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ALG. 2 FINAL EXAM REVIEW PACKET AND ANSWERS Multiple Choice Identify the choice that best completes the statement or answers the question. ____ ____ 1. An irrational number can ________ be expressed as a quotient of integers. a. always b. sometimes c. never 2. Use the vertical-line test to determine which graph represents a function. a. c. y 4 4 2 2 –4 –4 O –2 y 2 4 –2 x O 2 4 x 2 4 x –2 –2 –4 –4 b. –4 ____ –2 y 4 4 2 2 O 2 4 –4 x –2 O –2 –2 –4 –4 3. A biologist took a count of the number of migrating waterfowl at a particular lake, and recounted the lake’s population of waterfowl on each of the next six weeks. Week 0 1 2 3 4 5 6 Population 585 582 629 726 873 1,070 1,317 a. Find a quadratic function that models the data as a function of x, the number of weeks. b. Use the model to estimate the number of waterfowl at the lake on week 8. a. b. c. d. ____ d. y ; 1,614 waterfowl ; 2,679 waterfowl ; 1,961 waterfowl ; 2,201 waterfowl 4. Identify the graph of the complex number . a. c. Imaginary Axis Imaginary Axis 4 4 2 2 Real Axis –4 b. –2 O 2 Real Axis –4 4 –2 O –2 –2 –4 –4 d. Imaginary Axis –2 4 4 2 2 O 2 4 Imaginary Axis –4 –2 –2 –4 –4 2 15 a. integers, rational numbers, real numbers b. rational numbers, real numbers c. irrational numbers, real numbers d. rational numbers, irrational numbers, real numbers 5. Name the property of real numbers illustrated by the equation. ____ ____ O –2 To which sets of numbers does the number belong? ____ 6. a. b. c. d. Associative Property of Multiplication Distributive Property Commutative Property of Addition Associative Property of Addition a. b. c. d. Distributive Property Associative Property of Multiplication Commutative Property of Multiplication Associative Property of Addition 4 Real Axis Real Axis –4 2 7. Short Answer 8. Write the ordered pairs for the relation. Find the domain and range. 2 4 y 4 2 –4 –2 O 2 x 4 –2 –4 Solve the system by graphing. 9. 10. Determine whether the function is linear or quadratic. Identify the quadratic, linear, and constant terms. 11. 12. Identify the vertex and the axis of symmetry of the parabola. Identify points corresponding to P and Q. y 13. 8 4 Q –8 –4 O P 4 8 x –4 –8 14. Use a graphing calculator to solve the equation hundredth. 15. Simplify using the imaginary number i. Write the number in the form a + bi. . If necessary, round to the nearest 16. 17. Find . Simplify the expression. 18. 19. 20. Find the missing value to complete the square. Solve the quadratic equation by completing the square. 21. 22. Rewrite the equation in vertex form. 23. Use the Quadratic Formula to solve the equation. 24. 25. 26. Classify –3x5 – 2x3 by degree and by number of terms. 27. Classify –7x5 – 6x4 + 4x3 by degree and by number of terms. 28. Write the polynomial in standard form. 29. Use a graphing calculator to find a polynomial function to model the data. x 1 2 3 4 5 6 7 8 9 10 f(x) 12 4 5 13 9 16 19 16 24 43 30. Write the expression (x + 6)(x – 4) as a polynomial in standard form. 31. Write 4x3 + 8x2 – 96x in factored form. 32. Divide by x + 3. Divide using synthetic division. 33. 34. 35. In XYZ, Y is a right angle and nearest hundredth, if necessary. . Find cos X in fraction and in decimal form. Round to the Z 25 20 X Y 36. In XYZ, Y is a right angle and . Find sin Z in fraction and in decimal form. Round to the nearest hundredth, if necessary. Z 5 4 X Y Find the length x. Round to the nearest tenth. 37. 59 x 55 ° 38. x 42° 80 Find the angle measure to the nearest tenth of a degree. 39. 40. 