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File: Algebra2B 2011 Questions 8.12.11 6. Write the equation for the conic function below. Updated: August 12, 2011 Section 5.00 1. What are the four conic sections shapes generated from the intersection of a plane and a cone? Unless specific instructions are given below, perform the following conic operations: First state type of conic function, orientation and provide a sketch of the conic function and put equation in standard form. If conic is a Parabola find: a) Vertex (h,k) = b) p value and direction of opening c) Focus point (F = ) d) Equation of Directrix If conic is a Circle find: a) Center (h,k) = b) Radius (r = ) If conic is a Ellipse find: a) Center (h,k) = b) Vertices (V = ); as well as (U = ) c) Length of Major d) Length of Minor axis e) Foci ordered pairs (F = ) If conic is a Hyperbola find: a) Center (h,k) = b) Vertices (V = ); as well as (U = ) c) Length of Transverse axes d) Length of Conjugate axis e) Foci ordered pairs (F = ) f) Asmptotes: 7. 2. Explain how the formula of a circle is derived. Use an example to help illustrate your explanation. 11. 3. (x + 5)2 + (y – 3 )2 = 49 i) ii) iii) iv) 8. 9. 10. 12. 4. Complete the following table for circles with the given center and radius Center Radius (h,k) (r) Circle Equation a) (-5, 2) 6 b) (6, -3) 5 c) (4, -1) 8 d) (-5, 6) 4 13. 14. 15. 5. Complete the following table for circles with the given equations. a) b) c) d) 2 (x + 5) (x – 7)2 (x – 2)2 (x + 8)2 Equation + (y + 3)2 = 49 + (y – 2)2 = 81 + (y + 1)2 = 100 + (y – 4)2 = 4 Center (h, k) 16. Radius (r) Below are the equations of four conic sections. Indicate the conic type for each equation and sketch the parameters that are easily obtained. a) b) c) d) 17. Write the equation of a circle with center (0, 3) and a radius of 3. 18. Write the equation of a circle with center (-1, 2) and a radius of 4. 19. (x – 4)2 + y2 = 16 20. (x – 1)2 + (y + 2)2 = 25 21. Determine which of the four conic equation is/are functions. 22. Write the equation for the conic function below. 26. Write the equation for the conic function below. Section 5.07 \ 23. 27. (x – 5)2 + (y + 3)2 = 16 28. Write the equation for the conic function below. (Each square is 1 x 1) Write the equation for the conic function below. 29. 30. 24. 25. Write the equation of the ellipse with foci (0, +3) and y-intercepts +5. Write an equation of a circle with a diameter of 20 inches and a center at the point (4, -5) Section 5.10 Write the equation for the conic function below. 31. 32. 33. (x – 3)2 + (y + 2)2 = 25 y = 2x2 34. Write the equation for the conic function below. 39. Given the conic equation: y = -2(x – 3)2 – 4 Determine: a) Conic type: b) If any horizontal or vertical shift c) If any reflection d) If any vertical shrink or stretch e) The vertex 40. x2 = - 16y 41. y = ( 1/12) x2 42. 43. 35. Write the equation for the conic function below. 44. Below are the equations of four conic sections. Indicate the conic type for each equation and sketch the parameters. a) b) c) d) 45. 36. 37. 38. 46. y = 4(x + 3)2 + 5 a) Conic type: b) If any horizontal or vertical shift c) If any reflection d) If any vertical shrink or stretch e) The vertex f) The focus and directrix (sketch) 47. Where do the asymptotes of a hyperbola intersect the hyperbola? How is the hyperbola intersections defined? 48. Sketch and label: (x + 2)2 + (y – 3)2 = 16 49. Sketch and label: (x + 4)2 + (y – 1) = 25 50. Write the equation for the conic function below. Write the equation for the conic function below. Write the equation for the conic function below. 51. Sketch and label: (x – 5)2 + y2 = 16 52. Write the equation for the conic function below. 53. Write the equation of the ellipse with foci at ( +2, 0) and x-intercepts at +5 54. Write the equation of the parabola with focus at (0 , 6) and directrix y = -6 55. Write the equation for the conic function below. 56. 57. 58. 59. Write the equation for the conic function below. 60. Write an equation of a circle with a diameter of 10 inches and a center at the point (3, -6) Write an equation of a circle with a diameter of 16 inches and a center at the point (-2, 1) 61. 62. 63. 64. (x – 3)2 + (y + 1)2 = 5 Write the equation for the conic function below. 65. Write the equation for the conic function below. Write the equation for the conic function below. 66. 67. (Harder Problem) Algebra 2B: Unit 6 68. Find the least common denominator for the following rational expressions: 80. Simplify 69. Find the least common denominator for the following rational expressions: 81. Simplify 70. Find the least common denominator for the following rational expressions: 82. Simplify 83. Simplify 84. Simplify 85. Simplify 86. Simplify 87. Simplify 88. Simplify 89. Simplify 90. Simplify 71. 72. 