Unit 7 Quadratic Functions

Transcription

1 Algebra I Revised 11/16 Unit 7 Quadratic Functions Name: 1

2 CONTENTS 9.1 Graphing Quadratic Functions 9.2 Solving Quadratic Equations by Graphing Assessment 8.6 Solving x^2+bx+c=0 8.7 Solving ax^2+bx+c= / Assessment 9.5 Using the Quadratic Formula 9.4 Completing the Square Assessment Unit 7 Test Review Unit 7 Evaluation 2

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4 Guided Practice: 1) Use a table of values to graph y = 4x + 1. a. Find and graph the ordered pairs in the table and connect them with a smooth curve. x y b. What are the domain and range of this function? Domain: Range: 2) Use a table of values to graph y = 6x 7. a. Find and graph the ordered pairs in the table and connect them with a smooth curve. x y b. What are the domain and range of this function? Domain: Range: 4

5 Independent Practice: Use a table of values to graph each function. Determine the domain and range. 1. y = y = 4 3. y = 4x + 2 x y x y x y Domain: Domain: Domain: Range: Range: Range: 5

6 Symmetry and Vertices Parabolas have a geometric property called symmetry. That is, if the figure is folded in half, each half will match the other half exactly. The vertical line containing the fold line is called the axis of symmetry. The axis of symmetry contains the minimum or maximum point of the parabola, the vertex. Axis of Symmetry For the parabola y = a + bx + c, where a 0, the line x = is the axis of symmetry. Example: The axis of symmetry of y = + 2x + 5 is the line x = 1. Guided Practice: Find the vertex, the equation of the axis of symmetry, and the y-intercept. 6

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8 Guided Practice: 1) Find the equation for the axis of symmetry, vertex, y-intercept, domain & range, and identify vertex as maximum or minimum for the function y = 2 + 4x + 1. Then graph the function. a. Write the equation of the axis of symmetry. b. Find the coordinates of the vertex c. Identify the vertex as a maximum or a minimum and state the domain and range. d. Graph the function. 2) Find the equation for the axis of symmetry, vertex, y-intercept, domain & range, and identify vertex as maximum or minimum for the function below. 8

9 Independent Practice: Find the equation for the axis of symmetry, vertex, y-intercept, domain & range, and identify vertex as maximum or minimum for the functions below. Then graph each function. 1) 2) 3) 4) 9

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11 Guided Practice: Graph the function:. Step 1: Find the equation of the axis of symmetry. Step 2: Find the vertex and determine whether if it is a maximum or minimum. Step 3: Find the y-intercept. Step 4: Graph the function using the above information and a smooth curve. X Y 11

12 Independent Practice: Graph the function:. Step 1: Find the equation of the axis of symmetry. Step 2: Find the vertex and determine whether if it is a maximum or minimum. Step 3: Find the y-intercept. Step 4: Graph the function using the above information and a smooth curve. X Y 12

13 9-1 Practice ~ Graphing Quadratic Functions Use a table of values to graph each function. Determine the domain and range. 1. y = y = 6x y = 2 8x 5 X Y X Y X Y Domain: Domain: Domain: Range: Range: Range: Find the vertex, the equation of the axis of symmetry, and the y intercept of the graph of each function. 4. y = 9 5. y = 2 + 8x 5 6. y = 4 4x + 1 Vertex: Vertex: Vertex: Axis of Axis of Axis of Symmetry: Symmetry: Symmetry: y-intercept: y-intercept: y-intercept: 13

14 Consider each equation. Determine whether the function has a maximum or a minimum value (circle one). State the maximum or minimum value (vertex). What are the domain and range of the function? 7. y = 5 2x y = + 5x y = + 4x 9 Maximum or Minimum Maximum or Minimum Maximum or Minimum Vertex: Vertex: Vertex: Domain: Domain: Domain: Range: Range: Range: Graph each function. 10. f(x) = f(x) = 2 + 8x f(x) = 2 + 8x + 1 Concept Review: 14