41. In tenth. , is a right angle. Find the remaining sides and angles. Round your answers to the nearest 42. a = 3.4, c = 5.8 43. Use the Law of Sines. Find b to the nearest tenth. C 31 b 42° 64° B A 44. Use the Law of Sines. Find to the nearest tenth. C 69 67° B 72 45. Use the Law of Cosines. Find b to the nearest tenth. A C 60 b 48° B A 85 46. Use the Law of Cosines. Find to the nearest tenth of a degree. C 17 22 B A 30 47. In an experiment, a petri dish with a colony of bacteria is exposed to cold temperatures and then warmed again. a. Find a quadratic model for the data in the table. b. Use the model to estimate the population of bacteria at 9 hours. Time (hours) Population (1000s) 0 1 2 3 4 5 6 5.1 3.03 1.72 1.17 1.38 2.35 4.08 Graph the number on a number line. 48. 49. Simplify by combining like terms. 50. 51. Find the perimeter of the figure. Simplify the answer. x+y 2x 4x y 2x x Solve the equation. 52. 53. 54. Solve the equation or formula for the indicated variable. 55. , for t 56. , for U Solve the inequality. Graph the solution set. 57. 2 + 2k 8 58. 2(4y – 5) –10 Solve the compound inequality. Graph the solution set. 59. 4x – 5 < –17 or 5x + 6 > 31 60. Suppose Find the value of 61. Graph the equation and . . . Find the slope of the line through the pair of points. 62. y 8 4 –8 –4 O 4 8 x –4 –8 63. (6, 12) and (–6, –2) Write in standard form an equation of the line passing through the given point with the given slope. 64. slope = –8; (–2, –2) 65. Find the point-slope form of the equation of the line passing through the points (–6, –4) and (2, –5). Find the slope of the line. 66. 67. y 4 2 –4 –2 O 2 4 x –2 –4 Find an equation for the line: 68. through (–7, –4) and vertical. 69. Graph the equation of y = |x| translated 4 units up. Find the value of y for a given value of x, if y varies directly with x. 70. If y = 166 when x = 83, what is y when x = 23? 71. If y = 4.8 when x = 2.4, what is y when x = 2.05? 72. A balloon takes off from a location that is 158 ft above sea level. It rises 56 ft/min. Write an equation to model the balloon’s elevation h as a function of time t. Graph the absolute value equation. 73. Without graphing, classify each system as independent, dependent, or inconsistent. 74. Solve the system by the method of substitution. 75. Use the elimination method to solve the system. 76. 77. Solve the system of inequalities by graphing. 78. 79. Find a quadratic model for the set of values. 80. (–2, 8), (0, –4), (4, 68) 81. x –2 0 4 f(x) 1 –3 85 82. Write Factor the expression. 83. 84. 85. 86. in vertex form. 87. Solve the equation by finding square roots. 88. ALG. 2 FINAL EXAM REVIEW PACKET - 1 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: C 3. ANS: C 4. ANS: B 5. ANS: B 6. ANS: B 7. ANS: B SHORT ANSWER 1-1 Properties of Real Numbers 2-1 Relations and Functions 5-1 Modeling Data With Quadratic Functions 5-6 Complex Numbers 1-1 Properties of Real Numbers 1-1 Properties of Real Numbers 1-1 Properties of Real Numbers 8. ANS: {(–2, 5), (–1, 2), (0, 1), (1, 2), (2, 5)}; domain: {–2, –1, 0, 1, 2}; range: {1, 2, 5} 2-1 Relations and Functions 9. ANS: y 8 4 –8 –4 O 4 8 x –4 –8 (–5, –4) 3-1 Graphing Systems of Equations 10. ANS: y 4 2 –4 –2 O 2 4 –2 –4 no solutions 3-1 Graphing Systems of Equations 11. ANS: x linear function linear term: constant term: –6 5-1 Modeling Data With Quadratic Functions 12. ANS: quadratic function quadratic term: linear term: constant term: –6 5-1 Modeling Data With Quadratic Functions 13. ANS: (–1, –2), x = –1 P'(0, –1), Q'(–3, 2) 5-1 Modeling Data With Quadratic Functions 14. ANS: 0.87, –2.07 5-5 Quadratic Equations 15. ANS: 5-6 Complex Numbers 16. ANS: 5-6 Complex Numbers 17. ANS: 41 5-6 Complex Numbers 18. ANS: 