Factor Quadratic Expressions

Transcription

1 Factor Quadratic Expressions BLM 6... BLM 6 Factor Quadratic Expressions

2 Get Ready BLM 6... Graph Quadratic Relations of the Form y = a(x h) + k. Sketch each parabola. Label the vertex, the axis of symmetry, and the y-intercept. Use a graphing calculator to check your results. a) y = (x 3) b) y = (x ) 3 c) y = 3(x + ) d) y = (x + 3) + 4. Sketch each parabola. Label the vertex, the axis of symmetry, and the y-intercept. Use a graphing calculator to check your results. a) y = ( x+ 4) + 5 b) y = ( x+ ) 3 3 c) y = 0.75(x 3) + d) y = 0.4(x ) 0.6 Square Roots 3. Find the square roots of each number, where possible. Round to the nearest tenth, if necessary. a) 5 b) 49 c) 85 d) Use the order of operations to evaluate each expression. a) ± b) ± 0 36 c) ± 8 4(3)( 3) d) ± 7 4()(3) Factor Quadratic Expressions 5. Factor. a) x 3x 8 b) 4x c) 9x 30x 4 d) 5x + 70xy + 49y e) 8x 9x f) x + 6x + 56 g) 5x + 70x 5 h) 5x 6. Factor, if possible. a) x + 7x + 3 b) 6t 7t 3 c) y 7y + 5 d) 0x x e) 3z 3z 4 f) 4x 9 g) 4x x + 9 h) w + 9w + 0 Translate From Words to Algebra 7. Translate each phrase into an algebraic expression. a) three more than five times a number b) the difference between x and y c) the product of one number and one more than the same number d) the average of x and y e) the sum of three consecutive even numbers 8. Write an equation to represent each sentence, using two unknowns. a) A number is twice a second number. b) The sum of Ray s age and Toni s age is 54. c) The price of a hamburger is $3 more than half the price of a hot dog. d) The width of a rectangle is 5 units less than the length. BLM 6 Get Ready

3 Section 6. Practice Master BLM Use algebra tiles to rewrite each relation in the form y = (x h) + k by completing the square. a) y = x + 4x + 5 b) y = x 0x + 7 c) y = x + x + 6. Determine the value of c that makes each expression a perfect square. a) x + 8x + c b) x + x + c c) x 0x + c d) x 30x + c 3. Match each graph with the appropriate equation. a) y = (x 3) + b) y = (x + ) + 4 c) y = (x ) + 5 d) y = (x + 3) 4. Rewrite each relation in the form y = a(x h) + k by completing the square. Then, sketch a graph of the relation, labelling the vertex, the axis of symmetry, and two other points on the graph. a) y = x x + b) y = x + x + 4 c) y = x + 3x+ 5 d) y =.5x + 6x 5 5. Find the maximum or minimum point for each parabola by completing the square. a) y = x + 4x 4 b) y = x + x + 7 c) y = x 4x 7 d) y = 3x 30x The path of a golf ball can be modelled by the equation h = d + d 3, where d represents the horizontal distance, in metres, that the ball travels and h represents the height of the ball, in metres, above the ground. What is the maximum height of the golf ball and at what horizontal distance does it occur? 7. Angie sold 00 tickets for the holiday concert at $0 per ticket. Her committee is planning to increase the prices this year. Their research shows that for each $ increase in the price of a ticket, 60 fewer tickets will be sold. a) Determine the revenue relation that describes the ticket sales. b) What should the selling price per ticket be to maximize revenue? c) How many tickets will be sold at the maximum revenue? d) What is the maximum revenue? BLM 6 4 Section 6. Practice Master