73. Simplify Simplify Simplify 74. Simplify 75. Simplify 76. Simplify 77. Simplify 78. Simplify 79. Simplify 91. Simplify 92. Simplify 93. Simplify 103. Simplify 104. Simplify 105. Simplify 94. Simplify 106. Simplify 95. Simplify 107. Simplify 96. Simplify 108. Simplify 97. Simplify 109. Simplify 98. Simplify 110. Simplify 99. Simplify 100. Simplify 101. Simplify 102. Simplify 111. Simplify 112. How many solutions does each of the following equations have? 2x3 + 3x2 – 18x – 27 = 0 3x2 + 4x – 6 = 0 6x4 + 3x3 + 2x + 4 = 0 ____ ____ ____ 113. Look at the factored form of P(x) and determine the solutions for x? P(x) = (x + 7) (x + 3) (x – 4) 114. Look at the factored form of P(x) and determine the solutions for x? 126. Solve if possible and list any restrictions to the domain of the variable. P(x) = (x + 3) (x – 5) (x + 1) (x – 2) 115. Look at the factored form of P(x) and determine the solutions for x? P(x) = x(x + 1) (x – 7) 127. Solve if possible and list any restrictions to the domain of the variable. 116. Look at the factored form of P(x) and determine the solutions for x? P(x) = (x – 2) (x – 3) (x – 4) 117. Look at the factored form of P(x) and determine the solutions for x? 128. Solve if possible and list any restrictions to the domain of the variable. P(x) = (x + 2) (x + 3) (x + 4) 118. Look at the factored form of P(x) and determine the solutions for x? P(x) = -2(x – 2) (x – 3) (x – 4) 119. Look at the factored form of P(x) and determine the solutions for x? f(x) = (x – 4) (x + 2) (x – 1) 129. Solve if possible and list any restrictions to the domain of the variable. 130. Solve if possible and list any restrictions to the domain of the variable. 120. Solve if possible and list any restrictions to the variable. 121. Solve if possible and list any restrictions to the variable. 122. Solve if possible and list any restrictions to the variable. 131. Solve if possible and list any restrictions to the domain of the variable. 132. Solve if possible and list any restrictions to the domain of the variable. 123. Solve if possible and list any restrictions to the variable. 133. Solve if possible and list any restrictions to the domain of the variable. 124. Solve if possible and list any restrictions to the variable. 134. State whether the graph is of odd degree or even degree; positive or negative; then state the number of relative minima and relative maxima. 125. Solve if possible and list any restrictions to the variable. (If necessary use quadratic equation to find the exact solution) 135. Draw a graph representing all positive even degree polynomials. Give two examples. 136. Draw a graph representing all negative even degree polynomials. Give two examples. 137. Draw a graph representing all positive odd degree polynomials. Give two examples. 138. Draw a graph representing all negative odd degree polynomials. Give two equation examples. 139. Factor, then list all the solutions. P(x) = x2 + x – 6 140. Factor, then list all the solutions. P(x) = -x2 – x – 6 141. Factor, then list all the solutions. P(x) = 2x2 + 2x – 12 142. Factor by grouping, then list all the solutions. x3 – 2x2 – x + 2 = 0 143. Factor by grouping, then list all the solutions. x3 – 6x2 – x + 6 = 0 144. Factor by grouping, then list all the solutions. -x + 4x + x – 4 = 0 3 2 145. Find the remainder when dividing. Check your answer by substitution. -x3 + 2x – 6 = 0 ÷ by (x – 3) 146. Find the remainder when dividing. Check your answer by substitution. x3 – 4x + 1 = 0 ÷ by (x – 2) 147. Find the remainder when dividing. Check your answer by substitution. x3 + 2x + 1 = 0 by (x – 5) 148. Use Descartes’ Rule to analyze the possible SIGNS of the zeros of the following function. Next use the Rational Zeros Theorem to identify the possible RATIONAL zeros. P(x) = 2x3 + 5x2 – 3x – 4 149. Use Descartes’ Rule to analyze the possible SIGNS of the zeros of the following function. Next use the Rational Zeros Theorem to identify the possible RATIONAL zeros. P(x) = x + 3x – 5x – 35 3 2 150. Use Descartes’ Rule to analyze the possible SIGNS of the zeros of the following function. Next use the Rational Zeros Theorem to identify the possible RATIONAL zeros. P(x) = 2x3 - 4x2 + 5x – 14 151. Use Descartes’ Rule to analyze the possible SIGNS of the zeros of the following function. Next use the Rational Zeros Theorem to identify the possible RATIONAL zeros. f(x) = 2x3 + 3x2 – 5x – 15 152. Use Descartes’ Rule to analyze the possible SIGNS of the zeros of the following function. Next use the Rational Zeros Theorem to identify the possible RATIONAL zeros. P(x) = 3x2 + 4x – 6 153. Use Descartes’ Rule to analyze the possible SIGNS of the zeros of the following function. Next use the Rational Zeros Theorem to identify the possible RATIONAL zeros. f(x) = 3x4 – 11x3 + 10x – 4 154. Use Descartes’ Rule to analyze the possible SIGNS of the zeros of the following function. Next use the Rational Zeros Theorem to identify the possible RATIONAL zeros. P(x) = 6x4 + 3x3 + 2x + 4 155. Use Descartes’ Rule to analyze the possible SIGNS of the zeros of the following function. Next use the Rational Zeros Theorem to identify the possible RATIONAL zeros. P(x) = x5 – 3x2 + 1 156. Use Descartes’ Rule to analyze the possible SIGNS of the zeros of the following function. Next use the Rational Zeros Theorem to identify the possible RATIONAL zeros. P(x) = x7 + 37x5 – 6x2 + 12 157. Factor by grouping, then list all the solutions. P(x) = x3 – 2x2 – 9x + 18 158. Factor by grouping, then list all the solutions. “i" f(x) = x4 + 4x2 – 45 159. Factor by grouping, then list all the solutions. f(x) = x3 + 3x2 – 2x – 6 160. Factor by grouping, then list all the solutions. f(x) = x3 – x2 – 3x + 3 161. What are all the asymptotes in this expression? 162. What are all the asymptotes in this expression? 