Pre-Calculus Semester 1 Exam Review PDF

Name: ________________________ Class: ___________________ Date: __________ ID: A Pre-Calculus Semester 1 Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Perform the indicated operation. 3 4...

Name: ________________________ Class: ___________________ Date: __________ ID: A Pre-Calculus Semester 1 Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Perform the indicated operation. 3 4  14 21 5 a. 14 17 b. 42 23 c. 42 19 d. 42 1 e. 2 1 Name: ________________________ ID: A Numeric Response 2. Due to the curvature of the earth, the maximum distance D that you can see from the top of a tall building of height h is estimated by the formula D 2rh  h 2. where r = 3960 mi is the radius of the earth and D and h are also measured in miles. How far can you see from the observation deck of the Toronto CN Tower, 1,135 ft above the ground? Please give the answer to one decimal place. __________ miles 3. Perform the division and simplify. 9y 2  25 3y 2  2y  5  3y 2  16y  35 y 2  6y  7 4. A small–appliance manufacturer finds that the profit P (in dollars) generated by producing x microwave 1 ovens per week is given by the formula P  x(200  x) provided that 0  x  60. 10 How many ovens must be manufactured in a given week to generate a profit of $750? P = __________ ovens per week 5. An executive in an engineering firm earns a monthly salary plus a Christmas bonus of $8,400. If she earns a total of $97,200 per year, what is her monthly salary? 2 Name: ________________________ ID: A 1 1 6. A 14 -foot ladder leans against a building. The base of the ladder is 2 ft from the building. How high up 6 6 the building does the ladder reach? __________ ft 7. A jeweler has five rings, each weighing 18 g, made of an alloy of 20% silver and 80% gold. He decides to melt down the rings and add enough silver to reduce the gold content to 60%. How much silver should he add? __________ g 8. Evaluate the expression and write the result in the form a  bi. 5 75 1 3 9. The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If D 0 is the original amount of carbon-14 and D is the amount remaining, then the artifact's age A (in years) is given by  D  A  8267ln  .   D 0  Find the age of an object if the amount D of carbon-14 that remains in the object is 71% of the original amount D 0. Please round your answer to the nearest integer. A  __________ years 10. Evaluate the expression. log 3 6  log 3 8  log 3 108 3 Name: ________________________ ID: A 11. Evaluate the expression. log 4 64 600 12. Find the solution of the exponential equation, correct to four decimal places. 3 5x  2  4 x = __________ 13. Solve the logarithmic equation for x. Please round the answer to four decimal places, if necessary. 2  ln(7  x)  0 x = __________ 14. How long will it take for an investment of $3,000 to double in value if the interest rate is 6.5% per year, compounded continuously? Please round your answer to two decimal places. __________ years 15. A 10-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by m(t)  10e 0.087t where m(t) is measured in grams. After how many days is there only 5 g remaining? Please round your answer to the nearest integer. __________ days Short Answer 16. Perform the indicated operation. 7 2 8 1 1  2 5 17. Express the interval in terms of inequalities. (3, 6] 18. List the elements of the given set that are rational numbers.  24 1   0,9,43, ,0.535, 6 ,1.25, , 3 6   7 5  19. Use properties of real numbers to write the following expression without parentheses. (4a)(4x  5y  3z) 4 Name: ________________________ ID: A 20. Perform the indicated operation. 4 5  13 13 21. Select the correct statement. 22. Find A  B if A = { 6, 3, 10, 12, 2 }, and B = { 10, 7, –3, 12 }. 23. Find A  B if A = { 5, 7 }, B = { 7, 6 }. 24. Express the interval x  (2,6) as an inequality. 25. Express the inequality 2  x  1 in interval notation. 26. Evaluate the expression | | –10 | – | –9 | |. 27. Evaluate the expression.  13  2     2  13  28. Complete the following table. x 1 x 11 a 1.1 a 0.11 a 29. Write the radical expression using exponents. 1 13 30. Write the exponential expression using radicals. 9  2 10 31. Evaluate the expression. 3 3 2 32. Evaluate the expression. 3 150 3 180 33. Simplify the expression. 72  50 5 Name: ________________________ ID: A 34. Simplify the expression.  8 4   1 4 6   9x y   x y    3  35. Simplify the expression. a 8 b 7 a 6 b 4 36. Simplify the expression.  