it to make a pen

with the maximum area. A barn 40’ by 100’ is to be used as a corner of the pen. Which equation would she use?

a. 𝐴 =5000𝑥 −𝑥2

b. 𝐴 =5070𝑥 −𝑥2

c. 5140𝑥 − 𝑥2

d. 𝐴 =10,000𝑥 −𝑥2

In Questions 7 – 9, consider the following graph to the right:


7. The rate of change from 𝑥 = 4 to 𝑥 = 8 is

a. 3 b. 0.75

c. 0 d. -3

8. The greatest rate of change is between

a. 𝑥 = −7 and 𝑥 = −6 b. 𝑥 = −1 and 𝑥 = 3

c. 𝑥 = 1 and 𝑥 = 2 d. 𝑥 = 8 and 𝑥 = 9

9. The rate of change from 𝑥 = −5 to 𝑥 = −4 is the same as the rate of change from

a. 𝑥 = −6 to 𝑥 = −5 b. 𝑥 = −3 to 𝑥 = −2

c. 𝑥 = 6 to 𝑥 = 7 d. 𝑥 = 3 to 𝑥 = 5

In Questions 10 – 13, refer to the graph below: 10. There is a local max at

a. −6 b. −3 c. −1 d. 1

11. In terms of concavity, the point at 𝑥 = −5 is

a. concave up. b. concave down. c. a local minimum. d. none of the above.

12. The rate of change is positive on the interval

a. (−4, −1.5) b. (−2, 1) c. (−1, 3) d. (1, 3)

13. Given 𝑓(𝑥) = 24𝑥6 + 18𝑥3 + 6𝑥2, the function is

a. an odd function b. an even function

c. neither an odd or even function d. both an odd and even function

Determine if the statements in questions 14 – 16 are true or false.

14. The function 𝑓(𝑥) = −3𝑥5 + 4𝑥3 − 7𝑥 is symmetrical over the 𝑥-axis True False

15. Any function with odd exponents in all terms is symmetrical over the 𝑦-axis. True False

16. All equations that are symmetrical over the 𝑥-axis are functions True False


17. The period of the graph to the right is:

a. 𝜋 b. 𝜋

2

c. 2𝜋 d. 4𝜋

18. Identify the transformations on the function 𝑓(𝑥) = 3𝑥4 + 15.

a. shifted up 15; shifted left 3 b. shifted up 3; shifted right 15

c. vertical stretch; shifted up 15 d. vertical shrink; shifted up 15

19. Describe the transformation of the parent function f(x) = x2 to g(x) = x2 – 1

a. shift left 1

b. shift right 1

c. shift down 1

d. shift up 1

20. Describe the transformation of the parent function f(x) = |x| to g(x) = | x+1|

a. shift left 1

b. shift right 1

c. shift down 1

d. shift up 1

21. Describe the transformation of the parent function f(x)= [x] to g(x)= [2x]

a. horizontal stretch of scale factor 2

b. horizontal stretch of scale factor 1/2

c. vertical stretch of scale factor 2

d. vertical stretch of scale factor 1/2

22. Describe the transformation of the parent function f(x)= 1

x to g(x)=

2

x

a. horizontal stretch of scale factor 2

b. horizontal stretch of scale factor 1/2

c. vertical stretch of scale factor 2

d. vertical stretch of scale factor ½

23. Describe the transformation of the parent function f(x)= log(x) to g(x)= log(-x)

a. horizontal reflection

b. vertical reflection

c. does not affect f(x) since it is symmetrical

d. not possible because log(x) is undefined for negatives

24. The order of the following transformation of ℎ(𝑥) = √𝑥 to ℎ(𝑥) = 4√3 − 𝑥 + 5 is

a. Slide 3 right, stretch 4 vertically, slide 5 up

b. Slide 3 left, stretch 4 vertically, slide 5 up

c. Reflect over the y-axis, slide 3 right, stretch 4 vertically, slide 5 up

d. Reflect over the y-axis, slide 3 left, stretch 4 vertically, slide 5 up

http://www.google.com/imgres?hl=en&sa=X&biw=1366&bih=674&tbm=isch&tbnid=zF_J-3rlbJMS0M:&imgrefurl=http://www.dimi.uniud.it/~gorni/Mma/Mma.html&docid=aoH6M5gus1a6yM&imgurl=http://users.dimi.uniud.it/~gianluca.gorni/Immagini/periodicGraph.GIF&w=288&h=177&ei=KgtSUce-EpS4yAGj8oCoCg&zoom=1&ved=1t:3588,r:3,s:0,i:94&iact=rc&dur=7421&page=1&tbnh=141&tbnw=230&start=0&ndsp=16&tx=153&ty=81


25. 𝑎(𝑥) = {2𝑥 − 1 𝑖𝑓 𝑥 ≤ 3

−4𝑥 + 2 𝑖𝑓 𝑥 > 3 , find a(3).

a. 5

b. -10

c. 5 or -10

d. Undefined

26. 𝑏(𝑥) = {3𝑥 + 2𝑎 𝑖𝑓 𝑥 < 24𝑎 − 𝑥 𝑖𝑓 𝑥 ≥ 2

, find 𝑎 such that b(x) is continuous

a. -8

b. -4

c. 4

d. 8

Extended Response – Completely answer each question showing all work. 27. The number of people entering the exciting new amusement park, “Math World – HD in 3D” is given by the

following equation, where 𝑡 is the amount of time, in hours, after the park opens.

𝑒(𝑡) {𝑡2 + 30 0 < 𝑡 ≤ 6𝑡 + 10 6 < 𝑡 ≤ 12

a. If the park opened at 10 am, how many people entered at 1 pm?

b. During what hour did the most number of people enter the park? How many people entered during that hour?

c. What is the rate of change in people entering from 12 pm to 2pm? 28. Brenda decides to save her spare change in a jar. The initial amount in the jar, 𝐽(0) is $20.00 and after one

week it is 𝐽(7) = 23.50. a. How much money did Brenda save during the first week?

b. At what rate is Brenda saving money?

c. If Brenda continued to save at a linear rate, how much would she have on 𝐽(14)?


29. Let ℎ(𝑥) = 4 − 𝑥2 a. Describe the end behaviors of ℎ(𝑥).

b. Describe the intervals of increase and decrease of ℎ(𝑥).

30. Given the function of f(x) as shown at the right

a. a(x) = 2f(3 - x), graph a(x).

b. b(x) = x2 , c(x) =b(f(x)). Graph c(x).

c. Graph f-1(x).

31. ℎ(𝑥) = {2𝑥2 + 𝑥𝑚 𝑖𝑓 𝑥 ≤ 3

𝑥𝑚2 𝑖𝑓 𝑥 > 3

a. Find the values of m that make h(x) continuous.

b. For which value of m is the rate of change about h(3) the closest?

c. Find h(3) in terms of m.

32. People enter a park at a rate of 𝐸(𝑡) = {8𝑡 𝑖𝑓 𝑡 ≤ 62𝑡 𝑖𝑓 𝑡 > 6

where t is the number of hours after opening.

People leave the park at a rate of 𝐿(𝑡) = {4𝑡 𝑖𝑓 𝑡 ≤ 6

3𝑡 + 𝑐 𝑖𝑓 𝑡 > 6. The park is open 12 hours a day.

a. Write an, P(t) equation for the rate of change in the number of people in the park in terms of E(t)

and L(t).

b. Create the piecewise function for P(t).

c. Find c so that there is no one in the park at closing.

d. Does the answer in part c make sense? Explain.

alg ii analyzing and transforming functions 1...

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