Final Exam Practice. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

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1 Name: Class: Date: ID: A Final Exam Practice Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Compared to the graph of the base function f(x) = x, the graph of the function g(x) + 5 = x is translated A 5 units to the right C 5 units down B 5 units up D 5 units to the left 2. Compared to the graph of the base function f(x) = x, the graph of the function g(x) = x + 9 is translated A 9 units to the right C 9 units down B 9 units up D 9 units to the left 3. What is the equation of the transformed function, g(x), after the transformations are applied to the graph of the base function f( x) = x 2, shown in blue, to obtain the graph of g(x), shown in red? A B g( x) + 3 = ( x 5) 2 C g( x) 5 = ( x + 3) 2 g( x) = ( x + 3) 2 5 D g( x) = ( x 5)

2 Name: ID: A 4. The two functions in the graph shown are reflections of each other. Select the type of reflection(s). A a reflection in the line y = x C a reflection in the y-axis B a reflection in the x-axis and the y-axis D a reflection in the x-axis 5. When a function is reflected in the x-axis, the coordinates of point (x, y) become A (x, y) C ( x, y) B ( x, y) D (x, y) 2

3 Name: ID: A 6. Which of the graphs shown below represents the base function f(x) = x 2 and the stretched function g(x) = 1 5 x2? A C B D 3

4 Name: ID: A 7. Which is the graph of the function f(x) = ( x 6) 2 + 3? A C B D 8. In the graph shown, which transformations must be applied to the blue curve to obtain the red curve? A B C D a reflection in the x-axis and a translation of 5 units down a reflection in the y-axis and a translation of 5 units up a reflection in the x-axis and a translation of 5 units up a reflection in the y-axis and a translation of 5 units down 4

5 Name: ID: A 9. Which of the following graphs represents the graph of the function f( x) = x transformed to f( x) = 2 2x ? A C B D 5

6 Name: ID: A 10. When the function f( x) = x is transformed to f( x) = 4 x , the graph looks like A C B D 11. Which of the following functions is the correct inverse for the function f(x) = 3x + 5? A f 1 (x) = 1 3 x 5 3 C f 1 (x)= 1 3 x 5 3 B f 1 (x) = 1 3 x+ 5 3 D f 1 (x) = 1 3 x Which of the following functions is the correct inverse for the function f(x) = 9 2 x + 6? A f 1 (x) = 2 9 x+ 4 3 C f 1 (x) = 2 9 x 4 3 B f 1 (x)= 9 2 x+ 4 3 D f 1 (x) = 9 2 x Which of the following functions is the correct inverse for the function f(x) = x 2 + 7, {x x 0, x R}? A f 1 (x) = ( x 7) 2 C f 1 (x) = x 7 B f 1 (x) = x + 7 D f 1 (x) = x + 7 6

7 Name: ID: A 14. Which graph represents the inverse of the graph shown? A C B D 7

8 Name: ID: A 15. Which graph represents the inverse of the function shown? A C B D 16. Compared to the graph of the base function f(x) = x, the graph of the function g(x) = x 2 is translated A 2 units up C 2 units to the left B 2 units to the right D 2 units down 17. Compared to the graph of the base function f(x) = x, the graph of the function g(x) + 8 = x is translated A 8 units to the left C 8 units to the right B 8 units up D 8 units down 8

9 Name: ID: A 18. Compared to the graph of the base function f(x) = x, the graph of the function g(x) = x 5 is translated A 5 units down C 5 units right B 5 units left D 5 units up 19. When b < 0, the function g(x) = bx has what relationship to the base function f(x) = x? A f(x) is stretched horizontally by a factor of 1/ b B f(x) is stretched horizontally by a factor of 1/ b and reflected in the y-axis C f(x) is stretched vertically by a factor of b D f(x) is stretched vertically by a factor of b and reflected in the x-axis 20. In the following graph, what transformations must be applied to the blue curve to obtain the red curve? A a reflection in the x-axis, a vertical translation 5 units up, and a horizontal translation 3 units to the right B a reflection in the x-axis, a vertical translation 5 units down, and a horizontal translation 3 units to the right C a reflection in the x-axis, a vertical translation 3 units up, and a horizontal translation 5 units to the left D a reflection in the x-axis, a vertical translation 3 units up, and a horizontal translation 5 units to the right 9

10 Name: ID: A 21. The two functions in the graph shown are reflections of each other. Select the type of reflection(s). A a reflection in the y-axis C a reflection in the line y = x B a reflection in the x-axis and the y-axis D a reflection in the x-axis 22. Which is the graph of the square root of the function f(x) = (x 5) 2 2? A C B D 10

11 Name: ID: A 23. Which of the following functions is the correct inverse for the function f(x) = A f 1 (x) = x + 2 C f 1 (x) = x B f 1 (x) = x + 2 D f 1 (x) = ( x 2) 2 x 2, {x x 0, x R}? 24. Which graph represents the square root of the graph shown? A C B D 11

12 Name: ID: A 25. Which graph shows the graphical solution to the radical equation 0 = 2 (x 5) 2? A C B D 26. Which radical equation can be solved using the graph shown below? A 4 x = x + 2 C x + 2 = 4 + x B 4 x = x + 2 D 4 + x = x

13 Name: ID: A 27. What is the solution to the radical equation 0 = x + 9 3? A 18 C 18 B 36 D What is the solution to the radical equation 0 = 2 2(x + 4) 8? A 4 C 4 B 12 D Which of the following is a polynomial function? A y = 4x 4 + 4x 3 7x 2 + 9x C g( x) = x + 4 B f( x) = 4 x 7 D y = 4x + 9 x Which graph represents an odd-degree polynomial function with two x-intercepts? A C B D 31. If 9x 3 + 9x is divided by 6x + 5, then the restriction on x is A x 6 5 C x 5 6 B x 5 6 D x

14 Name: ID: A 32. If 2x 3 6x 2 + 5x 7 is divided by x 7 to give a quotient of 2x 2 20x 135 and a remainder of 952, then which of the following is true? A ( x 7)( 2x 2 20x 135) = 952 B 2x 3 6x 2 + 5x 7 = ( x 7)( 2x 2 20x 135) 952 C ( x 7)( 2x 2 20x 135) = 952 D 2x 3 6x 2 + 5x 7= ( x 7)( 2x 2 20x 135) When P( x) = 5x 3 2x + 2 is divided by 5x 2, the remainder is A x 2 + x + 12 C P(5 / 2) = 601 / 8 5 B P( 2) = 34 D P(2 / 5) = 38 / For a polynomial P(x), if P( 6) = 0, then which of the following must be a factor of P(x)? A x 2 6 C x B x + 6 D x Which of the following binomials is a factor of x x x + 18? A x 2 C x 1 B x 9 D x Determine the value of k so that x + 2 is a factor of x x x + k. A k = 1 C k = 14 B k = 14 D k = Which of the following is the fully factored form of x 3 + 2x 2 23x 60? A ( x + 3) ( x 4) ( x + 5) C ( x 3) ( x + 4) ( x + 5) B ( x + 3) ( x + 4) ( x 5) D ( x 3) ( x 4) ( x 5) 38. Which of the following is the fully factored form of x 3 + 9x 2 4x 36? A ( x 2) 2 ( x + 9) C ( x + 2) ( x 2) ( x 9) B ( x 2) 2 ( x 9) D ( x + 2) ( x 2) ( x + 9) 39. One root of the equation x 3 + 7x 2 33x 135 = 0 is A 3 C 9 B 3 D 5 14

15 Name: ID: A 40. Which of the following graphs of polynomial functions corresponds to a cubic polynomial equation with roots 4, 1, and 3? A C B D 15

16 Name: ID: A 41. Which of the following graphs of polynomial functions corresponds to a polynomial equation with zeros 6 (multiplicity of 2) and 1 (multiplicity of 2)? A C B D 42. Determine the equation of a circle with centre at (3, 3) and radius 10. A (x 3) 2 + (y + 3) 2 = 100 C (x 3) 2 + (y + 3) 2 = 10 B (x 3) 2 + (y + 3) 2 = 20 D (x 3) 2 + (y + 3) 2 = If the angle θ is 5000 in standard position, it can be described as having made A rotations C rotations B rotations D rotations 44. If the angle θ is 1600 in standard position, in which quadrant does it terminate? A quadrant III C quadrant II B quadrant IV D quadrant I 16

17 Name: ID: A 45. A ball is riding the waves at a beach. The ball s up and down motion with the waves can be described using Ê the formula h = 2.3sin πt ˆ where h is the height, in metres, above the flat surface of the water and t is the Á 3, time, in seconds. What is the height of the ball, to the nearest hundredth of a metre, after t = 17 s? A 0.87 m C 1.99 m B 2.66 m D 1.99 m 46. A tricycle has a front wheel that is 30 cm in diameter and two rear wheels that are each 12 cm in diameter. If the front wheel rotates through a angle of 32, through how many degrees does each rear wheel rotate, to the nearest tenth of a degree? A 32.0 C 80.0I B 40.0I D 160.0I 47. The point P(0.391, 0.921) is the point of intersection of a unit circle and the terminal arm of an angle θ in standard position. What is the equation of the line passing through the centre of the circle and the point P? Round the slope to two decimal places. A y = 2.36x C y = 2.36x B y = 0.42x D y = 2.36x Which function, where x is in radians, is represented by the graph shown below? A y = cos x C y = cos x B y = sin x D y = sin x 49. The period (in degrees) of the graph of y = cos 4x is A 270 C 90 B 180 D 45 17

18 Name: ID: A 50. Which function is represented by the graph shown below, where θ is in radians? A y = 5 4 sin( 2x) C y = 2 cos( 5 4 x) B y = 2 sin( 5 4 x) D y = 5 4 cos( 2x) 51. The graph of y = sin x can be obtained by translating the graph of y = cos x A π 4 units to the right C π 3 units to the right B π 2 units to the right D π units to the right Ê Ê 52. What is the period of the sinusoidal function y = cos 8 x π ˆˆ Á 2 Á 2? A 1 8 π C 1 4 π B 4π D 1 2 π 53. Which of the following is not an asymptote of the function f( θ) = tanθ? A x = 7 2 π C x = 5 2 π B x = 9 2 π D x = π 54. Which function has zeros only at θ = nπ,n I? A y = tan(θ π) C y = tan(θ 7 6 π) B y = tan( θ π) D y = tan(θ π) 18

19 Name: ID: A 55. Given the trigonometric function y = tanx, which is the x-coordinate at which the function is undefined? A 9 2 π C 1 3 π B 7 6 π D 3 4 π 56. Given the trigonometric function y = tanx, find the value of the y-coordinate of the point with x-coordinate A 3 C 1 B 1 D undefined 57. What are the solutions for sin 2 x 1 = 0 in the interval 0 x 360? 2 A x = 45 and 225 and 315 and 135 C x = 90 and 270 and 225 B x = 30 and 210 and 135 D x = 60 and 240 and 45 Use the following information to answer the questions. The height, h, in metres, above the ground of a car as a Ferris wheel rotates can be modelled by the function Ê h( t) = 18cos πt ˆ + 19, where t is the time, in seconds. Á What is the radius of the Ferris wheel? A 9 m C 19 m B 18 m D 36 m 59. How long does it take for the wheel to revolve once? A π 80 s C 160 s B 80 s D 80 π s 60. What is the minimum height of a car? A 19 m C 160 m B 9 m D 80 m 61. What is the maximum height of a car? A 19 m C 160 m B 80 m D 31 m 19

20 Name: ID: A Use the following information to answer the questions. The height, h, in centimetres, of a piston moving up and down in an engine cylinder can be modelled by the function h( t) = 14sin ( 80πt) + 14, where t is the time, in seconds. 62. What is the period? 7 A 40 s C 1 40 s B 8 s D 1 14 s 63. Which expression is equivalent to A B cos θ ( 1 + sinθ ) 1 + sin 2 θ cos θ 1 sinθ cos θ 1 + sinθ? C D 1 sinθ cos θ 1 + sinθ cos θ 64. Which expression is equivalent to tanθ + cotθ? A 1 C 1 cos θ sinθ B cos θ sinθ D Ê 2 sinθ ˆ Á cos θ tana tanb 65. Which expression is equivalent to 1 + tana tanb? A tan(a + B) C tan(a B) B cot(a + B) D cot(a B) 66. Simplify sin168 cos 143 cos 168 sin143. Round your answer to the nearest hundredth. A 0.47 C 0.75 B 1.15 D What is the general solution, in degress, to the equation 2cos x cos 2x 2sinx sin2x = 1? A n and n. where n I C n and n, where n I B n, where n I D n and n, where n I 68. What is the general solution, in radians, to the equation (4cos 2 2θ + 1) sin 1 3 θ = 0? A 2πn where n I C 3πn where n I B no solution D π 3 n where n I 69. Which set of properties does the function y = 2 x have? A no x-intercept, no y-intercept C no x-intercept, y-intercept is 1 B x-intercept is 1, no y-intercept D x-intercept is 0, y-intercept is 0 20

