Chapter 6 Diagnostic Test

Transcription

1 Chapter 6 Diagnostic Test STUDENT BOOK PAGES Consider the quadratic relation y = x 2 6x + 3. a) Use partial factoring to locate two points with the same y-coordinate on the graph. b) Determine the coordinates of the vertex. c) Sketch the graph. 2. Sketch the graph of each quadratic relation. a) y = (x 3)(x + 1) c) y = 4(x + 2) b) y = 2x x 12 d) y = 3 1 (x + 2)(x 4) 3. Factor each expression, if possible. a) x 2 5x 24 c) 9x 2 9x 10 b) x 2 4x + 8 d) 2x 2 + 7x Expand and simplify. a) (2x 5)(x + 7) c) (2m 3)(2m + 3) b) (3 2q) 2 d) (5c 3)(11 c) 5. For each quadratic relation, determine the zeros, the y-intercept, the equation of the axis of symmetry, the vertex, and the equation in standard form. a) b) 6. For each quadratic relation, determine the equation of the axis of symmetry, the vertex, and the zeros (if any). a) y = 2(x + 4)(x + 8) c) y = 3x(x 4) + 12 b) y = (x + 1) d) y = 3 1 (x 5) Use graphing technology to graph each quadratic relation. Then explain why the relation cannot be written as y = a(x r)(x s), where r and s are integers. a) y = 2x 2 + 6x + 7 b) y = x 2 8x + 5 Chapter 6 Diagnostic Test 411

2 Chapter 6 Diagnostic Test Answers 1. a) y = x(x 6) + 3; (0, 3), (6, 3) c) b) (3, 6) 2. The points that are labelled may vary, e.g., a) c) b) d) 3. a) (x + 3)(x 8) b) cannot be factored c) (3x + 2)(3x 5) d) (2x + 1)(x 4) 4. a) 2x 2 + 9x 35 b) 9 12q + 4q 2 c) 4m 2 9 d) 5c c a) 3, 5; 7.5; x = 1; (1, 8); y = 0.5x 2 x 7.5 b) 3, 4; 12; x = 3.5; (3.5, 0.25); y = x 2 + 7x a) x = 6; ( 6, 8); 4, 8 b) x = 1; ( 1, 9); 2, 4 c) x = 2; (2, 0); 2 d) x = 5; (5, 2); no zeros 7. a) b) There are no x-intercepts, so the relation cannot be factored. The x-intercepts are not integers, so r and s are not integers. If students have difficulty with the questions in the Diagnostic Test, it may be necessary to review the following topics: graphing a quadratic relation given in standard form, factored form, and vertex form using technology to graph a quadratic relation determining the equation of the axis of symmetry, the vertex, and the zeros of a quadratic relation expanding and simplifying expressions factoring quadratic expressions 412 Principles of Mathematics 10: Chapter 6 Diagnostic Test Answers

3 Lesson 6.1 Extra Practice STUDENT BOOK PAGES Use each graph to determine the roots of the quadratic equation, where y = 0. a) 4. Solve by graphing. Round your answers to two decimal places. a) x 2 6x 4 = 0 b) x 2 7x + 5 = 2 c) 4x 2 12x + 15 = 0 d) 32x 4x 2 = 52 b) 2. Determine whether 3 is a root of each equation. a) (x 3)(x + 6) = 0 b) x 2 + 5x + 6 = 0 c) 5x x + 5 = 11 d) 4x 2 = Solve by factoring. a) 2x 2 + x 10 = 0 b) x 2 2x 9 = 6 c) 3x x 15 = 5 d) 3x 2 6x = 0 5. The height, h, in metres, of a football can be modelled by h = 5t t, where t is the time, in seconds, after kickoff. The football lands on the ground without being caught. How long is the football in the air? 6. Tracy runs a small business doing tax returns. If she charges a fee of F dollars per tax return, the amount of money she will earn, E, in dollars, each 1 tax season can be modelled by E = F(480 F). 5 a) How much will Tracy earn if she charges $375 per tax return? b) If Tracy wants to earn at least $9000, between what two amounts should she charge? 7. a) Use algebra to determine the points of intersection of the line y = 4 3x and the parabola y = 3x 2 + 4x 2. b) Verify your answers for part a) using a graphing calculator. Lesson 6.1 Extra Practice 413