9 6  2  2 7  4 x y  x y      8 12 x y 37. Simplify the expression. Assume the letters denote any real numbers. 3 x3y6 38. Simplify the expression. Assume the letters denote any real numbers. 7 7 a5b a9b 39. Simplify the expression. 10  5 8 4   3 x y z    Eliminate any negative exponents. Assume that all variables are positive numbers. 40. Simplify the expression.    2 3  3  4 2    a b   x b   3 2     x y   3 1     2 3   a y  Eliminate any negative exponents. Assume that all variables are positive numbers. 41. State whether the polynomial is a monomial, binomial, or trinomial, then list its terms and state its degree. 9x 7  6x 4 42. Perform the indicated operations and simplify.   4(3t  2)   t 2  3   3t(t  3)   6 Name: ________________________ ID: A 43. Perform the indicated operations and simplify. (6  4y) 2 44. Perform the indicated operations and simplify.  1   1    c    c    e  e 45. Factor the trinomial. x 2  8x  15 46. Factor the expression by grouping terms. 3x 2  x 2  18x  6 47. Factor completely. 3x 2  13x  12 48. Find the domain of the expression. x 4  x 3  7x 49. Simplify the expression. x2 x2  4 50. Simplify the expression. t3 t3  t2  9 t2  9 51. Simplify the expression. 2y 2  5y  7 y 2  6y  7  4y 2  49 2y 2  7y  49 52. Simplify the expression. 1 1 y4 1 1 y4 7 Name: ________________________ ID: A 53. Rationalize the denominator. 4 3 6 54. A large pond is stocked with fish. The fish population P is modeled by the formula P  2t  7 t  269, where t is the number of days since the fish were first introduced into the pond. How many days will it take for the fish population to reach 416? 55. Solve the equation. z 6  z7 9 63 56. Solve the equation. 1 8  3 x 5x 57. Solve the equation. (t – 5) 2 = (t + 5) 2 + 160 58. Solve the equation P = 8l + 5w for l. 59. Solve the equation by completing the square. x 2  5x  6  0 60. Find all real solutions of the equation. 8 16 2    0 7 49 61. Find all real solutions of the equation. x2 1 x  30 62. Write the sum of three consecutive integers using l, where l = the middle integer of the three. 63. An executive in an engineering firm earns a monthly salary plus a Christmas bonus of $9,000. If she earns a total of $103,800 per year, what is her monthly salary? 64. Find the slope and y-intercept of the line and draw its graph. 3x  2y  12 65. Find the slope and y-intercept of the line and draw its graph. 2x  3y  1  0 66. Evaluate the function f (x) = –5x + 4 at f (–7). 8 Name: ________________________ ID: A f(a  h)  f(a) 67. For the function f x   3x 2  7, find , h  0. h 68. Find the domain of the following function: f(x)  3x,  6  x  8 x6 69. What is the domain of the function f(x)  ? x2  9 70. Sketch the graph of the function by first making a table of values. f(x)  x  5,  3  x  3 71. Sketch the graph of the function by first making a table of values. f(x)  x 2  3 72. Sketch the graph of the function by first making a table of values. f(x) |2x  2| 73. Sketch the graph of the piecewise defined function.  x if x  0 f(x)    x  3 if x  0 74. Sketch the graph of the piecewise defined function.  2 if x  2   f(x)   x if  2  x  2    2 if x  2 75. Determine whether the equation defines y as a function of x. x 2  (y  1) 2  16 76. Select the correct graph and domain of the function f(x)    25  x 2 .   77. Select the correct graph and domain of the function f(x)  4x  1. 78. Find f (g (x)) for f (x) = 7x + 7 and g(x) = 10x – 4. 79. A function f is given. f(x)  x5 Sketch the graph of f. Use the graph of f to sketch the graph of f 1. Find f 1. 9 Name: ________________________ ID: A 80. A one-to-one function is given. 1 f(x)  1  x 4 Find the inverse of the function. Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line y  x. 81. Find the inverse function of f (x) = 7x + 28. 2  7x 82. Find the inverse function of f(x) . 9  5x 1 1 83. Use a Graphing Calculator to determine the end behavior of P(x)   x 3  x 2  12x. 9 3 84. Use a Graphing Calculator to determine the end behavior of P(x)  x 4  5x 2  5x  5. 85. Sketch the graph of the function. P(x) = (x – 2)(x + 2)(x – 3) 86. Sketch the graph of the function. P(x) = (x – 1) 2(x – 3) 87. Sketch the graph of the function. P(x)  x 3  3x 2  4x 88. Determine the end behavior of the graph of the function. y = 8x 3 – 7x 2 + 3x + 7 10 Name: ________________________ ID: A 89. The graph of a polynomial function is given. Find the coordinates of all local extrema. 