174. Factor the polynomial given one of its factors. List all zeros and their multiplicities. P(x) = x3 – 6x2 + 11x – 6; x – 3 163. What are all the asymptotes in this expression? 175. Factor the polynomial given one of its factors. List all zeros and their multiplicities. P(x) = 6x3 + 19x2 + 2x – 3; x + 3 164. Use the Remainder Theorem and synthetic division to find: (a) f(-2) and (b) f(2) in the following function. Check your answer using substitution. f(x) = 3x4 – x2 + 2x – 6 165. Use the Remainder Theorem and synthetic division to find (a) f(-3) and (b) f(2) in the following function. Check your answer using substitution. f(x) = 2x + x – 10x – 5 3 2 166. Use the Remainder Theorem and synthetic division to find f(-4) and f(3) in the following function. Check your answer using substitution. 176. Factor the polynomial given one of its factors. List all zeros and their multiplicities. x4 + 2x3 – 7x2 – 20x – 12; (x + 2) 2 177. List all zeros. (Use the Quadratic formula to find other zeros) x3 – 4x2 + 21x – 34 = 0; x – 2 178. Factor the polynomial given one of its factors. List all zeros. **** P(x) = x3 + x2 – 4x – 24; -2 + 2i is a zero 179. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). f(x) = -2x3 – 8x2 + 7 167. Use the Remainder Theorem and synthetic division to find f(-6) and f(3) in the following function. Check your answer using substitution. 180. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). f(x) = -x3 – 5x2 + 8 168. Factor the polynomial given one of its factors. List all zeros. x3 – 7x + 6 ; x – 2 169. Factor the polynomial given one of its factors. List all zeros. x + 6x – x – 30; x + 5 3 2 181. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 182. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 170. Factor the polynomial given two of its factors. List all zeros. x3 + 2x2 – x – 2; (x – 1),(x + 1) 171. Factor the polynomial given two of its factors. List all zeros. (Use the Quadratic formula to find other zeros) x4 + 2x3 + 2x2 – 2x – 3; (x + 1), (x – 1) 183. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 184. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 172. One zero is given for the following polynomial. Including the given list all zeros. (Use the Quadratic formula to find other zeros) P(x) = x3 + 2x2 – 3x + 20; x + 4 173. Factor the polynomial given one of its factors. List all zeros and their multiplicities. P(x) = x3 – 3x – 2; x + 1 185. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 186. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). Then graph. 197. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 187. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 198. Solve if possible. 199. Solve if possible. 188. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 200. Solve if possible. 189. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). **** 201. Given the graph of the following function determine all asymptotes. 190. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 191. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 192. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 193. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 194. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 195. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 196. Simplify, if possible and determine any asymptotes or points of discontinuity (“holes”). 202. Given the graph of the following function determine all asymptotes. 203. Given the graph of the following function determine all asymptotes. 204. Given the graph of the following function determine all asymptotes. 205. Given the graph of the following function determine all asymptotes. 206. Given the graph of the following function determine all asymptotes. Algebra 2B Unit 7 NO CALCULATOR SECTION ================================= Rewrite with negative exponents: 207. 2y3 /x5 Use logarithmic properties to write the following expressions in logarithmic form. 229. 25 = 32 230. 4-2 = 1/16 231. 34 = 81 53 = 125 208. 1 209. 3 232. 210. 2x3/y2 233. 23 = 8 ================================= Sketch a graph of the following exponential functions. Indicate the y-intercept. Determine whether any of these equations are exponential growth functions or decay functions? 234. y = 3x /x4 211. 2ab3c5/d2 ================================= Rewrite with positive exponents. 212. 5/3x -3 213. 2x -2/y -3 214. 215. 235. y = -3x 2a-4b-3 236. y = (2/3)x 5x2 y -3 237. y = -(2/3)x 238. y = (1/2)x 239. y = 35x 240. y = (1/4)x 241. y = (1/2)2x 216. 3x2y -3 /7z -5 ================================= Write the following expressions in reduced simplified, radical form. 217. x -3/4 2/3 218. 16 219. 5 2/3 220. x2/5 221. 72/3 222. x 3/4 223. x 2/3 ================================= Write the following expressions in exponential form. 224. 225. 242. y = 2x ================================= First state the inverse function of each of the following. Next sketch and label a graph of both the function and inverse. 243. y = log 4 x 244. y = log2 x ================================= What is the domain and range of the following? 245. g(x) = 3x 246. g(x) = 5x ================================= 247. Make a table of the following functions then explain how they relate to each other. y = 5x and y = log5 x ================================= 226. 