Chapter 4: Quadratic Functions and Factoring 4.1 Graphing Quadratic Functions in Stand

Transcription

1 Chapter 4: Quadratic Functions and Factoring 4.1 Graphing Quadratic Functions in Stand VOCAB: a quadratic function in standard form is written y = ax 2 + bx + c, where a 0 A quadratic Function creates a U-shaped graph called a parabola The vertex is the lowest/highest point of the parabola Axis of symmetry is the line that divides the graph into two symmetric parts The roots of the function (answers) are the two points that cross the x-axis (x-intercepts), when y = 0 1

2 Practice Graphing Graph of a quadratic function if a is positive, the parabola opens up if a is negative, the parabola opens down b The vertex has an x coordinate of 2a b The axis of symmetry is written x = 2a 2

3 Practice Graphing Graph of a quadratic function STEP 1: find the x coordinate of the vertex by using b vertex = 2a STEP 2: Make a table of values STEP 3: Plot the points and create the parabola 3

4 Graph: y = x 2 2x 3 4

5 Graph: y = 2x 2 x +2 5

6 Graph: y = 2x 2 7x 8 6

7 4.2 Graphing Quadratic Functions in Vertex form standard form: vertex form: The graph of y = a(x h) 2 + k is the parabola y = ax 2 translated horizontally h units and vertically k units y y = a(x h) 2 +k Characteristics of the graph: y = ax 2 the vertex is (h,k) The axis of symmetry is x = h The graph opens up if a>0 and down if a<0 (h,k) k x (0,0) h 7

8 Graph a quadratic function in vertex form: y = a(x-h) 2 +k Step 1: Identify h, a and k. Use a to determine if the parabola opens up or down STEP 1 Step 2: Plot the vertex (h,k) and draw the axis of symmetry STEP 2 Step 3: Evaluate the function for two values of x. STEP 3 Step 4: Draw parabola through plotted points STEP 4 8

9 Example 2 Graph a quadratic function in vertex form: y = a(x-h) 2 +k Step 1: Identify h, a and k. Use a to determine if the parabola opens up or down STEP 1 Step 2: Plot the vertex (h,k) and draw the axis of symmetry STEP 2 Step 3: Evaluate the function for two values of x. STEP 3 Step 4: Draw parabola through plotted points STEP 4 9

10 Intercept form: y=a(x p)(x q) Characteristics of the graph y=a(x p)(x q): The x intercepts are p and q The axis of symmetry is halfway between (p,0) and (q,0) it has equation y = ax 2 The graph opens up if a>0 and down if a <0 y x (p,0) (q,0) 10

11 Example 1 Graph a quadratic function in intercept form: y = a(x-p)(x-q) Step 1: Identify the x-intercepts STEP 1 Step 2: Find the coordinates of the vertex STEP 2 Step 3: Draw parabolua through the vertex and x-intercepts STEP 3 STEP 4 11

12 Example 1 Graph a quadratic function in intercept form: y = a(x-p)(x-q) Step 1: Identify the x-intercepts STEP 1 Step 2: Find the coordinates of the vertex STEP 2 Step 3: Draw parabolua through the vertex and x-intercepts STEP 3 STEP 4 12

13 Example: Change intercept form to standard form 13

14 Example: Change intercept form to standard form 14

15 Example: Change vertex form to standard form 15

16 Example: Change vertex form to standard form 16

17 4.3 Solve 2 + x bx + c = 0 by factoring ac b 17

18 8 6 18

19 Factor x 2 5x

20 Factor x 2 + 7x

21 Factor x 2 1x 6 21

22 Solve x 2 + 3x 4 = 0 22

23 Solve x 2 + 5x 14 = 0 23

24 Solve x x = 27 24

25 Special Factoring Patterns Difference of Squares (DOS): a 2 b 2 = (a b)(a+b) Perfect Square Trinomial: a 2 + 2ab + b 2 = (a+b) 2 a 2 2ab + b 2 = (a b) 2 Example 1: 4x 2 9 Example 2: x 2 + 6x + 9 Example 3: x 2 8x

26 You try: 1. x x x x 2 18x

27 4.4 Solve 2 ax + bx + c = 0 by factoring Factor 2x 2 3x 20 27

28 Factor 3x x 4 28

29 Factor 7x 2 31x

30 Factor 6x 2 11x

31 Solve 5x 2 8x + 3 = 0 31

32 Solve 3x x + 15 = 0 32

33 Solve 7x x 30 = 0 33

34 Solve 2x 2 + 3x 5 = 0 34

35 Factor out monomials first 35

36 4.5 Solve Quadratic Equations by Finding S SQUARE ROOT OF A NUMBER if b 2 = a, then b is a square root of a example: if 3 2 = 9, then 3 is a square root of 9. VOCAB: MATH IS DUD positive square root: negative square root: - positive/negative square root: ± radicand: the number or expression inside a radical symbol. 36

37 Simplifying radicals Properties of Radicals Product property ab = a b Quotient property a a b = b 37

38 An expression with radicals is simplest form if the following true 1.) No perfect square factors other than 1 are in the radicand 2.) No fractions are in the radicand 3.) No radicals appear in the denominator of a fraction 38

39 Examples 39

40 Examples 40

41 Examples: rationalize denominators of fractions 41

42 Examples: solve a quadratic 42

43 Examples: solve a quadratic 43

44 4.6 Perform Operations with Comple Numbers Not all quadratic equations have real-number solutions. For example: x 2 = -1 has no real-number solutions because the square of any real number x is never a negative number To overcome this problem mathematicians created an expanded system of numbers using the imaginary unit i. Defined as not that 2 i = -1. The imaginary unit can be used to write the square root of any negative number. Property The square root of a negative number 1. If r is a positive real number, then Example 2. By Property (1), it follows that 44