90. The graph of a polynomial function is given. Find the coordinates of all local extrema. 11 Name: ________________________ ID: A 91. The rabbit population on a small island is given by the function below, where t is the time in months since observation of the rabbit population began. P(t) = 140t – 0.35t 4 + 1500 What is the maximum population? Round your answer to the nearest integer number. 92. Find the quotient and remainder using long division. x 2  3x  36 x2 93. Find the quotient and remainder using long division. 2x 2  3x  3 2x 2  5x 94. Use synthetic division and the Remainder Theorem to evaluate P(3) , for P(x)  3x 2  2x  5. 95. Use synthetic division and the Remainder Theorem to evaluate P(2) , for P(x)  6x 5  4x 3  x  8. 96. Find a polynomial of degree 3 that has zeros 7, –7, and 6. 97. Find a polynomial of degree 3 that has zeros of 2, –4, and 4, and where the coefficient of x 2 is 6. 98. Find the polynomial of degree 4 whose graph is shown. 99. List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). U(x)  6x 5  6x 3  2x  12 12 Name: ________________________ ID: A 100. Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. P(x)  3x 3  x 2  2x  8 101. A polynomial P is given. Find all the real zeros of P. Sketch the graph of P. P(x)  x 3  4x 2  7x  10 102. A polynomial function P and its graph are given. P(x)  2x 3  x 2  2x  1 From the graph, determine which of the possible rational zeros actually turn out to be zeros. 103. Find the real and imaginary part of the complex number 8 – 3i. 104. Find the real and imaginary parts of the complex number. 8 105. Find the real and imaginary part of the complex number 3 10. 106. Evaluate the expression (4 + 9i)(11 – 10i) and write the result in the form a + bi. 4  10i 107. Evaluate the expression and write the result in the form a + bi. 2i 108. Evaluate the expression i 64 and write the result in the form a + bi.    109. Evaluate the expression  4  16   8  25  and write the result in the form a + bi.    110. Find all solutions of the equation x 2 + 49 = 0 and express them in the form a + bi. 13 Name: ________________________ ID: A 111. Find all solutions of the equation x 2 – 8 x + 25 = 0 and express them in the form a + bi. 112. A polynomial P is given. P(x)  x 4  5x 2  6 Factor P completely. 113. Find the polynomial P(x) of degree 3 with integer coefficients, and zeros 4 and 3i. 114. Find the polynomial P(x) of degree 4 with integer coefficients, and zeros 3  3i and 2 with 2, a zero of multiplicity 2. 1 115. Use transformations of the graph of y  to graph the rational function. x 3 s(x)   x2 x6 116. Find the x- and y-intercepts of the rational function r(x) . x6 x2  9 117. Find the vertical asymptote of the rational function r(x) . x8 1 118. Use transformations of the graph of y  to graph the rational function. x 3x  11 r(x)  x4 x2 119. Find the slant asymptote of the function y . x4 8x 3  2x 120. Determine the correct graph of the rational function r(x) . x2  2 121. A drug is administered to a patient and the concentration of the drug in the bloodstream is monitored. At time t > 0 (in hours since giving the drug), the concentration (in mg/L) is given by the equation: 50t c(t) . t 4 2 Graph the function c with a graphing device. What is the highest concentration of the drug? 122. Sketch the graph of the function by making a table of values. Use a calculator if necessary. g(x)  9 x 14 Name: ________________________ ID: A 123. Sketch the graph of the function by making a table of values. Use a calculator if necessary.  1  x f(x)     8    124. Graph the function, not by plotting points, but by starting from the graph in the figure. State the domain, range, and asymptote. f(x)  9 x 125. Graph the function, not by plotting points, but by starting from the graph in the figure. State the domain, range, and asymptote. f(x)  11 x  3 15 Name: ________________________ ID: A 126. Find the exponential function f(x)  a x whose graph is given. 127. State the domain of the function f(x)  3 x. 128. Determine the domain and range of the function h(x)  5  2 x. 129. Radioactive iodine is used by doctors as a tracer in diagnosing certain thyroid gland disorders. This type of iodine decays in such a way that the mass remaining after t days is given by the function m(t)  7e 0.066t where m(t) is measured in grams. Find the mass at time t = 10. 