Precalculus Student Solution Chapter 2

34 Chapter 2 Try It 2.1. m= 4−3 = 1 = − 1 0 − 2 −2 2 ; decreasing because m < 0. 2.2. m = 1, 868 − 1, 442 = 426 = 142 people per year 2, 012 − 2, 009 3 2.3. y − 2 = − 2(x + 2) ; y = − 2x − 2 2.4. y − 0 = − 3(x − 0) ; y = − 3x 2.5. y = − 7x + 3 35 2.6. H(x) = 0.5x + 12.5 2.7. 2.8. Possible answers include ( − 3, 7), ( − 6, 9), or ( − 9, 11). 2.9. 2.10. (16, 0) 2.11. a. f (x) = 2x ; b. g(x) = − 1 x 2 2.12. y = – 1x + 6 3 2.13. a. b. (0, 5) (5, 0) 36 c. d. e. f. 2.14. Slope -1 Neither parallel nor perpendicular Decreasing function Given the identity function, perform a vertical flip (over the t-axis) and shift up 5 units. C(x) = 0.25x + 25, 000 ; The y-intercept is (0, 25, 000). If the company does not produce a single doughnut, they still incur a cost of $25,000. 2.15. 41,100; 2020 2.16. 21.15 miles 2.17. 54 ° F 2.18. 150.871 billion gallons; extrapolation Section Exercises 1. Terry starts at an elevation of 3000 feet and descends 70 feet per second. 3. 3 miles per hour 5. d(t) = 100 − 10t 7. Yes. 9. No. 11. No. 13. No. 15. Increasing. 17. Decreasing. 19. Decreasing. 21. Increasing. 23. Decreasing. 25. 3 27. –1 3 29. 4 5 31. f (x) = − 1 x + 7 2 2 33. y = 2x + 3 35. y = − 1 x + 22 3 3 37. y = 4x + 4 5 39. −5 4 41. y = 2x + 1 3 43. y = − 2x + 3 45. y=3 47. Linear, g(x) = − 3x + 5 37 49. Linear, f (x) = 5x − 5 51. Linear, g(x) = − 25 x + 6 2 53. Linear, f (x) = 10x − 24 55. f (x) = − 58x + 17.3 57. 59. a. a = 11, 900 ; b = 1001.1 b. q(p) = 1000p − 100 61. 63. x = − 16 3 65. x=a 67. d x − ad y= c− a c−a 69. $45 per training session. 71. The rate of change is 0.1. For every additional minute talked, the monthly charge increases by $0.1 or 10 cents. The initial value is 24. When there are no minutes talked, initially the charge is $24. 38 73. The slope is −400. This means for every year between 1960 and 1989, the population dropped by 400 per year in the city. 75. c. 77. The slopes are equal; y-intercepts are not equal. 79. The point of intersection is (a, a). This is because for the horizontal line, all of the y coordinates are a and for the vertical line, all of the x coordinates are a. The point of intersection will have these two characteristics. 81. First, find the slope of the linear function. Then take the negative reciprocal of the slope; this is the slope of the perpendicular line. Substitute the slope of the perpendicular line and the coordinate of the given point into the equation y = mx + b and solve for b. Then write the equation of the line in the form y = mx + b by substituting in m and b. 83. neither parallel or perpendicular 85. perpendicular 87. parallel 89. ( – 2, 0) ; (0, 4) 91. ( 1 , 0) 5 ; (0, 1) 93. (8, 0) ; (0, 28) 95. Line 1 : m=8 Line 1 : m= –1 2 Line 2 : m=2 m= –2 Line 2 : m= –2 97. 99. Line 1 : 101. g(x) = 3x − 3 103. p(t) = − 1 t + 2 3 105. (−2, 1) 107. ⎛ 17 5 ⎞ ⎝− 5 , 3 ⎠ 109. F 111. C 113. A 115. Line 2 : m= –6 Neither Perpendicular Parallel 39 117. 119. 40 121. 123. 125. 41 127. 129. 42 131. 133. 43 135. a) g(x) = 0.75x − 5.5 ; b) 0.75; c) (0, − 5.5) 137. y=3 139. x= −3 141. no point of intersection 143. (2, 7) 145. ( – 10, –5) 147. y = 100x − 98 149. x < 1999 x > 1999 201 201 151. Less than 3000 texts 153. Determine the independent variable. This is the variable upon which the output depends. 