45 example: 45

46 example: 46

47 Complex Numbers: A complex number written in standard form is a number a+bi where a and b are real numbers. The number a is the real part of the complex number and the bi is imaginary part. If b 0 then a + bi is an imaginary number. If a = 0 and b 0, then a+bi is a pure imaginary number. Complex Numbers: a + bi Real Numbers Imaginary Numbers Pure Imaginary Numbers 47

48 The sum and difference of complex numbe To add (subtract) two complex numbers, add (subtract) their real parts and their imaginary parts separately. 1. Sum of complex numbers: 2. Difference of complex numbers: examples: a: b: c: 48

49 examples: 1: 2: 3: 49

50 Multiply Complex Numbers: a: b: 50

51 Multiply Complex Numbers: 1: 2: 51

52 Complex Conjugates: Two complex numbers of the form a+bi and a bi are complex conjugates. The product of conjugates is real number. Divide Complex Numbers: use complex conjugates 52

53 Divide Complex Numbers: use complex conjugates 1: 2: 53

54 Plot Complex Numbers Plot the complex numbers in the same complex plane. a. b. c. d. 54

55 Find absolute values of complex numbers Absolute Vale of a Complex Number: The absolute value of a complex number z = a + bi, denoted is a nonnegative real number defined as This is the distance between z and the origin in the complex plane examples: a. b. 55

56 4.7 Complete the Square b x 2 + bx +( ) b 2 2 = (x + ) 2 2 To complete the square for the expression 2 x+ bx, add Diagram: x b x x x 2 bx x x 56

57 example 1: Solve a quadratic by finding square roots step 1: Write left side as binomial squared step 2: Take square root of each side. step 3: Solve for x 57

58 example 2: Make a perfect square trinomial Step 1: Find half of the coefficient of x Step 2: square the result of step 1 Step 3: Replace c with the result from step 2 58

59 example 3: Solve x 2 + bx + c (a = 1) step 1: write left side in the form x 2 +bx step 2: Complete the square and add it to both sides step 3: Write left side as a binomial squared step 4: Take square root of each side step 5: Solve for x (simplify if necessary) 59

60 example 4: Solve ax 2 + bx + c (a 1) step 1: divide everything by a step 2: write left side in the form x 2 + bx step 3: complete the square and add it to both sides step 4: write the left side as a binomial squared step 5: take square root of both sides step 6: solve for x, simplify if necessary 60

61 Extra Examples Solve the equation by completing the square 61

62 Write a quadratic function in vertex form step 1: complete the square and add it to both sides step 2: write beginning part as binomial squared step 3: solve for y 62

63 Write a quadratic function in vertex form 63

64 4.8 Use the quadratic formula and the dis Quadratic Formula: The solutions of the quadratic equation ax 2 +bx+c=0 are 64

65 Proving the Quadratic Formula! ax 2 +bx+c = 0 where a 0 65

66 Example 1: x 2 8x + 15 = 0 66

67 Example 2: 2x 2 + 6x + 2 = 1 67

68 Example 3: One solution 68

69 Example 4:One solution 69

70 Example 5: Imaginary Solution 70

71 Example 6: Imaginary Solution 71

72 Word Problems Objects Dropped: Objects Launched/Thrown: initial velocity initial height A juggler tosses a ball into the air. The ball leaves the juggler's hand 4 feet above the ground and has an initial vertical velocity of 40 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air? 72

73 Word Problems Objects Dropped: Objects Launched/Thrown: initial velocity initial height A basketball player passes the ball to a teammate. The ball leaves the player's hand 5 feet above the ground and has an initial vertical velocity of 55 feet per second. The teammate catches the ball when it returns to a height of 5 feet. How long is the ball in the air? 73

74 Word Problems The equation h = 16t t + 6 gives the height, h, in feet of a basketball as a function of t, in seconds a) What is the maximum height the ball reaches? b) At what time does the ball hit the ground? 74

75 Applications of the discrimina In the quadratic formula the expression inside the rad discriminant Discriminant b 2-4ac 75

76 Number of solutions of a quadratic Consider the quadratic equation ax 2 + bx + c = 0: If b 2 4ac is positive, then the equation has two solutions if b 2 4ac is zero, then the equation has one solution if b 2 4ac is negative, then the equation has no real solutions 76

77 Example: a. x 2 3x 4 b. x 2 + 2x 1 c. 2x 2 2x

78 Example: a. 3x 2 + 5x 1 b. x x 25 c. x 2 2x

79 Match the graph with the discriminant: a. b 2 4ac = 2 b. b 2 4ac = 0 c. b 2 4ac = 3 c A b 79

80 4.10 Write Quadratic Function Models Example 1: Write a quadratic function in vertex form y = a(x-h) 2 +K Vertex: (1, 2) Point: (3, 2) 80

81 Example 2: Write a quadratic function in vertex form y = a(x-h) 2 +K Vertex: (4, 5) Point: (2, 1) 81

82 Example 3: Write a quadratic function in intercept form y = a(x-p)(x-q) x intercepts: 2 and 5 Point: (6, 2) 82

83 Example 4: Write a quadratic function in intercept form y = a(x-p)(x-q) x intercepts: 1 and 4 Point: (3, 2) 83

84 Example 5: Write a quadratic function in Standard form y = ax 2 + bx + c points: ( 1, 3), (0, 4), (2, 6) 84

85 Example 6: Write a quadratic function in Standard form y = ax 2 + bx + c points: ( 1, 5), (0, 1), (2, 11) 85