16 Name: ________________________ ID: A 130. A 50-gallon barrel is filled completely with pure water (see the figure below). Salt water with a concentration of 0.5 lb/gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time t is given by   Q(t)  25  1  e 0.08t    where t is measured in minutes and Q(t) is measured in pounds. How much salt is in the barrel after 10 min? 131. If $1,000 is invested at an interest rate of 10% per year, compounded monthly, find the amount of the investment at the end of 4 years. 132. Express the equation in exponential form. log 4 16  2 133. Express the equation ln (x + 1) = 4 in exponential form. 134. Express the equation in logarithmic form. 3 4  81 135. Evaluate the expression log 7 343. 136. Evaluate the expression. log  10 137. Use the definition of the logarithmic function to find x. log x 81  4 17 Name: ________________________ ID: A 138. Identify the logarithmic function corresponding to the graph. 139. Identify the logarithmic function corresponding to the graph. 18 Name: ________________________ ID: A 140. Identify the graph of the function y = log 2 (x – 2) – 2 using the graph shown below. 141. Evaluate the expression. log 5 5 625 142. Evaluate the expression. log 3 189 – log 3 7 143. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power. log 4  x(x  9)  144. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.  x  log 9    8  145. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.  2   x  log a  7    yz   146. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.    a6  log 3   b c   19 Name: ________________________ ID: A 147. Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. log 18 2.3 148. Simplify. (log 3 7)(log 7 13) 149. Find the solution of the exponential equation, correct to four decimal places. e 2  3x  12 150. Find the solution of the exponential equation, correct to four decimal places. 12 x  5 x  4 151. Solve the equation. x 3 9x  9x  0 152. Solve the equation. e 2x  5e x  4  0 153. Solve the logarithmic equation for x. log x = 1 154. Solve the logarithmic equation for x. log 3 (4 – x) = 7 155. Find the time required for an investment of $3,000 to grow to $8,000 at an interest rate of 8% per year, compounded quarterly. 156. How long will it take for an investment of $1,000 to double in value if the interest rate is 7.5% per year, compounded continuously? 20 Name: ________________________ ID: A 157. A small lake is stocked with a certain species of fish. The fish population is modeled by the function 12 P 1  4e 0.8t where P is the number of fish in thousands and t is measured in years since the lake was stocked. After how many years will the fish population reach 5,000 fish? 158. The number of bacteria in a culture is modeled by the function n ( t ) = 400e 0.9t where t is measured in hours. What is the initial number of bacteria? 159. The fox population in a certain region has a relative growth rate of 5% per year. It is estimated that the population in 2000 was 22,000. Find a function n(t) that models the population t years after 2000. 21 Name: ________________________ ID: A 160. A lamp with a parabolic reflector is shown in the figure. The bulb is placed at the focus, and the focal diameter is 4 cm. Find the diameter d(C, D) of the opening 15 cm from the vertex. ( a = 2, b = 15 ) 161. In a suspension bridge, the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 800 m apart, and the lowest point of the suspension cables is 100 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the origin. ( d = 800 m, h = 100 m ) NOTE: This equation is used to find the length of the cable needed in the construction of the bridge. 162. Find the vertices and foci of the ellipse, and sketch the graph. 9x 2  16y 2  144 163. From the list below, find the graph matching the given equation. y2 x2  1 4 164. Find the foci, major, and minor axes of the ellipse. x 2 + 16y 2 = 1 22 Name: ________________________ ID: A 165. Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. y2 x2  1 36 49 166. Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. y2 x  2 1 5 167. Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. (x  3) 2  y2  1 4 168. Find the vertex, focus, and directrix of the parabola, and sketch the graph. (x  4) 2  4(y  1) 169. Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. (x  1) 2 (y  1) 2  1 16 9 170. Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. 9x 2  16y 2  36x  160y  292  0 171. Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. 