Algebra I Quadratic & Non-Linear Functions

Transcription

1 1

2 Algebra I Quadratic & Non-Linear Functions

3 Table of Contents Click on the topic to go to that section Key Terms Explain Characteristics of Quadratic Functions Graphing Quadratic Functions in Standard Form Graphing Quadratic Functions in Vertex and Intercept Form Transforming and Translating Quadratic Functions Comparison of Types of Functions 3

4 Key Terms Return to Table of Contents 4

5 Axis of Symmetry Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves 5

6 Parabolas Maximum: The y-value of the vertex if a < 0 and the parabola opens downward. Minimum: The y-value of the vertex if a > 0 and the parabola opens upward. (+ a) Max Min (- a) Parabola: The curve result of graphing a quadratic equation. 6

7 Quadratic Equation: An equation that can be written in the standard form ax 2 + bx + c = 0. Where a, b and c are real numbers and a does not = 0. Quadratic Function: Any function that can be written in the form y = ax 2 + bx + c. Where a, b and c are real numbers and a does not equal 0. Vertex: The highest or lowest point on a parabola. Zero of a Function: An x value that makes the function equal zero. Quadratics 7

8 Explain Characteristics of Quadratic Equations Return to Table of Contents 8

9 Quadratics A quadratic equation is an equation of the form ax 2 + bx + c = 0, where a is not equal to 0. The form ax 2 + bx + c = 0 is called the standard form of the quadratic equation. The standard form is not unique. For example, x 2 x + 1 = 0 can be written as the equivalent equation x 2 + x 1 = 0. Also, 4x 2 2x + 2 = 0 can be written as the equivalent equation 2x 2 x + 1 = 0. Why is this equivalent? 9

10 Writing Quadratic Equations Practice writing quadratic equations in standard form. (Simplify if possible.) Write 2x 2 = x + 4 in standard form. Answer 10

11 Writing Quadratic Equations Write 3x = x in standard form. Answer 11

12 Writing Quadratic Equations Write 6x 2 6x = 12 in standard form. Answer 12

13 Writing Quadratic Equations Write 3x 2 = 5x in standard form. Answer 13

14 Characteristics of Quadratic Functions 1. Standard form is y = ax 2 + bx + c, where a 0. y = 3x 2 + 4x 10 y = 5x 2 9 y = x y = x 2 + 5x

15 Characteristics of Quadratic Functions 2. The graph of a quadratic is a parabola, a u-shaped figure. 3. The parabola will open upward or downward. downward upward 15

16 Characteristics of Quadratic Functions 4. A parabola that opens upward contains a vertex that is a minimum point. A parabola that opens downward contains a vertex that is a maximum point. vertex vertex 16

17 Characteristics of Quadratic Functions 5. The domain of a quadratic function is all real numbers. 17

18 Characteristics of Quadratic Functions 6. To determine the range of a quadratic function, ask yourself two questions: > Is the vertex a minimum or maximum? > What is the y-value of the vertex? If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value. The range of this quadratic is [ 6, ) 18

19 Characteristics of Quadratic Functions If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value. The range of this quadratic is (,10] 19

20 Characteristics of Quadratic Functions 7. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form y = 2x 2 8x + 2 x = b 2a ( 8) x = = 2 2(2) x=2 Teacher Notes 20

21 Characteristics of Quadratic Functions 8. The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots or solutions and solution sets. Each quadratic function will have two, one or no real x-intercepts. 21

22 1 True or False: The vertex is the highest or lowest value on the parabola. True False 22

23 2 If a parabola opens upward then... A a>0 B a<0 C a=0 23

24 3 The vertical line that divides a parabola into two symmetrical halves is called... A discriminant B perfect square C axis of symmetry D vertex E slice 24

25 4 What is the equation of the axis of symmetry of the parabola shown in the diagram below? A x=0.5 B x=2 C x=4.5 D x=15 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

26 5 The height, y, of a ball tossed into the air can be represented by the equation y = x x + 3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola? A y=5 B y= 5 C x=5 D x= 5 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

27 6 What are the vertex and axis of symmetry of the parabola shown in the diagram below? A vertex: (1, 4); axis of symmetry: x = 1 B vertex: (1, 4); axis of symmetry: x = 4 C vertex: ( 4,1); axis of symmetry: x = 1 D vertex: ( 4,1); axis of symmetry: x = 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

