Class: Date: Algebra II/Trig Final Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short Answer 2. A manufacturer determines that the number of drills it can sell is given by the formula D = p 2 + 180p 285, where p is the price of the drills in dollars. a. At what price will the manufacturer sell the maximum number of drills? b. What is the maximum number of drills that can be sold?. The function y = 16t 2 + 486 models the height y in feet of a stone t seconds after it is dropped from the edge of a vertical cliff. How long will it take the stone to hit the ground? Round to the nearest hundredth of a second. 1
4. Zach wrote the formula w(w 1)(5w + 4) for the volume of a rectangular prism he is designing, with width w, which is always has a positive value greater than 1. Find the product and then classify this polynomial by degree and by number of terms. 5. The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since 1995. Find a cubic function to model the data and use it to estimate the number of cottonwoods planted in 2006. Years since 1995 1 5 7 9 Trees planted (in thousands) 1. 18. 70.5 177.1 57. 6. The dimensions in inches of a shipping box at We Ship 4 You can be expressed as width x, length x + 5, and height x 1. The volume is about 7.6 ft. Find the dimensions of the box in inches. Round to the nearest inch. 7. Ian designed a child s tent in the shape of a cube. The volume of the tent in cubic feet can be modeled by the equation s 64 = 0, where s is the side length. What is the side length of the tent? Find the roots of the polynomial equation. 8. x 4 5x + 11x 2 25x + 0 = 0 9. The formula for the volume of a sphere is V = 4 πr. Find the radius, to the nearest hundredth, of a sphere with a volume of 15 in.. 10. A garden has width 1 and length 7 1. What is the perimeter of the garden in simplest radical form? 11. A rope is 150 units long. The rope is cut into two pieces, so that the lengths of the pieces are in the ratio 2 :. What is the length of the longer piece expressed in simplest radical form? 12. An initial population of 895 quail increases at an annual rate of 7%. Write an exponential function to model the quail population. 1. The generation time G for a particular bacteria is the time it takes for the population to double. The bacteria t increase in population is shown by the formula G =, where t is the time period of the population. log a P increase, a is the number of bacteria at the beginning of the time period, and P is the number of bacteria at the end of the time period. If the generation time for the bacteria is 6 hours, how long will it take 8 of these bacteria to multiply into a colony of 7681 bacteria? Round to the nearest hour. 14. A group of college students are volunteering for Help the Homeless during their spring break. They are putting the finishing touches on a house they built. Working alone, Irina can paint a certain room in 7 hours. Paulo can paint the same room in 6 hours. Write an equation that can be used to find how long it will take them working together to paint the room. How many hours will it take them to paint the room? If necessary, round your answer to the nearest hundredth. Suppose Q and R are independent events. Find P(Q and R). 15. P(Q) = 4 5, P(R) = 4 11 2
16. If all possible results are equally likely, what is the probability that a spin of the spinner will land on an upper case letter or a consonant? 17. A grocery clerk sets up a display of 12-pack cartons of cola. There are 15 cartons at the base of the triangle and one at the top. How many cartons of cola are needed for the complete display? 18. A rope is swinging in such a way that the length of the arc is decreasing geometrically. If the first arc is 18 feet long and the third arc is 8 feet long, what is the length of the second arc? 19. A large asteroid crashed into a moon of a planet, causing several boulders from the moon to be propelled into space toward the planet. Astronomers were able to measure the speed of one of the projectiles. The distance (in feet) that the projectile traveled each second, starting with the first second, was given by the arithmetic sequence 26, 44, 62, 80,.... Find the total distance that the projectile traveled in seven seconds. 20. In June, Cory begins to save money for a video game and a TV he wants to buy in December. He starts with $20. Each month he plans to save 10% more than the previous month. How much money will he have at the end of December? 21. Each person in a group of students was identified by year and asked when he or she preferred taking classes: in the morning, afternoon, or evening. The results are shown in the table. Find the probability that the student preferred afternoon classes given he or she is a junior. Round to the nearest thousandth. When Do You Prefer to Take Classes? Freshman Sophomore Junior Senior Morning 19 2 6 16 Afternoon 17 1 15 Evening 8 14 9 7 22. The probability that a city bus is ready for service when needed is 84%. The probability that a city bus is ready for service and has a working radio is 67%. Find the probability that a bus chosen at random has a working radio given that it is ready for service. Round to the nearest tenth of a percent. Find the mean and standard deviation of the of data. Round to the nearest tenth. 2. 62, 7, 48, 67, 44, 58, 47, 47
