Download presentation

Presentation is loading. Please wait.

Published bySyed Swinton Modified over 2 years ago

1
Slide P- 1

2
Chapter P Prerequisites

3
P.1 Real Numbers

4
Slide P- 4 Quick Review

5
Slide P- 5 What you’ll learn about Representing Real Numbers Order and Interval Notation Basic Properties of Algebra Integer Exponents Scientific Notation … and why These topics are fundamental in the study of mathematics and science.

6
Slide P- 6 Real Numbers A real number is any number that can be written as a decimal. Subsets of the real numbers include: The natural (or counting) numbers: {1,2,3…} The whole numbers: {0,1,2,…} The integers: {…,-3,-2,-1,0,1,2,3,…}

7
Slide P- 7 Rational Numbers Rational numbers can be represented as a ratio a/b where a and b are integers and b ≠ 0. The decimal form of a rational number either terminates or is indefinitely repeating.

8
Slide P- 8 The Real Number Line

9
Slide P- 9 Order of Real Numbers Let a and b be any two real numbers. SymbolDefinitionRead a>ba – b is positivea is greater than b a

10
Slide P- 10 Trichotomy Property Let a and b be any two real numbers. Exactly one of the following is true: a b.

11
Slide P- 11 Example Interpreting Inequalities Describe the graph of x > 2.

12
Slide P- 12 Example Interpreting Inequalities Describe the graph of x > 2. The inequality describes all real numbers greater than 2.

13
Slide P- 13 Bounded Intervals of Real Numbers Let a and b be real numbers with a < b. Interval NotationInequality Notation [a,b]a ≤ x ≤ b (a,b)a < x < b [a,b)a ≤ x < b (a,b]a < x ≤ b The numbers a and b are the endpoints of each interval.

14
Slide P- 14 Unbounded Intervals of Real Numbers Let a and b be real numbers. Interval NotationInequality Notation [a,∞)x ≥ a (a, ∞)x > a (-∞,b]x ≤ b (-∞,b)x < b Each of these intervals has exactly one endpoint, namely a or b.

15
Slide P- 15 Graphing Inequalities x > 2 x < -3 (- ,-3] (2, ) -1< x < 5 (-1,5]

16
Slide P- 16 Properties of Algebra

17
Slide P- 17 Properties of Algebra

18
Slide P- 18 Properties of the Additive Inverse

19
Slide P- 19 Exponential Notation

20
Slide P- 20 Properties of Exponents

21
Slide P- 21 Example Simplifying Expressions Involving Powers

22
Slide P- 22 Example Converting to Scientific Notation Convert 0.0000345 to scientific notation.

23
Slide P- 23 Example Converting from Scientific Notation Convert 1.23 × 10 5 from scientific notation. 123,000

24
P.2 Cartesian Coordinate System

25
Slide P- 25 Quick Review Solutions

26
Slide P- 26 What you’ll learn about Cartesian Plane Absolute Value of a Real Number Distance Formulas Midpoint Formulas Equations of Circles Applications … and why These topics provide the foundation for the material that will be covered in this textbook.

27
Slide P- 27 The Cartesian Coordinate Plane

28
Slide P- 28 Quadrants

29
Slide P- 29 Absolute Value of a Real Number

30
Slide P- 30 Properties of Absolute Value

31
Slide P- 31 Distance Formula (Number Line)

32
Slide P- 32 Distance Formula (Coordinate Plane)

33
Slide P- 33 The Distance Formula using the Pythagorean Theorem

34
Slide P- 34 Midpoint Formula (Number Line)

35
Slide P- 35 Midpoint Formula (Coordinate Plane)

36
Slide P- 36 Find the distance and midpoint for the line segment joined by A(-2,3) and B(4,1). A(-2,3) B(4,1) = (1,2) Distance and Midpoint Example

37
Slide P- 37 Show that A(4,1), B(0,3), and C(6,5) are vertices of an isosceles triangle. A(4,1) B(0,3) C(6,5) Since d(AC) = d(AB), ΔABC is isosceles Example Problem