2x 2  y 2  2y  1 172. Find the vertex and focus of the parabola. y 2 = 36x – 216 23 Name: ________________________ ID: A 173. Find an equation for the conic whose graph is shown. 174. Find an equation for the conic whose graph is shown. 175. Which of the following equations represents a parabola? 176. Find the center and lengths of the major and minor axes of the ellipse. 36y 2 + 9x 2 – 108x – 72y + 36 = 0 177. Find the vertex and directrix of the parabola. x 2 – 4x – 28y – 248 = 0 24 Name: ________________________ ID: A 178. Place the correct symbol (, or =) in the blank. 11 (a) 5 __ 2 11 (b) 5 __  2 11 (c) 5.5 __ 2 179. Find the indicated set if A = {3, 4, 5, 6, 7, 8, 9} B = {4, 6, 8, 10} C = {8, 9, 10, 11} (a) A  B (b) A  B  C 180. Find the indicated set if       A   x | x  5  B   x | x  2  C   x |  3  x  6        (a) A  C (b) A  B 181. Express the set in interval notation. (a) (b) 182. Evaluate the expression. 3 3 (a) 27 2  1  (b)    4 25 Name: ________________________ ID: A 183. Police use the formula s  30fd to estimate the speed s (in mi/h) at which a car is traveling if it skids d feet after the brakes are applied suddenly. The number f is the coefficient of friction of the road, which is a measure of the "slipperiness" of the road. The table gives some typical estimates for f. Tar Concrete Gravel Dry 1.0 0.8 0.2 Wet 0.5 0.4 0.1 (a) If a car skids 55 ft on dry gravel, how fast was it moving when the brakes were applied? Please round the answer to the nearest integer. __________ mi/h (b) If a car is traveling at 40 mi/h, how far will it skid on dry concrete? Please round the answer to the nearest foot. __________ feet 184. Simplify the expression. 50  72 185. Simplify the expression and eliminate any negative exponent(s). 2 2  2u 2 v 3   5u 2 v      186. Simplify the expression and eliminate any negative exponent(s).  2 3 3  4  x y z     4 2 6  x y z    26 Name: ________________________ ID: A 187. Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers.   1  4   3a        2   3   4b  188. Rationalize the denominator. 7 x 189. Rationalize the denominator. x 2 5 y 190. Factor the trinomial. 25x 2  20x  12  __________ 191. Factor the trinomial. 2(a  b) 2  3(a  b)  2  __________ 192. Perform the indicated operations and simplify.  3     x  8x 2  2x  6    7x 2  2x  3      193. Perform the indicated operations and simplify.  3 3   3 3   2   2   2   2   x  y   x  y            194. Use a Special Factoring Formula to factor the expression. x 3  8y 3 195. Use a Special Factoring Formula to factor the expression. 216s 3  125t 6 196. Factor the expression completely. 4x 2  81 27 Name: ________________________ ID: A 197. Factor the expression completely.  2  2  4  3    4  3   x   x    198. Find the domain of the expression. 7t 2  8 2t  6 199. Perform the multiplication and simplify. x 2  2xy  y 2 4x 2  3xy  y 2  x2  y2 x 2  xy  2y 2 200. Perform the division and simplify. 2x 2  7x  4 x2  9 2x 2  9x  4 x 2  x  12 201. Use the formula h  16t 2  v 0 t to solve the problem. A ball is thrown straight upward at an initial speed of v 0  72 ft/s. (a) When does the ball reach a height of 80 ft? (b) When does it reach a height of 104 ft? (c) What is the greatest height reached by the ball? (d) When does the ball reach the highest point of its path? (e) When does the ball hit the ground? 202. The given equation is either linear or equivalent to a linear equation. Solve the equation. 1 5  1 x 2x 203. Solve the equation for x. ax  b  11 cx  d 28 Name: ________________________ ID: A 204. Solve the equation by factoring. 2y 2  5y  2  0 205. Find all real solutions of the quadratic equation. x2  17x  4  0 206. Find all real solutions of the equation. |3x|  7 207. Evaluate the piecewise defined function at the indicated values.  2  x  4x if x  3  f(x)   x if  3  x  1    9 if x  1 (a) Evaluate f(4). f(4) = __________  7  (b) Evaluate f   .  2   7  f     __________  2 (c) Evaluate f(3). f(3)  __________ (d) Evaluate f(0). f(0)  __________ (e) Evaluate f(35). f(35)  __________ 208. Find the domain of the function. (x  6) 2 f(x)  8x  7 29 Name: ________________________ ID: A 209. Use f(x)  2x  6 and g(x)  5  x 2 to evaluate the expression. (a) f  g(1)   __________ (b) g  f(1)   ___________ 210. Find the domain of the function. f(x)  x  8x 211. Assume f is a one-to-one function. (a) If f(8) = 1, find f 1 (1) (b) If f(2) = –7, find f 1 (7). 