4 Lesson 6.1 Extra Practice Answers 1. a) 3, 3 b) 5, 2 2. a) no b) yes c) yes d) no 1 3. a) 2, 2 2 b) 5, 3 4 c), a) $7875 b) between $ and $ a), 2, ( 3, 13) 3 b) d) 0, 2 4. a) 0.61, 6.61 b) 0.46, 6.54 c) 3.95, 0.95 d) 2.27, s 414 Principles of Mathematics 10: Lesson 6.1 Extra Practice Answers

5 Lesson 6.3 Extra Practice STUDENT BOOK PAGES Write each relation in vertex form by completing the square. a) y = x 2 12x b) y = 3x 2 + 6x a) Complete the square to write the relation y = 5x 2 20x 13 in vertex form. b) Graph the relation. 3. Copy and complete this area model. Then express the relation y = x x + 8 in vertex form. 4. Benoît has a home business making fancy gift boxes. If he sells each box for x dollars, his monthly profit, P, in dollars, can be modelled by P = 50x x 520. a) What is the maximum profit that Benoît can make per month? b) How much should he charge to make this profit? 5. Kara wants to make a square and a rectangle with a piece of flexible wire that is 102 cm long. The length of the rectangle will be twice its width. Kara decides to represent the length of wire, measured in centimetres, that she will use for the square as 12x. a) Write expressions for the dimensions of the square and the rectangle in centimetres. b) Show that the combined area of the square and the rectangle, in square centimetres, is 17(x 2 8x + 34). c) Determine the minimum combined area that Kara can make. What are the lengths of wire that she must use for each shape? 6. a) Determine the vertex and the equation of the axis of symmetry of the quadratic relation y = 3x 2 12x + 5 by completing the square. b) How does doubling the coefficient of x in the relation affect the position of the axis of symmetry? 7. a) Continue the following pattern of perfect-square trinomials for three more terms: 100x 2 20x + 1, 100x 2 40x + 4, b) Express 100x 2 140x + 65 as the sum of a perfect-square expression and the square of an integer. Lesson 6.3 Extra Practice 415

6 Lesson 6.3 Extra Practice Answers a) y = (x 6) 2 36 b) y = 3(x + 1) a) y = 5(x + 2) b) 4. a) $92.50 b) $ a) 3x by 3x, (34 4x) by (17 2x) b) Answers may vary, e.g., A = (3x) 2 + (34 4x)(17 2x) A = 9x (34) 34(2x) 17(4x) + 8x 2 A = 17x 2 17(8x) + 17(34) A = 17(x 2 8x + 34) c) 306 cm 2 ; square: 48 cm; rectangle: 54 cm 6. a) (2, 7), x = 2 b) Answers may vary, e.g., after doubling, the axis of symmetry is twice as far from the y-axis. 7. a) 100x 2 60x + 9, 100x 2 80x + 16, 100x 2 100x + 25 b) (10x 7) y = (x + 5.5) Principles of Mathematics 10: Lesson 6.3 Extra Practice Answers

7 Chapter 6 Mid-Chapter Review Extra Practice STUDENT BOOK PAGES Solve each quadratic equation by factoring. a) x 2 13x + 30 = 0 b) x 2 = 6(12 x) c) (x + 1) 2 = 16 d) x(x + 4) = Solve each quadratic equation by factoring or by using a graphing calculator. Round your answers to two decimal places, if necessary. a) 2x 2 + 9x 5 = 0 b) x = 3x c) 3x 2 5x = 7 d) 3(x 2 5) = 4x 3. A rectangular theatre stage, with an area of 84 m 2, is 5 m longer than it is wide. How long is the stage? 4. Determine the value of c that makes each expression a multiple of a perfect-square trinomial. a) x 2 + 8x + c b) 2x x + c c) 5x 2 60x + c d) 7x 2 42x + c 6. Armin charges $50/h to repair computers. If he charges for x hours per month, his monthly profit, P, in dollars, can be modelled by the relation P = x x a) What is Armin s maximum monthly profit? b) How many hours should he work each month to achieve this profit? 7. A suborbital space plane is designed to give space tourists a brief experience of zero-g flight. According to the flight plan, when the engine shuts down at an altitude of m, the space plane gains altitude at a speed of 1250 m/s. Its height, t seconds after engine shutdown, can be modelled by 5t t a) What is the maximum altitude of the space plane? b) The period of freefall lasts from engine shutdown to re-entry. Assuming that the space plane starts to re-enter the atmosphere at the same altitude of m, how long is the space plane in freefall? 5. Complete the square to express each relation in vertex form. Then describe the transformations that must be applied to the graph of y = x 2 to graph the relation. a) y = x x + 29 b) y = 3x 2 18x + 14 Chapter 6 Mid-Chapter Review Extra Practice 417