4 Section 6. Practice Master BLM Solve for x. a) (x + 6)(x + ) = 0 b) (x 4)(x + 7) = 0 c) x(x 9) = 0 d) (3x + )(x ) = 0 e) (5x + 6)(4x + 3) = 0 f) (3x )(0x 3) = 0. Solve and check. a) x + 4x = 0 b) x + 5x + 6 = 0 c) 6x + x = 0 d) 5x 9x 4 = 0 e) 3x + x = 0 f) 9x 6 = 0 3. Solve. a) x + 5x = 4 b) 9c = 49 c) 4a + a = 9 d) 6x = 3x e) x + 5 = x f) x 5x = 4. A rectangle has dimensions x + and x + 5, both measured in centimetres. Determine the value of x so that the area is 7 cm. 5. The area of the rectangle shown in the diagram is 36 cm. What are its dimensions? 6. Write a quadratic equation in factored form, using integers, for each situation. a) The roots of the equation are 3 and 4. b) The roots of the equation are and Write a quadratic equation in the form x + bx + c = 0 with roots and Write a quadratic equation in the form ax + bx + c = 0, where a, b, and c are integers and the roots are 5 and A picture that measures 0 cm by 5 cm is to be surrounded by a mat. The mat is to be the same width on all sides of the picture. The area of the mat is to be twice the area of the picture. What is the width of the mat? nn ( 3) 0. A regular polygon with n sides has diagonals. Find the number of sides of a regular polygon that has 44 diagonals.. Solve. a) x + 9x 5 = 0 b) x x 3= 0 4 c) x + x + 0 = d) x x = 9 3 e) x + 7x = 0 4 f) x x = Solve. z a) = x b) = 0 3y + 7 c) = 5 5n d) = 3 4 BLM 6 5 Section 6. Practice Master

5 Section 6.3 Practice Master BLM (page ). Factor to find the x-intercepts. a) y = x + 5x + 6 b) y = 4x + x + 9 c) y = x 8x d) y = x 4x e) y = 6x 7x 4 f) y = x 0x b). Find the x-intercepts and the vertex for each quadratic relation. Sketch a graph for each relation. a) y = x 6 b) y = x 6x + 7 c) y = 3x + x 5 d) y = 5 x 3. Find the zeros and the vertex of each quadratic relation, and then sketch its graph. Check your results using a graphing calculator. a) y = 4x 6x 9 b) y = 5x 3x + c) y = 4x 35x 4 c) 4. Write an equation in the form y = ax + bx + c to represent each parabola. a) BLM 6 7 Section 6.3 Practice Master

6 d) BLM (page ) 7. A roadway on a bridge is supported by two towers with cables that join them as pictured below. 5. A parabola has a vertex of (, 8) and one x-intercept is 3. a) Find the equation of the parabola in the form y = a(x h) + k. b) Find the other x-intercept. c) Find the y-intercept. 6. The path of a soccer ball can be defined by the relation h = 0.05d + d, where h represents the height, in metres, and d represents the horizontal distance, in metres, that the ball travels before it hits the ground. a) Find the d-intercepts. b) Sketch a graph of the relation. c) For what values of d is the relation invalid? Explain. d) What is the maximum height? e) How far will the ball have travelled horizontally at its maximum height? The cables hang in a parabolic shape that can be represented by the equation y = x + 4. a) Use a graphing calculator to graph the relation. b) Identify the minimum or maximum value and the coordinates of the vertex. c) Write an equation for the axis of symmetry. d) Identify the y-intercept and the x-intercepts. BLM 6 7 Section 6.3 Practice Master