212. Draw the graph of f and use it to determine whether the function is one-to-one. f(x) | x|| x  8| f is __________ 213. Find the inverse function of f. f(x)  2  4x 214. Graph the polynomial y  x 4  11x 2  18. (a) From the graph, determine how many local maxima the polynomial has. (b) From the graph, determine how many local minima the polynomial has. 30 Name: ________________________ ID: A 215. An open box is to be constructed from a piece of cardboard 24 cm by 35 cm by cutting squares of side length x from each corner and folding up the sides, as shown in the figure. (a) Express the volume V of the box as a function of x. V(x)  __________ (b) What is the domain of V? (Use the fact that length and volume must be positive.) __________ < x < __________ (c) Draw a graph of the function V and use it to estimate the maximum volume for such a box. Please round the answer to the nearest whole number. V  __________ cm3 216. Find all rational zeros of the polynomial. P(x)  x 4  2x 3  8x 2  18x  9 217. Find all rational zeros of the polynomial. P(x)  x 3  3x  2 218. Perform the addition and write the result in the form a  bi.  1  3  2i  7  i   3  219. Find all solutions of the equation and express them in the form a  bi. 1 2 x  5x  25  0 2 220. A polynomial P is given. Factor P completely into linear factors with complex coefficients. P(x)  x 3  7x 2  4x  28 P(x)  __________ 31 Name: ________________________ ID: A 221. Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3, and zeros 0 and 7i 222. Find all zeros of the polynomial. P(x)  x 3  5x 2  16x  80 223. Find the intercepts and asymptotes. 3x(x  2) r(x)  (x  1)(x  6) (a) Determine the x-intercept(s). (b) Determine the y-intercept(s). (c) Determine the vertical asymptote(s). (d) Determine the horizontal asymptote(s). 32 Name: ________________________ ID: A 224. Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. d Under such conditions the population follows a logistic growth model P(t)  where c, d, and k are 1  ke ct positive constants. For a certain fish population in a small pond d = 1,080, k = 11, c = 0.2, and t is measured in years. The fish were introduced into the pond at time t  0. (a) How many fish were originally put in the pond? Please round the answer to the nearest integer. P(0) = __________ (b) Find the population after 9 years. Please round the answer to the nearest integer. P(9) = __________ (c) Find the population after 19 years. Please round the answer to the nearest integer. P(19) = __________ (d) Find the population after 32 years. Please round the answer to the nearest integer. P(32) = __________ (e) Evaluate P(t) for large values of t. What value does the population approach as t ? Please round the answer to the nearest integer. P()  __________ (f) Does the graph shown confirm your calculations? 33 Name: ________________________ ID: A 225. If $3,000 is invested in an account for which interest is compounded quarterly, find the amount of the investment at the end of 3 years for the following interest rates. (a) 6%. Please give the answer to two decimal places. A(3) = __________ 1 (b) 6 %. Please give the answer to two decimal places. 2 A(3) = __________ (c) 7%. Please give the answer to two decimal places. A(3) = __________ (d) 8%. Please give the answer to two decimal places. A(3) = __________ 226. Find the domain of the function. f(x)  log 8 (x  3) 227. Use the Laws of Logarithms to expand the expression. Assume A > 0 and B > 0.   log 3  AB 5    228. Use the Laws of Logarithms to expand the expression. Assume x > 0, y > 0 and z > 0.  2   x  log a  4    yz   229. Simplify.  log 5   log 2   7  5  230. Find an equation for the parabola that has its vertex at the origin and satisfies the given condition. Focus F(8, 0). 34 Name: ________________________ ID: A 231. Find the equation of the parabola whose graph is shown. 35 Name: ________________________ ID: A 232. Find the equation of the parabola whose graph is shown. 36 Name: ________________________ ID: A 233. Find the equation of the parabola whose graph is shown. 234. A satellite is in an elliptical orbit around the earth with the center of the earth at one focus. The height of the satellite above the earth varies between 150 mi and 430 mi. Assume the earth is a sphere with radius 3960 mi. Find an equation for the path of the satellite with the origin at the center of the earth. 37 Name: ________________________ ID: A 235. Perform the indicated operation.  1   4   2  1  6   5   38