15 9-1 Skills Practice ~ Graphing Quadratic Functions Use a table of values to graph each function. State the domain and the range. 1. y = 4 2. y = y = 2x 6 X Y X Y X Y Domain: Domain: Domain: Range: Range: Range: Find the vertex, the equation of the axis of symmetry, and the y intercept of the graph of each function. 4. y = 2 8x y = + 4x y = 3 12x + 3 Vertex: Vertex: Vertex: Axis of Axis of Axis of Symmetry: Symmetry: Symmetry: y-intercept: y-intercept: y-intercept: 15

16 Consider each equation. Determine whether the function has a maximum or a minimum value (circle one). State the maximum or minimum value (vertex). What are the domain and range of the function? 7. y = 2 8. y = 2x 5 9. y = + 4x 1 Maximum or Minimum Maximum or Minimum Maximum or Minimum Vertex: Vertex: Vertex: Domain: Domain: Domain: Range: Range: Range: Graph each function and determine the domain and range. 10. f(x) = 2x f(x) = 2 + 4x f(x) = 2 4x + 6 Domain: Domain: Domain: Range: Range: Range: 16

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18 Application Problems: Guided Practice: Emily is competing in the javelin throw. The height of the javelin can be modeled by the equation, where y represents the height in feet of the javelin after x seconds. a) Graph the path of the javelin. b) At what height is the javelin thrown? c) What is the maximum height of the javelin? Independent Practice: A juggler is tossing a ball into the air. The height of the ball in feet can be modeled by the equation, where y represents the height of the ball at x seconds. a) Graph this equation. b) At what height is the ball thrown? c) What is the maximum height of the ball? 18

19 Application Problems Practice 1. OLYMPICS Olympics were held in 1896 and have been held every four years except 1916, 1940, and The winning height y in men s pole vault at any number Olympiad x can be approximated by the equation y = x Complete the table to estimate the pole vault heights in each of the Olympic Games. Round your answers to the nearest tenth. Year Olympiad (x) Height (y inches) Source: National Security Agency 2. PHYSICS Mrs. Capwell s physics class investigates what happens when a ball is given an initial push, rolls up, and then back down an inclined plane. The class finds that y = + 6x accurately predicts the ball s position y after rolling x seconds. On the graph of the equation, what would be the y value when x = 4? 3. ARCHITECTURE A hotel s main entrance is in the shape of a parabolic arch. The equation y = + 10x models the arch height in feet y for any distance x from one side of the arch. Use a graph to determine its maximum height. 4. SOFTBALL Olympic softball gold medalist Michele Smith pitches a curveball with a speed of 64 feet per second. If she throws the ball straight upward at this speed, the ball s height h in feet after t seconds is given by h = t. Find the coordinates of the vertex of the graph of the ball s height and interpret its meaning. 19

20 5. BASEBALL The equation h = x + 3 describes the path of a baseball hit into the outfield, where h is the height of the ball in feet and x is the horizontal distance the ball travels. a. What is the equation of the axis of symmetry? b. What is the maximum height reached by the baseball? c. An outfielder catches the ball three feet above the ground. How far has the ball traveled horizontally when the outfielder catches it? 6. GEOMETRY Teddy is building the rectangular deck shown below. a. Write an equation representing the area of the deck y. b. What is the equation of the axis of symmetry? c. Graph the equation and label its vertex. 20

21 The solutions of a quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found by graphing the related quadratic function f(x) = a + bx + c and finding the x-intercepts or zeros of the function. 21

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23 Guided Practice: Solve each equation by graphing. 1) + 4x + 3 = 0 2) 6x + 9 = 0 X Y X Y 4) X Y X Y 23

