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Slide P- 1. Chapter P Prerequisites P.1 Real Numbers.

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Presentation on theme: "Slide P- 1. Chapter P Prerequisites P.1 Real Numbers."— Presentation transcript:

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, <, ≥, and ≤ are inequality symbols.

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 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 = (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) = Complex Numbers

106 Slide P- 106 Show: (3 + 2i) (3 – 2i) = (3 + 2i) (3 – 2i) = (-2i) + 2i i (-2i) = 3 2 – 6i + 6i – 2 2 i 2 = 3 2 – 2 2 (-1) = = = 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 x < 8 x > Solve |x – 2| < 1 -1 < x – 2 < 1 1 < x < Solving Absolute Value Inequalities

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

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

117 Slide P- 117 Example Solving an Absolute Value Inequality

118 Slide P- 118 Example Solving a Quadratic Inequality

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) 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, ∞) 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] 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


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