8 Chapter 6 Mid-Chapter Review Extra Practice Answers 1. a) 3, 10 b) 6, 12 c) 3, 5 d) 2, 6 2. a) 0.5, 5 b) no solutions c) about 0.91, about 2.57 d) about 1.67, 3 6. a) $3950 b) 90 h 7. a) m b) 250 s, or 4 min 10 s m 4. a) 16 b) 18 c) 180 d) a) y = (x + 7) 2 20; translation 7 units left and 20 units down b) y = 3(x 3) 2 13; vertical stretch by a factor of 3, translation 3 units right and 13 units down 418 Principles of Mathematics 10: Chapter 6 Mid-Chapter Review Extra Practice Answers

9 Lesson 6.4 Extra Practice STUDENT BOOK PAGES a) Solve the equation 2x 2 27x 80 = 0 by factoring. b) Solve the same equation using the quadratic formula. c) Identify one advantage of each strategy over the other. 2. Solve each equation using the quadratic formula. a) x 2 16 = 49 b) 3x(x 5) = 5(15 3x) 3. Determine the roots of each equation. Round the roots to two decimal places, if necessary. a) 2x 2 10 = 0 b) 3(x 3) = 0 c) 3(x + 3) 2 75 = 0 d) 0.5(x 0.5) = 0 6. A baseball is bunted so that its height, h, in metres, after t seconds is modelled by h = 4.9t t a) How long is the baseball in the air, to the nearest tenth of a second? b) How long is the baseball above 25 m, to the nearest tenth of a second? 7. A landscape architect is planning a circular flower bed that will be surrounded by a ring of paving stones, 25 cm wide. The area of the ring of paving stones will be half the area of the flower bed. Determine the diameter of the flower bed, to the nearest centimetre. 4. Solve each equation. Round your solutions to two decimal places, if necessary. a) 2x 2 10x 6 = 0 b) (x 1)(2x + 3) = 0 c) 2(x 2 + 1) = 7x d) 2x(2 x) = 1 x 5. Calculate the value of x, to two decimal places. Lesson 6.4 Extra Practice 419

10 Lesson 6.4 Extra Practice Answers 1. a) b) 16, 2.5 c) Answers may vary, e.g., factoring makes it easier to check that the solutions are correct; the quadratic formula can be used directly, without checking all the factor pairs of a) about 8.06, about 8.06 b) 5, 5 3. a) 2.24, 2.24 b) 0.55, 5.45 c) 2, 8 d) 1.91, a) 5.54, 0.54 b) 1, 1.5 c) 0.31, 3.19 d) 0.22, a) 4.8 s b) 1.9 s cm 420 Principles of Mathematics 10: Lesson 6.4 Extra Practice Answers