24 Independent Practice: Solve each equation by graphing. 1) X Y 2) X Y 3) + 7x + 12 = 0 4) x 12 = 0 5) 4x + 5 = 0 24

25 Independent Practice (continued): Solve each equation by graphing. X Y X Y 25

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27 Guided Practice: Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth x + 9 = 0 2. x 4 = x + 3 = 0 4. A goalie kicks the soccer ball with an upward velocity of 55 feet per second and his foot meets the ball 2 feet off the ground. The quadratic function represents the height of the ball h in feet after t seconds. Approximately how long is the ball in their air? Independent Practice: Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth

28 Independent Practice (continued): Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth Ricky built a model rocket. Its flight can be modeled by the equation where h is the height of the rocket in feet after t seconds. About how long was Ricky s rocket in the air? 28

29 9-2 Practice ~ Solving Quadratic Equations by Graphing Solve each equation by graphing. 1. 5x + 6 = w + 9 = b + 4 = 0 Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth p = = 10m v = 7 29

30 9-2 Skills Practice ~ Solving Quadratic Equations by Graphing Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. 1. 2x + 3 = c + 8 = a = n = p + 2 = x 3 = d = 3 8. = 4h 30

31 9-2 Application Problems ~ Solving Quadratic Equations by Graphing 1. NUMBER THEORY Two numbers have a sum of 2 and a product of 8. The quadratic equation + 2n + 8 = 0 can be used to determine the two numbers. a. Graph the related function f(n) = + 2n + 8 and determine its x-intercepts. b. What are the two numbers? 2. DESIGN A footbridge is suspended from a parabolic support. The function h(x) = + 9 represents the height in feet of the support above the walkway, where x = 0 represents the midpoint of the bridge. a. Graph the function and determine its x-intercepts. b. What is the length of the walkway between the two supports? 3. FARMING In order for Mr. Moore to decide how much fertilizer to apply to his corn crop this year, he reviews records from previous years. His crop yield y depends on the amount of fertilizer he applies to his fields x according to the equation y = + 4x Graph the function, and find the point at which Mr. Moore gets the highest yield possible. 31

32 4. LIGHT Ayzha and Jeremy hold a flashlight so that the light falls on a piece of graph paper in the shape of a parabola. Ayzha and Jeremy sketch the shape of the parabola and find that the equation y = 3x 10 matches the shape of the light beam. Determine the zeros of the function. 5. FRAMING A rectangular photograph is 7 inches long and 6 inches wide. The photograph is framed using a material that is x inches wide. If the area of the frame and photograph combined is 156 square inches, what is the width of the framing material? 6. ENGINEERING The shape of a satellite dish is often parabolic because of the reflective qualities of parabolas. Suppose a particular satellite dish is modeled by the following equation. 0.5 = y a. Find the zeros of this function by graphing. b. On the coordinate plane above, translate the parabola so that there is only one zero. Label this curve A. c. Translate the parabola so that there are no zeros. Label this curve B. 32

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34 Guided Practice: Factor each polynomial. 1) 2) 3) 4) 5) 6) 34

35 Independent Practice: Factor each polynomial. 1) 2) 3) 4) 5) 6) 35

36 Guided Practice: Factor each polynomial. 1) 2) Independent Practice: Factor each polynomial. 1) 2) 3) 4) 5) 6) 36

37 Guided Practice: Solve each equation by factoring. Check your solutions. 1) 2) 37

38 Independent Practice: Solve each equation by factoring. Check your solutions. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) Thinking Questions: Factor each polynomial. 13) 14) 15) 16) 38

39 17) The height of a parallelogram is 18 centimeters less than its base. If the area is 175 square centimeters, what is its height? (Hint: ) 18) A triangle has an area of 36 square feet. If the height of the triangle is 6 feet more than its base, what are its height and base? (Hint: ) 19) A rectangle has an area represented by square feet. If the length is x+2 feet, what expression represents the width of the rectangle? 20) Tina bought a frame for a photo, but the photo is too big for the frame. Tina needs to reduce the width and length of the photo by the same amount. The area of the photo should be reduced to half the original area. If the original photo is 12 inches by 16 inches, what will be the dimensions of the smaller photo? 21) The width of a high school soccer field is 45 yards shorter than its length. a) Define a variable, and write an expression for the area of the field. b) The area of the field is 9000 square yards. Find the dimensions. 39