28 7 What is the domain and range of the quadratic function? A B C D 28

29 8 The equation y = x 2 + 3x 18 is graphed on the set of axes below. Based on this graph, what are the roots of the equation x 2 + 3x 18 = 0? A 3 and 6 B 0 and 18 C 3 and 6 D 3 and 18 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

30 9 The equation y = x 2 2x + 8 is graphed on the set of axes below. Based on this graph, what are the roots of the equation x 2 2x + 8 = 0? A 8 and 0 B 2 and 4 C 9 and 1 D 4 and 2 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

31 10 What is the domain and range of the quadratic function? A B C D 31

32 Graphing Quadratic Functions in Standard Form Return to Table of Contents 32

33 Graph by Following Six Steps: Step 1 - Find Axis of Symmetry Step 2 - Find Vertex Step 3 - Find Y intercept Step 4 - Find two more points Step 5 - Partially graph Step 6 - Reflect 33

34 Step 1 - Find the Axis of Symmetry What is the Axis of Symmetry? Hint Axis of Symmetry 34

35 Step 1 Find the Axis of Symmetry Graph y = 3x 2 6x + 1 Formula: a = 3 b = -6 x = b 2a x = ( 6) = 6 = 1 2(3) 6 The axis of symmetry is x = 1. 35

36 Step 2 Find the vertex by substituting the value of x (the Axis of Symmetry) into the equation to get y. y = 3x 2 6x + 1 a = 3, b = 6 and c = 1 y = 3(1) 2 + 6(1) + 1 y = y = 2 Vertex = (1, 2) 36

37 Step 3 Find y-intercept. What is the y-intercept? Hint y-intercept 37

38 Step 3 Graph y = 3x 2 6x + 1 Step 3 - Find y-intercept. The y-intercept is always the c value, because x = 0. y = ax 2 + bx + c y = 3x 2 6x + 1 c = 1 The y-intercept is 1 and the graph passes through (0,1). 38

39 Step 4 Graph y = 3x 2 6x + 1 Step 4 - Find two points Choose different values of x and plug in to the equation find points. Let's pick x = 1 and x = 2 y = 3x 2 6x + 1 y = 3( 1) 2 6( 1) + 1 y = y = 10 ( 1,10) y = 3x 2 6x + 1 y = 3( 2) 2 6( 2) + 1 y = 3(4) y = 25 ( 2,25) 39

40 Step 5 Step 5 - Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points. 40

41 Step 6 Step 6 - Reflect the points across the axis of symmetry. Connect the points with a smooth curve. (4,25) 41

42 11 What is the axis of symmetry for y = x 2 + 2x 3 (Step 1)? A 1 B 1 42

43 12 What is the vertex for y = x 2 + 2x 3 (Step 2)? A ( 1, 4) B (1, 4) C ( 1,4) 43

44 13 What is the y-intercept for y = x 2 + 2x 3 (Step 3)? A 3 B 3 44

45 14 What is an equation of the axis of symmetry of the parabola represented by y = x 2 + 6x 4? A x = 3 B y = 3 C x = 6 D y = 6 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

46 Graph Graph y = 2x 2 6x

47 Graph Graph f(x) = x 2 4x

48 Graph Graph y = 3x

49 Solve Equations On the set of axes below, solve the following system of equations graphically for all values of x and y. y = x 2 4x + 12 y = 2x + 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

50 Solve Equations On the set of axes below, solve the following system of equations graphically for all values of x and y. y = x 2 6x + 1 y + 2x = 6 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

51 15 The graphs of the equations y = x 2 + 4x 1 and y + 3 = x are drawn on the same set of axes. At which point do the graphs intersect? A (1,4) B (1, 2) C ( 2,1) D ( 2,5) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

52 Graphing Quadratic Functions in Vertex and Intercept Form Return to Table of Contents 52

53 Quadratic Equation Forms Standard Form of a Quadratic Equation is y = ax 2 + bx + c. Vertex Form of a Quadratic Equation is y = a (x h) 2 + k, where (h, k) is the vertex and x = h is the axis of symmetry. Intercept Form of a Quadratic Equation is y = a (x p) (x q), where (p, 0) and (q, 0) are the roots (or zeros) of the graph. 53

54 Graph in Vertex Form Graph in Vertex Form by Following Five Steps: Step 1 - Draw the axis of symmetry, x = h Step 2 - Plot the vertex, (h, k) Step 3 - Find two more points Step 4 - Partially graph Step 5 - Reflect 54

