Algebra I. Slide 1 / 137. Slide 2 / 137. Slide 3 / 137. Quadratic & Non-Linear Functions. Table of Contents

Transcription

1 Slide 1 / 137 Slide 2 / 137 Algebra I Quadratic & Non-Linear Functions Table of Contents Slide 3 / 137 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

2 Slide 4 / 137 Key Terms Return to Table of Contents Axis of Symmetry Slide 5 / 137 Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves Parabolas Slide 6 / 137 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. Parabola: The curve result of graphing a quadratic equation. (+ a) Min Max (- a)

3 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. Quadratics Slide 7 / 137 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. Slide 8 / 137 Explain Characteristics of Quadratic Equations Return to Table of Contents Quadratics Slide 9 / 137 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?

4 Writing Quadratic Equations Slide 10 / 137 Practice writing quadratic equations in standard form. (Simplify if possible.) Write 2x 2 = x + 4 in standard form. Writing Quadratic Equations Slide 11 / 137 Write 3x = x in standard form. Writing Quadratic Equations Slide 12 / 137 Write 6x 2 6x = 12 in standard form.

5 Writing Quadratic Equations Slide 13 / 137 Write 3x 2 = 5x in standard form. Characteristics of Quadratic Functions Slide 14 / 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 20 4 Characteristics of Quadratic Functions Slide 15 / The graph of a quadratic is a parabola, a u-shaped figure. 3. The parabola will open upward or downward. upward downward

6 Characteristics of Quadratic Functions Slide 16 / 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 Characteristics of Quadratic Functions Slide 17 / The domain of a quadratic function is all real numbers. Characteristics of Quadratic Functions Slide 18 / 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, )

7 Characteristics of Quadratic Functions Slide 19 / 137 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] Characteristics of Quadratic Functions Slide 20 / 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 b x = 2a y = 2x 2 8x + 2 ( 8) x = = 2 2(2) x=2 Characteristics of Quadratic Functions Slide 21 / 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.

8 Slide 22 / True or False: The vertex is the highest or lowest value on the parabola. True False 2 If a parabola opens upward then... Slide 23 / 137 A a>0 B a<0 C a=0 3 The vertical line that divides a parabola into two symmetrical halves is called... Slide 24 / 137 A discriminant B perfect square C axis of symmetry D vertex E slice

9 4 What is the equation of the axis of symmetry of the Slide 25 / 137 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, 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? Slide 26 / 137 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, Slide 27 / 137

10 Slide 28 / 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? Slide 29 / 137 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, The equation y = x 2 2x + 8 is graphed on the set of axes below. Slide 30 / 137 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, 2011.

11 Slide 31 / 137 Slide 32 / 137 Graphing Quadratic Functions in Standard Form Return to Table of Contents Graph by Following Six Steps: Slide 33 / 137 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

12 Step 1 - Find the Axis of Symmetry Slide 34 / 137 What is the Axis of Symmetry? Axis of Symmetry Slide 35 / 137 Slide 36 / 137

13 Step 3 Find y-intercept. What is the y-intercept? Slide 37 / 137 y-intercept Step 3 Slide 38 / 137 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). Step 4 Slide 39 / 137 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)

14 Step 5 Slide 40 / 137 Step 5 - Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points. Step 6 Step 6 - Reflect the points across the axis of symmetry. Connect the points with a smooth curve. Slide 41 / 137 (4,25) 11 What is the axis of symmetry for y = x 2 + 2x 3 (Step 1)? Slide 42 / 137 A 1 B 1

15 12 What is the vertex for y = x 2 + 2x 3 (Step 2)? Slide 43 / 137 A ( 1, 4) B (1, 4) C ( 1,4) 13 What is the y-intercept for y = x 2 + 2x 3 (Step 3)? Slide 44 / 137 A 3 B 3 14 What is an equation of the axis of symmetry of the parabola represented by y = x 2 + 6x 4? Slide 45 / 137 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, 2011.

16 Graph Slide 46 / 137 Graph y = 2x 2 6x + 4 Graph Slide 47 / 137 Graph f(x) = x 2 4x + 5 Graph Slide 48 / 137 Graph y = 3x 2 7

17 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 Slide 49 / 137 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011 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 Slide 50 / 137 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 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? Slide 51 / 137 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, 2011

18 Slide 52 / 137 Graphing Quadratic Functions in Vertex and Intercept Form Return to Table of Contents Quadratic Equation Forms Slide 53 / 137 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. Graph in Vertex Form Slide 54 / 137 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

19 Slide 55 / 137 Slide 56 / 137 Step 3 Slide 57 / 137 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)

20 Step 4 Slide 58 / 137 Graph the axis of symmetry, the vertex, and two other points. Step 5 Slide 59 / 137 Reflect the points across the axis of symmetry. Connect the points with a smooth curve. 16 What is the axis of symmetry of the equation y = (x + 5) 2 + 1? Slide 60 / 137