227. 228. ================================= Which of the following is NOT an exponential function? What type of function is it? 248. f(x) = x3 f(x) = 3x 2x f(x) = 5 f(x) = (3/4)(x+1) 249. f(x) = 4x f(x) = x4 f(x) = 34x f(x) = (1/2)(x+2) 250. f(x) = 5x f(x) = 54x f(x) = x5 f(x) = (1/2)(x+2) ================================= State the base of the following: 251. y = log 3 x 252. y = log x 253. y = Ln x ================================= Solve for x or simplify. 254. Solve: 2 x = 8 255. Solve: 2 x = 16 256. Solve: 3 x = 27 257. Solve: 2 2x = 4 2x + 4 258. Solve: 5 2x = 25 3x – 4 Write the following expressions in exponential form. 287. log 8 4 = 2/3 288. log 5 125 = 3 ================================= Solve for x in each of the following using logarithmic properties. 289. Solve: log x 9 = 2 290. Solve: log2 32 = x 291. Solve: log4 x = 3 292. Solve: log x 216 = 3 293. 294. Solve: log5 x = 2 Solve: log6 x = 2 295. 296. Solve: log10 (x2 – 3) = log10 6 Simplify: log3 (1/81) 297. Simplify: log2 128 298. Solve: log x 125 = 3 299. Solve: log 4 x = 3 300. Solve: log 7 (1/49) = x 259. 260. 261. 262. 263. Solve: 3 2x = 9 2x + 4 Solve: 3 4x = 3 5 – x Solve: 5 x - 3 = 1/25 Solve: 9 x = 81x + 4 Solve: 8 4 – 2x = 4x + 2 264. 265. Solve: 3x – 2 = Solve: 2 x = 4 5/2 266. 267. 268. Solve: 3 3x = 813x – 4 Solve: 8 = 2 3x+1 Solve: 5 x = 1/125 301. Solve: log x (1/64) = -3 302. Solve: log 4 x = -2 303. Solve: log 9 27 = x 269. 270. Solve: 16 x = 8 x + 1 Solve: 2 2x = 1/16 304. Solve: logx 9 = -2 305. Solve: logx 8 = -3 271. Simplify: 16 -3/4 306. Solve: logx 625 = -4 272. Solve: 16 = 2 3x + 1 307. Solve: log 9 x = 3/2 273. Simplify: 3 /33 308. Solve: log 7 (x2 – 6) = log 7 x 274. Simplify: (8 -1/ 3)2 275. Simplify: (4 ) 276. Simplify: (22 ) (23 ) 277. Simplify: 32 4/5 278. Simplify: 27 4/3 311. Solve: 3log5 x – log5 4 = log5 16 279. Simplify: 16 -1/4 312. Solve: 2log3 x – log3 2 = log3 x 280. Solve: x 2 = 25 313. Solve: 2log2 x + log21 = log2 4 281. Solve: 5x = 25 314. Solve: 2log2 x – log21 = log2 9 315. Solve: log3 (x+7) – log3 (x-1) = 2 316. Solve: log2 (3x + 1) – log2 x = 2 -1 5x (1 – x) =4 309. Solve: log 3 (3x – 5) = log3 (x + 7) ================================= Solve for x for each of the following using logarithmic properties. 310. Solve: log3 2 + log3 7 = log3 x x–3 282. Solve: (2 ) 16 283. Solve: 284. Solve: 4x = 317. Solve: log5 x + 3 = log5 (x – 20) + 4 285. Solve: 64 = 23x + 1 318. Solve: log2 x + 5 = 8 – log2 (x + 7) 286. Simplify: (2-1) (43) (8-1) 319. Solve: Ln x – Ln (x – 4) = Ln 2 320. Simplify: log2 8 + log2 2 = 8x + 2 ================================= 321. Solve: x = 4(5/2) 322. Solve: log2 (3x + 5) – log2 x = 3 323. Solve: 2log3 6 – (1/4)log3 16 = log3 x 324. Solve: log x + log (x + 3) = 1 325. Simplify: log2 4 + log2 2 ================================= Use logarithmic properties to express the following as a single logarithm whose coefficient is 1: 326. log2 x + log2 5 347. A bank account starts with $100 and has an annual interest rate of 4%. (a) Start a data table for this problem for the first 3 years. (b) Determine the type of equation this is and then write the equation (given any time limit). (c) How much money will be in the account in 12 years? 327. 3log(x 2 – 4) + (1/3)log(x + 2) – 4log(x + 7) 328. 2log a – 3log b – 5log c 329. 2log a + log b - 3log c - log d 330. 3log5 a – (1/2)log5 b + log5 c 331. (1/2)log3 4 + log3 5 – log3 x 348. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after 1985. (a) Start a data table for this problem for the first 3 years. (b) Determine the type of equation this is and then write the equation (given any time limit). (c) How many cell phone subscribers were in Centerville in 1994? 332. log3 x + 2log3 b – (1/2)log3 c 349. 333. 3x = 64 334. 5x = 12 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ A CALCULATOR CAN BE USED FOR THE FOLLOWING SECTIONS (a) Convert the following to: x = logarithmic form (Do Not use the calculator up to this point). (b) Using your calculator solve for x (Round your answer to the thousandths place) 335. 2x = 3 336. 4x = 12 337. 5x = 15 338. 3x – 2 = 4 339. 6 x + 2 = 17.2 For the following use your calculator solve for x (Round your answer to the thousandths place) 340. 53x = 8x – 1 341. 4x = 13x – 3 342. 2x – 7 x + 2 = 0 343. 3 2x = 13 x– 5 344. 7 2x + 1 = 3 3x – 2 345. 12x = 5 2x + 4 ================================= 346. Evaluate each of the following expressions: Round answer to the thousandths place. A. log3 54 B. log 49 C. ln 23 D. log5 22 E. log4 5 - log4 3 ================================= Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players remain after 5 rounds? (a) Complete a data table for this problem for the first 3 rounds (b) Determine the type of equation this is and then write the equation (given any time limit). (c) How many players will there be after 5 rounds? 350. A teacher has a class of 100 students simulate a half life of a radioactive substance by having each student roll a die. If they roll a 1 they must sit down. Each time the teacher records the number of students standing and records this. (a) Complete a data table for this problem for the first 2 rounds for those students still standing (simulating that they are radioactive) (b) Determine the type of equation this is and then write the equation (given any time limit). (c) How many students will be standing after 10 rounds? 