11 Lesson 6.5 Extra Practice STUDENT BOOK PAGES a) Determine the roots of 2x 2 9x 35 = 0 using the strategy of your choice. b) What do your results for part a) tell you about the graph of y = 2x 2 9x 35? c) Predict whether the discriminant of the equation in part a) is positive, negative, or zero. Justify your prediction. d) Confirm your prediction by calculating the value of D. 2. Calculate the discriminant of each equation, and state the number of real roots. a) x 2 + 5x 3 = 0 b) 4x(x 1) + 1 = 0 c) 2x = 3x d) 3x(x + 5) = x How many times does each relation cross or touch the x-axis? Justify your answer in terms of the graph of the relation. a) y = (x 3) 2 7 b) y = 2x 2 13 c) y = 5(x + 2) 2 d) y = (x + 10) Determine whether the vertex of each parabola lies above, below, or on the x-axis. Explain your answers. a) y = x 2 + 7x + 5 b) y = (x + 2) 2 + 2x + 7 c) y = 4x 2 20x + 25 d) y = 5x(x 3) For what value(s) of k does the relation y = 4x 2 kx + 9 have each number of zeros? a) two zeros b) one zero c) no zeros 7. Suppose that a and c have opposite signs in the quadratic relation y = ax 2 + bx + c. a) What does this tell you about the y-intercept and the direction of opening of the graph of the relation? b) Use sketches, without a grid, to show why the relation must have two roots. c) Give an algebraic reason why the discriminant must be positive when a and c have opposite signs. 4. The support rails of a suspension bridge over a river are modelled by the relation h = x 2, where x is the distance, in metres, from the middle of the bridge and h is the height, in metres, above the river. Explain why, in terms of the bridge and the river, the discriminant of the relation must be negative. Lesson 6.5 Extra Practice 421

12 Lesson 6.5 Extra Practice Answers 1. a) 7, 2.5 b) The parabola crosses the x-axis twice. c) From parts a) and b), the parabola has two roots. The discriminant must be positive. d) D = 361; my prediction is correct. 7. Answers may vary, e.g., a) Either the y-intercept is positive and the graph opens downward, or the y-intercept is negative and the graph opens upward. b) Answers may vary slightly, e.g., 2. a) 37; two real roots c) 31; no real roots b) 0; one real root d) 289; two real roots 3. Answers may vary, e.g., a) 2; the vertex is below the x-axis, and the graph opens upward. b) 0; the vertex is below the x-axis, and the graph opens downward. c) 1; the vertex is on the x-axis. d) 2; the vertex is above the x-axis, and the graph opens downward. 4. Answers may vary, e.g., the bridge is above the river, so the height of the rail is always positive. Therefore, the relation has no zeros. 5. Answers may vary, e.g., a) Above the x-axis; the relation opens downward and, because the discriminant is positive, has two zeros. b) Above the x-axis; the relation opens upward and, because the discriminant is negative, has no zeros. c) On the x-axis; the discriminant is zero, so the relation has one zero. d) Below the x-axis; the relation opens downward and, because the discriminant is negative, has no zeros. 6. a) k < 12 or k > 12 b) k = ±12 c) 12 < k < 12 In each case, the vertex is farther from the x-axis than the y-intercept, so the graph must cross the x-axis twice. c) Since a and c have opposite signs, ac < 0, so 4ac > 0. Therefore, b 2 4ac > Principles of Mathematics 10: Lesson 6.5 Extra Practice Answers

13 Lesson 6.6 Extra Practice STUDENT BOOK PAGES For each relation, explain what each coordinate of the vertex represents and what the zeros represent. a) a relation that models the height, h, of a diver who jumps upward from a board, after time t b) a relation that models the monthly profit, P, earned for a given number of clients per month, x c) a relation that models the area, A, of a rectangle with a fixed perimeter and width w For questions 2 to 7, round your answers to two decimal places, if necessary. 2. The height, h, in metres, of a football is modelled by h = 1 + 5t(4 t), where t is the time, in seconds, after the football is punted. a) What is the maximum height of the football? b) How long is the football in the air? 3. Carly is diving from the 10 m board at her local pool. During her descent underwater, her depth, d, in metres, is modelled by d = 7.5t t, where t is the time, in seconds, after she enters the water. a) What is Carly s maximum depth? b) How long does Carly take to reach this depth? 6. Darya attempts a high jump of 2.22 m. Her centre of mass is initially at a height of 1.0 m, passes 1.8 m after 0.2 s, and passes 1.8 m again after 0.8 s. a) The height of Darya s centre of mass, in metres, is modelled by y = a(t h) 2 + k. Show that h = 0.5, 0.25a + k = 1, and 0.09a + k = 1.8. b) Write the relation that models the height of Darya s centre of mass in vertex form. c) Does Darya clear the attempted height? Explain. 7. Rashid is planning six cubicles for an office. He wants each cubicle to have the same dimensions, and he has chosen an L-shaped layout. He has written the relation P = 9l + 10w for the length of cubicle wall needed. If he has 72 m of cubicle wall to use, what dimensions will maximize the area of each cubicle? 4. The sum of the squares of three consecutive positive odd integers is 515. Determine the integers. 5. Karsten, a tennis pro, is attempting a drop shot so that the ball just clears the net. The ball is 1.3 m above the ground when Karsten hits it. The ball reaches its maximum height, 3.1 m, after it has travelled 2.5 m horizontally. a) Write an equation to model this situation. b) The height of the net is 0.92 m in the middle of the court. Assuming that Karsten s shot is aimed down the centre line, how far away from the net must he be to make the shot? Lesson 6.6 Extra Practice 423