55 Step 1 Step 1 - Find the Axis of Symmetry Graph y = 2(x 4) 2 3 y = 2(x 4) 3 The axis of symmetry is x = 4. 55

56 Step 2 Plot the Vertex Graph y = 2(x 4) 2 3 y = 2(x 4) 2 3 The Vertex is (4, 3). 56

57 Step 3 Find two points Graph y = 2(x 4) 2 3 Choose different values of x and plug in to the equation to find points. Let's pick x = 2 and x = 3 y = 2(x 4) 2 3 y = 2(2 4) 2 3 y = 2(2) 2 3 y = 8 (2, 5) y = 2(x 4) 2 3 y = 2(3 4) 2 3 y = 2(1) 2 3 y = 1 (3, 1) 57

58 Step 4 Graph the axis of symmetry, the vertex, and two other points. 58

59 Step 5 Reflect the points across the axis of symmetry. Connect the points with a smooth curve. 59

60 16 What is the axis of symmetry of the equation y = (x + 5) 2 + 1? 60

61 17 What is the vertex of y = (x + 5) 2 + 1? A (5,1) B (1,5) C ( 5,1) D (5, 1) 61

62 18 What is the axis of symmetry of the equation of y =.5(x - 7) 2 + 3? 62

63 Graph y Graph y = (x + 5)

64 Graph f(x) Graph f(x) =.5(x 5) f(x) 64

65 Graph in Intercept Form Graph in Intercept Form by Following Four Steps Step 1 - Draw the axis of symmetry Step 2 - Find and Plot the vertex Step 3 - Plot the roots Step 4 - Graph 65

66 Graph in Intercept Form Step 1 - Find the Axis of Symmetry Graph y = 1 / 3 (x + 2)(x 4) Use the formula x = p + q 2 x = The axis of symmetry is x = 1. 66

67 Graph in Intercept Form y = 3 (x + 2) (x 4) y = 3 (1 + 2) (1 4) y = 1 /3 (3) ( 3) y = 3 Step 2 - Find and Plot the Vertex Graph y = 1 / 3 (x + 2)(x 4) 67

68 Graph in Intercept Form y = a(x p)(x q) y = 1 /3(x + 2)(x 4) p = 2 and q = 4 Step 3 - Find and Plot the Roots Graph y = 1 /3(x + 2)(x 4) ( 2, 0) and (4, 0) are the roots 68

69 Graph in Intercept Form Step 4 - Connect the points with a smooth curve. 69

70 19 What is the axis of symmetry of the equation y = 4(x 5) (x + 4)? 70

71 20 What is the vertex of the equation y = 4(x 5)(x + 4)? A (0.5, 77) B (0.5, 81) C (0.5,81) D ( 81,0.5) 71

72 21 What are the roots of the equation y = 4(x 5)(x+4)? A (0 5),(0,4) B (0,5),(0, 4) C ( 5,0),(4,0) D (5,0),( 4,0) 72

73 Graph in Intercept Form Graph y = 2(x 3)(x+5) 73

74 Graph in Intercept Form Graph y =.5(x 2)(x 8) 74

75 Graph in Intercept Form Graph f(x) = (x + 1) 2 4 f(x) 75

76 Graph in Intercept Form Graph f(x) = 0.5(x + 3)(x 3) f(x) 76

77 Transforming and Translating Quadratic Functions Return to Table of Contents 77

78 Quadratic Functions The quadratic parent function is f(x) = x 2. The graph of all other quadratic functions are transformations of the graph of f(x) = x 2. y = x 2 x x

79 Quadratic Functions The quadratic parent function is f(x) = x 2. How is f(x) = x 2 into f(x) = 2x 2? y = 2x 2 y = x 2 x

80 Quadratic Functions The quadratic parent function is f(x) = x 2. How is f(x) = x 2 into f(x) =.5x 2? y = x 2 y = x x

81 What does "a" do in y = ax 2 + bx + c? How does a>0 effect the parabola? How does a<0 effect the parabola? y = x 2 y = x 2 81

82 What does "a" do in y = ax 2 + bx + c? How does your conclusion about "a" change as a changes? 1 y = x 2 2 y = 3x 2 y = x 2 y = 1x 2 y = 3x 2 y = x

83 What does "a" do in y = ax 2 + bx + c? If a > 0, the graph opens up. If a < 0, the graph opens down. If the absolute value of a is > 1, then the graph of the function is narrower than the graph of the parent function. If the absolute value of a is < 1, then the graph of the function is wider than the graph of the parent function. 83

84 Absolute Value Consider an "absolute value" number line to compare a parabola to parent function. wider narrower y = 0.5x 2 y = 1.75x parent function y = x 2 84