38
Slide P- 38 P is a point on the y-axis that is 5 units from the point Q (3,7). Find P. P Q(3,7) (0,y) y = 3, y = 11 The point P is (0,3) or (0,11) Example

39
Slide P- 39 Prove that the diagonals of a rectangle are congruent. Coordinate Proofs Given ABCD is a rectangle. Prove AC = BD A(0,0) B(0,a) D(b,0) C(b,a) Since AC= BD, the diagonals of a square are congruent

40
Slide P- 40

41
Slide P- 41 Standard Form Equation of a Circle

42
Slide P- 42 Standard Form Equation of a Circle

43
Slide P- 43 Example Finding Standard Form Equations of Circles

44
P.3 Linear Equations and Inequalities

45
Slide P- 45 Quick Review

46
Slide P- 46 What you’ll learn about Equations Solving Equations Linear Equations in One Variable Linear Inequalities in One Variable … and why These topics provide the foundation for algebraic techniques needed throughout this textbook.

47
Slide P- 47 Properties of Equality

48
Slide P- 48 Linear Equations in x A linear equation in x is one that can be written in the form ax + b = 0, where a and b are real numbers with a ≠ 0.

49
Slide P- 49 Operations for Equivalent Equations

50
Slide P- 50 Example Solving a Linear Equation Involving Fractions

51
Slide P- 51 Linear Inequality in x

52
Slide P- 52 Properties of Inequalities

53
P.4 Lines in the Plane

54
Slide P- 54 Quick Review

55
Slide P- 55 What you’ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables Parallel and Perpendicular Lines Applying Linear Equations in Two Variables … and why Linear equations are used extensively in applications involving business and behavioral science.

56
Slide P- 56 Slope of a Line

57
Slide P- 57 Slope of a Line

58
Slide P- 58 Example Finding the Slope of a Line Find the slope of the line containing the points (3,-2) and (0,1).

59
Slide P- 59 Point-Slope Form of an Equation of a Line

60
Slide P- 60 Point-Slope Form of an Equation of a Line

61
Slide P- 61 Slope-Intercept Form of an Equation of a Line The slope-intercept form of an equation of a line with slope m and y-intercept (0,b) is y = mx + b.

62
Slide P- 62 Forms of Equations of Lines General form: Ax + By + C = 0, A and B not both zero Slope-intercept form: y = mx + b Point-slope form: y – y 1 = m(x – x 1 ) Vertical line: x = a Horizontal line: y = b

63
Slide P- 63 Graphing with a Graphing Utility To draw a graph of an equation using a grapher: 1. Rewrite the equation in the form y = (an expression in x). 2. Enter the equation into the grapher. 3. Select an appropriate viewing window. 4. Press the “graph” key.

64
Slide P- 64 Viewing Window

65
Slide P- 65 Parallel and Perpendicular Lines

66
Slide P- 66 Example Finding an Equation of a Parallel Line or y = mx + b

67
Slide P- 67 Determine the equation of the line (written in standard form) that passes through the point (-2, 3) and is perpendicular to the line 2y – 3x = 5. Example

68
P.5 Solving Equations Graphically, Numerically, and Algebraically

69
Slide P- 69 Quick Review Solutions

70
Slide P- 70 What you’ll learn about Solving Equations Graphically Solving Quadratic Equations Approximating Solutions of Equations Graphically Approximating Solutions of Equations Numerically with Tables Solving Equations by Finding Intersections … and why These basic techniques are involved in using a graphing utility to solve equations in this textbook.

71
Slide P- 71 Example Solving by Finding x-Intercepts

72
Slide P- 72 Example Solving by Finding x-Intercepts

73
Slide P- 73 Zero Factor Property Let a and b be real numbers. If ab = 0, then a = 0 or b = 0.

74
Slide P- 74 Quadratic Equation in x A quadratic equation in x is one that can be written in the form ax 2 + bx + c = 0, where a, b, and c are real numbers with a ≠ 0.

