Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter P Prerequisites
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.1 Real Numbers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 4 Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 5 Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 6 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 7 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,…}
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 8 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 9 The Real Number Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 10 Order of Real Numbers Let a and b be any two real numbers. SymbolDefinitionRead a>ba – b is positivea is greater than b a<ba – b is negativea is less than b a≥ba – b is positive or zeroa is greater than or equal to b a≤ba – b is negative or zeroa is less than or equal to b The symbols >, <, ≥, and ≤ are inequality symbols.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 11 Trichotomy Property Let a and b be any two real numbers. Exactly one of the following is true: a b.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 12 Example Interpreting Inequalities Describe the graph of x > 2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 13 Example Interpreting Inequalities Describe the graph of x > 2. The inequality describes all real numbers greater than 2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 14 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 15 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 16 Properties of Algebra
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 17 Properties of Algebra
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 18 Properties of the Additive Inverse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 19 Exponential Notation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 20 Properties of Exponents
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 21 Example Simplifying Expressions Involving Powers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 22 Example Simplifying Expressions Involving Powers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 23 Example Converting to Scientific Notation Convert to scientific notation.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 24 Example Converting to Scientific Notation Convert to scientific notation.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 25 Example Converting from Scientific Notation Convert 1.23 × 10 5 from scientific notation.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 26 Example Converting from Scientific Notation Convert 1.23 × 10 5 from scientific notation. 123,000
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.2 Cartesian Coordinate System
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 28 Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 29 Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 30 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 31 The Cartesian Coordinate Plane
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 32 Quadrants
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 33 Absolute Value of a Real Number
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 34 Properties of Absolute Value
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 35 Distance Formula (Number Line)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 36 Distance Formula (Coordinate Plane)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 37 The Distance Formula using the Pythagorean Theorem
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 38 Midpoint Formula (Number Line)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 39 Midpoint Formula (Coordinate Plane)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 40 Standard Form Equation of a Circle
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 41 Standard Form Equation of a Circle
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 42 Example Finding Standard Form Equations of Circles
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 43 Example Finding Standard Form Equations of Circles
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.3 Linear Equations and Inequalities
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 45 Quick Review
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 47 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 48 Properties of Equality
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 49 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 50 Operations for Equivalent Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 51 Example Solving a Linear Equation Involving Fractions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 52 Example Solving a Linear Equation Involving Fractions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 53 Linear Inequality in x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 54 Properties of Inequalities
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.4 Lines in the Plane
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 56 Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 57 Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 58 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 59 Slope of a Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 60 Slope of a Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 61 Example Finding the Slope of a Line Find the slope of the line containing the points (3,-2) and (0,1).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 62 Example Finding the Slope of a Line Find the slope of the line containing the points (3,-2) and (0,1).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 63 Point-Slope Form of an Equation of a Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 64 Point-Slope Form of an Equation of a Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 65 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 66 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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 67 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 68 Viewing Window
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 69 Parallel and Perpendicular Lines
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 70 Example Finding an Equation of a Parallel Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 71 Example Finding an Equation of a Parallel Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.5 Solving Equations Graphically, Numerically, and Algebraically
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 73 Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 74 Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 75 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 76 Example Solving by Finding x-Intercepts
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 77 Example Solving by Finding x-Intercepts
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 78 Zero Factor Property Let a and b be real numbers. If ab = 0, then a = 0 or b = 0.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 79 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 80 Completing the Square
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 81 Quadratic Equation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 82 Example Solving Using the Quadratic Formula
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 83 Example Solving Using the Quadratic Formula
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 84 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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 85 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 86 Example Solving by Finding Intersections
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 87 Example Solving by Finding Intersections
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.6 Complex Numbers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 89 Quick Review
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 91 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 92 Complex Number 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 93 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 94 Example Multiplying Complex Numbers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 95 Example Multiplying Complex Numbers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 96 Complex Conjugate
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 97 Discriminant of a Quadratic Equation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 98 Example Solving a Quadratic Equation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 99 Example Solving a Quadratic Equation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.7 Solving Inequalities Algebraically and Graphically
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 101 Quick Review
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 103 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 104 Solving Absolute Value Inequalities
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 105 Example Solving an Absolute Value Inequality
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 106 Example Solving an Absolute Value Inequality
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 107 Example Solving a Quadratic Inequality
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 108 Example Solving a Quadratic Inequality
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 109 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 110 Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 111 Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 112 Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 113 Chapter Test Solutions