7 Section 6.4 Practice Master BLM Use the quadratic formula to solve each equation. Express answers as exact roots. a) x + 7x + 5 = 0 b) y 6y + 3 = 0 c) x 3x + = 0 d) x 5x = 0 e) y + y 5 = 0. Solve using the quadratic formula. Express answers as exact roots and as approximate roots, to the nearest hundredth. a) x + 5x + = 0 b) x 3x = 0 c) x x 3 = 0 d) x + 7x + = 0 e) x 5x = 0 f) x x 4 = 0 g) 0 = x + x 7 h) x x = 5 3. Find the x-intercepts, to the nearest hundredth; the vertex; and the equation of the axis of symmetry of each quadratic relation. Sketch the parabola that each relation defines. a) y = 3x + 6x + 4 b) y = x + 4x For each quadratic relation, state the coordinates of the vertex, the direction of opening, and the number of x-intercepts. a) y = (x ) + 3 b) y = (x + 5) Sketch a graph representing a quadratic relation with each condition. a) two x-intercepts b) one x-intercept c) no x-intercepts 6. Solve each quadratic equation. Leave your answers as exact roots, if necessary. a) c 6c = 7 b) x(x 3) = 7 c) 3x(x 4) = (x + ) d) 3y (5y + )(y 3) = 3 7. The hypotenuse of a right triangle measures 0 cm. The sum of the lengths of the legs is 8 cm. Find the length of each leg. 8. A rectangular skating rink measures 30 m by 0 m. It is doubled in area by extending each side of the rink by the same amount. Determine by how much each side was extended. 9. a) Find the width, w, in metres, of the Canadian flag on the Peace Tower in Ottawa by solving the equation 8w + 8w 8 = 0. b) The height of the Peace Tower is 90 m. If an object is thrown downward from this height at 5 m/s, the approximate time, t, in seconds, the object takes to reach the ground can be found by solving the equation 5t 5t + 90 = 0. Find the time taken, to the nearest tenth of a second. 0. A rectangular piece of tin 50 cm by 40 cm is made into a lidless box of base area 875 cm by cutting squares of equal sizes from the corners and bending up the sides. a) Find the side length of each removed square. b) Find the volume of the box.. Sipapu Natural Bridge is in Utah. Find the horizontal distance, x, in metres, across this natural arch at the base by solving the equation 0.04x + 3.8x = 0. BLM 6 8 Section 6.4 Practice Master

8 Section 6.5 Practice Master BLM A model rocket is launched from the deck in Jim s backyard and the path followed by the rocket can be modelled by the relation h = 5t + 00t + 5, where h, in metres, is the height that the model rocket reaches after t seconds. a) What is the height of the deck? b) What is the height of the model rocket after s? c) What is the maximum height reached by the model rocket? d) How long did the model rocket take to reach this height? e) How long was the model rocket above 00 m, to the nearest second? f) Estimate how long the model rocket was in the air.. A harbour ferry service has about riders per day for a fare of $. The port authority wants to increase the fare to help with increasing operational costs. Research has shown that for every $0.0 increase in the fare the number of riders will drop by The port authority established a relation defined by R = 000p p , where R represents the revenue from fares and p represents the number of $0.0 increases in the fare. a) What increase in the fare will maximize the revenue? b) What is the new fare? c) What is the revenue that will be received from the new fare? 3. A rectangular lawn measures 30 m by 40 m. Jason is cutting the lawn from the outside perimeter in toward the centre by cutting strips along the entire perimeter first, then continuing as he cuts toward the centre. How wide is the strip that has been cut along the outside when the area is half cut? 4. The hypotenuse of a right triangle measures 3 cm. The legs of the triangle differ by 7 cm. Find the length of each leg. 5. A triangle has an area of 308 cm. If the base is cm more than three times the height of the triangle, find the base and height of the triangle. 6. The sum of the squares of four consecutive integers is 630. Find the integers. 7. Twice the width of a rectangle is 3 m more than the length. If the area of the rectangle is 09 m, find the dimensions of the rectangle. 8. The playing field at the local high school measures 40 m by 50 m. By increasing this rectangular area by the same amount on all sides, the new area will be exactly double the area of the field. By how much was each dimension increased, to the nearest metre? BLM 6 0 Section 6.5 Practice Master