75
Slide P- 75 Completing the Square

76
Slide P- 76 Quadratic Equation

77
Slide P- 77 Example Solving Using the Quadratic Formula

78
Slide P- 78 Solving Quadratic Equations Algebraically These are four basic ways to solve quadratic equations algebraically. 1. Factoring 2. Extracting Square Roots 3. Completing the Square 4. Using the Quadratic Formula

79
Slide P- 79 Agreement about Approximate Solutions For applications, round to a value that is reasonable for the context of the problem. For all others round to two decimal places unless directed otherwise.

80
Slide P- 80 Example Solving by Finding Intersections

81
Slide P- 81 Example Solving by Finding Intersections

82
P.6 Complex Numbers

83
Slide P- 83 Quick Review

84
Slide P- 84 What you’ll learn about Complex Numbers Operations with Complex Numbers Complex Conjugates and Division Complex Solutions of Quadratic Equations … and why The zeros of polynomials are complex numbers.

85
Slide P- 85 Find two numbers whose sum is 10 and whose product is 40. x = 1 st number 10 – x = 2 nd number x(10 – x) = 40 Complex Numbers

86
Slide P- 86 x(10 – x) = 40 10x – x 2 = 40 x 2 – 10x = -40 x 2 – 10x + 25 = -40 +25 (x – 5) 2 = -15 Complex Numbers

87
Slide P- 87 Complex Numbers

88
Slide P- 88 The imaginary number i is the square root of –1. Complex Numbers

89
Slide P- 89 Imaginary numbers are not real numbers, so all the rules do not apply. Example: The product rule does not apply: Complex Numbers

90
Slide P- 90 If a and b are real numbers, then a + bi is a complex number. a is the real part. bi is the imaginary part. The set of complex numbers consist of all the real numbers and all the imaginary numbers Complex Numbers

91
Slide P- 91 A complex number is any number that can be written in the form a + bi, where a and b are real numbers. The real number a is the real part, the real number b is the imaginary part, and a + bi is the standard form. Complex Numbers

92
Slide P- 92 Examples of complex numbers: 3 + 2i 8 - 2i 4 (since it can be written as 4 + 0i). The real numbers are a subset of the complex numbers. -3i (since it can be written as 0 – 3i). Complex Numbers

93
Slide P- 93 Complex Numbers

94
Slide P- 94 Complex Numbers

95
Slide P- 95 * i -1 -i 1 -i 1 i 1 i -1 1 i -i i -i 1 i -1 1 Complex Numbers

96
Slide P- 96 Evaluate: Complex Numbers

97
Slide P- 97 Addition and Subtraction of Complex Numbers If a + bi and c + di are two complex numbers, then Sum: (a + bi ) + (c + di ) = (a + c) + (b + d)i, Difference: (a + bi ) – (c + di ) = (a - c) + (b -d)i.

98
Slide P- 98 Example Multiplying Complex Numbers

99
Slide P- 99 Example Multiplying Complex Numbers

100
Slide P- 100 Complex Conjugate

101
Slide P- 101 Discriminant of a Quadratic Equation

102
Slide P- 102 Example Solving a Quadratic Equation

103
Slide P- 103 Example Solving a Quadratic Equation

104
Slide P- 104 When dividing a complex number by a real number, divide each part of the complex number by the real number. Complex Numbers

105
Slide P- 105 The numbers (a + bi ) and (a – bi ) are complex conjugates. The product (a + bi )·(a – bi ) is the real number a 2 + b 2. Show: (3 + 2i) (3 – 2i) = 3 2 + 2 2. Complex Numbers

106
Slide P- 106 Show: (3 + 2i) (3 – 2i) = 3 2 + 2 2. (3 + 2i) (3 – 2i) = 3. 3 + 3(-2i) + 2i. 3 + 2i (-2i) = 3 2 – 6i + 6i – 2 2 i 2 = 3 2 – 2 2 (-1) = 3 2 + 2 2 = 9 + 4 = 13 Complex Numbers

107
Slide P- 107 When dividing a complex number by a complex number, multiply the denominator and numerator by the conjugate of the denominator. Complex Numbers

108
Slide P- 108 Complex Numbers

109
Slide P- 109 Complex Numbers

110
P.7 Solving Inequalities Algebraically and Graphically

111
Slide P- 111 Quick Review

112
Slide P- 112 What you’ll learn about Solving Absolute Value Inequalities Solving Quadratic Inequalities Approximating Solutions to Inequalities Projectile Motion … and why These techniques are involved in using a graphing utility to solve inequalities in this textbook.