5-6 Complex Numbers 19. ANS: 5-6 Complex Numbers 20. ANS: 1 5-7 Completing the Square 21. ANS: 5-7 Completing the Square 22. ANS: 5-7 Completing the Square 23. ANS: 5-7 Completing the Square 24. ANS: 1 , 2 5 5-8 The Quadratic Formula 25. ANS: 1 8 5-8 The Quadratic Formula 26. ANS: quintic binomial 6-1 Polynomial Functions 27. ANS: quintic trinomial 6-1 Polynomial Functions 28. ANS: 6-1 Polynomial Functions 29. ANS: f(x) = 0.08x4 – 1.73x3 + 12.67x2 – 34.68x + 35.58 6-1 Polynomial Functions 30. ANS: x2 + 2x – 24 6-2 Polynomials and Linear Factors 31. ANS: 4x(x – 4)(x + 6) 6-2 Polynomials and Linear Factors 32. ANS: , R –93 6-3 Dividing Polynomials 33. ANS: 6-3 Dividing Polynomials 34. ANS: , R –38 6-3 Dividing Polynomials 35. ANS: 14-3 Right Triangles and Trigonometric Ratios 36. ANS: 14-3 Right Triangles and Trigonometric Ratios 37. ANS: 48.3 14-3 Right Triangles and Trigonometric Ratios 38. ANS: 72.0 14-3 Right Triangles and Trigonometric Ratios 39. ANS: 11.7° 14-3 Right Triangles and Trigonometric Ratios 40. ANS: 86.1° 14-3 Right Triangles and Trigonometric Ratios 41. ANS: 82.8° 14-3 Right Triangles and Trigonometric Ratios 42. ANS: = 54.1°, = 35.9°, b = 4.7 14-3 Right Triangles and Trigonometric Ratios 43. ANS: 23.1 14-4 Area and the Law of Sines 44. ANS: 73.8 14-4 Area and the Law of Sines 45. ANS: 63.2 14-5 The Law of Cosines 46. ANS: 33.9 14-5 The Law of Cosines 47. ANS: a. b. 13,830 bacteria 5-1 Modeling Data With Quadratic Functions 48. ANS: –5 –4 –3 –2 –1 0 1 2 3 4 5 1-1 Properties of Real Numbers OBJ: 1-1.1 Graphing and Ordering Real Numbers 49. ANS: –5 –4 –3 –2 –1 0 1 2 3 4 5 1-1 Properties of Real Numbers 50. ANS: 1-2 Algebraic Expressions 51. ANS: 10x + 2y 1-2 Algebraic Expressions 52. ANS: 1 2 2 1-3 Solving Equations 53. ANS: 1 x = 1 or x = 2 3 1-5 Absolute Value Equations and Inequalities 54. ANS: 4 4 i, i 3 3 5-6 Complex Numbers 55. ANS: 1-3 Solving Equations 56. ANS: 1-3 Solving Equations 57. ANS: k3 –8 –6 –4 –2 0 2 4 6 8 4 6 8 1-4 Solving Inequalities 58. ANS: y0 –8 –6 –4 –2 0 2 1-4 Solving Inequalities OBJ: 1-4.1 Solving and Graphing Inequalities 59. ANS: x < –3 or x > 5 –8 –6 –4 –2 0 2 4 6 8 1-4 Solving Inequalities OBJ: 1-4.2 Compound Inequalities 60. ANS: 4 2 7 2-1 Relations and Functions 61. ANS: y 4 2 –4 –2 O –2 –4 2-2 Linear Equations 62. ANS: 4 2-2 Linear Equations 63. ANS: 7 6 2-2 Linear Equations 64. ANS: 8x + y = –18 2-2 Linear Equations 65. ANS: 1 y + 4 = (x + 6) 8 2-2 Linear Equations 66. ANS: 1 2 2-2 Linear Equations 67. ANS: 0 2 4 x 2-2 Linear Equations OBJ: 2-2.2 Writing Equations of Lines 68. ANS: x = –7 2-2 Linear Equations 69. ANS: y 6 4 2 –6 –4 –2 O –2 2 4 6 x –4 –6 2-6 Families of Functions 70. ANS: 46 2-3 Direct Variation 71. ANS: 4.1 2-3 Direct Variation 72. ANS: h = 56t + 158 2-4 Using Linear Models OBJ: 2-4.1 Modeling Real-World Data 73. ANS: y 16 12 8 4 –8 –4 O 4 8 x –4 2-5 Absolute Value Functions and Graphs 74. ANS: dependent 3-1 Graphing Systems of Equations 75. ANS: (0, –5) 3-2 Solving Systems Algebraically 76. ANS: (5, 3) 3-2 Solving Systems Algebraically 77. ANS: (0, –2) 3-2 Solving Systems Algebraically 78. ANS: y 6 4 2 –6 –4 –2 O –2 2 4 6 x –4 –6 3-3 Systems of Inequalities 79. ANS: y 4 2 –4 –2 O 2 4 x –2 –4 3-3 Systems of Inequalities 80. ANS: 5-1 Modeling Data With Quadratic Functions 81. ANS: 5-1 Modeling Data With Quadratic Functions 82. ANS: 5-3 Translating Parabolas 83. ANS: 5-4 Factoring Quadratic Expressions 84. ANS: 5-4 Factoring Quadratic Expressions 85. ANS: 5-4 Factoring Quadratic Expressions 86. ANS: 5-4 Factoring Quadratic Expressions 87. ANS: 5-4 Factoring Quadratic Expressions 88. ANS: 7, – 7 5-5 Quadratic Equations