9 Chapter 6 Review BLM 6... (page ) 6. Minima and Maxima. Rewrite each relation in the form y = a(x h) + k by completing the square. Use algebra tiles or a diagram to support your solution. a) y = x + 6x + 3 b) y = x + 4x c) y = x + 8x + 7 d) y = x + 0x 5. Find the vertex of each quadratic relation. Sketch a graph of the relation, labelling the vertex, the axis of symmetry, and two other points. a) y = x + 4x 7 b) y = x + 6x + c) y = x + x + 4 d) y = x + 6x Use a graphing calculator to find the minimum or maximum value for each quadratic relation. Round your answer to the nearest tenth, if necessary. a) y = x + 3x + 5 b) y = 0.3x + 0.9x + 9 c) y = x x d) y = x + 8x The path of a basketball can be modelled by the equation h = 0.06d + 0.6d + 3, where h represents the height, in metres, of the ball above the ground and d represents the horizontal distance, in metres, that the ball travels. a) What is the maximum height reached by the ball? b) What horizontal distance has the ball travelled when it reaches this maximum height? 6. Solve Quadratic Equations 5. Solve by factoring. Check your solutions. a) x + x 5 = 0 b) m 3m + 36 = 0 c) 4y 8y 5 = 0 d) 5c 8c = 0 6. Solve. a) y + y = 8 b) 5x = 8x 3 c) 4z = d) 0m 40m = 0 e) 8x 40 = x f) 8x + 39x = 5 7. Write a quadratic equation in the form ax + bx + c = 0, where a, b, and c are integers, given the following roots. a) 5 and 3 b) and Graph Quadratics Using the x-intercepts 8. Find the x-intercepts and the vertex of each quadratic relation. Sketch each graph. a) y = x + 6x + 9 b) y = 5x 9 c) y = x + 4x + d) y = x + x + 3 e) y = x + 4x + 48 f) y = 0x 5 BLM 6 Chapter 6 Review

10 9. Write an equation in the form y = ax + bx + c to represent each parabola. a) d) BLM 6... (page ) b) 0. A parabola has a vertex ( 3, 4) and one x-intercept is. Find the other x-intercept and the y-intercept. 6.4 The Quadratic Formula. Use the quadratic formula to solve each equation. Express your answers as exact results. a) x + 5x + = 0 b) 3x + x = 0 c) x 6x + 4 = 0 d) 5x 3x 4 = 0 e) x + 3x 7 = 0 f) 3x x = 0 g) x + x 5 = 0 h) 0 = 3x + 3x + c). For each quadratic relation, state the coordinates of the vertex and the direction of opening and determine the number of x-intercepts. a) y = 3(x + ) + b) y = ( x+ ) + 3 c) ( 3) y = x 3 d) y = 3(x + 4) BLM 6 Chapter 6 Review

11 3. A toy rocket is launched from a platform that is m off the ground at an initial velocity of 7.4 m/s. The height, h, in metres, of the rocket t seconds after it is launched is given by the equation h = 4.9t + 7.4t +. a) How long will it take the toy rocket to reach a height of 9 m, to the nearest tenth of a second? b) When will the toy rocket fall back to the height of 9 m, to the nearest tenth of a second? c) Using your answers from parts a) and b), find the time when the rocket will reach its maximum height and determine this maximum height. Round to the nearest tenth. 6.5 Solve Problems Using Quadratic Equations 4. If the product of two consecutive even numbers is 8648, what are the two numbers? 5. A garden against the wall of a house is to be surrounded on three sides by a total of 336 m of fencing. What dimensions of the garden will result in an area of 4 m? 6. If part of a photograph is used to fill an available space in a book or magazine, the photograph is said to be cropped. A photograph that was originally 5 cm by 0 cm is cropped by removing the same width from the top and the left side. Cropping reduces the area of the photograph by 46 cm. What are the dimensions of the cropped photograph? BLM 6... (page 3) 7. A set of p non-collinear points (points not in a straight line) can be connected by a p p maximum of line segments. a) Find the number of non-collinear points that can be connected by a maximum of 55 line segments. b) Is it possible for a set of non-collinear points to be connected by a maximum of 40 line segments? 8. The acceleration due to gravity on Earth is 9.8 m/s. A tennis ball is hit into the air at an initial velocity of 5 m/s from a height of 0.7 m above the ground. a) Write an equation for the height, h, in metres, of the tennis ball in terms of the time, t, in seconds, it has been in the air. b) Find the height of the tennis ball.5 s after it was hit, to the nearest tenth of a metre. c) Find the maximum height of the tennis ball and when it occurs. Round to the nearest tenth. 9. Need-a-Ride is a car rental agency that rents 400 cars a week at $80 per car. Industry research has shown that for every $ increase in rental price, an agency will rent eight fewer cars. a) Total revenue is the product of the price per rental and the number of vehicles rented. Write an expression to represent the revenue for the rental agency. b) Find the maximum revenue. c) For this revenue, how many cars are rented and how much is the rental price per car? BLM 6 Chapter 6 Review