85 22 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. y =.3x 2 A B C D up, wider up, narrower down, wider down, narrower 85

86 23 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. y = 4x 2 A B C D up, wider up, narrower down, wider down, narrower 86

87 24 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. y = 4x x + 45 A B C D up, wider up, narrower down, wider down, narrower 87

88 25 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. A up, wider y = 2 x 3 2 B C D up, narrower down, wider down, narrower 88

89 26 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. A B C D up, wider up, narrower down, wider down, narrower y = x

90 What does "c" do in y = ax 2 + bx + c? y = x y = x y = x 2 y = x 2 2 y = x 2 5 y = x

91 What does "c" do in y = ax 2 + bx + c? "c" moves the graph up or down the same value as "c." "c" is the y-intercept, when the graph is in standard form. 91

92 27 Without graphing, what is the y-intercept of the given equation? y = x

93 28 Without graphing, what is the y-intercept of the given equation? y = x

94 29 Without graphing, what is the y-intercept of the given function? f(x) = 3x x 9 94

95 30 Without graphing, what is the y-intercept of the given equation? y = 2x 2 + 5x 95

96 31 Choose all that apply to the following quadratic: f(x) =.7x 2 4 A B C D opens up opens down wider than parent function narrower than parent function E y-intercept of y = 4 F y-intercept of y = 2 G y-intercept of y = 0 H y-intercept of y = 2 I y-intercept of y = 4 J y-intercept of y = 6 96

97 32 Choose all that apply to the following quadratic: f(x) = 4 3 x 2 6x A B C D opens up opens down wider than parent function narrower than parent function E y-intercept of y = 4 F y-intercept of y = 2 G y-intercept of y = 0 H y-intercept of y = 2 I y-intercept of y = 4 J y-intercept of y = 6 97

98 33 The diagram below shows the graph of y = x2 c. Which diagram shows the graph of y = x 2 c? A B C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,

99 What does "d" do in y = (x d) 2? y = (x + 4) 2 y = x 2 y = (x - 4) 2 99

100 What does "d" do in y = (x d) 2? "d" moves the graph left or right, the same value as "d." In this form "d" is the x-intercept. 100

101 34 Without graphing, what is the x-intercept of the given equation? A (2, 0) y = (x + 2) 2 B ( 2,0) 101

102 35 Without graphing, what is the x-intercept of the given function? A (7,0) B ( 7,0) f(x) = (x 7) 2 102

103 36 Without graphing, what is the x-intercept of the given function? f(x) = (x + 11) 2 103

104 37 For the following quadratic, choose all that apply... f(x) = 5x 2 3x + 6 A opens up B opens down C wider than the parent function D narrower than the parent function E x-intercept is (6,0) F x-intercept is ( 6,0) G x-intercept is ( 3,0) H y-intercept is (0,6) I y-intercept is (0, 6) J y-intercept is (0, 3) 104

105 38 For the following quadratic, choose all that apply... 1 f(x) = 4 (x + 6) 2 A opens up F x-intercept is (-6,0) B opens down G x-intercept is (-3,0) C wider than the parent function H y-intercept is (0,6) D narrower than the parent function I y-intercept is (0,-6) E x-intercept is (6,0) J y-intercept is (0.-3) 105

106 39 For the following question, choose all that apply... f(x) = 2(x + 5) A opens up B opens down C wider thant the parent function D narrower than the parent function E vertex is ( 5,9) F vertex is (5, 9) G vertex is ( 5, 9) H vertex is ( 2,5) 106

107 Comparison of Types of Functions Return to Table of Contents 107

108 Review Thus far we have learned: 1. The characteristics of a quadratic function. 2. How to graph a quadratic function. 3. Transformations and translations of quadratic functions. Next we will compare at least two different functions to each other. We will look at Quadratic, Linear, and Exponential Functions in terms of y-intercept, rate of change at an interval, maximum or minimum at an interval, and evaluate at a point. 108

109 Comparison Functions In the next few questions we will compare key features of the two functions below. The linear function represented by the table and the quadratic function represented by the graph. x g(x) f(x) x 109

110 Compare y-intercepts Compare the y-intercepts of the functions. f(x) x g(x) x The y-intercept of g(x) is (0,1) and f(x) is (0, 1) g(x) > f(x) 110

111 Compare Functions Compare the functions at f(2). f(x) x g(x) x Referring to the graph for f(x), f(2) is approximately 3. Referring to the table of values for g(x) is between 1 and 3. At f(2) g(x) < f(x) 111