155. To determine the initial value, find the output when the input is equal to zero. 157. 6 square units 159. 20.012 square units 161. 2,300 163. 64,170 165. P(t) = 75, 000 + 2500t 167. (–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000. 169. Ten years after the model began. 44 171. W(t) = 7.5t + 0.5 173. (−15, 0) : The x-intercept is not a plausible set of data for this model because it means the baby weighed 0 pounds 15 months prior to birth. (0, 7.5) : The baby weighed 7.5 pounds at birth. 175. At age 5.8 months. 177. C(t) = 12, 025 − 205t 179. (58.7, 0) : In roughly 59 years, the number of people inflicted with the common cold would be 0. (0,12,025) : Initially there were 12,025 people afflicted by the common cold. 181. 2064 183. y = − 2t+180 185. In 2070, the company’s profit will be zero. 187. y = 30t − 300 189. (10, 0) In 1990, the profit earned zero profit. 191. Hawaii 193. During the year 1933 195. $105,620 197. 696 people; 4 years; 174 people per year; 305 people; P(t) = 305 + 174t ; 2219 people 199. C(x) = 0.15x + 10 ; The flat monthly fee is $10 and there is an additional $0.15 fee for each additional minute used; $113.05 201. P(t) = 190t + 4360 ; 6640 moose 203. R(t) = 16 − 2.1t ; 5.5 billion cubic feet; During the year 2017 205. More than 133 minutes 207. More than $42,857.14 worth of jewelry 209. $66,666.67 211. When our model no longer applies, after some value in the domain, the model itself doesn’t hold. 213. We predict a value outside the domain and range of the data. 215. The closer the number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data. 217. 61.966 years 219. No. 221. No. 223. Interpolation. About 60 ° F. 225. C 227. B 231. 45 233. Yes, trend appears linear because r = 0.985 and will exceed 12,000 near midyear, 2016, 24.6 years since 1992. 235. y = 1.640x + 13.800 , r = 0.987 237. y = − 0.962x + 26.86, r = − 0.965 239. y = − 1.981x + 60.197 ; r = − 0.998 241. y = 0.121x − 38.841, r = 0.998 243. (−2, −6), (1, −12), (5, −20), (6, −22), (9, −28) ; y = −2x−10 245. (189.8, 0) 247. If 18,980 units are sold, the company will have a profit of zero dollars. y = 0.00587x + 1985.41 249. y = 20.25x − 671.5 251. y = − 10.75x + 742.50 Review Exercises 253. Yes 255. Increasing. 257. y = − 3x + 26 259. 3 261. y = 2x − 2 263. Not linear. 265. parallel 267. (–9, 0); (0, –7) 269. Line 1: m = − 2; Line 2: m = − 2; Parallel 271. y = − 0.2x + 21 46 273. 275. 250. 277. 118,000. 279. y = − 300x + 11, 500 281. a) 800; b) 100 students per year; c) P(t) = 100t + 1700 283. 18,500 285. $91,625 287. Extrapolation. 47 289. 291. Midway through 2024. 293. y = − 1.294x + 49.412; r = − 0.974 295. Early in 2022 297. 7,660 Practice Test 298. Yes. 300. Increasing 302. y = −1.5x − 6 304. y = − 2x − 1 306. No. 308. Perpendicular 310. (−7, 0) ; (0, − 2) 312. y = − 0.25x + 12 314. 316. 150 318. 165,000 320. y = 875x + 10, 675 322. a) 375; b) dropped an average of 46.875, or about 47 people per year; c) y = − 46.875t + 1250 48 324. 326. Early in 2018 328. y = 0.00455x + 1979.5 330. r = 0.999