14 Lesson 6.6 Extra Practice Answers 1. a) vertex: maximum height, h, and time, t, when the maximum height is reached; first zero: no meaning; second zero: time when the diver enters the water b) vertex: maximum profit, P, and number of clients per month, x, for the maximum profit; zeros: break-even points c) vertex: maximum area, A, and width, w, for the maximum area; zeros: dimensions of the rectangles with zero length or width 2. a) 21 m b) 4.05 s 3. a) 6.82 m b) 0.95 s 4. 11, 13, a) h = 0.29(x 2.5) b) 5.24 m a) h = = 0.5; 1 = a(0 0.5) 2 + k, so a + k = 1; 1.8 = a( ) 2 + k, so 0.09a + k = 1.8 b) h = 5(t 0.5) c) Yes. The maximum height of Darya s centre of mass is greater than 2.22 m m by 3.6 m 424 Principles of Mathematics 10: Lesson 6.6 Extra Practice Answers

15 Chapter 6 Review Extra Practice STUDENT BOOK PAGES Solve each equation by factoring. Check your solutions with graphing technology. a) x = 7x b) 2x 2 + 9x 35 = 0 c) 17(x + 5) = 5(x 2 + 9) d) 3x(x 6) = 5(2 x) 2. The quadratic relation y = 5x x + c has a minimum value of 0. What is the value of c? Explain. 3. Write each relation in vertex form by completing the square. Then describe the transformations that must be applied to the graph of y = x 2 to graph the relation. a) y = x 2 12x + 50 b) y = 4x x c) y = 0.25x x d) y = 2x x 15 For questions 4 to 8, round your answers to two decimal places, if necessary. 4. A bridge with a parabolic inner arch is being constructed over a canal. The arch will just touch the water on each side of the canal, which is 27.2 m wide. The arch must have a clearance of 6.4 m at a horizontal distance of 9.5 m from either bank of the canal. a) Write a relation, in factored form, to model the height, h, in metres, of the bridge at a horizontal distance of x metres from one bank of the canal. b) Rewrite your relation for part a) in vertex form by expanding and completing the square. c) How high above the water will the arch be, at its highest point, to the nearest hundredth of a metre? 5. Solve each equation using the quadratic formula. a) x 2 7x + 9 = 0 b) 2.5x 2 = 16x c) 2x(x + 3) = 14(1 x) d) 2x(2x 5) = Kaneisha owns a business that produces and sells handmade carriage clocks. She models her monthly revenue, in dollars, as R = 0.2x x, where x is the price, in dollars, of each clock. She models her monthly costs, in dollars, as C = x. She calculates her monthly profit, in dollars, as P = R C. Between what prices should Kaneisha charge to break even? 7. The height of a football, h, in metres, can be modelled by h = 5t t + 1.5, where t is the time, in seconds, after it is punted. a) Write an equation for the time(s) at which the football is c metres high. Determine an expression for the discriminant, D, of this equation in terms of c. b) Use your expression for the discriminant to determine the maximum height of the football. Explain your reasoning. c) How many realistic solutions does your equation from part a) have for each case below? Explain your reasoning. i) 1.50 c < ii) 0 c < 1.50 iii) c < 0 8. The hourly cost, C, in dollars, of running a printing press is modelled as C = x x + 965, where x is the number of pages that are printed per hour. a) How many pages should be printed each hour to minimize the cost? b) What rates of printing will keep the cost below $800? Chapter 6 Review Extra Practice Answers 425