12 Chapter 6 Practice Test BLM (page ). Graph each quadratic relation by completing the square. Label the vertex, the axis of symmetry, and two other points. a) y = x 8x + b) y = x 4x c) y = x 6x+. Explain the process for finding the vertex of a quadratic relation by completing the square. Include an example in your explanation. 3. Solve each quadratic relation by factoring. a) x + x 6 = 0 b) 4x = 0 c) 9b + b + 4 = 0 d) x = 3x e) x 63 = x f) 6m m 0 = 0 g) 6x x 45 = 0 h) 6d = 5 4. Find the x-intercepts, the axis of symmetry, and the vertex of each quadratic relation. Then, graph each relation, labelling fully. a) y = 4x 8x 5 b) y = x + x + 4 c) y = x + 4x 5. Write an equation in the form y = ax + bx + c to represent each relation. a) b) 6. Use the quadratic formula to solve, if possible. Leave your answers as exact roots, if necessary. a) 3x x + = 0 b) x + 7x + 5 = 0 c) 3x 6x 5 = 0 d) 4x + 5x = 0 e) 3t 7 = t f) 5y + 6y 7 = 0 7. Use an appropriate method to find the exact roots of each equation, if possible. a) 6x + x = 0 b) 3x 5x + 5 = 0 c) 4x 6 = 0 d) x 3x + 5 = 0 e) x + 5x 3 = 0 f) 7x 9x + = 0 8. Jennifer is a high jumper. Her path can be modelled by the function h =.5d +.5, where h is her jump height, in metres, and d represents where she started and ended her jump, in metres, with d. a) Graph her jump path. b) What is her maximum jump height? c) Write a relation for a jumper who reached a maximum height of 3 m, but started and ended the jump at the same points as Jennifer. BLM 6 3 Chapter 6 Practice Test

13 9. Two pillars support the archway entrance of a castle. The archway can be modelled by the relation h = 0.05d + d, where h represents the height, in feet, of the arch above the top of the pillars and d represents the horizontal distance, in feet, between the pillars. a) How far apart are the pillars? b) What is the maximum height of the arch above the pillars? c) If the pillars are 5 ft tall, what is the height of the top of the arch above the ground? 0. The cost, C, in thousands of dollars, to produce items at a clothing manufacturer is given by the relation C = x 9x + 00, where x represents the number of items produced, in hundreds. The revenue, R, in thousands of dollars, that these items generate is given by the relation R = x 0x + 50, where x represents the number of items sold, in hundreds. a) Profit is defined as the difference between the revenue and the cost. Using P = R C, develop a profit relation for the company. b) Determine the zeros of the profit relation. c) How many items should be produced to maximize profit? d) What will the maximum profit be? BLM (page ). Louise kicked a football that followed a path given by h = 4.9t + 6t + 0.5, where h represents the height, in metres, above the ground and t represents the time, in seconds, after she kicked the ball. a) Find the zeros of the relation, to the nearest hundredth, using the quadratic formula. Interpret their meaning. b) How long after the ball was kicked did it reach its maximum height? c) What was the maximum height?. A rectangle s length is 4 m more than double its width. Find the length and the width if the diagonal of the rectangle measures 6 m. 3. The area of the front cover of a book is 73 cm and the length is 8 cm greater than the width. What are the dimensions of the cover? 4. A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform width. The combined area of the lawn and the flower bed is 65 m. What is the width of the flower bed? 5. The height of a triangle is cm more than the base. The area of the triangle is 0 cm. Find the base, to the nearest hundredth of a centimetre. 6. The size of a television screen or a computer monitor is usually stated as the length of the diagonal. A screen has a 38-cm diagonal. The width of the screen is 6 cm more than the height. Find the dimensions of the screen, to the nearest tenth of a centimetre. BLM 6 3 Chapter 6 Practice Test