21 Name: ID: A 70. Which choice best describes the function y = 6 x? A both increasing and decreasing C increasing B decreasing D neither increasing nor decreasing 71. Which set of properties is correct for the function y = 1 x Ê ˆ? Á 9 A domain {x x R}, range {y y > 0, y R} C domain {x x R}, range {y y 0, y R} B domain {x x R}, range {y y 0, y R} D domain {x x R}, range {y y < 0, y R} 72. Which exponential equation matches the graph shown? A y = 1 x Ê ˆ Ê C y = 1 ˆ Á 8 Á 8 B y = 8 x D y = 8 x 73. A bacteria colony initially has 1500 cells and doubles every week. Which function can be used to model the population, p, of the colony after t days? x A B p( t) = 1500( 3) t C p( t) = 1500( 2) t 7 p( t) = 1500( 2) t D p( t) = 1500( 3) t 7 21

22 Name: ID: A 74. To the nearest year, how long would an investment need to be left in the bank at 5%, compounded annually, for the investment to triple? A 15 years C 28 years B 26 years D 23 years 75. Which function results when the graph of y = 6 x is translated 2 units down? A y = 6 x 2 C y = 6 x 2 B y = 6 x + 2 D y = 6 x What is the exponential equation for the function that results from the transformations listed being applied to the base function y = 9 x? a reflection in the y-axis a vertical stretch by a factor of 6 a horizontal stretch by a factor of 7 A y = 7( 9) x 6 C y = 7( 9) x 6 B y = 6( 9) x 7 D y = 6( 9) x 7 22

23 Name: ID: A 77. Which graph represents the function y = 2 7 x Ê ˆ? Á 9 A C B D 78. Which equation can be used to model the given information, where the population has been rounded to the nearest whole number? Year (x) Population (y) A y = 100( 1.04) x C y = 100( 1.04) x 1 B y = 100( 1.4) x D y = 100( 1.4) x 1 23

24 Name: ID: A 79. Solve for x, to one decimal place = 5 x A C B 11.1 D Solve for x. ( 36) = 216 x + 7) A 0.3 C 6 B 7 D 3.0 t Ê 1 ˆ The half-life of a radioactive element can be modelled by M = M 0 Á 32, where M 0 is the initial mass of the element; t is the elapsed time, in hours; and M is the mass that remains after time t. The half-life of the element is A 11 h C 18 h B 10 h D 9 h 82. Another way of writing 5 5 = 3125 is A log 5 5 = 3125 C log = 5 B log = 5 D log 5 5 = Another way of writing 7 3 = is Ê 1ˆ A log 7 Á 3 = 343 C log Ê 1 ˆ 7 Á 343 = 3 Ê B log 3 1 ˆ Á 7 = 343 D log Ê 1 ˆ 7 Á 343 = Compared to the graph of the base function y = log 10 x, the graph of the function y = log 10 x + 4 is translated A 4 units to the left C 4 units up B 4 units down D 4 units to the right 24

25 Name: ID: A 85. Which graph represents the function y = 3log 3 [(x 2)] 3? A C B D 86. Which if the following is equivalent to the expression log 4 sw 10 y? A log 4 s + 10log 4 w + log 4 y C log 4 s + log 4 w + 10log 4 y B 10log 4 s 10log 4 w + log 4 y D 10log 4 s + log 4 w + log 4 y 87. Solve 8 x = 486. Round your answer to two decimal places. A 3.59 C 1.78 B 2.97 D What is true about the behaviour of the function f(x) = 4x + 5 as x 5 (right to left)? 4 A f(x) C f(x) 0 B f(x) + D f(x) is undefined 25

26 Name: ID: A What is the x-intercept of f(x) = 2x + 4? A There is no x-intercept. C 2 B Which graph represents the function f(x) = A D 0 4 x 9 5? C B D 26

27 Name: ID: A 91. Which function represents the graph shown below? A f(x) = C f(x) = x 8 x B f(x) = 9 x D f(x) = 9 x Which of the following functions has a slant asymptote when graphed? A B f(x) = 5x 3 10x 2 15x x 2 3x 4 f(x) = 5x 3 10x 2 15x x 2 2x 3 C D f(x) = 5x 3 10x 2 15x x 2 3x all of the above 93. Which function has vertical asymptotes with equations x = 9 and x = 6 7? 7x + 6 A f(x) = x x B f(x) = 7x x C f(x) = 7x 2 69x D f(x) = x x Which function has a point of discontinuity at x = 3? x 3 x 3 A f(x) = C f(x) = 2x 2 2x 12 x 2 6x 12 x + 3 x + 3 B f(x) = D f(x) = x 2 6x 12 x 2 6x

28 Name: ID: A 95. Which graph represents f(x) = A x 3 5x 2 23x + 24? C B D 96. Which function has a y-intercept of 8 27? 8 A f(x) = x 2 12x 27 8 B f(x) = ( 8x + 3)(x + 9) C f(x) = D 8 x x + 27 all of the above 97. Which function has a horizontal asymptote with equation y = 2 7? A f(x) = 2x 3 7x + 8 B f(x) = 7x + 8 2x 3 C f(x) = 7x 3 2x + 8 D f(x) = 2x 3 7x

29 Name: ID: A 98. Which function has an x-intercept of 1 3? A f(x) = 6x 2 5x 3 B f(x) = 5x 3 6x 2 C f(x) = 6x 2 5x 3 D f(x) = 5x 2 6x What is the equation for the vertical asymptote of the graph of the function shown? A x = 2 C y = 7 B x = 3 D y = Which function has a graph in the shape of a parabola? A f(x) = (x 3)2 (x 7) (x 3)(x 7) B f(x) = (x 3)2 (x 7) x 3 C f(x) = D x 3 (x 3) 3 (x 7) none of the above 101. What are the x-intercepts of the graph of f(x) = x 2 + 7x 18 x x + 35? A 7, 5 C 7, 5 B 2, 9 D 2, 9 29

30 Name: ID: A 102. Solve the equation 0 = 6x x 3 8x 2 graphically. A no solution C 0 B x = 1 D x = Given the functions f( x) = x 2 3 and g( x) = 9 x, determine the equation for the combined function y = f( x) + g( x). A y = x 2 27x 12 C y = x x + 6 B y = x 2 x + 6 D y = x 2 x Given the functions f( x) = x 2 8 and g( x) = 2 x 2, determine the equation for the combined function y = f( x) + g( x). A 6 2x 2 C 16 x 4 B 10 D 4 x 30

31 Name: ID: A For the following question(s), assume that x is in radians, if applicable Given the functions f( x) = cosx and g( x) = x, a graph of the combined function h( x) = f ( x) + g( x) most likely resembles A C B D 106. Given the functions f( x) = 9 x and g( x) = 9sinx, what is the range of the composite function h( x) = f ( x)g( x)? A {y 9 y 9, y R} C {y y R} B cannot be determined D {y y > 9, y R} 31

32 Name: ID: A 107. Shown is the graph of h( x) = f(g(x)), where f( x) = sinx and g( x) is a function of the form g( x) = a(x + b). What equation represents g( x)? A B g( x) = 1 2 ( x + 9) C g( x) = 2( x + 9) g( x) = 2(x 9) D g( x) = 1 (x 9) 2 32

33 Name: ID: A 108. Given the functions f( x) = 0.6 x and g( x) = cosx, the graph of the combined function h( x) = f ( x)g( x) most likely resembles A C B D 33

34 Name: ID: A 109. Given the functions f( x) = x 2 4 and g( x) = x 4, a graph of the combined function h( x) = f( x) g( x) resembles A C most likely B D 34

35 Name: ID: A 110. An equation for the graph shown is most likely A 4 x + cosx C 4 x + sinx B 4 x cosx D 4 sinx 35

36 Name: ID: A 111. An equation for the graph shown is most likely A f( x) = cosx x C sinx cosx B 2 sinx D f( x) = sinx x 112. Given the functions f( x) = x + 3 and g( x) = 1, what is the simplified form of (f û g)(x)? x 3 A (f û g)(x) = 8 x, x 0 C (f û g)(x) = 3x 8 x 3, x 3 B (f û g)(x) = 3x 8 x + 3, x 3 D (f û g)(x) = 1 x, x Given f( x) = 9x 2 + 7x and g( x) = 2 x, determine 5g ( x) f ( x). A 45x x 2 C 45x 2 + 6x 5 B 9x 2 + 6x + 2 D 9x 2 12x

37 Name: ID: A 114. Given the functions f( x) = x and g( x) = 1 ( 3 x 5 ), which of the following is most likely the graph of y = f(g(x))? A C B D 37

38 Name: ID: A 115. Given the functions f( x) = logx and g( x) = x + 4, which of the following is most likely the graph of h( x) = f(g(x))? A C B D 116. Solve for the variable: P = 20 5 r A 5 C 60 B 2 D An orchestra has 2 violinists, 3 cellists, and 4 harpists. Assume that the players of each instrument have to sit together, but they can sit in any position in their own group. In how many ways can the conductor seat the members of the orchestra in a line? A 144 C 24 B 72 D For a mock United Nations, 6 boys and 7 girls are to be chosen. If there are 12 boys and 9 girls to choose from, how many groups are possible? A C 960 B D For which of the following terms is a = 55 in the expansion of (x + y) 11? A ax 3 y 8 C ax 2 y 9 B ax 8 y 3 D ax 11 38

39 Name: ID: A 120. The leadership committee at a high school has 4 grade 10 students, 2 grade 11 students, and 6 grade 12 students. This year, 12 grade 10, 8 grade 11, and 10 grade 12 students applied for the committee. How many ways are there to select the committee? A C 733 B D Short Answer 1. Create a graph of g(x) = f( x 1) + 2 for each base function given, using transformations. a) f(x) = x 2 b) f(x) = x 2. Determine the equation, in standard form, of each parabola after being transformed from f(x) = x 2 by the given translations. a) 4 units to the right and 3 units up b) 2 units to the left and 1 unit up c) 2 units down and 7 units to the left 39

40 Name: ID: A 3. Given the graph of a function, sketch the resulting graph after the specified transformation. a) reflection in the x-axis b) reflection in the y-axis c) reflection in the x-axis and the y-axis 4. Determine the equation of the function g(x) after the indicated reflection. a) f(x) = ( x 1) 2 + 2, in the x-axis b) f(x) = x + 1, in the y-axis 5. a) Sketch the graph of g(x) = 2f( 2x)for each base function. i) f(x) = x ii) f(x) = x 2 iii) f(x) = x b) Write the equation for g(x) to represent a single stretch that results in the same graph as in each function in part a). c) Describe how each stretch affects the domain and range for each function. 40

41 Name: ID: A 6. For each g(x), describe, in the appropriate order, the combination of transformations that must be applied to the base function f(x) = x. a) g(x) = 2( x + 1) 2 b) g(x) = 2 x 3 4 c) g(x) = x For each of the following, describe the combination of transformations that must be applied to the graph of f(x) = x 2 (shown in blue) to obtain the graph of g(x) (shown in red). a) b) c) 41