40 8-6 Practice ~Factoring + bx + c Factor each polynomial a h x g w y b n t z d x q x r g jk mv 56 40

41 8-6 Practice ~ Solving + bx + c = 0 Solve each equation. Check the solutions x + 42 = p 84 = k 54 = b 64 = n = h = t = z = = 5y = 18a 29. = 16u r + = Find all values of k so that the trinomial + kx 35 can be factored using integers. 41

42 32. Construction A construction company is planning to pour concrete for a driveway. The length of the driveway is 16 feet longer than its width, w. a) Write an expression for the area of the driveway. b) Find the dimensions of the driveway if it has an area of 260 square feet. 33. Web Design Janel has a 10-inch by 12-inch photograph. She wants to scan the photograph, then reduce the result by the same amount in each dimension, length and width, to post on her web site. Janel wants the area of the image to be one eighth of the original photograph. a) Write an equation to represent the area of the reduced image. b) Find the dimensions of the reduced image. 42

43 8-6 Word Problem Practice ~ Solving + bx + c = 0 1. CONSTRUCTION A construction company is planning to pour concrete for a driveway. The length of the driveway is 20 feet longer than its width w. a. Write an expression for the area of the driveway. b. Find the dimensions of the driveway if it has an area of 300 square feet. 2. COMPACT DISCS A rectangular compact disc jewel case has a width 2 centimeters greater than its length. The area for the front cover is 168 square centimeters. Write and solve the equation to find the length of the case. 3. MATH GAMES Fiona and Greg play a number guessing game. Greg gives Fiona this hint about his two secret numbers, The product of the two consecutive positive integers that I am thinking of is 11 more than their sum. What are Greg s numbers? 43

44 4. BRIDGE ENGINEERING A car driving over a suspension bridge is supported by a cable hanging between the ends of the bridge. Since its shape is parabolic, it can be modeled by a quadratic equation. The height above the road bed of a bridge s cable h in inches measured at distance d in yards from the first tower is given by h = 36d If the driver of a car looks out at a height of 49 inches above the roadbed, at what distance(s) from the tower will the driver s eyes be at the same height as the cable? 5. MONUMENTS Susan is designing a pyramidal stone monument for a local park. The design specifications tell her that the height needs to be 9 feet, the width of the base must be 5 feet less than the length, and the volume should be 150 cubic feet. Recall that the volume of a pyramid is given by V = Bh, where B is the area of the base and h is the height. Write and solve an equation to find the width of the base of the monument. 44

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46 Guided Practice: Factor each trinomial. 1) 2) 46

47 Guided Practice: Factor each trinomial, if possible. If the polynomial cannot be factored using integers, write prime. 1) 2) Independent Practice: Factor each trinomial. If the polynomial cannot be factored using integers, write prime. 1) 2) 3) 5) 6) 8) 9) 11) 12) 47

48 Guided Practice: A person throws a ball upward from a 506 foot tall building. The ball s height h in feet after t seconds is given by the equation. The ball lands on a balcony that is 218 feet above the ground. How many seconds was it in the air? Independent Practice: When Jerry shoots a free throw, the ball is 6 feet from the floor and has an initial upward velocity of 20 feet per second. The hoop is 10 feet from the floor. a) Use the vertical motion model to determine an equation that models Jerry s free throw. b) How long is the basketball in the air before it reaches the hoop? c) Ray shoots a free throw that is 5 foot 9 inches from the floor with the same initial upward velocity. Will the ball be in the air more or less time? Explain. 48

49 Independent Practice (continued) - Solve each equation. Confirm your answers using a graphing calculator. 1) 2) 3) 5) 6) 7) Ben dives from a 36-foot platform into a pool. The equation models his dive. How long will it take Ben to reach the water? 8) Ken throws the discus at a school track meet. a) What is the initial height of the discus? b) After how many seconds does the discus hit the ground? 49