14 Chapter 6 Test BLM (page ). Graph each quadratic relation by completing the square. Label the vertex and the axis of symmetry. a) y = x + 8x 5 b) y = 3x x + 8 c) y = 5x + 0x d) y = x + 6x Write an equation in the form y = ax + bx + c to represent each relation. a). Find an equation for the axis of symmetry for each quadratic relation. a) y = x x b) y = x x + 3 c) y = x x 3. Compare your answers in question. 4. Solve each quadratic equation by factoring. a) x + x 5 = 0 b) x + 5x 3 = 0 c) 6x + 3x 5 = 0 d) x + 90x 4 = 0 5. Solve each quadratic equation, if possible, writing your solutions as exact values and to two decimal places. Check your answers using a graphing calculator. a) x 7x + 4 = 0 b) 3x + 5x = 0 c) x 3x = 7 d) 5x 6 = x e) x + x = f) 0 = 8x + 9x 4 g) x = 5x + 4 h) 3 = 7x 5x b) 7. A right triangle has a hypotenuse of 58 cm, with legs that differ by cm. Find the length of each leg. BLM 6 4 Chapter 6 Test

15 8. Write a quadratic equation in the form ax + bx + c = 0, where a, b, and c are integers, for each situation. a) The roots of the equation are 6 and 3 4. b) The roots of the equation are 3 and. 9. How many x-intercepts does each relation have? Explain your answer. a) y = 3(x ) 4 b) y = (x + 4) c) y = (x + ) + 3 d) ( 4) y = x+ 0. Use an appropriate method to find the roots of each equation. Round your answers to the nearest tenth. a) 3x + x = 0 b) x + x + 4 = 0 c) 5x 6x 5 = 0 d) x + x + 7 = 0. The Ambassador Bridge in Windsor, Ontario, uses suspension cables that cause the roadway to have a parabola shape. The road surface spans a length of 800 m and has a maximum height of 46 m above the road surface at each end. a) Sketch a graph of the relation that can be used to describe the roadway, using the y-axis as the axis of symmetry and placing the highest point of the roadway at the origin. b) Determine an equation for the quadratic relation that describes the road surface. c) If the road surface is to pass each of two support pillars at a height of 30 m, how far from each end should the supports be placed, to the nearest metre? BLM (page ). The height, h, in metres, of a football is given by the relation h = 0.0d + d, where d represents the horizontal distance, in metres, that the ball travels. a) How far does the ball travel before it hits the ground? b) At what horizontal distance is the ball at its maximum height? c) What is the maximum height? d) If the kick is to go through the goal posts for a field goal, the ball needs to be at a height of at least 3 m when it reaches the goal posts. Will the kick be successful if the kicker is 4 m away? e) What is the maximum distance away from the goal posts the kicker can be and still have a successful kick, to the nearest tenth of a metre? 3. Each figure is made using toothpicks. a) An expression for the number of toothpicks in terms of the figure number, n, is n(n + 3). If the pattern continues, what is the figure number of the figure with 70 toothpicks? b) Can there be a figure in the pattern with 380 toothpicks? c) An expression for the number of small squares in terms of the figure number is nn ( ). If the pattern continues, what is the figure number of the figure with 630 squares? d) Can there be a figure in the pattern with 70 squares? BLM 6 4 Chapter 6 Test

16 BLM Answers BLM (page ) Get Ready. a) d) b). a) c) b) Chapter 6 Practice Masters Answers

17 c) d) BLM (page ) 8. a) Let x represent the first number and y represent the second number: x = y. b) Let r represent Ray s age and t represent Toni s age: r + t = 54. c) Let h represent the price of a hamburger and d represent the price of a hot dog: h = d + 3. d) Let w represent the width and l represent the length: w = l 5. Section 6. Practice Master. a) y = (x + ) + b) y = (x 5) 8 c) y = (x + ) + 5. a) 6 b) 36 c) 00 d) 5 3. a) D b) B c) A d) C 4. Labelled points may vary. Examples are shown on the graphs. a) y = (x 3) a) 5, 5 b) 7, 7 c) not possible d) 6.3, a) ±9 b) ±8 c) ±0 d) ±5 5. a) (x + 3)(x 6) b) (x )(x + ) c) 3(3x + )(x 4) d) (5x + 7y) e) (6x + )(3x ) f) (x + 4)(x 7) g) 5(x 9)(x 5) h) (5x )(5x + ) 6. a) (x + 3)(x + ) b) (t 3)(3t + ) c) (y )(y 5) d) (x )(5x + ) e) not possible f) (x 3)(x + 3) g) (x 3) h) (w + 5)(w + ) 7. a) Let x represent the number: 5x + 3. b) x y c) Let x represent the number: x(x + ). x + y d) e) Let x represent the first even number: x + x + + x + 4. b) y = (x ) + 5 Chapter 6 Practice Masters Answers