42 Name: ID: A 8. For each function f(x), i) determine f 1 (x) ii) graph f(x) and its inverse a) f(x) = 5 2 x 3 b) f(x) = 3( x 2) Determine the equation of each radical function, which has been transformed from f(x) = x by the given translations. a) vertical stretch by a factor of 5, then a horizontal translation of 6 units right b) horizontal stretch by a factor of 1, then a vertical translation of 4 units down 6 c) horizontal reflection in the y-axis, then a vertical translation of 9 units up and horizontal translation of 2 units right d) horizontal stretch by a factor of 1 3, vertical reflection in the x-axis, and vertical stretch by a factor of Sketch the graph of f(x) = 2x and use it to sketch the graph of y = f(x). 11. Solve the equation 3x 6 = 12 graphically. 12. Jim states that the equations x 2 2 Ê ˆ = 25 and x Á = 25 have the same solution. Is he correct? Justify your reasoning. 13. A student designs a special container as part of an egg drop experiment. She believes that the container can withstand a fall as long as the speed of the container does not exceed 80 ft/s. She uses the equation v = (v 0 ) 2 + 2ad to model the velocity, v, in feet per second, as a function of constant acceleration, a, in feet per second squared and the drop distance, d, in feet. Assuming the student s specifications are correct, will the egg break if the student drops the egg from shoulder height (5 ft) off a building 80 ft high? What is the maximum height the egg can be dropped from? (Note: The acceleration due to gravity is 32 ft/s 2.) 14. Solve the equation x 3 4 = 2 graphically. 15. Factor fully. a) x 3 + 6x x + 6 b) 4x 3 11x 2 3x c) x Factor fully. a) x 2 (x 2)(x + 2) + 3x + 6 b) 16x 4 (x + 1) 2 42

43 Name: ID: A 17. Factor 2x 3 + 5x 2 14x 8 fully 18. Solve. a) 3x 3 + 2x 2 8x + 3 = 0 b) 2x 3 + x 2 10x 5 = 0 c) 5x 4 = 7x Solve by factoring. a) x 4 + 3x 2 28 = 0 b) 2x 4 54x = Solve by graphing using technology. Round answers to one decimal place. a) x 3 7 > 0 b) (x + 14) A child swings on a playground swing set. If the length of the swing s chain is 3 m and the child swings through an angle of π, what is the exact arc length through which the child travels? A 3-m ladder is leaning against a vertical wall such that the angle between the ground and the ladder is π 3. What is the exact height that the ladder reaches up the wall? Ê 23. Given that sinx = cos π ˆ and that x lies in the first quadrant, determine the exact measure of angle x. Á Without using a calculator, determine two angles between 0 and 360 that have a cosecant of 2 3. Include an explanation of how you determined the two angles. 25. Given a circle of diameter 21 cm, determine the arc length subtended by a central angle of 1.2 radians. 26. Angles A and B are located in the first quadrant. If sin A = of sec A + sec B. 2 2 and cos B = 3 2, determine the exact value 27. Determine the exact measures for all angles where tanθ = 3 in the domain 180 θ A grandfather clock shows a time of 7 o clock. What is the exact radian measure of the angle between the hour hand and the minute hand? 29. Explain how you could graph the function y = cos x given a table of values containing ordered pairs for the function y = sinx. 43

44 Name: ID: A 30. Describe the transformations that, when applied to the graph of y = cos x, result in the graph of È y = 2cos 1 Ê 8 x π ˆ Á 3 ÎÍ A pebble is embedded in the tread of a rotating bicycle wheel of diameter 60 cm. If the wheel rotates at 4 revolutions per second, determine a relationship between the height, h, in centimetres, of the pebble above the ground as a function of time, t, in seconds. 32. A population, p, of bears varies according to p( t) = cos t, where t is the time, in years, and angles are measured in radians. a) What are the maximum and minimum populations? b) What is the first interval, in years and months, over which the population is increasing? 33. A girl jumps rope such that the height, h, in metres, of the middle of the rope can be approximated by the equation h = 0.7sin ( 72t + 9) , where t is the time, in seconds. a) What is the amplitude of this function? b) How many revolutions of the rope does the girl make in 1 min? 34. Use a counterexample to show that cos(x + y) = cos x + cos y is not an identity. 35. What is the solution for 2cos x 3 = 0 for 0 x 2π? 36. Solve cot 2 θ + cot θ = 0. State the solution in general form. 37. Solve sec 2 θ 2tanθ 3 = 0. State the general solution to the nearest degree. 44

45 Name: ID: A 38. a) Determine the type of function shown in each graph. i) ii) iii) b) Describe what you would expect to see in the first differences column of a table of values for each graph in part a). 39. Sketch the graph of an exponential function with all of the following characteristics: domain {x x R} range {y y > 0, y R} y-intercept of 3 no x-intercept the function is always decreasing 45

46 Name: ID: A 40. For the function y = 1 2 ( 3) x 2, a) describe the transformations of the function when compared to the function y = 3 x b) sketch the graph of the given function and y = 3 x on the same set of axes c) state the domain, the range, and the equation of the asymptote 41. Write the equation for the function that results from each transformation or set of transformations applied to the base function y = 5 x. a) reflect in the y-axis b) shift 3 units to the right c) shift 1 unit down and 4 units to the left d) reflect in the x-axis and shift 2 units down 42. Match each exponential scatter plot with the corresponding equation of its curve of best fit. a) b) c) i) y = 2( 1.6) x ii) y = 40( 0.6) x iii) y = 10( 1.8) x Ê 43. Solve for n: 9 n 1 = 1 ˆ Á 3 4n Graph the function f(x) = log(x + 2) 1. Identify the domain, the range, and the equation of the vertical asymptote. 46

47 Name: ID: A 45. Given log , find the value of log Solve the equation 6 3x + 1 = 2 2x 3. Leave your answer in exact form. 47. a) Determine an equation in the form f(x) = y-intercept of 1 8. b) Sketch the graph of the function. 1 for a function with a vertical asymptote at x = 2 and a kx c 47

48 Name: ID: A Consider the function f(x) = 4x 5. a) Determine the key features of the function: i) domain and range ii) intercepts iii) equations of any asymptotes b) Sketch the graph of the function. 48

49 Name: ID: A 49. Consider the function f(x) = 3x + 8 x 2. a) Determine the key features of the function: i) domain and range ii) intercepts iii) equations of any asymptotes b) Sketch the graph of the function. x Consider the function f(x) = x 2 x 12. a) Determine the key features of the function: i) domain and range ii) intercepts iii) equations of any asymptotes b) Sketch the graph of the function. 51. Given the functions f( x) = x + 1 and g( x) = x 2 + 3x + 1, determine a simplified equation for h( x) = f ( x) + g( x). 52. Given the functions f( x) = x 2 4 and g( x) = x 2 3x + 2, determine a simplified equation for h( x) = f( x) g( x). 49

50 Name: ID: A Ê x Á 53. a) Graph the functions f( x) = sin( x) and g( x) = 2. b) Use the graphs to graph the function h( x) = f ( x)g( x). ˆ 54. Determine the equation(s) of the vertical asymptote(s) of the function y = 0.9x 2x Given the functions f( x) = x 2 7 and g( x) = 2 x 3, what is the value of f(g(2))? 50

51 Name: ID: A 56. Joe wants to travel from his home to school. The school is 6 blocks east and 6 blocks north. How many routes can Joe take from his house to school if he only moves east and north. Ê a 57. Use the binomial theorem to expand 2 b ˆ Á Simplify the expression (2n + 2)! (2n 2)!0! A neon sign with the words Espresso Coffee on it has 5 letters burnt out. In how many ways can you select 3 good letters and 2 burnt-out letters? 60. A math teacher is preparing a quiz for all of the students in grade 12. She wants to give each student the same questions, but have each student s questions appear in a different order. If there are 128 students in the grade 12 class, what is the least number of questions the quiz must contain so everyone gets a test with the questions in a different order. Problem 1. An object falls to the ground from a height of 25 m. The height, h, in metres, of the object above the ground can be modelled by the function h(t) = 1 2 at2 + 25, where a is the acceleration due to gravity, in metres per second squared, and t is the time, in seconds. a) Write an equation for the height of the object on Earth given a = 9.8 m/s 2. b) Write an equation for the height of the object on Mars given a = 3.7 m/s 2. c) Graph both functions on the same set of axes. d) What scale factor can be applied to the Earth function to transform it to the Mars function? 51

52 Name: ID: A 2. The base function f(x) = x is reflected in the x-axis, stretched horizontally by a factor of 2, compressed vertically by a factor of 1, and translated 3 units to the left and 5 units down. 3 a) Write the equation of the transformed function g(x). b) Graph the original function and the transformed function on the same set of axes. c) Which transformations must be done first but in any order? d) Which transformations must be done last but in any order? 3. The cost of renting a car for a day is a flat fee of $50 plus $0.12 for each kilometre driven. Let C represent the total cost of renting a car for a day if it is driven a distance, x, in kilometres. a) Write the total cost function for the car rental. b) Determine the inverse of this function. c) What does this inverse function represent? d) Give an example of how this function can be used. 4. For f(x) = 5 x and g(x) = 2 6(x + 2) 3, do the following. a) Graph f(x) and g(x) on the same set of axes. b) Determine the domain and range of each function. c) Explain which transformations would need to be applied to the graph of f(x) to obtain the graph of g(x). 5. Two groups of students are conducting a lab to determine the relationship between the period, p, in seconds, of a pendulum and the length, l, in metres, of the string. The curves of best fit from the experiment are shown on the graph. a) When asked the type of function that could be used to model their findings, both groups argue that a radical function can be used. Do you agree with each group? b) How do these graphs differ from the graph of f(x) = x? c) Write a function to approximate the graph for each group. d) What may have caused the differences in the data between the two groups? Justify your answer in terms of transformations. 52

53 Name: ID: A 6. The kinetic energy (energy of motion), E, in joules, of an object is given by the equation E = 1 2 mv 2, where m represents the mass of the object, in kilograms, and v represents its speed, in metres per second. a) Determine the general equation for the velocity of a mass as a function of its kinetic energy. b) Find the speed of an object of mass 12 kg moving with a kinetic energy of i) 200 J ii) 420 J c) Graph the function if the mass is 12 kg. d) John conducts an experiment and graphs the data, resulting in the graph below. What is the mass of the object? 7. Factor 2x 4 7x 3 41x 2 53x 21 fully. 8. Show that x + a is a factor of the polynomial P(x) = (x + a) 4 + (x + c) 4 (a c) The height of a square-based box is 4 cm more than the side length of its square base. If the volume of the box is 225 cm 3, what are its dimensions? 10. Solve x 3 + 5x 2 8x algebraically and graphically. 11. Determine an equation in expanded form for the polynomial function represented by the graph. 53

54 Name: ID: A 12. Two billiard balls collide and then separate from one another at the same, constant speed. Assume the billiard table is frictionless. The angle between the balls is 1.25 radians. After 2 s, the distance between the balls is 1 m. How fast are the balls moving, to the nearest hundredth of a metre per second? 13. To support a new 2.5-m wall in the construction of a home, the carpenters nail a piece of wood from the top of the wall to the floor, with the piece of wood forming the hypotenuse of the right triangle it makes with the wall and floor. The piece of wood is nailed to the ground such that it makes a 30 angle with the floor. a) Represent this situation with a diagram. b) Which trigonometric ratio can be used to determine the length of the piece of wood? c) Determine the length of the piece of wood. 14. a) Without using a calculator, determine two angles between 0 and 360 that have a sine ratio of 1 2. b) Use a calculator and a diagram to verify your answers to part a). 15. When a pendulum that is 0.5 m long swings back and forth, its angular displacement, θ, in radians, from rest position is given by θ = 1 4 sin Ê π 2 t ˆ where t is the time, in seconds. At what time(s) during the first 4 s is the Á, pendulum displaced 1 cm vertically above its rest position? (Assume the pendulum is at its rest position at t = 0.) 16. The table shows the hours of daylight measured on the first day of each month, over a 1-year period in a northern Ontario city. Month Hours of Daylight (h:min) 1 8:25 2 9: : : : : : : : : : :00 a) Graph the table data. b) Use the graph and the table to develop a sinusoidal model to represent the information. c) Graph the model on the same set of axes as the data. Comment on the fit. d) Use your model to estimate the number of hours of daylight, to the nearest tenth of an hour, on January 15, and verify the solution using the graph. 54