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52 The solutions of a + bx + c = 0, where a 0, are given by Quadratic Formula x = Guided Practice: Solve each equation using the quadratic formula. Round to the nearest tenth if necessary. 1) + 2x = 3 2) 6x 2 = 0 Independent Practice: Solve each equation using the quadratic formula. Round to the nearest tenth if necessary. 1. 3x + 2 = x = 16 a = a = b = b = c = c = x = x = 6 a = a = b = b = c = c = 52

53 x = x 5 = 0 a = a = b = b = c = c = x = x = 5 a = a = b = b = c = c = x 15 = x = 24 a = a = b = b = c = c = x = x 4 = 0 a = a = b = b = c = c = 53

54 Hold that thought You will be taught how to Complete the Square in the next section. 54

55 Guided Practice: Solve each equation below using any method. Be sure to show your work. Round to the nearest tenth if necessary. 1) 2) 55

56 Guided Practice: State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. 1) x = 4 2) 2 + 3x = 4 3) 4) Independent Practice: State the value of the discriminant for each equation. Then determine the number of real solutions of the equation x 3 = x 8 = x 9 = = x x = x + 10 = 0 56

57 = x 8. 6 = 11x x + 9 = = 6x = x + 4 = x = x + 4 = x = 14 57

58 9-5 Skills Practice ~Solving Quadratic Equations by Using the Quadratic Formula Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary = 0 2. x 20 = x 36 = x + 30 = x = x = x + 22 = x + 3 = x 7 = x = x + 4 = x = x 3 = x 6 = 0 58

59 State the value of the discriminant for each equation. Then determine the number of real solutions of the equation x + 3 = x + 1 = x + 10 = x + 7 = x 7 = x + 25 = x 8 = x + 12 = x + 10 = x + 3 = Practice ~ Solving Quadratic Equations by Using the Quadratic Formula Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary x 3 = x + 7 = x + 6 = x + 7 = x 5 = x + 10 = x = x = x = 4 59

60 = 8x x = x = x 1.5 = x + = = State the value of the discriminant for each equation. Then determine the number of real solutions of the equation x + 16 = x + 12 = x = x = = 12x x = x 0.5 = x = = x 1 60

61 25. CONSTRUCTION A roofer tosses a piece of roofing tile from a roof onto the ground 30 feet below. He tosses the tile with an initial downward velocity of 10 feet per second. a. Write an equation to find how long it takes the tile to hit the ground. Use the model for vertical motion, H = 16 + vt + h, where H is the height of an object after t seconds, v is the initial velocity, and h is the initial height. (Hint: Since the object is thrown down, the initial velocity is negative.) b. How long does it take the tile to hit the ground? 26. PHYSICS Lupe tosses a ball up to Quyen, waiting at a third-story window, with an initial velocity of 30 feet per second. She releases the ball from a height of 6 feet. The equation h = t + 6 represents the height h of the ball after t seconds. If the ball must reach a height of 25 feet for Quyen to catch it, does the ball reach Quyen? Explain. (Hint: Substitute 25 for h and use the discriminant.) 61

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63 Guided Practice: 1) Find the value of c that makes + 2x + c a perfect square trinomial. Step 1 Find of 2 Step 2 Square the result of Step 1 Step 3 Add the result of Step 2 to + 2x Factor the perfect square trinomial 2) Find the value of c that makes - 8x + c a perfect square trinomial Independent Practice: Find the value of c that makes each trinomial a perfect square x + c x + c 3. 4x + c 4. 8x + c x + c x + c 7. 3x + c 8. 15x + c 63

64 Since few quadratic expressions are perfect square trinomials, the method of completing the square can be used to solve some quadratic equations. Use the following steps to complete the square for a quadratic expression of the form a + bx. Guided Practice: Solve each equation by completing the square. 1) + 6x + 3 = 10 2) - 12x + 3 = 8 Independent Practice: Solve each equation by completing the square. Round to the nearest tenth if necessary. 1. 4x + 3 = x = x 9 = x = x 5 = x = 9 64