24. Susan keeps track of the number of tickets sold for each play presented at The Community Theater. Within how many standard deviations of the mean do all the values fall? 17, 14, 91, 61, 150, 155, 110, 148, 90, 169, 67, 61 25. The bar graph shows the rents paid per month for apartments in an urban neighborhood. The curve shows that the rents are normally distributed. Estimate the percent of apartment residents who pay from $600 to $749 per month. 26. A set of data with a mean of 56 and a standard deviation of. is normally distributed. Find the values that are 2 standard deviations from the mean. 27. The numbers of cookies in a shipment of bags are normally distributed, with a mean of 64 and a standard deviation of 4. What percent of bags of cookies will contain between 60 and 68 cookies? 28. Find the measure of an angle between 0º and 60º coterminal with an angle of 110º in standard position. 29. Find the exact value of cos 00º and sin 00º. 0. 20º Write the measure in radians. Express the answer in terms of π. Write the measure in degrees. 1. 7π 4 radians 4
2. Use the circle below. Find the length s to the nearest tenth.. Write a cosine function for the graph. Graph the function in the interval from 0 to 2π. Ê 4. y = 2 cos x π ˆ Á 6 + 2 5. Suppose tanθ = 8. Find cotθ. 15 6. Find the exact value of csc 15º. If the expression is undefined, write undefined. 7. Find the exact value of sec ( 270º). If the expression is undefined, write undefined. Solve the equation. 8. 9x 2 + 16 = 0 9. x 2 + 18x + 81 = 25 40. x + 10 7 = 5 5
41. For f( x) = 5x + 1, find f( 4). Determine whether the function is linear or quadratic. Identify the quadratic, linear, and constant terms. 42. y = (x + 1)(6x 6) 6x 2 4. Identify the vertex and the y-intercept of the graph of the function y = (x + 2) 2 + 5. Write the equation of the parabola in vertex form. 44. vertex (0, ), point ( 4, 45) Factor the expression. 45. 15x 2 21x 46. x 2 + 14x + 48 47. x 2 6x + 8 48. x 2 2x 6 49. 9x 2 16 50. x + 216 51. c 512 52. x 4 20x 2 + 64 Solve the equation by finding square roots. 5. x 2 = 21 54. Simplify 175 using the imaginary number i. Simplify the expression. 55. ( 1 + 6i) + ( 4 + 2i) 56. (2 5i) ( + 4i) 57. ( 6i)( 6i) 58. (2 + 5i)( 1 + 5i) 59. Classify x 5 2x by degree and by number of terms. 60. Classify 7x 5 6x 4 + 4x by degree and by number of terms. 61. Write the polynomial 6x 2 9x + in standard form. 6
62. Write the expression (x + 6)(x 4) as a polynomial in standard form. 6. Write 4x + 8x 2 96x in factored form. 64. Find the zeros of y = x(x )(x 2). Then graph the equation. 65. Write a polynomial function in standard form with zeros at 5, 4, and 1. Divide using synthetic division. 66. (x 4 + 15x 77x 2 + 1x 6) (x 4) 67. Solve 125x + 4 = 0. Find all complex roots. 68. Find all the real square roots of 9 16. Find the real-number root. 69. 125 4 Multiply and simplify if possible. 70. 6 2 Divide and simplify. 71. 72. 162 2 90x 18 2x Add if possible. 4 7. 2 2x 74. 4 x 4 + 6 2x + 5 10x Simplify. 75. 5 6 + 6 5 1 76. 9 1 7
Multiply. Ê ˆ 77. 7 2 Á Ê Á 8 + 2 ˆ Ê ˆ 78. 8 2 Á Ê Á 9 + 5 ˆ Graph the function. 79. y = x + 1 80. y = x + 81. Suppose you invest $1600 at an annual interest rate of 4.6% compounded continuously. How much will you have in the account after 4 years? 82. How much money invested at 5% compounded continuously for years will yield $820? Write the equation in logarithmic form. 8. 6 4 = 1, 296 Evaluate the logarithm. 84. log 24 Write the expression as a single logarithm. 85. log 4 log 2 86. Use the properties of logarithms to evaluate log 9 + log 6 log 4. 87. Solve 15 2x = 6. Round to the nearest ten-thousandth. 88. Solve log(4x + 10) =. 89. Solve ln(2x 1) = 8. Round to the nearest thousandth. Simplify the rational expression. State any restrictions on the variable. 90. q 2 + 11q + 24 q 2 5q 24 91. The Sears Tower in Chicago is 1454 feet tall. The function y = 16t 2 + 1454 models the height y in feet of an object t seconds after it is dropped from the top of the building. a. After how many seconds will the object hit the ground? Round your answer to the nearest tenth of a second. b. What is the height of the object 5 seconds after it is dropped from the top of the Sears Tower? 8
Multiply or divide. State any restrictions on the variables. 92. 4a 5 7b 4 2b 2 2a 4 9. x 2 16 x 2 + 5x + 6 x 2 + 5x + 4 x 2 2x 8 Add or subtract. Simplify if possible. 94. 95. 7 a + 8 + 7 a 2 64 b 2 2b 8 b 2 + b 2 6 b 1 Solve the equation. Check the solution. 96. g + 4 g 2 = g 5 g 8 97. 6 x 2 9 1 x = 1 98. The volume in cubic feet of a workshop s storage chest can be expressed as the product of its three dimensions: V(x) = x x 2 x +. The depth is x + 1. a. Find linear expressions with integer coefficients for the other dimensions. b. If the depth of the chest is 6 feet, what are the other dimensions? Essay 99. The table shows the number of squirrels in a particular forest t years after a forest fire. Number of Squirrels Years Squirrels 0 0 1 60 2 120 240 4 480 5 960 a. Explain how the population of squirrels is changing each year. b. Write a function to model the situation. Explain what each number represents. 9