351. A species of bacteria doubles every ten minutes. (a) Starting out with only one bacterium, how many bacteria would be present after one hour? (b) Determine the type of equation this is and then write the equation (given any time limit). 352. The half-life of carbon 14 is 5,700 years. After 5,700 years, the amount of carbon 14 left in the body is half of the original amount. (a) Complete a data table for this problem for the first 3 half lives. (b) Determine the type of equation this is and then write the equation (given any time limit). (c) How much C-14 will there be after 30,000 years? Algebra 2B Unit 8A C. Looking at the original question above, one can see that: Sn = a1 + a1 ( r ) + a1 ( r )2 + a1 ( r )3 + a1 ( r )4 Arithmetic Series and Sums 353. Make up a sequence starting at some number greater than two (a1). Add some number (d) to your original number, and continue to add the same number. Record your beginning number and the six subsequent numbers. Lastly total: a1 + a2 + a3… an = Sn n = 7 ; d = ____ a1 a2 = a3 = a4 = a5 = a6 = a7 = = ___ a1 + d = ___ a1 + d + d = ___ a1 + d + d + d = ___ a1 + d + d + d + d = ___ a1 + d + d + d + d + d = ___ a1 + d + d + d + d + d + d = ___ = an Sn = ____ A. How does the total number of d’s in an relate to what n is? B. Using only your numbers for a1, d, and n determine a formula to calculate the last number (an). Check that your formula works for your values. C. Now look at the variables: a1, an and n. Determine a formula to calculate the total (Sn) using these variables. (Hint: Average). Check first that your formula works for your values. D. Modify the formula determined in part C (Sn = ), by replacing an with what you found in part B. Do not bother simplifying this new formula. ================================= Geometric Series and Sums 354. Make up a sequence starting at some number greater than two (a1). Multiply your original number by some number (r). Continue to multiply your answer by the same number. Record your beginning number and the 4 subsequent numbers. Lastly total: a1 + a2 + a3… an = Sn n = 5 ; r = ____ a1 a2 = a3 = a4 = a5 = = ___ a1 ( r ) = ___ a1 ( r )( r ) = ___ a1 ( r )( r )( r ) = ___ a1 ( r )( r )( r ) ( r ) = ___ = an Sn = ____ A. How does the total number of r’s in an relate to what n is? B. Using only your numbers for a1, r, and n determine a formula to calculate the last number (an). Check that your formula works for your values. From this start show how to derive the formula for finding the sum of a geometric sequence. Check that your formula works for your values. ================================= Determine whether the following sequences are arithmetic or geometric - next determine d or r: 355. 3, 5, 7, 9, … 356. 2, 6, 18, 54, 162, … 357. 4, 7, 10, 13, … 358. 1, - 1/2 , 1/4 , - 1/8 , … 359. 1, 1, 1, 1, … 360. 1, 3, 6, 10, 15, … ================================= Determine: (a) The first 4 terms and a10, and a15. (b) The sequence type including d or r. (c) The formula for an using a1. (d) Solve for the exact value of a10, using the formula for an. 361. an = 3n 362. an = n – 2 363. an = (n – 3)/3 364. a1 = 3, 365. a1 = -2, d = -3 366. a1 = -3, r = -2 367. a1 = -54, 368. an = 2n + 1 d =5 r = 2/3 369. an = (n – 1)/ n ================================= Determine: (a) The formula for an. (b) Solve for the exact indicated an value using the formula for an. 370. -12, -7, -2, 3, … ; a21 371. 10, 7, 4, 1, … ; a95 372. 2, 2 1/4, 2 1/2, 2 3/4, … ; a43 373. 5, -10, 20, - 40, … ; a7 Find the indicated term of the given sequence 388. a1 = 4, d = 3 ; a81 389. 374. /2, 1/20, 1/200, 1/2000… ; a9 a1 = 15, r = - 2/3 ; a6 (fraction) 1 375. -1, - 4, -16, -64, … ; a6 376. 3, 7, 11, 15, … ; a10 377. 9, 3, 1, 1/3 , … ; a20 378. 8, 4, 2, 1, 1/2 , 1/4 , … ; a15 379. 12, 9, 6, 3, 0, -3, … ; a25 380. 2, 9, 16, 23, … ; a51 381. 4, 2, 1, 1/2 , … ; a7 382. 3, -1, -5, -9, … ; a21 383. 3, -6, 12, -24, … ; a8 ================================= 384. Given that: a3 = 8, and a16 = 47. Find: a) d b) a 1 c) The formula for an d) Write the first five terms of the sequence. e) Find a61 390. 391. ================================= Terry accepts a job, starting with an hourly wage of $14.25, and is promised a raise of $0.25 per hour every 2 months for 5 years. Find: a) a 1 b) d c) n d) a n e) Terry’s hourly wage at the end of 5 years. Given: a3 = - 4 and d = -2 . Find: f) a 1 g) The formula for an h) a 56 ================================= Arithmetic Sums Find and evaluate the sum 392. 393. 394. 385. 386. 387. Given that: a5 = 19, and a19 = 75. Find: a) d b) a 1 c) The formula for an d) Write the first five terms of the sequence. e) Find a55 Given that: a3 = 3, and a6 = -81. Find: a) r b) a 1 c) The formula for an d) Write the first five terms of the sequence. Given that: a4 = 375, and a7 = 46875 Find: a) r b) a 1 c) The formula for an d) Write the first five terms of the sequence. ================================= 395. 396. ================================= Write in summation (sigma) notation with k = 1 397. 3 + 6 + 9 + 12 398. 12 + 8 + 4 + 0 + (-4) 399. 1+3+5+7+9 ================================= 400. 