65 7. + 8x = = 2x x + 11 = = 5x 11. = 22x x = x = x = x = 16 Guided Practice: Solve each equation by completing the square. Round to the nearest tenth if necessary. 1) 2) 3) 65

66 Independent Practice: Solve each equation by completing the square. Round to the nearest tenth if necessary = x x + 2 = = 40x

67 Guided Practice: The price p in dollars for a particular stock can be modeled by the quadratic equation where t represents the number of days after the stock is purchased. When is the stock worth $60? Independent Practice: 1) The product of two consecutive even integers is 224. Find the integers. 2) The product of two consecutive negative odd integers is 483. Find the integers. 3) Find all values of c that make a perfect square trinomial. 4) Find all values of c that make a perfect square trinomial. 67

68 9.1 ~ Graphing Quadratic Functions Unit 7 Evaluation Review Use a table of values to graph each function. Determine the domain and range, equation of the axis of symmetry, and state the maximum or minimum (vertex). 1. y = y = 4 3. y = 3x y = y = 4x 4 6. y = + 2x ~ Solving Quadratic Equations by Graphing Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth x + 12 = 0 8. x 12 = x + 5 = 0 68

69 (continued) Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth x + 9 = x 4 = x + 6 = x 1 = x + 3 = x 4 =0 8.6 Solving x 2 + bx + c = 0 Factor each polynomial x m r x x x t p x x a y 8 69

70 28. 2x y m x a y xy ab xy 7 Solve each equation. Check the solutions x + 3 = y + 4 = m + 9 = = x x = x + 36 = = 7t 44. = 9p x + = = 5x 47. = 11a y + 15 = = 24 10x a = = 10b 16 70

71 Use the formula h = vt 16 to solve each problem. 52. FOOTBALL A punter can kick a football with an initial velocity of 48 feet per second. How many seconds will it take for the ball to first reach a height of 32 feet? 53. ROCKET LAUNCH If a rocket is launched with an initial velocity of 1600 feet per second, when will the rocket be 14,400 feet high? 8.7 ~ Solving ax 2 + bx + c = 0 Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime x m r x x x a y t x p x y x m

72 x a y + 2 Solve each equation. Check the solutions x 3 = n 5 = d 7 = = x x = x 10 = = 11k = 21p x + 9 = = 8x = 65a y 3 = x = 3 + 7x a + 5 = b = 10b The difference of the squares of two consecutive positive odd integers is 24. Find the integers. 72

73 88. GEOMETRY The length of a rectangular conservatory garden in Charlotte, North Carolina \\] is 20 yards greater than its width. The area is 300 square yards. What are the dimensions? 89. GEOMETRY A rectangle with an area of 24 square inches is formed by cutting strips of equal width from a rectangular piece of paper. Find the dimensions of the new rectangle if the original rectangle measures 8 inches by 6 inches. 9.4 Solving Quadratic Equations by Completing the Square Find the value of c that makes each trinomial a perfect square x + c x + c 92. 4x + c 93. 8x + c x + c x + c 96. 3x + c x + c x + c 73

74 Solve each equation by completing the square. Round to the nearest tenth if necessary x + 3 = x = x 9 = x = x 5 = x = x = = 2x x + 11 = = 5x 109. = 22x x = x = x = x = = x x + 2 = = 40x

75 9.5 Solving Quadratic Equations by Using the Quadratic Formula Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary x + 2 = x = x = x = x = x 5 = x = x = x 15 = x = x = x 4 = x + 4 = x + 2 = 0 75

76 State the value of the discriminant for each equation. Then determine the number of real solutions of the equation x 3 = x 8 = x 9 = = x x = x + 10 = = x = 11x x + 9 = = 6x = x + 4 = x = x + 4 = x = 14 76