112 Compare Rate of Change Compare the rate of change over the interval 3 x 0 x g(x) f(x) x To find the rate of change of the functions, find the slope at the intervals. 112

113 Compare Rate of Change Compare the rate of change over the interval 3 x 0 y slope(m) = = x y 2 y 1 x 2 y 1 g(x) x g(0) g( 3) 1 ( 0.5) = = = 0 ( 3) f(x) x f(0) f( 3) 1 2 = = = 1 0 ( 3) At the interval g(x) > f(x) 113

114 40 Compare the y-intercepts of the functions. A g(x)<h(x) B g(x)>h(x) C g(x)=h(x) g(x) x x f(x)

115 41 Compare the y-intercepts of the functions. A g(x)<h(x) B g(x)>h(x) C g(x)=h(x) g(x) x x f(x)

116 42 Compare the rate of change of the functions over the interval 2 x 2. g(x) A g(x)<h(x) B g(x)>h(x) C g(x)=h(x) x x f(x)

117 43 Compare the y-intercepts of the functions. A g(x)<h(x) B g(x)>h(x) C g(x)=h(x) g(x) x x h(x)

118 44 Compare g(2) and h(2). A g(x)<h(x) B g(x)>h(x) C g(x)=h(x) g(x) x x h(x)

119 45 Compare the minimum value g(x) to f(x). A g(x)<f(x) B g(x)>f(x) C g(x)=f(x) 1 f(x) = (x 3)(x + 4) 4 g(x) Hint x 119

120 46 Compare the maximum value of h(x) to f(x). A h(x)<f(x) B h(x)>f(x) C h(x)=f(x) h(x) = 7(x 12) f(x) = 14x x 57 Hint 120

121 47 Compare the rate of change of the functions on the interval A g(x)<f(x) B g(x)>f(x) C g(x)=f(x) x f(x)

122 Compare Functions In the next few questions we will compare key features of the two functions below. The linear function represented by the graph and the exponential function represented by the table. g(x) x x h(x)

123 Compare y-intercepts Compare the y-intercepts of the functions. g(x) x x h(x) The y-intercept of g(x) is (0,1). The y-intercept of h(x) is (0,1). The y-intercepts of the functions are equal. 123

124 Compare Rate of Change Compare the rate of change over the interval 0 x 1. g(x) x h(x)

125 Compare Rate of Change Compare the rate of change over the interval 0 x 1. g(x) x g(1) g(0) 4 1 = = = h(x) x f(1) f(0) 3 1 = = = At the interval g(x) > h(x) 125

126 Compare Rate of Change Compare the rate of change over the interval 1 x 2. g(x) x x h(x)

127 Compare Rate of Change Compare the rate of change over the interval 1 x 2. g(x) x g(2) g(1) 7 4 = = = h(x) x h(2) h(1) 9 3 = = = At the interval g(x) < h(x) 127

128 48 Compare the y-intercepts of the functions. g(x) A g(x)<f(x) x B g(x)>f(x) x f(x)

129 49 Compare the rate of change over the interval -2 x 0 A g(x)<f(x) B g(x)>f(x) g(x) x x f(x)

130 50 Compare the rate of change over the interval 0 x 2 A g(x)<f(x) B g(x)>f(x) g(x) x x f(x)

131 51 Compare the y-intercepts of the functions. A h(x)<f(x) h(x) B h(x)>f(x) x f(x) = 1 / 5 x

132 52 Compare the rate of change over the interval 0 x 1 A h(x)<f(x) h(x) B h(x)>f(x) x f(x) = 1 / 5 x

133 Next Four Problems The next four problems are comparisons between linear, quadratic, and exponential functions. 133

134 53 Compare the y-intercepts of the functions. A g(x)<h(x)<f(x) B g(x)<f(x)<h(x) C h(x)<g(x)<f(x) D h(x)<f(x)<g(x) E f(x)<g(x)<h(x) F f(x)<h(x)<g(x) g(x) h(x) f(x) 134

135 54 Which function has the greatest rate of change over the given interval? 2 x 1 A g(x) B h(x) C f(x) g(x) Answer h(x) f(x) 135

136 55 Which function has the greatest rate of change over the given interval? 1 x 1 A g(x) B h(x) C f(x) g(x) Answer h(x) f(x) 136

137 56 Which function has the greatest rate of change over the given interval? 0 x 1 A g(x) B h(x) C f(x) g(x) Answer h(x) f(x) 137