21 17 What is the vertex of y = (x + 5) 2 + 1? A (5,1) Slide 61 / 137 B (1,5) C ( 5,1) D (5, 1) 18 What is the axis of symmetry of the equation of y =.5(x - 7) 2 + 3? Slide 62 / 137 Graph y Slide 63 / 137 Graph y = (x + 5) 2 + 1

22 Graph f(x) Slide 64 / 137 Graph f(x) =.5(x 5) f(x) Graph in Intercept Form Slide 65 / 137 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 Slide 66 / 137

23 Slide 67 / 137 Slide 68 / 137 Graph in Intercept Form Slide 69 / 137 Step 4 - Connect the points with a smooth curve.

24 19 What is the axis of symmetry of the equation y = 4(x 5) (x + 4)? Slide 70 / What is the vertex of the equation y = 4(x 5)(x + 4)? Slide 71 / 137 A (0.5, 77) B (0.5, 81) C (0.5,81) D ( 81,0.5) 21 What are the roots of the equation y = 4(x 5)(x+4)? Slide 72 / 137 A (0 5),(0,4) B (0,5),(0, 4) C ( 5,0),(4,0) D (5,0),( 4,0)

25 Graph in Intercept Form Slide 73 / 137 Graph y = 2(x 3)(x+5) Graph in Intercept Form Slide 74 / 137 Graph y =.5(x 2)(x 8) Graph in Intercept Form Slide 75 / 137 Graph f(x) = (x + 1) 2 4 f(x)

26 Graph in Intercept Form Slide 76 / 137 Graph f(x) = 0.5(x + 3)(x 3) f(x) Slide 77 / 137 Transforming and Translating Quadratic Functions Return to Table of Contents Quadratic Functions Slide 78 / 137 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

27 Quadratic Functions Slide 79 / 137 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 Quadratic Functions Slide 80 / 137 The quadratic parent function is f(x) = x 2. How is f(x) = x 2 into f(x) =.5x 2? y = x 2 1 y = x 2 2 x What does "a" do in y = ax 2 + bx + c? Slide 81 / 137 How does a>0 effect the parabola? How does a<0 effect the parabola? y = x 2 y = x 2

28 What does "a" do in y = ax 2 + bx + c? Slide 82 / 137 How does your conclusion about "a" change as a changes? 1 y = x 2 2 y = 3x 2 y = x 2 1 y = 1x 2 y = 3x 2 y = x 2 2 What does "a" do in y = ax 2 + bx + c? Slide 83 / 137 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. Absolute Value Slide 84 / 137 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

29 22 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. Slide 85 / 137 y =.3x 2 A B C D up, wider up, narrower down, wider down, narrower 23 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. Slide 86 / 137 y = 4x 2 A B C D up, wider up, narrower down, wider down, narrower 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 Slide 87 / 137 A B C D up, wider up, narrower down, wider down, narrower

30 25 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. y = 2 x 3 2 A up, wider B C D up, narrower down, wider down, narrower Slide 88 / Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function. Slide 89 / 137 A B C D up, wider up, narrower down, wider down, narrower y = 7 x 5 2 What does "c" do in y = ax 2 + bx + c? Slide 90 / 137 y = x y = x y = x 2 y = x 2 2 y = x 2 5 y = x 2 9

31 What does "c" do in y = ax 2 + bx + c? Slide 91 / 137 "c" moves the graph up or down the same value as "c." "c" is the y-intercept, when the graph is in standard form. 27 Without graphing, what is the y-intercept of the given equation? Slide 92 / 137 y = x Without graphing, what is the y-intercept of the given equation? y = x 2 6 Slide 93 / 137

32 29 Without graphing, what is the y-intercept of the given function? Slide 94 / 137 f(x) = 3x x 9 30 Without graphing, what is the y-intercept of the given equation? Slide 95 / 137 y = 2x 2 + 5x 31 Choose all that apply to the following quadratic: Slide 96 / 137 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

33 32 Choose all that apply to the following quadratic: Slide 97 / 137 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 Slide 98 / 137 What does "d" do in y = (x d) 2? Slide 99 / 137 y = (x + 4) 2 y = x 2 y = (x - 4) 2

34 What does "d" do in y = (x d) 2? Slide 100 / 137 "d" moves the graph left or right, the same value as "d." In this form "d" is the x-intercept. 34 Without graphing, what is the x-intercept of the given equation? A (2, 0) y = (x + 2) 2 Slide 101 / 137 B ( 2,0) 35 Without graphing, what is the x-intercept of the given function? Slide 102 / 137 f(x) = (x 7) 2 A (7,0) B ( 7,0)

35 36 Without graphing, what is the x-intercept of the given function? Slide 103 / 137 f(x) = (x + 11) 2 37 For the following quadratic, choose all that apply... Slide 104 / 137 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) 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) Slide 105 / 137

36 39 For the following question, choose all that apply... Slide 106 / 137 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) Slide 107 / 137 Comparison of Types of Functions Return to Table of Contents Review Slide 108 / 137 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.