16 Chapter 6 Review Extra Practice Answers 1. a) 3, 4 b) 7, ; answers may vary, e.g., only a perfect-square trinomial could have a minimum value of zero because it has only one root, so 5x x + c = 5(x 2 + 8x + 16) = 5x x a) y = (x 6) ; translation 6 units right and 14 units up b) y = 4(x + 6) 2 ; vertical stretch by a factor of 4, translation 6 units left c) y = 0.25(x 7) ; reflection in the x-axis, vertical compression by a factor of 0.25, translation 7 units right and 19 units up d) y = 2(x 2.5) 2 2.5; reflection in the x-axis, vertical stretch by a factor of 2, translation 2.5 units right and 2.5 units down 4. a) h = 0.04x(x 27.2) b) h = 0.04(x 13.6) c) 7.40 m c) 1.6, 5 5. a) about 1.70, about 5.30 b) 7, 0.6 c) about 0.66, about d) about 3.55, about between $97.87 and $ d) 2, a) 5t t c = 0; D = c b) m; answers may vary, e.g., there is exactly one solution for the time to reach maximum height, and the equation has exactly one solution for c = c) i) D > 0, so there are two solutions; both solutions are realistic, since the height is between the starting value of 1.5 m and the maximum height. ii) D > 0, so there are two solutions; only one solution is realistic, since the height is less than the starting value of 1.5 m. iii) D > 0, so there are two solutions; neither solution is realistic, since the height cannot be negative. 8. a) 6000 b) between 1000 and pages per hour 426 Principles of Mathematics 10: Chapter 6 Review Extra Practice Answers

17 Chapters 4 6 Cumulative Review Extra Practice STUDENT BOOK PAGES What is the factored form of 12a 2 7a 10? A. (6a 2)(2a + 5) B. (4a 5)(3a + 2) C. (4a + 2)(3a 5) D. (4a + 2)(3a 2) 2. What is the factored form of 16c 2 56c + 49? A. (4c 7) 2 B. (4c + 7)(4c 7) C. (4c + 7) 2 D. (8c 7)(2c 7) 3. What is the factored form of 81x 2 y 2 49? A. (9xy 7) 2 B. (9x 7y)(9x + 7y) C. (9xy + 7) 2 D. (9xy 7)(9xy + 7) 4. The following transformations were applied to the graph of y = x 2 : a reflection in the x-axis, a vertical compression by a factor of 2 1, and a translation 3 units right and 7 units down. What is the equation of the transformed graph? A. y = 2 1 (x 3) B. y = (x 7) C. y = 2(x 3) D. y = (x 3) The vertex of a parabola is ( 4, 5), and the parabola passes through ( 6, 3). What is the equation of the parabola? A. y = 11(x + 4) B. y = 2(x + 4) C. y = 2(x 4) D. y = 11(x 5) What is the vertex form of the quadratic relation y = 4x(x 5) 10? A. y = 4(x + 2.5) B. y = 4(x 2.5) 2 15 C. y = 4(x 2.5) 2 35 D. y = 4(x 2.5) Which values of x are solutions to the equation 8x 2 2x 3 = 0? 1 3 A. x =, x = B. x =, x = C. x =, x = D. x =, x = Which expression is not a perfect-square trinomial? A. 9x 2 6x + 1 B. x x + 25 C. x 2 x + 1 D. 4x 2 12x What is the minimum value of the quadratic equation y = 2x x + 17? A. 81 B. 81 C. 32 D Which equation is equivalent to x 2 + 8x + 15? A. y = (x + 8) 2 1 B. y = (x + 4) 2 1 C. y = (x + 4) D. y = (x + 2) Which value of b will make the quadratic equation 6x 2 + bx + 5 have no real solutions? A. 11 B. 17 C. 4 D. 12 Chapters 4 6 Cumulative Review Extra Practice 427

18 Chapters 4 6 Cumulative Review Extra Practice Answers 1. B 2. A 3. D 4. D 5. B 6. C 7. B 8. C 9. A 10. B 11. C 428 Principles of Mathematics 10: Chapters 4 6 Cumulative Review Extra Practice Answers