55 Name: ID: A 17. Wilson places a measuring tape on a pillar of a dock to record the water level in his local coastal community. He finds that a high tide of 1.77 m occurs at 5:17 a.m., and a low tide of 0.21 m occurs at 11:38 a.m. a) Estimate the period of the fluctuation of the water level. b) Estimate the amplitude of the pattern. c) Predict when the next two high tides will occur. d) Predict when the next two low tides will occur. 18. The graph of y = cos x is transformed so that the amplitude becomes 2 and the x-intercepts coincide with the maximum values. a) What is the equation of the transformed function? b) What phase shift of the transformed function will produce a y-intercept of 1? c) What is the equation of the function after the transformation in part b)? d) Verify your solution to part c) by graphing. 19. Prove the identity 1 + cos θ = sin2 θ 1 cos θ. Ê 20. Prove the identity sin π 2 x ˆ Á cot Ê π 2 + x ˆ Á = sinx. 21. Prove the identity 22. Prove the identity 1 cos 2θ + sin2θ 1 + cos 2θ + sin2θ = tanθ. cos 2 θ sin 2 θ cos 2 θ + sinθ cos θ = 1 tanθ. 23. An angle satisfies the relation ( sec θ) ( cotθ) = 1. a) Use the definition of the reciprocal trigonometric ratios to express the left side of the relation in terms of the sine and/or cosine ratios. b) Determine the value(s) for the angle. Do not use a calculator. c) Verify your answer to part b) using a calculator. d) Show your answer to part b) using a unit circle. 24. Prove the identity sin3θ + sinθ cos 3θ + cos θ = tan2θ. 25. Solve sin3x + sinx = cos x for 0 x 2π. 26. What is the general solution to tanx( csc x + 2) = 0? 27. A radioactive sample with an initial mass of 72 mg has a half-life of 10 days. a) Write a function to relate the amount remaining, A, in milligrams, to the time, t, in days. b) What amount of the radioactive sample will remain after 20 days? c) What amount of the radioactive sample was there 30 days ago? d) How long, to the nearest day, will it take for there to be 0.07 mg of the initial sample remaining? 55

56 Name: ID: A 28. a) Rewrite the function y = 2 2x in the form y = a(2) b(x h) + k. b) Describe the transformations that must be applied to the graph of y = 2 x to obtain the graph of the given function. c) Graph the function. d) Determine the equation of the function that results after the graph in part c) is reflected in the x-axis. e) Graph the function from part d). 29. Solve the equation x = 64 x Solve the equation 2 3x = A $ investment earns 5.25% interest, compounded quarterly. a) Determine the value of the investment in 5 years. b) How long will it take the original investment to double in value? 32. A chemist has a 20-mg sample of polonium-218. He needs approximately 81.5% of it for an experiment. Given that the half-life of polonium-218 is approximately 3.1 min, how many seconds will it take for the sample to decay to the desired mass? 33. An investment offers a bonus of 2% of the principal after being invested for 5 years. If $ is invested at 4.75%, compounded annually, for 10 years, describe how the graph of the investment with the bonus differs from the graph of the investment without the bonus. 34. Given log and log , find the value of log28. Ê 35. The stellar magnitude scale compares the brightness of stars using the equation m 2 m 1 = log b ˆ 1 b Á 2, where m 1 and m 2 are the apparent magnitudes (how bright the stars appear in the sky) of the two stars being compared, and b 1 and b 2 are their brightness (how much light they emit). a) The brightest appearing star in our sky, Sirius, has an apparent magnitude of 1.5. How much brighter does Sirius appear than Betelgeuse, whose apparent magnitude is 0.12? Round your answer to the nearest whole number. b) The Sun appears about times as bright in the sky as does Sirius. What is the apparent magnitude of the Sun, to the nearest tenth? 36. Prove that loga + loga 2 + loga 3 loga 6 = log Show that 1 log a b = log b a. 38. Prove that log q 5 p 5 = log q p. 56

57 Name: ID: A 39. a) Graph the function y = log 3 x. b) Graph the following functions on the same graph: y = log 3 3x y = log 3 9x y = log 3 27x c) Explain the effect of the constant k in the function y = log 3 kx. 40. For his dream car, Bruce invested $ at 7.8% interest, compounded monthly, for 5 years. After the 5 years, he still did not have enough money. How much longer will he have to invest the money at 5% interest, compounded daily, to have a total of $35 000? Round to the nearest tenth of a year. È 41. Sketch the graph of the function f(x) = 2log 1 2 ( x + 1) 1. Determine the x-intercept algebraically, to the ÎÍ nearest hundredth. 42. The time, t, in hours, that it takes Alistair to jog 5 km is inversely proportional to his average speed, v, in kilometres per hour. a) Write a function to represent the time as a function of the speed. b) Sketch the graph of this function. c) If Alistair jogs at 4.5 km/h, how long does it take him to complete a 5-km run, to the nearest minute? 43. The pressure exerted on the floor by the heel of someone s shoe is inversely proportional to the square of the width of the heel of the shoe. When Megumi wears 2-cm-wide heels, she exerts a pressure of 400 kpa. a) Determine a function to represent the pressure, p, exerted by Megumi if she wears heels of width w. b) Sketch the graph of this function. c) If she wears spike heels with a width of 0.5 cm, what pressure does she exert? 44. A photographer uses a light meter to measure the intensity of light from a flash bulb. The intensity, I, in lux, of the flash bulb is a function of the distance, d, in metres, from the light and can be represented by I(d) = 10 d 2, d > 0. a) Determine the following, to two decimal places: i) the intensity of light 3 m from the flash bulb ii) the average rate of change in the intensity of light for the interval 1 < d < 3 b) What does the sign of your answer to part a)ii) indicate about the light intensity? 57

58 Name: ID: A 45. Write an equation for the graph of the rational function shown. Explain your reasoning. 46. Write an equation for a rational function whose graph has all of the following features: vertical asymptote with equation x = 3 horizontal asymptote with equation y = 2 hole at x = 1 no x-intercepts 47. a) Use the asymptotes and intercepts to make a quick sketch of the function f(x) = x + 1 and its reciprocal x 5 g(x) = x 5 on the same set of axes. x + 1 b) Describe the symmetry in the graphs in part a). c) Determine the equation of the mirror line in your graph from part a). d) Determine intervals where f is positive and where f is negative. Determine intervals where g is positive and where g is negative. How do the two sets of intervals compare to each other? e) Does the pattern from part d) occur for all pairs of functions f(x) = x + b why or why not. x + d x + d and g(x) =, b d? Explain x + b 48. An airplane makes a 990-mi flight with a tail wind and returns, flying into the wind. The total flying time is 3 h 20 min, and the plane s airspeed is 600 mph. What is the wind speed? 58

59 Name: ID: A 49. A ski club charters a bus for a ski trip at a cost of $480. In an attempt to lower the bus fare per skier, the club invites non-members to go along. After five non-members join the trip, the fare per skier decreases by $4.80. How many club members are going on the trip? 50. Given the functions f( x) = 1 1 x and g ( x ) = sinx, determine the equation for h ( x ) = f(g(x)) Given the functions f( x) = x 2 3x 10 and g( x) = x 2 5x, graph the function h( x) = f( x) g( x) intercepts and identify any asymptotes and/or points of discontinuity.. Label all 52. What are the domain and range of the function y = 2 sin x, where x is in radians? 59

60 Name: ID: A 53. Use the graph of the combined function f( x) = 2 x x 2 to determine an approximate solution to the inequality 2 x > x Jenny and Jimmy are a married couple who work at the same store. Jimmy s total weekly salary, in dollars, if he sells x items is given by S( x) = x, and Jenny s total weekly salary, in dollars, if she sells x items is given by S( x) = x. a) Assuming that they sell the same number of items in a week, what is the minimum number of items they have to sell so that Jenny s weekly salary is at least $100 more than Jimmy s? b) Assuming that they sell the same number of items in a week, what is the minimum number of items they each need to sell to make their combined weekly salary greater than $1000? 60

61 Name: ID: A 55. The heights, h, of two balls, in metres, for a horizontal distance of x metres are shown in the graph. What was the difference in height of the two balls when the horizontal distance was 0 m? 56. The dimensions of a window are shown. a) What function in simplest form represents the area of the entire window? b) If the width, x, of the window is 1.2 m, what is the total area of the window, to the nearest tenth of a square metre? 7! 57. The number of different permutations using all of the 1-digit numbers of a set is given by 3!2!. a) What is the smallest number that can be created that meets these conditions? Explain your reasoning. b) What is the difference between the largest number and the smallest number? Explain your reasoning. 61

62 Name: ID: A 58. To win the grand prize in lottery A, a player must select all six of the winning numbers drawn from the numbers 1 to 49. To win in lottery B, a player must select all seven of the winning numbers drawn from 1 to 49. Bernadette argues that the chances of randomly selecting the winning number for lottery A are seven times as good as winning for lottery B. Create an argument to agree or disagree with this statement. 59. Tanya goes to a fast food stand at the beach. There are 4 types of burgers, 3 sizes of French fries, and either orange pop or root beer to drink. a) Create a tree diagram to show the possible choices of lunch if one of each item can be selected. b) In how many ways can Tanya buy 2 burgers, 2 fries, and 2 drinks for her and her friend? c) If Tanya does not like root beer and her friend does not like orange pop, how many possible choices are there? 62

63 Name: ID: A 60. On a Saturday, Charlie has to go to the library to study for a few hours, and then to the school to play a volleyball game. a) How many routes are there for Charlie to go from home to the library if she only moves south and east? b) How many routes are there for her to go from home to school moving only south and east? c) Assuming Charlie moves south and east going from home to school and north and west going from school to home, how many routes are there for her to complete the round trip? d) If Charlie could walk each route from home to school in 40 min, how long would it take her and her 22 classmates to walk all of the routes? Consider only the route from home to school, not the round trip. 61. Prove the identity 1 + tanθ 1 + cotθ = 1 tanθ cot θ Solve the equation log x x =

64 Final Exam Practice Answer Section MULTIPLE CHOICE 1. ANS: C PTS: 1 DIF: Average OBJ: Section 1.1 NAT: RF2 TOP: Horizontal and Vertical Translations KEY: vertical translation 2. ANS: D PTS: 1 DIF: Easy OBJ: Section 1.1 NAT: RF2 TOP: Horizontal and Vertical Translations KEY: horizontal translation 3. ANS: B PTS: 1 DIF: Average OBJ: Section 1.1 NAT: RF2 TOP: Horizontal and Vertical Translations KEY: horizontal translation vertical translation 4. ANS: D PTS: 1 DIF: Average OBJ: Section 1.2 NAT: RF5 TOP: Reflections and Stretches KEY: reflection 5. ANS: A PTS: 1 DIF: Easy OBJ: Section 1.1 NAT: RF5 TOP: Reflections and Stretches KEY: reflection 6. ANS: A PTS: 1 DIF: Average OBJ: Section 1.2 NAT: RF3 RF5 TOP: Reflections and Stretches KEY: graph vertical stretch reflection 7. ANS: C PTS: 1 DIF: Easy OBJ: Section 1.3 NAT: RF4 TOP: Combining Transformations KEY: graph horizontal translation vertical translation 8. ANS: C PTS: 1 DIF: Average OBJ: Section 1.3 NAT: RF4 TOP: Combining Transformations KEY: graph vertical translation reflection 9. ANS: D PTS: 1 DIF: Difficult OBJ: Section 1.3 NAT: RF4 RF5 TOP: Combining Transformations KEY: graph vertical translation horizontal translation stretch reflection 10. ANS: C PTS: 1 DIF: Average OBJ: Section 1.3 NAT: RF4 RF5 TOP: Combining Transformations KEY: graph vertical translation horizontal translation stretch reflection 11. ANS: A PTS: 1 DIF: Easy OBJ: Section 1.4 NAT: RF6 TOP: Inverse of a Relation KEY: inverse of a function function notation 12. ANS: A PTS: 1 DIF: Average OBJ: Section 1.4 NAT: RF6 TOP: Inverse of a Relation KEY: inverse of a function function notation 13. ANS: C PTS: 1 DIF: Average OBJ: Section 1.4 NAT: RF6 TOP: Inverse of a Relation KEY: inverse of a function function notation 14. ANS: D PTS: 1 DIF: Average OBJ: Section 1.4 NAT: RF6 TOP: Inverse of a Relation KEY: graph inverse of a function 15. ANS: D PTS: 1 DIF: Easy OBJ: Section 1.4 NAT: RF6 TOP: Inverse of a Relation KEY: graph inverse of a function 1