401. The first term in the arithmetic series is 3, the last term is 136, and the sum is 1,390. What are the first four terms? 408. Find the sum of the first 15 terms in the arithmetic sequence: 4, 7, 10, 13, … 409. Find the sum of the first 20 terms in the arithmetic sequence: 5, 11, 17, 23, … 410. A stack of telephone poles has 30 poles in the bottom row. There are 29 poles in the second row, 28 in the next row, and so on. Use the formula for Sn to find how many poles are in the stack if there are 5 poles in the top row? How many poles will be in a stack of telephone poles if there are 50 poles in the first layer, 49 in the second, and so on, with 6 in the top layer? Find the sum of the first one hundred terms in the progression: -6, -2, 2, … ================================= Find and evaluate the sum 402. 411. 403. 412. ================================= 404. Carl Gauss (1777-1855) was the greatest mathematician of his time. When he was in elementary school his teacher wanted to keep students busy by asking them to add the numbers 1 to 100. Within seconds, Gauss had found the answer. Show how to find the sum of the first 100 natural numbers. (1 + 2 + 3 + … + 99 + 100) Theaters are often built with more seats per row as the rows move toward the back. Suppose that the first balcony of a theater has 28 seats in the first row, 32 in the second, 36 in the third, and so on, for 20 rows. How many seats are in the first balcony altogether? ================================= More Geometric Sums 413. Use the formula for Sn to find the sum of the first 10 terms in the geometric sequence: 16, 32, 64, 128, … 414. 405. You are building a staircase out of cubes. 1 step = 1 cube 2 steps = 2 cubes 3 steps = 3 cubes How many cubes does it take to build a staircase that is: 25 cubes high? 406. If your first step is: 1 step = 3 cube 2 steps = 4 cubes 3 steps = 5 cubes Use the formula for Sn to find how many cubes does it take to build a staircase that is 7 cubes high? 407. If your first step is: 1 step = 50 cube 2 steps = 51 cubes 3 steps = 52 cubes How many cubes does it take to build a staircase that has 78 cubes in the last row? 415. Use the formula for Sn to find the sum of the first 7 terms of the geometric sequence: 3, 15, 75, 375, … Quite a while ago, a king wished to reward a royal mathematician, and asked him what he desired. The mathematician, who by the way appeared both modest and humble, replied that if just one grain of rice was placed on a square of an ordinary 8 by 8 chess board, and then two grains of rice in the next square, and so forth, doubling the previous amount of rice, until the last square on the chessboard was reached, then he would be totally content with the total sum of all the grains of rice. How much rice was this? ================================= Algebra 2B Unit 8B 416. There are 5 girls and 3 boys in my family. In how many different ways can my mother choose a girl and a boy to do the dishes? 417. 418. There are 4 sopranos and 7 altos in the choir. How many different soprano/alto duets can be formed? The menu at the cafeteria lists 3 different sandwiches, 6 different drinks, and 5 varieties of chips. How many different sandwich / chip / drink meals are possible? 426. A social security number is a 9-digit number like 522-77-0823. (a) How many different social security numbers can there be? (b) There are about 275 million people in the U.S.. Can each person have a unique social security number? Explain 427. A U.S. postal zip code is a five digit number. (a) How many zip codes are possible if any of the digits 0 to 9 can be used? (b) If each post office has its own zip code, how many possible post offices can there be? ================================ Permutations – an ordered arrangement of objects 419. When creating your fall class schedule you discover that there is a choice of 5 sections of English, 3 sections of Math, 4 sections of Science, and 2 sections of History. How many schedule arrangements are possible? 420. How many different 3-digit code symbols can be formed with the letters A, B, C with repetition (that is, allowing letter to be repeated)? 421. How many different 3-digit code symbols can be formed with the letters A, B, C , D, and E with repetition (that is, allowing letter to be repeated)? 422. How many different 5-digit code symbols can be formed with the letters A, B, C , and D if we allow a letter to occur more than once? 423. Using only the odd digits (1, 3, 5, 7, 9) how many different 3-digit numbers can you form (repetitions are allowed)? 424. How would one find the total number of license plate possibilities given that a license plate is three letters followed by three digits? 425. How many 7-digit phone number can be formed with the digits 0-9, assuming that the first number cannot be 0 or 1? 428. Three finalists: Sue, Tim, Pat are in the annual bake off contest. (a) Show all the different orders that they can finish? (b) How many total different ways is this? (c) Show this same answer using factorial notation. (d) Show this same answer using permutation notation. Explain what each number / symbol stands for. 429. (a) Show all the different orders that the letters in the word MATH be arranged. (b) How many total different ways is this? (c) Show this answer using factorial notation. (d) Show this same answer using permutation notation. Explain what each number / symbol stands for. 430. (a) Using factorial notation to show how many different ways 8 packages be placed in 8 mailboxes, one package in a box? (b) Show this same answer using permutation notation. 