113
Slide P- 113 Solving Absolute Value Inequalities

114
Slide P- 114 Solve 2x – 3 < 4x + 5 -2x < 8 x > -4 -5 -4 -3 Solve |x – 2| < 1 -1 < x – 2 < 1 1 < x < 3 0 1 2 3 4 Solving Absolute Value Inequalities

115
Slide P- 115 Solve -1 < 3 – 2x < 5 -4 < -2x < 2 2 > x > -1 -1 < x < 2 Solve |x – 1| > 3 -3 > x – 1 or x – 1 > 3 -2 > x or x > 4 x 4 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 Solving Absolute Value Inequalities

116
Slide P- 116 |2x – 6| < 4 -4 < 2x – 6 < 4 2 < 2x < 10 1< x < 5 -1 0 1 2 3 4 5 ( ) |3x – 1| > 2 3x – 1 2 3x 3 x 1 -1 0 1 2 3 4 5 ] [ Solving Absolute Value Inequalities

117
Slide P- 117 Example Solving an Absolute Value Inequality

118
Slide P- 118 Example Solving a Quadratic Inequality -2 -1 +++0--------0+++

119
Slide P- 119 Solve x 2 – x – 20 < 0 1.Find critical numbers (x + 4)(x - 5) < 0 x = -4, x = 5 2. Test Intervals (-∞,-4) (-4,5) and (5, ∞) 3. Choose a sample in each interval x = -5 (-5) 2 – (-5) – 20 = Positive x = 0 (0) 2 - (0) - 20 = Negative x = 6 (6) 2 – 3(6) = Positive Solution is (-4,5) -4 5 +++0-------0+++ Example Solving a Quadratic Inequality

120
Slide P- 120 Solve x 2 – 3x > 0 1.Find critical numbers x(x - 3) > 0 x = 0, x = 3 2. Test Intervals (-∞,0) (0,3) and (3, ∞) 3. Choose a sample in each interval x = -1 (-1) 2 – 3(-1) = Positive x = 1 (1) 2 - 3(1) = Negative x = 4 (4) 2 – 3(4) = Positive Solution is (-∞,0) or (3, ∞) 0 3 +++0------0+++ Example Solving a Quadratic Inequality

121
Slide P- 121 Solve x 3 – 6x 2 + 8x < 0 1.Find critical numbers x(x 2 – 6x + 8) < 0 x(x – 2)(x – 4) x = 0, x = 2, x = 4 2. Test Intervals (-∞,0) (0,2) (2,4) and (4, ∞) 3. Choose a sample in each interval x = -5 (-5) 3 – 6(-5) 2 + 8(-5) = Negative x = 1 (-1) 3 – 6(-1) 2 + 8(-1) = Positive x = 3 (3) 3 – 6(3) 2 + 8(3) = Negative x = 5 (5) 3 – 6(5) 2 + 8(5) = Positive Solution is (-∞,0] U [2,4] 0 2 4 -----0++0----0+++ Example Solving a Quadratic Inequality

122
Slide P- 122 Projectile Motion Suppose an object is launched vertically from a point s o feet above the ground with an initial velocity of v o feet per second. The vertical position s (in feet) of the object t seconds after it is launched is s = -16t 2 + v o t + s o.

123
Slide P- 123 Chapter Test

124
Slide P- 124 Chapter Test

Similar presentations

Presentation is loading. Please wait....

OK

1 Radio Maria World. 2 Postazioni Transmitter locations.

1 Radio Maria World. 2 Postazioni Transmitter locations.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on cross docking logistics Ppt on bluetooth controlled robot car Ppt on kindness is contagious Ppt on physical methods of gene transfer in plants Ppt on artificial intelligence and neural networks Ppt on chemical properties of metals and nonmetals Ppt on computer graphics by baker Ppt on phonetic transcription vowels Free ppt on obesity Ppt on electrical appliances and their uses