18 c) ( 3) x y = + BLM (page 3) 5. 9 cm by 4 cm 6. Answers may vary. For example: a) (x 3)(x + 4) = 0 b) (x )(x + 5) = 0 7. x + x 8 = 0 8. Answers may vary. For example: 5x + 7x = cm 0.. a), b), 6 c) 0, 7 d) 3, 6 e), 0 f), 3. a) 6, 6 b) 4, 4 c), d) 5, 5 d) y =.5(x ) + Section 6.3 Practice Master. a) 3, 3 b) c) 0, 8 d) 6, e), 7 3 f) 0, 5 3. a) 4, 4; (0, 6) 5. a) maximum point: (, 4) b) minimum point: ( 3, ) c) maximum point: ( 4, ) d) minimum point: (5, ) 6. The maximum height of 5 m occurs at a horizontal distance of 3 m. 7. a) R = (0 + x)(00 60x), which simplifies to R = 0x + 00x b) $30 c) 900 d) $7 000 b) 7, ; ( 3, 6) Section 6. Practice Master. a), 6 b) 7, 4 c) 0, 9 d), 6 3 e), f) , 3 0. a) 7, 3 b), 6 c), 3 d), 4 e) 0, 4 f), a) 4, 7 b), 7 3 c) 3 3 d) 0, e), 5 f), 4. cm Chapter 6 Practice Masters Answers

19 c), 5; (, 7) 3. a), 9 ; (, 5) BLM (page 4) d) 5, 5; (0, 5) b) 5 ( ), 6; 6, Chapter 6 Practice Masters Answers

20 c) 3 4, 8; ( 35, 84) a) y = x x + 6 b) y = 3x + 75 c) y = x x + 6 d) y = 4x 4x a) y = (x ) + 8 b) c) 6 6. a) 0, 40 b) Section 6.4 Practice Master. a) 7 ± 9 d) 0, 5. a) 5 ± 7 b) 3 ± 3 c) ± 3 7± 4 d) e) 5 ± 33 f) ± 7 ± 9 g) h) ± b) 3± 6 c) 3 ± 5 e) ± 4 4 ; 0.44, 4.56 ; 3.30, 0.30 ;.30,.30 ; 0.30, 6.70 ; 5.37, 0.37 ;.56,.56 ;.9, 3.9 ;.79, a) no x-intercepts; (, ); x = BLM (page 5) c) The relation is invalid for d < 0 and d > 40, because negative heights have no meaning in this context. d) 0 m e) 0 m 7. a) b) ± 3 ; (, 9); x = b) minimum value: 4 m; (0, 4) c) x = 0 d) 4; no x-intercepts Chapter 6 Practice Masters Answers

21 4. a) (, 3), upward, none b) ( 5, 4), downward, two 5. Answers will vary. 6. a) 3 ± 3 b) 3 ± 65 4 c) 4± 3 d) 3 7 or cm and cm 8. 5 m on each side 9. a).5 m b) 3.8 s 0. a) 7.5 cm b) cm 3. 8 m Section 6.5 Practice Master. a) 5 m b) 95 m c) 55 m d) 0 s e) 6 s f) 0 s. a) $0.0 b) $.0 c) $ m 4. cm and 5 cm 5. base 44 cm, height 4 cm 6.,, 3, 4 or 4, 3,, 7. width m, length 9 m 8. 6 m Chapter 6 Review. a) y = (x + 3) 6 b) y = (x + ) 5 c) y = (x + 4) 9 d) y = (x + 5) 30. Labelled points may vary. Examples are shown on the graphs. a) ( 7, 56) b) (3, 0) c) ( 3, 4) BLM (page 6) Chapter 6 Practice Masters Answers