65 16. ANS: D PTS: 1 DIF: Easy OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: vertical translation 17. ANS: D PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: vertical translation 18. ANS: C PTS: 1 DIF: Easy OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: horizontal translation 19. ANS: B PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: horizontal stretch reflection 20. ANS: D PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: graph horizontal translation vertical translation reflection 21. ANS: A PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: reflection 22. ANS: D PTS: 1 DIF: Average OBJ: Section 2.2 NAT: RF13 TOP: Square Root of a Function KEY: graph 23. ANS: C PTS: 1 DIF: Easy OBJ: Section 2.1 Section 2.2 NAT: RF13 TOP: Radical Functions and Transformations Square Root of a Function KEY: inverse of a radical function 24. ANS: D PTS: 1 DIF: Average OBJ: Section 2.2 NAT: RF13 TOP: Square Root of a Function KEY: graph square root of a function 25. ANS: A PTS: 1 DIF: Average OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: graphical solution 26. ANS: B PTS: 1 DIF: Easy OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: graphical solution 27. ANS: D PTS: 1 DIF: Average OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: algebraic solution 28. ANS: C PTS: 1 DIF: Difficult OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: algebraic solution 29. ANS: A PTS: 1 DIF: Easy OBJ: Section 3.1 NAT: RF12 TOP: Characteristics of Polynomial Functions KEY: polynomial function 30. ANS: B PTS: 1 DIF: Average OBJ: Section 3.1 NAT: RF12 TOP: Characteristics of Polynomial Functions KEY: odd-degree x-intercepts 31. ANS: C PTS: 1 DIF: Average OBJ: Section 3.2 NAT: RF12 TOP: The Remainder Theorem KEY: restriction 32. ANS: B PTS: 1 DIF: Average OBJ: Section 3.2 NAT: RF11 TOP: The Remainder Theorem KEY: quotient remainder 2

66 33. ANS: D PTS: 1 DIF: Difficult + OBJ: Section 3.2 NAT: RF11 TOP: The Remainder Theorem KEY: remainder theorem remainder 34. ANS: D PTS: 1 DIF: Easy OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factor theorem factor 35. ANS: D PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factor theorem integral zero theorem factor 36. ANS: C PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factor theorem factor 37. ANS: B PTS: 1 DIF: Easy OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factored form factor theorem factor 38. ANS: D PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factored form factor theorem factor 39. ANS: A PTS: 1 DIF: Average OBJ: Section 3.4 NAT: RF12 TOP: Equations and Graphs of Polynomial Functions KEY: polynomial equation roots 40. ANS: B PTS: 1 DIF: Average OBJ: Section 3.4 NAT: RF12 TOP: Equations and Graphs of Polynomial Functions KEY: polynomial equation roots graph 41. ANS: C PTS: 1 DIF: Average OBJ: Section 3.4 NAT: RF12 TOP: Equations and Graphs of Polynomial Functions KEY: polynomial equation zeros graph multiplicity 42. ANS: A PTS: 1 DIF: Difficult + OBJ: Section 4.2 NAT: T2 TOP: Unit Circle KEY: unit circle unit circle equation 43. ANS: B PTS: 1 DIF: Average OBJ: Section 4.1 NAT: T1 TOP: Angles and Angle Measure KEY: rotations standard position NOT: Mixed numbers 44. ANS: C PTS: 1 DIF: Average OBJ: Section 4.1 NAT: T1 TOP: Angles and Angle Measure KEY: rotations standard position 45. ANS: C PTS: 1 DIF: Easy OBJ: Section 4.4 NAT: T4 TOP: Introduction to Trigonometric Equations KEY: trigonometric ratios 46. ANS: C PTS: 1 DIF: Difficult + OBJ: Section 4.1 NAT: T1 TOP: Angles and Angle Measure KEY: arc length degrees 47. ANS: A PTS: 1 DIF: Difficult OBJ: Section 4.3 NAT: T3 TOP: Trigonometric Ratios KEY: unit circle trigonometric ratios 48. ANS: A PTS: 1 DIF: Easy OBJ: Section 5.1 NAT: T4 TOP: Graphing Sine and Cosine Functions KEY: graph sinusoidal function 49. ANS: C PTS: 1 DIF: Easy OBJ: Section 5.1 NAT: T4 TOP: Graphing Sine and Cosine Functions KEY: period sinusoidal function 50. ANS: D PTS: 1 DIF: Average OBJ: Section 5.1 NAT: T4 TOP: Graphing Sine and Cosine Functions KEY: function amplitude period sinusoidal function 3

67 51. ANS: B PTS: 1 DIF: Easy OBJ: Section 5.2 NAT: T4 TOP: Transformations of Sinusoidal Functions KEY: translation primary trigonometric function 52. ANS: C PTS: 1 DIF: Average OBJ: Section 5.2 NAT: T4 TOP: Transformations of Sinusoidal Functions KEY: period sinusoidal function 53. ANS: D PTS: 1 DIF: Easy OBJ: Section 5.3 NAT: T4 TOP: The Tangent Function KEY: asymptote tangent function 54. ANS: B PTS: 1 DIF: Difficult + OBJ: Section 5.3 NAT: T4 TOP: The Tangent Function KEY: zeros transformation 55. ANS: A PTS: 1 DIF: Average OBJ: Section 5.3 NAT: T4 TOP: The Tangent Function KEY: undefined tangent function 56. ANS: A PTS: 1 DIF: Average OBJ: Section 5.3 NAT: T4 TOP: The Tangent Function KEY: coordinate tangent function 57. ANS: A PTS: 1 DIF: Difficult + OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: quadratic trigonometric equation 58. ANS: B PTS: 1 DIF: Easy OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: amplitude sinusoidal function 59. ANS: C PTS: 1 DIF: Average OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: period sinusoidal function 60. ANS: B PTS: 1 DIF: Average OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: minimum sinusoidal function 61. ANS: D PTS: 1 DIF: Average OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: maximum sinusoidal function 62. ANS: C PTS: 1 DIF: Easy OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: period sinusoidal function 63. ANS: C PTS: 1 DIF: Average OBJ: Section 6.1 NAT: T6 TOP: Reciprocal, Quotient, and Pythagorean Identities KEY: trigonometric identity 64. ANS: C PTS: 1 DIF: Average OBJ: Section 6.1 NAT: T6 TOP: Reciprocal, Quotient, and Pythagorean Identities KEY: trigonometric identity 65. ANS: C PTS: 1 DIF: Average OBJ: Section 6.2 NAT: T6 TOP: Sum, Difference, and Double-Angle Identities KEY: tangent sum identities difference identities 66. ANS: D PTS: 1 DIF: Average OBJ: Section 6.2 NAT: T6 TOP: Sum, Difference, and Double-Angle Identities KEY: sum identities difference identities evaluate 67. ANS: C PTS: 1 DIF: Difficult OBJ: Section 6.4 NAT: T6 TOP: Solving Trigonometric Equations Using Identities KEY: double-angle identities general solutions 4

68 68. ANS: C PTS: 1 DIF: Difficult OBJ: Section 6.4 NAT: T6 TOP: Solving Trigonometric Equations Using Identities KEY: double-angle identities general solutions 69. ANS: C PTS: 1 DIF: Easy OBJ: Section 7.1 NAT: RF9 TOP: Characteristics of Exponential Functions KEY: intercepts exponential function 70. ANS: C PTS: 1 DIF: Easy OBJ: Section 7.1 NAT: RF9 TOP: Characteristics of Exponential Functions KEY: increasing decreasing 71. ANS: A PTS: 1 DIF: Average OBJ: Section 7.1 NAT: RF9 TOP: Characteristics of Exponential Functions KEY: domain range 72. ANS: A PTS: 1 DIF: Average OBJ: Section 7.1 NAT: RF9 TOP: Characteristics of Exponential Functions KEY: equation graph exponential function 73. ANS: C PTS: 1 DIF: Average OBJ: Section 7.2 NAT: RF9 TOP: Transformations of Exponential Functions KEY: modelling exponential growth 74. ANS: D PTS: 1 DIF: Easy OBJ: Section 7.3 NAT: RF10 TOP: Solving Exponential Equations KEY: compound interest 75. ANS: C PTS: 1 DIF: Easy OBJ: Section 7.2 NAT: RF9 TOP: Transformations of Exponential Functions KEY: transformations of exponential functions 76. ANS: B PTS: 1 DIF: Easy OBJ: Section 7.2 NAT: RF9 TOP: Transformations of Exponential Functions KEY: transformations of exponential functions 77. ANS: A PTS: 1 DIF: Average OBJ: Section 7.2 NAT: RF9 TOP: Transformations of Exponential Functions KEY: graph transformations of exponential functions 78. ANS: A PTS: 1 DIF: Difficult OBJ: Section 7.1 NAT: RF9 TOP: Characteristics of Exponential Functions KEY: modelling exponential function 79. ANS: D PTS: 1 DIF: Average OBJ: Section 7.3 NAT: RF10 TOP: Solving Exponential Equations KEY: exponential equation systematic trial 80. ANS: B PTS: 1 DIF: Average OBJ: Section 7.3 NAT: RF10 TOP: Solving Exponential Equations KEY: exponential equation equate exponents 81. ANS: D PTS: 1 DIF: Difficult OBJ: Section 7.3 NAT: RF10 TOP: Solving Exponential Equations KEY: half-life exponential decay 82. ANS: B PTS: 1 DIF: Easy OBJ: Section 8.1 NAT: RF7 TOP: Understanding Logarithms KEY: logarithm exponential function NOT: Draft 83. ANS: C PTS: 1 DIF: Easy OBJ: Section 8.1 NAT: RF7 TOP: Understanding Logarithms KEY: logarithm exponential function NOT: Draft 5

69 84. ANS: C PTS: 1 DIF: Easy OBJ: Section 8.2 NAT: RF8 TOP: Transformations of Logarithmic Functions KEY: vertical translation transformation 85. ANS: D PTS: 1 DIF: Average OBJ: Section 8.2 NAT: RF8 TOP: Transformations of Logarithmic Functions KEY: horizontal translation vertical translation vertical stretch horizontal stretch 86. ANS: A PTS: 1 DIF: Easy OBJ: Section 8.3 NAT: RF9 TOP: Laws of Logarithms KEY: product law laws of logarithms 87. ANS: B PTS: 1 DIF: Easy OBJ: Section 8.4 NAT: RF10 TOP: Logarithmic and Exponential Equations KEY: exponential equation 88. ANS: B PTS: 1 DIF: Average OBJ: Section 9.1 NAT: RF14 TOP: Exploring Rational Functions Using Transformations KEY: reciprocal of linear function behaviour at non-permissible values 89. ANS: A PTS: 1 DIF: Easy OBJ: Section 9.1 NAT: RF14 TOP: Exploring Rational Functions Using Transformations KEY: reciprocal of linear function x-intercept 90. ANS: B PTS: 1 DIF: Average OBJ: Section 9.1 NAT: RF14 TOP: Exploring Rational Functions Using Transformations KEY: reciprocal of linear function graph from function 91. ANS: A PTS: 1 DIF: Average OBJ: Section 9.1 NAT: RF14 TOP: Exploring Rational Functions Using Transformations KEY: reciprocal of linear function function from graph 6

70 92. ANS: A PTS: 1 DIF: Difficult + OBJ: Section 9.1 NAT: RF14 TOP: Exploring Rational Functions Using Transformations KEY: slant asymptote hole factor 93. ANS: B PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: reciprocal of quadratic function vertical asymptote 94. ANS: A PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: reciprocal of quadratic function vertical asymptote 95. ANS: C PTS: 1 DIF: Difficult OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: rational function discontinuity hole 96. ANS: C PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: reciprocal of quadratic function y-intercept 97. ANS: D PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: linear expressions in numerator and denominator horizontal asymptote 7

71 98. ANS: C PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: linear expressions in numerator and denominator x-intercept 99. ANS: B PTS: 1 DIF: Easy OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: quadratic denominator vertical asymptote 100. ANS: B PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: hole 101. ANS: B PTS: 1 DIF: Average OBJ: Section 9.3 NAT: RF14 TOP: Connecting Graphs and Rational Equations KEY: rational function x-intercept 102. ANS: B PTS: 1 DIF: Difficult + OBJ: Section 9.3 NAT: RF14 TOP: Connecting Graphs and Rational Equations KEY: rational equation graph 103. ANS: D PTS: 1 DIF: Easy OBJ: Section 10.1 NAT: RF1 TOP: Sums and Differences of Functions KEY: add functions subtract functions 104. ANS: B PTS: 1 DIF: Easy OBJ: Section 10.1 NAT: RF1 TOP: Sums and Differences of Functions KEY: subtract functions add functions 105. ANS: A PTS: 1 DIF: Difficult OBJ: Section 10.1 NAT: RF1 TOP: Sums and Differences of Functions KEY: add functions graph subtract functions 106. ANS: C PTS: 1 DIF: Easy OBJ: Section 10.2 NAT: RF1 TOP: Products and Quotients of Functions KEY: multiply functions range 107. ANS: A PTS: 1 DIF: Difficult OBJ: Section 10.3 NAT: RF1 TOP: Composite Functions KEY: composite functions transformations graph 8