431. (a) Show all the different 3-digit code symbols that can be formed with the letters A, B, C without repetition (that is, using each letter only once)? (b) How many total different ways is this? (c) Show this same answer using factorial notation. (d) Show this same answer using permutation notation. Explain what each number / symbol stands for. 432. On one Saturday night, all 6 employees decided to line up and sing “Happy Birthday” to a customer. (a) Using factorial notation to show how many different ways could they have lined up? (b) Show this same answer using permutation notation. 433. (a) Using factorial notation to show how many different ways can 5 people can line up for a photograph? (b) Show this same answer using permutation notation. 434. (a) Using factorial notation to show how many different ways 5 starters on the basketball team be assigned to their positions? (b) Show this same answer using permutation notation. 435. 436. 437. (a) Using factorial notation to show how many different ways 7 athletes be arranged in a straight line? (b) Show this same answer using permutation notation. (a) Using factorial notation to show how many different ways 6 classes be scheduled during a 6-period day? (b) Show this same answer using permutation notation.. (a) Using factorial notation to show how many different ways 9 starters of a baseball team be placed in their positions? (b) Show this same answer using permutation notation. ============================= 438. (a) Show all the different 3-digit code symbols that can be formed with the letters A, B, C and D without repetition (that is, using each letter only once)? (b) How many total different ways is this? (c) Show this same answer using permutation notation. Explain what each number / symbol stands for. 439. (a) Show all the different ways can you arrange the letters of MATH taking two at a time? (b) How many total different ways is this? (c) Show this same answer using permutation notation. Explain what each number / symbol stands for. 440. (a) If six baseball teams must play each other team twice. Demonstrate with a chart showing each team’s opponents. (b) How many total different ways is this? (c) Show this same answer using permutation notation. Explain what each number / symbol stands for. 441. Using permutation notation show how many ways can you arrange 3 books on a book shelf from a group of 7 books? Explain what each number / symbol stands for. 442. Using permutation notation show how many different ways you can arrange the letters of the word COMPUTER taking 4 at a time? Explain what each number / symbol stands for. 443. Using permutation notation show how many different ways you can arrange the odd digits (1, 3, 5, 7, 9) for a 3-digit numbers (when repetitions are NOT allowed)? Explain what each number / symbol stands for. 444. Using permutation notation show how many different 3-digit code symbols can be formed with the letters A, B, C, D, E and F without repetition (that is, using each letter only once)? Explain what each number / symbol stands for. 445. The flags of many nations consist of three vertical stripes. For example, the flag of Ireland has its first stripe green, second white, and third orange. Suppose that the following colors are available: black, yellow, red, blue, white, gold, orange, pink, purple. Using permutation notation show how many different flags of 3 colors can be made without repetition of colors? This assumes that the order in which the stripes appear is considered. Explain what each number / symbol stands for. 446. A president and a vice-president are to be selected from a 7-member student council committee. Using permutation notation show how many different ways can the selection be made? (Each member may be a president or a vice-president, but not both). 447. 12 students are in a race. Using permutation notation show how many different ways possible that they can win the 1st, 2nd and 3rd place trophies? 448. A baseball manager arranges the batting order as follows: The 4 infielders will bat first. Then the 3 outfielders, the catcher, and the pitcher will follow, not necessarily in that order. How many different batting orders are possible? 452. Find the total number of distinguishable permutations to the last three questions: Permutations of Sets with Nondistinguishable Objects. Question 33: How many distinguishable permutations are possible using this set of 3 marbles - 1 which is blue and 2 of which are red. Question 34: How many distinguishable permutations are possible using this set of 4 marbles - 2 which are blue and 2 of which are red. Question 35: How many distinguishable permutations are possible using this set of 5 marbles - 3 which are blue and 2 of which are red. 453. Consider a set of 7 marbles, 4 of which are blue and 3 of which are red. Although the marbles are all different, when they are lined up, one red marble will look just like any other red marble and are nondistinguishable, similarly, the blue marbles are also non-distinguishable from each other. How many distinguishable permutations are possible using this set of 7 marbles? 