37 Comparison Functions Slide 109 / 137 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 Compare y-intercepts Compare the y-intercepts of the functions. f(x) Slide 110 / 137 x g(x) x The y-intercept of g(x) is (0,1) and f(x) is (0, 1) g(x) > f(x) Compare Functions Compare the functions at f(2). Slide 111 / 137 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)

38 Compare Rate of Change Slide 112 / 137 Compare the rate of change over the interval 3 x 0 f(x) x g(x) x To find the rate of change of the functions, find the slope at the intervals. Compare Rate of Change Slide 113 / 137 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) g(0) g( 3) 1 ( 0.5) = = = x 0 ( 3) f(x) f(0) f( 3) 1 2 = = = 1 x 0 ( 3) At the interval g(x) > f(x) 40 Compare the y-intercepts of the functions. A g(x)<h(x) g(x) Slide 114 / 137 B g(x)>h(x) C g(x)=h(x) x x f(x)

39 41 Compare the y-intercepts of the functions. A g(x)<h(x) g(x) Slide 115 / 137 B g(x)>h(x) C g(x)=h(x) x x f(x) Compare the rate of change of the functions over the interval 2# x # 2. g(x) Slide 116 / 137 A g(x)<h(x) B g(x)>h(x) C g(x)=h(x) x x f(x) Compare the y-intercepts of the functions. Slide 117 / 137 A g(x)<h(x) B g(x)>h(x) C g(x)=h(x) g(x) x x h(x)

40 44 Compare g(2) and h(2). Slide 118 / 137 A g(x)<h(x) B g(x)>h(x) C g(x)=h(x) g(x) x x h(x) Compare the minimum value g(x) to f(x). Slide 119 / 137 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 46 Compare the maximum value of h(x) to f(x). Slide 120 / 137 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

41 47 Compare the rate of change of the functions on the interval Slide 121 / 137 A g(x)<f(x) B g(x)>f(x) C g(x)=f(x) x f(x) Compare Functions Slide 122 / 137 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) Compare y-intercepts Slide 123 / 137 Compare the y-intercepts of the functions. g(x) x h(x) 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.

42 Compare Rate of Change Slide 124 / 137 Compare the rate of change over the interval 0 x 1. g(x) x h(x) Compare Rate of Change Slide 125 / 137 Compare the rate of change over the interval 0 x 1. g(x) g(1) g(0) 4 1 = = = 3 x h(x) f(1) f(0) 3 1 = = = 2 x At the interval g(x) > h(x) Compare Rate of Change Slide 126 / 137 Compare the rate of change over the interval 1 x 2. g(x) x x h(x)

43 Compare Rate of Change Slide 127 / 137 Compare the rate of change over the interval 1 x 2. g(x) g(2) g(1) 7 4 = = = 3 x h(x) h(2) h(1) 9 3 = = = 6 x At the interval g(x) < h(x) 48 Compare the y-intercepts of the functions. A g(x)<f(x) B g(x)>f(x) g(x) x Slide 128 / 137 x f(x) Compare the rate of change over the interval -2 x 0 A g(x)<f(x) g(x) x Slide 129 / 137 B g(x)>f(x) x f(x)

44 50 Compare the rate of change over the interval 0 x 2 A g(x)<f(x) g(x) x Slide 130 / 137 B g(x)>f(x) x f(x) Compare the y-intercepts of the functions. A h(x)<f(x) h(x) Slide 131 / 137 B h(x)>f(x) x f(x) = 1 / 5x Compare the rate of change over the interval 0 x 1 Slide 132 / 137 A h(x)<f(x) h(x) B h(x)>f(x) x f(x) = 1 / 5x +6

45 Next Four Problems Slide 133 / 137 The next four problems are comparisons between linear, quadratic, and exponential functions. 53 Compare the y-intercepts of the functions. Slide 134 / 137 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) 54 Which function has the greatest rate of change over the given interval? 2 x 1 Slide 135 / 137 A g(x) B h(x) C f(x) g(x) h(x) f(x)

46 55 Which function has the greatest rate of change over the given interval? 1 x 1 Slide 136 / 137 A g(x) B h(x) C f(x) g(x) h(x) f(x) 56 Which function has the greatest rate of change over the given interval? 0 x 1 Slide 137 / 137 A g(x) B h(x) C f(x) g(x) h(x) f(x)