72 108. ANS: B PTS: 1 DIF: Difficult OBJ: Section 10.2 NAT: RF1 TOP: Products and Quotients of Functions KEY: multiply functions graph 109. ANS: C PTS: 1 DIF: Average OBJ: Section 10.2 NAT: RF1 TOP: Products and Quotients of Functions KEY: divide functions graph 110. ANS: A PTS: 1 DIF: Average OBJ: Section 10.1 NAT: RF1 TOP: Sums and Differences of Functions KEY: add functions graph 111. ANS: D PTS: 1 DIF: Difficult OBJ: Section 10.2 NAT: RF1 TOP: Products and Quotients of Functions KEY: divide functions graph 112. ANS: C PTS: 1 DIF: Average OBJ: Section 10.2 NAT: RF1 TOP: Composite Functions KEY: composite functions notation 113. ANS: D PTS: 1 DIF: Average OBJ: Section 10.1 NAT: RF1 TOP: Sums and Differences of Functions KEY: add functions 114. ANS: D PTS: 1 DIF: Average OBJ: Section 10.3 NAT: RF1 TOP: Composite Functions KEY: composite functions transformations graph 115. ANS: C PTS: 1 DIF: Average OBJ: Section 10.3 NAT: RF1 TOP: Composite Functions KEY: composite functions transformations graph 116. ANS: B PTS: 1 DIF: Easy OBJ: Section 11.1 NAT: PC2 TOP: Permutations KEY: permutations 117. ANS: D PTS: 1 DIF: Average OBJ: Section 11.1 NAT: PC2 TOP: Permutations KEY: fundamental counting principle 118. ANS: B PTS: 1 DIF: Difficult OBJ: Section 11.2 NAT: PC3 TOP: Combinations KEY: combinations 119. ANS: C PTS: 1 DIF: Average OBJ: Section 11.3 NAT: PC4 TOP: The Binomial Theorem KEY: binomial expansion binomial theorem 120. ANS: A PTS: 1 DIF: Difficult OBJ: Section 11.2 NAT: PC3 TOP: Combinations KEY: combinations fundamental counting principle 9

73 SHORT ANSWER 1. ANS: a) The graph of f(x) = x 2 is shown in blue and the graph of g(x) = ( x 1) is shown in red. b) The graph of f(x) = x is shown in blue and the graph of g(x) = x is shown in red. PTS: 1 DIF: Average OBJ: Section 1.3 NAT: RF4 TOP: Combining Transformations KEY: translation 2. ANS: a) g(x) = ( x 4) = x 2 8x = x 2 8x + 19 b) g(x) = ( x + 2) = x 2 + 4x = x 2 + 4x + 5 c) g(x) = ( x + 7) 2 2 = x x = x x + 47 PTS: 1 DIF: Average OBJ: Section 1.3 NAT: RF4 TOP: Combining Transformations KEY: translation 10

74 3. ANS: a) b) c) PTS: 1 DIF: Difficult + OBJ: Section 1.2 NAT: RF3 RF5 TOP: Reflections and Stretches KEY: graph reflection 4. ANS: a) g( x) = f(x) = ( x 1) 2 2 b) g(x) = f( x) = x + 1 = x + 1 PTS: 1 DIF: Average OBJ: Section 1.3 NAT: RF4 RF5 TOP: Combining Transformations KEY: reflection translation 11

75 5. ANS: a) i) The graph of f(x) = x is shown in blue and the graph of g(x) = 2(2x) is shown in red. ii) The graph of f(x) = x 2 is shown in blue and the graph of g(x) = 2(2x) 2 is shown in red. iii) The graph of f(x) = x is shown in blue and the graph of g(x) = 2 2x is shown in red. b) i) g(x) = 4x (vertical stretch by a factor of 4) ii) g(x) = 8x 2 (vertical stretch by a factor of 8) iii) g(x) = 4 x (vertical stretch by a factor of 4) c) The stretches do not affect the domain or range of any of the functions. PTS: 1 DIF: Average OBJ: Section 1.2 NAT: RF3 TOP: Reflections and Stretches KEY: stretch graph 12

76 6. ANS: a) a reflection in the x-axis, a horizontal compression by a factor of 1, and then a translation of 1 unit to the 2 left and 2 units down b) a vertical stretch by a factor of 2, and then a translation of 3 units to the right and 4 units down c) reflections in the x-axis and the y-axis, a vertical compression by a factor of 1, and then a translation of 5 2 units to the right and 1 unit up PTS: 1 DIF: Difficult + OBJ: Section 1.3 NAT: RF4 RF5 TOP: Combining Transformations KEY: translation stretch reflection 7. ANS: a) a reflection in the x-axis, and then a translation of 2 units to the right and 3 units up b) a vertical stretch by a factor of 2, and then a translation of 1 unit to the left and 1 unit down c) a vertical compression by a factor of 1, a reflection in the x-axis, and then a translation of 2 units to the 2 right and 2 units up PTS: 1 DIF: Average OBJ: Section 1.3 NAT: RF4 TOP: Combining Transformations KEY: graph transformation 13

77 8. ANS: a) i) y = 5 2 x 3 x = 5 2 y 3 x + 3 = 5 2 y 2x + 6 = 5y y = 2 5 x f 1 (x) = 2 5 x ii) The graph of f(x) is shown in blue and the graph of f 1 (x) is shown in red. b) i) y = 3(x 2) 2 3 x = 3(y 2) 2 3 x + 3 = 3(y 2) 2 ± x x = (y 2) 2 = y 2 y = 2 ± x x + 3 f 1 (x) = 2 ± 3 ii) The graph of f(x) is shown in blue and the graph of f 1 (x) is shown in red. 14

78 PTS: 1 DIF: Difficult OBJ: Section 1.4 NAT: RF6 TOP: Inverse of a Relation KEY: inverse of a function graph function notation 9. ANS: Substitute values into the general equation g(x) = a b(x h) + k. a) g(x) = 5 x 6 b) g(x) = 6x 4 c) g(x) = (x 2) + 9 or g(x) = x d) g(x) = 2 3 or 1 x 1 3 g(x) = 2 3 3x PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: transformations 15

79 10. ANS: The graph of y = f(x) is shown in black, and the graph of y = f(x) is shown in blue. PTS: 1 DIF: Difficult OBJ: Section 2.2 NAT: RF13 TOP: Square Root of a Function KEY: graph square root of a function 11. ANS: PTS: 1 DIF: Average OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: graphical solution 12. ANS: No, Jim is not correct. x 2 2 Ê ˆ = 25 has two possible solutions of ±25, and x Á = 25 has only one solution, +25. PTS: 1 DIF: Difficult OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: algebraic solution 16

80 13. ANS: The velocity can be calculated by using the height from which the egg is dropped. The height is 80 ft plus the height of the student, or 85 ft. v = (v 0 ) 2 + 2ad = 0 + 2(32)(85) = 5440 v = Since the speed is less than 80 ft/s, the egg will not crack. The maximum height is limited by a velocity of 80 ft/s, so v = (v 0 ) 2 + 2ad 80 = 0 + 2(32)d 80 = 64d 6400 = 64d 100 = d The maximum height is 100 ft. PTS: 1 DIF: Average OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: algebraic solution 14. ANS: PTS: 1 DIF: Difficult OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: graphical solution 17

81 15. ANS: a) The possible factors are (x ± 1), (x ± 2), and (x ± 3). Try x = 1 using the factor theorem. P(x) = x 3 + 6x x + 6 P( 1) = ( 1) 3 + 6( 1) ( 1) + 6 = P( 1) = 0 Thus, (x + 1) is a factor of P(x). Use synthetic division to find the quadratic factor Thus, x 3 + 6x x + 6 = (x + 1)(x 2 + 5x + 6) = (x + 1)(x + 2)(x + 3) b) 4x 3 11x 2 3x = x(4x 2 11x 3) = x(4x 2 12x + x 3) = x[4x(x 3) + (x 3)] = x(x 3)(4x + 1) c) x 4 81 = (x 2 9)(x 2 + 9) = (x 3)(x + 3)(x 2 + 9) PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factor theorem factor NOT: A variety of factoring techniques is required. 18

82 16. ANS: a) x 2 (x 2)(x + 2) + 3x + 6 = x 2 (x 2)(x + 2) + 3(x + 2) = (x + 2)[x 2 (x 2) + 3] = (x + 2)(x 3 2x 2 + 3) Use the factor theorem on the second factor. Try x = 1. P(x) = x 3 2x P( 1) = ( 1) 3 2( 1) P( 1) = 0 Divide. = The quotient is not factorable. Thus, x 2 (x 2)(x + 2) + 3x + 6 = (x + 2)(x + 1)(x 2 3x + 3) b) 16x 4 (x + 1) 2 = (4x 2 ) 2 (x + 1) 2 = [4x 2 (x + 1)][4x 2 + (x + 1)] = (4x 2 x 1)(4x 2 + x + 1) PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factor theorem factor integral zero theorem grouping rational zero theorem NOT: A variety of factoring techniques is required. 19

83 17. ANS: Possible values of x in the factor theorem are ±1, ± 1, ±2, ±4, and ±8. 2 Try x = 2. P(x) = 2x 3 + 5x 2 14x 8 P(2) = 2(2) 3 + 5(2) 2 14(2) 8 = P(2) = 0 Thus, x 2 is a factor of P(x). Divide Thus, 2x 3 + 5x 2 14x 8 = (x 2)(2x 2 + 9x + 4) = (x 2)(2x 2 + x + 8x + 4) = (x 2)[x(2x + 1) + 4(2x + 1)] = (x 2)(2x + 1)(x + 4) PTS: 1 DIF: Difficult + OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factor theorem factor integral zero theorem grouping rational zero theorem NOT: A variety of factoring techniques is required. 20

84 18. ANS: a) Try x = 1 in the factor theorem. P(x) = 3x 3 + 2x 2 8x + 3 P(1) = 3(1) 3 + 2(1) 2 8(1) + 3 = P(1) = 0 Thus, x 1 is a factor. Use synthetic division to find another factor Another factor is 3x 2 + 5x 3. Thus, 3x 3 + 2x 2 8x + 3 = 0 (x 1)(3x 2 + 5x 3) = 0 x = 1 or 3x 2 + 5x 3 = 0 Use the quadratic formula to find the other solutions. 3x 2 + 5x 3 = 0 x = 5 ± 52 4(3)( 3) 2(3) = 5 ± The solutions are x = 6 b) 2x 3 + x 2 10x 5 = 0, 1, x 2 (2x + 1) 5(2x + 1) = 0 (x 2 5)(2x + 1) = 0 x = 5, 5, 1 2 c) 5x 4 = 7x 2 2 5x 4 7x = 0 5x 4 5x 2 2x = 0 5x 2 (x 2 1) 2(x 2 1) = 0 (x 2 1)(5x 2 2) = 0 x = 1, 1, 2 5,

85 PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: polynomial equation factor theorem factor roots integral zero theorem NOT: A variety of factoring techniques is required. 19. ANS: a) x 4 + 3x 2 28 = 0 (x 2 + 7)(x 2 4) = 0 (x 2 + 7)(x 2)(x + 2) = 0 b) 2x 4 54x = 0 2x(x 3 27) = 0 x = 2, 2 2x(x 3)(x 2 + 3x + 9) = 0 x = 0, 3 PTS: 1 DIF: Average OBJ: Section 3.3 Section 3.4 NAT: RF11 TOP: The Factor Theorem Equations and Graphs of Polynomial Functions KEY: polynomial equation factor theorem factor integral zero theorem rational zero theorem roots NOT: A variety of factoring techniques is required. 22

86 20. ANS: a) x > 1.9 b) x 13 PTS: 1 DIF: Difficult + OBJ: Section 3.4 NAT: RF12 TOP: Equations and Graphs of Polynomial Functions KEY: technology inequality graph roots 21. ANS: a = θr = π 9 (3) a = π 3 The child travels through an arc length of π 3 m. PTS: 1 DIF: Easy OBJ: Section 4.1 NAT: T1 TOP: Angles and Angle Measure KEY: arc length 23