454. How many distinguishable code symbols can be formed from the letters of the word MATHEMATICS? 455. How many distinguishable code symbols can be formed from the letters of the word BUSINESS? ================================= Permutations of Sets with Nondistinguishable Objects 449. Consider a set of 3 marbles, 1 of which is blue and 2 of which are red. Although the marbles are all different, when they are lined up, one red marble will look just like any other red marble and are non-distinguishable. (a) First show all the distinguishable permutations possible using this set of 3 marbles, than (b) state the total number of distinguishable permutations. 450. Consider a set of 4 marbles, 2 of which are blue and 2 of which are red. Although the marbles are all different, when they are lined up, one red marble will look just like any other red marble and are non-distinguishable, similarly, the blue marbles are also nondistinguishable from each other. (a) First show all the distinguishable permutations possible using this set of 4 marbles, than (b) state the total number of distinguishable permutations. 451. Consider a set of 5 marbles, 3 of which are blue and 2 of which are red. Although the marbles are all different, when they are lined up, one red marble will look just like any other red marble and are non-distinguishable, similarly, the blue marbles are also nondistinguishable from each other. (a) First show all the distinguishable permutations possible using this set of 3 marbles, than (b) state the total number of distinguishable permutations. Suppose the expression a2b3c4 is rewritten without exponents. In how many ways can this be done? ================================ 456. Combination – a selection without regard to the order 457. (a) If seven soccer teams must play each other one time only. Demonstrate with a chart showing each team’s opponents (b) How many total different ways is this? (c) Show this same answer using combination notation. Explain what each number / symbol stands for. 458. (a) Show all the combinations of three letters taken from the set of 5 letters:{A, B, C, D, E} (b) How many total different ways is this? (c) Show this same answer using combination notation. 459. In how many ways can three numbers be chosen from 10 numbers? . 460. There are 8 finalists in a trivia contest. If each finalist plays a round against each of the other finalists, how many rounds will have been played in all? (Use combination notation). 461. Yummy Ice Cream shop serves 12 flavors of ice cream. How many different flavored double scoop cones can they make? (No two scoops of the same flavor). 462. Two of the 6 players on a basketball team are to be selected as co-captains. In how many ways can the selection be made? 463. Students are asked to choose 4 numbers from 1-10 for a school game of chance. How many different combinations are possible? 464. A principal wants to start a peer counseling group to work with students in need. He needs to narrow down his choice to six students from a group of nine students. How many ways can a group of six be selected? Use Combination notation to find the total number of diagonals (connecting two vertices – not on the perimeter) that can be drawn in each of the following figures: Check your answers by drawing the total number of diagonals for each figure. a) Triangle b) Quadrilateral c) Pentagon d) Hexagon e) Heptagon f) Octagon g) Determine a quadratic equation for determining this total number of ydiagonals given the total number of xsides. =============================== 466. How many committees can be formed from a group of 5 governors and 7 senators if each committee consists of 3 governors and 4 senators? 467. A bag of marbles contains 4 red marbles, 5 green marbles, and 8 blue marbles. How many ways can 2 red marbles, 1 green marble, and 2 blue marbles be chosen? You can do this long-hand or use your calculator to get the answer but show work. 468. A bucket of flowers contains 6 red carnations, 5 white daisies, and 7 yellow tulips. How many bouquets could be created so that each bouquet has 3 red carnations, 1 white daisy, and 2 yellow tulips? =============================== 469. Suppose that 3 people are selected at random from a group that consists of 6 men and 4 women. What is the probability that 1 man and 2 women are selected? 470. 471. 465. Suppose that 2 cards are drawn from deck of 52 cards. What is the probability that both of them are spades? ============================== Eight cards are drawn from a standard deck of 52 cards. How many 8-card hands having 5 cards of one suit and 3 cards of another suit can be formed? You may use a calculator but show all work. 472. To win the jackpot in the Colorado lottery you must pick 6 correct numbers out of 49 total numbers. How many 6-number combinations are possible? (For this problem work does not have to be shown if you put down what you entered into your calculator and the result.) 473. How many 5-card poker hands are possible with a 52-card deck? (For this problem work does not have to be shown if you put down what you entered into your calculator and the result.)