22 d) (8, 67) b) 3 5, 3 ; (0, 9) 5 BLM (page 7) c) 3, 7; (, 5) 3. a) minimum: 3.9 b) minimum: 8.3 c) minimum: 0. d) maximum: 3 4. a) 4.5 m b) 5 m 5. a) 5, 3 b) 4, 9 c), 5 d) 6. a) 4, 3 b), c), 5 d) 0, 4 5 e), 4 f), Answers may vary. For example: a) x x 5 = 0 b) 5x x = 0 8. a) 3; ( 3, 0), d) 4, 8; ( 6, 4) Chapter 6 Practice Masters Answers

23 e) 4, 6; (, 50) BLM (page 8) 8. a) h = 4.9t + 5t b) 7. m c) 3.6 m;.6 s 9. a) R = (400 8x) (80 + x) or R = 6(x 5) b) $3 400 c) 360; $90 Chapter 6 Practice Test. Points graphed may vary. Examples are shown on the graphs. a) y = (x 4) + 6 f), ; (0, 5) b) y = (x + ) 7 9. a) y = x + 6x b) y = 3x + 6x 5 c) y = x 8 d) y = x x ; 5 5± 7. a) d) 3 ± 89 0 ± 4 g) ± 3 b) 6 3± 65 e) 4 h) 3 ± c) 3± 3 f) ± a) (, ), upward, none b) (, 3), downward, two c) ( 3, 0), upward, one d) ( 4, ), downward, none 3. a) 0.5 s b) 3. s c).8 s, 7.4 m 4. 9 and 94 or 94 and m by 68 m 6. 3 cm by 8 cm 7. a) b) No. c) ( 6) 3 y = x + Chapter 6 Practice Masters Answers

24 . Answers may vary. For example: For the quadratic relation y = ax + bx + c, complete the square to rewrite the relation as y = a(x h) + k. The vertex of the relation is (h, k). 3. a) 3, b), c) d) 0, a) e) 7, 9 f), 5 3 g) 3, 5 h) 5, c) BLM (page 9) 5. a) y = x 6x b) y = x 6x a) not possible b), c) 3 ± 6 3 d), e) ± 3± f) a) 3, b) not possible c), 5± 37 d) not possible e) f), 7 8. a) b) b).5 m c) h = 3d a) 40 ft b) 0 ft c) 5 ft 0. a) P = x + 9x + 50 b) 6, 5 c) 950 items d) $ a) 0.0, 5.3 b).65 s c) m. 4 m by 0 m 3. cm by 3 cm m cm cm by 9.7 cm Chapter 6 Practice Masters Answers

25 Chapter 6 Test. a) y = (x + 4) d) ( 6) 5 y = x + BLM (page 0) b) y = 3(x ) 4 c) y = 5(x + ) 6. a) x = 3 b) x = 3 c) x = 3 3. All three relations have the same axis of symmetry of x = a) 3, 5 b), 3 c) 3, 5 d), 7 5. a) 7 ± 7 ;.78, b) c) 3 ± 37 ; 4.54,.54 d) e) not possible f) 5± 37 6 ± 3 5 3± 4 ; 0.8,.85 ;.3, 0.9 ; 0.8, 0.78 g) 5 ± 7 ; 0.8, 0.4 h) not possible 4 6. a) y = 3x + 6x + 9 b) y = x x cm and 4 cm 8. Answers may vary. For example: a) 4x + x 8 = 0 b) 6x x = 0 9. a) two, as the vertex is below the x-axis and the parabola points upward b) zero, as the vertex is below the x-axis and the parabola points downward c) two, as the vertex is above the x-axis and the parabola points downward d) one, as the vertex is on the x-axis 0. a).6,.3 b).,.7 c).8, 0.6 d) 3.,. Chapter 6 Practice Masters Answers

26 . a) BLM (page ). a) 50 m b) 5 m c).5 m d) yes e) 46.8 m away 3. a) 5 b) No. c) 35 d) No. b) y = x c) 86 m Chapter 6 Practice Masters Answers