87 22. ANS: Use the trigonometry of right triangles. The hypotenuse is the length of the ladder, or 3 m. The angle between the ladder and the ground is π. The opposite side to the angle is the height the ladder reaches up the wall. Let 3 this height be h. h 3 = sin Ê π ˆ Á 3 Ê h = 3sin π ˆ Á 3 h = The height the ladder reaches up the wall is m. PTS: 1 DIF: Average OBJ: Section 4.3 NAT: T3 TOP: Trigonometric Ratios KEY: special angles trigonometric ratios 23. ANS: Ê sinx = cos π ˆ Á 5 Ê sinx = cos π 2 3π ˆ Á 10 Ê sinx = sin 3π ˆ Á 10 x = 3π 10 PTS: 1 DIF: Average OBJ: Section 4.4 NAT: T4 TOP: Introduction to Trigonometric Equations KEY: equivalent trigonometric expression exact value 24. ANS: Since csc θ = 2 3, sinθ = 3 2. Since sin60 = 3, the reference angle is 60. The ratio is negative in 2 quadrants III and IV.This means that the angle can be found by looking for reflections of 60 that lie in these quadrants. quadrant III: = 240 quadrant IV: = 300 PTS: 1 DIF: Average OBJ: Section 4.3 NAT: T2 TOP: Trigonometric Ratios KEY: reference angle reciprocal trigonometric ratios unit circle 24

88 25. ANS: r = d 2 = 21 2 = 10.5 a = rθ a = (10.5)(1.2) = 12.6 The arc length is 12.6 cm. PTS: 1 DIF: Average OBJ: Section 4.1 NAT: T1 TOP: Angles and Angle Measure KEY: arc length central angle 26. ANS: sin A = 2 2 cos B = A = π B = π 4 6 Ê sec A + sec B = sec π ˆ Á 4 + sec Ê π ˆ Á = = = = = PTS: 1 DIF: Average OBJ: Section 4.3 NAT: T3 TOP: Trigonometric Ratios KEY: exact value reciprocal trigonometric ratios 27. ANS: The tangent ratio is negative in quadrants II and IV. In quadrant II for the domain 0 θ 180, θ = 120. In quadrant IV for the domain 180 θ 0, θ = 60. PTS: 1 DIF: Difficult OBJ: Section 4.3 NAT: T3 TOP: Trigonometric Ratios KEY: primary trigonometric ratios exact value 25

89 28. ANS: 5π 6 PTS: 1 DIF: Easy OBJ: Section 4.1 NAT: T1 TOP: Angles and Angle Measure KEY: radian 29. ANS: You would need to subtract π or 90 from each x-value for y = sin x and plot the points using the 2 corresponding y-values. The zeros of the sine function would become the maximum or minimum values of the cosine function. PTS: 1 DIF: Easy OBJ: Section 5.2 NAT: T4 TOP: Graphing Sine and Cosine Functions KEY: phase shift 30. ANS: a reflection in the x-axis, a vertical stretch by a factor of 2, a horizontal stretch by a factor of 8, a phase shift of π 3 to the right, and a vertical translation of 1 unit up PTS: 1 DIF: Average OBJ: Section 5.2 NAT: T4 TOP: Transformations of Sinusoidal Functions KEY: transformations sinusoidal function 31. ANS: Solutions may vary. Sample solution: The amplitude is half the diameter, or 30 cm. The maximum height of the pebble is 60 cm, so the vertical displacement must be 30 cm. The wheel rotates at 4 revolutions per second, so the period is 1 s. Thus, the 4 value of b is 2π or 8π. 1 4 Thus, the relationship between the height of the pebble above the ground and time is h = 30sin( 8πt) + 30 PTS: 1 DIF: Easy OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: sinusoidal function modelling 26

90 32. ANS: a) From the function, the maximum and minimum populations are or 280 bears and or 220 bears. b) Graph the function p( t) = cos t. The graph is first increasing over the interval [3.14,6.28] or [π, 2π], or from approximately 3 years months to 6 years months. PTS: 1 DIF: Average OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: sinusoidal function maximum minimum 33. ANS: a) 0.7 m b) Since b = 72 and period = 2π 2π, then period = b 72 or π s. The number of revolutions of the rope is the 36 reciprocal of the period, or 36, or rev/s. Multiply by 60 to get 688 revolutions/min. π PTS: 1 DIF: Difficult OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: period amplitude sinusoidal function 27

91 34. ANS: Answers may vary. Sample answer: Use x = 0 and y = π 2 : L.S. = cos(x + y) Ê = cos 0 + π ˆ Á 2 = cos π 2 R.S. = cos x + cos y = cos 0 + cos π 2 = = 1 = 0 L.S. R.S. Thus, cos(x + y) = cos x + cos y is not an identity. PTS: 1 DIF: Average OBJ: Section 6.3 NAT: T6 TOP: Proving Identities KEY: counterexample 35. ANS: 2cos x 3 = 0 2cos x = 3 cos x = 3 2 Ê ˆ 3 x = cos 1 2 Á = π 6 Since cosine is also positive in quadrant IV, another solution is 2π π 6 = 11π 6. PTS: 1 DIF: Easy OBJ: Section 6.4 NAT: T6 TOP: Solving Trigonometric Equations Using Identities KEY: exact value 36. ANS: cot 2 θ + cotθ = 0 cot θ( cotθ + 1) = 0 cotθ = 0 or cotθ = 1 θ = π 2 or θ = 3π 4 Since the period for cot θ is π, the solution in general form is θ = π 2 + nπ and θ = 3π 4 + nπ, where n I. PTS: 1 DIF: Difficult OBJ: Section 6.4 NAT: T6 TOP: Solving Trigonometric Equations Using Identities KEY: exact value general solutions 28

92 37. ANS: sec 2 θ 2tanθ 3 = 0 (1 + tan 2 θ) 2tanθ 3 = 0 tan 2 θ 2tanθ 2 = 0 Use the quadratic formula. tanθ = 2 ± 12 2 = 1 ± or θ = tan 1 (2.732) or tan 1 ( 0.732) 70 or 36 Since the period for tanθ is 180, a positive solution corresponding to 36 is or 144. The general solution is n and n, where n I. PTS: 1 DIF: Difficult OBJ: Section 6.4 NAT: T6 TOP: Solving Trigonometric Equations Using Identities KEY: exact value general solutions quadratic formula 38. ANS: a) i) quadratic ii) exponential iii) linear b) i) successive values would be increasing by a constant amount ii) successive values would be increasing by a constant factor iii) all values would be constant PTS: 1 DIF: Average OBJ: Section 7.1 NAT: RF9 TOP: Characteristics of Exponential Functions KEY: linear quadratic exponential function 29

93 39. ANS: Answers may vary. Ê Sample answer: The graph of y = 3 1 ˆ Á 3 x : PTS: 1 DIF: Average OBJ: Section 7.1 Section 7.2 NAT: RF9 TOP: Characteristics of Exponential Functions Transformations of Exponential Functions KEY: domain range intercepts exponential function 40. ANS: a) a vertical compression by a factor of 1 and a translation of 2 units to the right 2 b) The graph of y = 3 x is shown in blue and the graph of y = 1 2 ( 3) x 2 is shown in red. c) domain {x x R}, range {y y > 0, y R}, y = 0 PTS: 1 DIF: Average OBJ: Section 7.2 NAT: RF9 TOP: Transformations of Exponential Functions KEY: graph transformations of exponential functions 30

94 41. ANS: a) y = 5 x b) y = 5 x 3 c) y = 5 x d) y = 5 x 2 PTS: 1 DIF: Average OBJ: Section 7.2 NAT: RF9 TOP: Transformations of Exponential Functions KEY: equation transformations of exponential functions 42. ANS: a) ii) b) i) c) iii) PTS: 1 DIF: Average OBJ: Section 7.1 NAT: RF9 TOP: Characteristics of Exponential Functions KEY: graph modelling exponential function 43. ANS: 9 n 1 = 1 4n 1 Ê ˆ Á 3 Ê 32 Á ˆ n 1 = Ê 3 1 Á ˆ 4n 1 3 2n 2 = 3 1 4n Equate the exponents: 2n 2 = 1 4n 6n = 3 n = 1 2 PTS: 1 DIF: Average OBJ: Section 7.3 NAT: RF10 TOP: Solving Exponential Equations KEY: change of base 31

95 44. ANS: domain: {x x > 2,x R} range: {y y R} equation of vertical asymptote: x = 2 PTS: 1 DIF: Difficult OBJ: Section 8.2 NAT: RF8 TOP: Transformations of Logarithmic Functions KEY: transformation vertical translation asymptote graph 45. ANS: log 2 14 = log 2 (2 7) = log log = PTS: 1 DIF: Difficult OBJ: Section 8.3 NAT: RF9 TOP: Laws of Logarithms KEY: power law laws of logarithms 46. ANS: 6 3x + 1 = 2 2x 3 log(6 3x + 1 ) = log(2 2x 3 ) (3x + 1) log6 = (2x 3) log2 3x log6 + log6 = 2x log2 3log2 x(3log6 2log2) = 3log2 log6 x = (3log2 + log6) 3log6 2log2 PTS: 1 DIF: Average OBJ: Section 8.3 Section 8.4 NAT: RF9 TOP: Laws of Logarithms Logarithmic and Exponential Equations KEY: exponential equation laws of logarithms 32

96 47. ANS: a) f(x) = b) 1 4x 8 PTS: 1 DIF: Average OBJ: Section 9.1 NAT: RF14 TOP: Exploring Rational Functions Using Transformations KEY: reciprocal of linear function vertical asymptote y-intercept graph 33

97 48. ANS: Ï a) i) x x 5 Ô Ì 4, x R Ô, {y y 0, y R} ÓÔ Ô ii) x-intercept: none, y-intercept: 3 5 iii) x = 5 4, y = 0 b) PTS: 1 DIF: Average OBJ: Section 9.2 Section 9.3 NAT: RF14 TOP: Analysing Rational Functions Connecting Graphs and Rational Equations KEY: reciprocal of linear function key features graph 34

98 49. ANS: a) i) {x x 2, x R}, {y y 3, y R} ii) x-intercept: 8, y-intercept: 4 3 iii) x = 2, y = 3 b) PTS: 1 DIF: Average OBJ: Section 9.2 Section 9.3 NAT: RF14 TOP: Analysing Rational Functions Connecting Graphs and Rational Equations KEY: linear expressions in numerator and denominator key features graph 35

99 50. ANS: a) i) {x x 3,x 4, x R}, {y y 0, y R} ii) x-intercept: none, y-intercept: 1 4 iii) x = 4, y = 0 Ê b) Note the hole at the point 3, 1 ˆ Á 7. PTS: 1 DIF: Difficult OBJ: Section 9.2 Section 9.3 NAT: RF14 TOP: Analysing Rational Functions Connecting Graphs and Rational Equations KEY: rational function key features graph 51. ANS: h( x) = f ( x) + g( x) = x x 2 + 3x + 1 = x 2 + 4x + 2 PTS: 1 DIF: Easy OBJ: Section 10.1 NAT: RF1 TOP: Sums and Differences of Functions KEY: add functions 36

100 52. ANS: h( x) = f ( x) g( x) = x 2 4 x 2 3x + 2 = ( x 2) ( x + 2) ( x 2) ( x 1) = ( x + 2) ( x 1), x 2, x 1 PTS: 1 DIF: Average OBJ: Section 10.2 NAT: RF1 TOP: Products and Quotients of Functions KEY: divide functions restrictions 53. ANS: f(x) is in blue, g(x) is in red, and h(x) is in black. PTS: 1 DIF: Difficult OBJ: Section 10.2 NAT: RF1 TOP: Products and Quotients of Functions KEY: multiply functions graph 37

101 54. ANS: 2x 2 3 = 0 2x 2 = 3 x 2 = 3 2 x = ± x = ± PTS: 1 DIF: Difficult + OBJ: Section 10.2 NAT: RF1 TOP: Products and Quotients of Functions KEY: divide functions vertical asymptotes 55. ANS: g( 2) = 2 ( 2) 3 = 6 f( 6) = ( 6) 2 7 = 29 f(g(2)) = 29 PTS: 1 DIF: Average OBJ: Section 10.3 NAT: RF1 TOP: Composite Functions KEY: composite functions evaluate 56. ANS: The number of different routes can be calculated using the following permutation: 12! 6!6! = 924 Joe can take one of 924 different routes to travel from home to school. PTS: 1 DIF: Average OBJ: Section 11.1 NAT: PC1 TOP: Permutations KEY: permutations 57. ANS: a 2 b 4 4 Ê ˆ Ê aˆ = Á 3 4 C 0 b 0 3 Ê ˆ Ê aˆ + Á 2 Á 3 4 C 1 b 1 2 Ê ˆ Ê aˆ + Á 2 Á 3 4 C 2 b 2 1 Ê ˆ Ê aˆ + Á 2 Á 3 4 C 3 b 3 Ê ˆ Ê aˆ + Á 2 Á 3 4 C 4 Á 2 = 1 a 4 Ê ˆ b 0 Ê ˆ + 4 a 3 Ê ˆ b 1 Ê ˆ + 6 a 2 Ê ˆ b 2 Ê ˆ + 4 a 1 Ê ˆ b 3 Ê ˆ + 1 a 0 Ê ˆ b 4 Ê ˆ Á 2 Á 3 Á 2 Á 3 Á 2 Á 3 Á 2 Á 3 Á 2 Á 3 = a 4 16 a 3 b 6 + a 2 b 2 6 2ab b 4 81 PTS: 1 DIF: Average OBJ: Section 11.3 NAT: PC4 TOP: The Binomial Theorem KEY: binomial expansion binomial theorem 0 Ê b ˆ Á

102 58. ANS: (2n + 2)! (2n 1)!0! = (2n 2)(2n 1)(2n)(2n + 1)(2n + 2) (2n 2)(2n 1) 1 = (2n)(2n + 1)(2n + 2) = 2n(4n 2 + 6n + 2) = 8n n 2 + 4n PTS: 1 DIF: Average OBJ: Section 11.1 NAT: PC1 TOP: Permutations KEY: factorial 59. ANS: There are 14 letters in total. 5 of these are burnt out. ( 9 C 3 )( 5 C 2 ) = = 840 There are 840 ways to select 3 good letters and 2 burnt-out letters. PTS: 1 DIF: Average OBJ: Section 11.1 NAT: PC3 TOP: Combinations KEY: combinations 60. ANS: Assume there are two questions on the test. They could appear in 2! possible ways. This can be carried on for additional questions. For 3 questions, there are 3! = 6 possible orders. For 4 questions, there are 4! = 24 possible orders. For 5 questions, there are 5! = 120 possible orders. For 6 questions, there are 6! = 720 possible orders. There must be 6 questions for everyone to get a test with the questions in a different order. PTS: 1 DIF: Average OBJ: Section 11.1 NAT: PC1 TOP: Permutations KEY: fundamental counting principle 39

103 PROBLEM 1. ANS: a) h(t) = 4.9t b) h(t) = 1.85t c) The Earth function is shown in blue and the Mars function is shown in red. d) The scale factor that can be applied to the Earth function to transform it to the Mars function is = 1.85 PTS: 1 DIF: Difficult OBJ: Section 1.2 NAT: RF3 TOP: Reflections and Stretches KEY: graph stretch function notation 2. ANS: a) g(x) = 1 3 b) 1 2 ( x + 3) 5 c) The reflection, horizontal stretch, and vertical compression must be done first, but can be done in any order. d) The translations to the left and down must be done last, but can be done in any order. PTS: 1 DIF: Difficult + OBJ: Section 1.3 NAT: RF4 TOP: Combining Transformations KEY: graph transformation function notation 40

104 3. ANS: a) C(x) = x b) C = x C 50 = 0.12x x = C f 1 (C) = C c) This function represents the distance the car can be driven for a given rental cost. d) Answers may vary. Sample answer: If you have $65 to spend on the car rental, for how many kilometres can you drive the car? f 1 (65) = 0.12 = 125 You can drive a total of 125 km for the $65 rental fee. PTS: 1 DIF: Average OBJ: Section 1.4 NAT: RF6 TOP: Inverse of a Relation KEY: inverse of a function function notation 4. ANS: a) b) f(x): Domain: {x x 0, x R}; Range: {y y 0, y R} g(x): Domain: {x x 2, x R}; Range: {y y 3, y R} c) The vertical stretch changes from 5 to 2, which means a vertical compression by a factor of 2. There is a 5 horizontal compression by a factor of 1. The graph is reflected in both the x-axis and the y-axis. The graph is 6 translated 2 units left and 3 units down. PTS: 1 DIF: Difficult OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: graph transformations domain range 41

105 5. ANS: a) In both cases, the functions are in the shape of a transformed radical function. b) In both cases, the function is stretched horizontally. The graph created by group 1 has a larger horizontal stretch than the graph created by group 2. Vertical and horizontal translations are applied to both graphs. c) Answers may vary. Sample answer: Group 1: f(x) = 2(x 1) Group 2: g(x) = 3(x 1) d) Answers may vary. Sample answer: Group 1 may have pushed the pendulum or Group 2 may have started the timer early. Students should note that the only difference between the two groups can be accounted for by a horizontal stretch in the function. PTS: 1 DIF: Difficult OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: graph transformations 42

106 6. ANS: a) E = 1 2 mv 2 2E = mv 2 2E m = v 2 2E v = m b) i) Substitute m = 12 and E = 200 in the equation. v = 2E m = 2(200) 12 v 5.8 The speed of the object is 5.8 m/s. ii) Substitute m = 12 and E = 420 in the equation. v = 2E m = 2(420) 12 v 8.4 The speed of the object is 8.4 m/s. c) When m = 12, v = 2E m = v = 2E 12 E 6 43

107 d) The point (20, 2) is on the graph. Substitute this into the equation v = v = 2E m 2E m and solve for m. 2 = 2(20) m 4 = 40 m m = 10 The mass of the object is 10 kg. PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: graph horizontal stretch 44

108 7. ANS: Let P(x) = 2x 4 7x 3 41x 2 53x 21. Test x = 1 in the factor theorem. P(x) = 2x 4 7x 3 41x 2 53x 21 P( 1) = 2( 1) 4 7( 1) 3 41( 1) 2 53( 1) 21 = P( 1) = 0 Thus, x + 1 is a factor. Divide to determine another factor Thus, P(x) = (x + 1)(2x 3 9x 2 32x 21) Now factor the cubic. Test x = 1 in the factor theorem. Let Q(x) = 2x 3 9x 2 32x 21. Q(x) = 2x 3 9x 2 32x 21 Q( 1) = 2( 1) 3 9( 1) 2 32( 1) 21 = Q( 1) = 0 Thus, x + 1 is a factor. Divide to determine another factor Thus, P(x) = (x + 1)(x + 1)(2x 2 11x 21) = (x + 1) 2 (2x 2 14x + 3x 21) = (x + 1) 2 [2x(x 7) + 3(x 7)] = (x + 1) 2 (x 7)(2x + 3) PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factor theorem integral zero theorem factor 45

109 8. ANS: By the factor theorem, x + a is a factor of P(x) if P( a) = 0. P( a) = ( a + a) 4 + ( a + c) 4 (a c) 4 P( a) = 0 = 0 + [ (a c)] 4 (a c) 4 = (a c) 4 (a c) 4 PTS: 1 DIF: Average OBJ: Section 3.3 NAT: RF11 TOP: The Factor Theorem KEY: factor theorem factor 9. ANS: Let x represent the side length of the base. Then, V(x) = x 2 (x + 4). Solve x 2 (x + 4) = 225. Since 25 is a factor of 225, and 25 is a square, try x = 5 as a solution. L.S. = x 2 (x + 4) = 5 2 (5 + 4) = 25(9) = 225 = R.S. Thus, x = 5 is a solution. The dimensions of the box could be 5 cm by 5 cm by 9 cm. Rewrite the equation in the form P(x) = 0. x 2 (x + 4) = 225 x 3 + 4x = 0 Since x = 5 is a solution, x 5 is a factor of P(x). Divide to find another factor x 3 + 4x = 0 (x 5)(x 2 + 9x + 45) = 0 x 2 + 9x + 45 = 0 has no real solutions. So the only solution is x = 5. The dimensions of the box are 5 cm by 5 cm by 9 cm. PTS: 1 DIF: Average OBJ: Section 3.3 Section 3.4 NAT: RF11 TOP: The Factor Theorem Equations and Graphs of Polynomial Functions KEY: factor theorem integral zero theorem polynomial equation root 46

110 10. ANS: Multiply the equation by 1 to clear the negative leading coefficient. x 3 + 5x 2 8x ( x 3 + 5x 2 8x + 4) 0 x 3 5x 2 + 8x 4 0 Factor x 3 5x 2 + 8x 4 using the factor theorem. Let P(x) = x 3 5x 2 + 8x 4. Try x = 1 in the factor theorem. P(x) = x 3 5x 2 + 8x 4 P(1) = (1) 3 5(1) 2 + 8(1) 4 = P(1) = 0 Thus, x 1 is a factor of P(x). Divide to find another factor Thus, x 3 5x 2 + 8x 4 0 (x 1)(x 2 4x + 4) 0 (x 1)(x 2) 2 0 Construct a table. x < 1 1 < x < 2 x > 2 x x 2 + x 2 + (x 1)(x 2)(x 2) + + The expression is equal to zero at x = 1 and x = 2. Thus, the solution is x 1 and x = 2. Use a graphing calculator to graph the corresponding polynomial function, y = x 3 + 5x 2 8x + 4. Then, use the Zero operation. 47

111 The zeros are 1 and 2. From the graph, x 3 + 5x 2 8x when x 1 or x = 2. PTS: 1 DIF: Difficult + OBJ: Section 3.3 Section 3.4 NAT: RF11 RF12 TOP: The Factor Theorem Equations and Graphs of Polynomial Functions KEY: polynomial inequality factor theorem factor integral zero theorem root 11. ANS: The graph has a single zero at x = 0 and a double zero at x = 2. The graph also passes through the point (3, 6). Thus, the graph is of the form y = ax(x 2) 2. Substitute the point (3, 6) to find a. y = ax(x 2) 2 6 = a(3)(3 2) 2 6 = 3a a = 2 Thus, y = 2x(x 2) 2 = 2x(x 2 4x + 4) y = 2x 3 8x 2 + 8x PTS: 1 DIF: Average OBJ: Section 3.4 NAT: RF12 TOP: Equations and Graphs of Polynomial Functions KEY: polynomial equation graph zeros 48

112 12. ANS: Use the cosine law. 1 2 = d 2 + d 2 2d ( d) cos = 2d 2 2d 2 cos 1.25 = d 2 ( 2 2cos 1.25) d = 1 2 2cos 1.25 Recall that s = d. The balls have travelled a distance, d, in metres, in a time, t, of 2 s. t s = 1 2 2cos Therefore, after 2 s, the billiard balls are moving at approximately 0.43 m/s. PTS: 1 DIF: Difficult + OBJ: Section 4.3 Section 4.4 NAT: T3 T5 TOP: Trigonometric Ratios Introduction to Trigonometric Equations KEY: trigonometric ratios radians cosine law 49

113 13. ANS: a) b) Use the sine ratio, since the side opposite the given angle is known and the hypotenuse is needed. c) sin 30 = 2.5 x x = 2.5 sin 30 = = 5 The length of the piece of wood is 5 m. PTS: 1 DIF: Easy OBJ: Section 4.3 Section 4.4 NAT: T3 T5 TOP: Trigonometric Ratios Introduction to Trigonometric Equations KEY: trigonometric ratios special angles trigonometric equations 50

114 14. ANS: a) Since sin 30 = 1, the reference angle is 30. The sine ratio is negative in quadrants III and IV. Look for 2 reflections of the 30 angle in these quadrants. quadrant III: = 210 quadrant IV: = 330 b) Using a calculator, sin 210 = 1 2 and sin 330 = 1 2. PTS: 1 DIF: Average OBJ: Section 4.2 Section 4.3 NAT: T2 T3 TOP: Unit Circle Trigonometric Ratios KEY: sine ratio reference angle unit circle 51

115 15. ANS: <fix tech art so 49 cm arrow goes only to bottom of right-angle triangle, not to black dot> Ê θ = cos 1 49ˆ Á Graph the function θ = 1 4 sin Ê π 2 t ˆ and determine the points at which θ = ± Á Therefore, during the first 4 s the pendulum is displaced 1 cm vertically at approximately 0.6 s and 1.4 s to one side and at approximately 2.6 s and 3.4 s to the other side. PTS: 1 DIF: Difficult OBJ: Section 5.4 NAT: T4 TOP: Equations and Graphs of Trigonometric Functions KEY: linear trigonometric equation 52