Chapter P Prerequisites

P.1 Real Numbers

Quick Review

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.

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,…}

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.

The Real Number Line

Order of Real Numbers Let a and b be any two real numbers. Symbol Definition Read a>b a – b is positive a is greater than b a<b a – b is negative a is less than b a≥b a – b is positive or zero a is greater than or equal to b a≤b a – b is negative or zero a is less than or equal to b The symbols >, <, ≥, and ≤ are inequality symbols.

Trichotomy Property Let a and b be any two real numbers. Exactly one of the following is true: a < b, a = b, or a > b.

Example Interpreting Inequalities Describe the graph of x > 2.

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

Bounded Intervals of Real Numbers Let a and b be real numbers with a < b. Interval Notation Inequality 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.

Unbounded Intervals of Real Numbers Let a and b be real numbers. Interval Notation Inequality 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.

Graphing Inequalities x > 2 (2,) x < -3 (-,-3] -1< x < 5 (-1,5]

Properties of Algebra

Properties of Algebra

Properties of the Additive Inverse

Exponential Notation

Properties of Exponents

Example Simplifying Expressions Involving Powers

Example Converting to Scientific Notation Convert 0.0000345 to scientific notation.

Example Converting from Scientific Notation Convert 1.23 × 105 from scientific notation. 123,000

Cartesian Coordinate System P.2 Cartesian Coordinate System

Quick Review Solutions

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.

The Cartesian Coordinate Plane

Quadrants

Absolute Value of a Real Number

Properties of Absolute Value

Distance Formula (Number Line)

Distance Formula (Coordinate Plane)

The Distance Formula using the Pythagorean Theorem

Midpoint Formula (Number Line)

Midpoint Formula (Coordinate Plane)

Distance and Midpoint Example 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)

Example Problem 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 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)

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

Standard Form Equation of a Circle

Standard Form Equation of a Circle

Example Finding Standard Form Equations of Circles

Linear Equations and Inequalities P.3 Linear Equations and Inequalities

Quick Review

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.

Properties of Equality

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.

Operations for Equivalent Equations

Example Solving a Linear Equation Involving Fractions

Linear Inequality in x

Properties of Inequalities

P.4 Lines in the Plane

Quick Review

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.

Slope of a Line

Slope of a Line

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

Point-Slope Form of an Equation of a Line

Point-Slope Form of an Equation of a Line

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.

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 – y1 = m(x – x1) Vertical line: x = a Horizontal line: y = b

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

Viewing Window

Parallel and Perpendicular Lines

Example Finding an Equation of a Parallel Line or y = mx + b

Example 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.

Solving Equations Graphically, Numerically, and Algebraically

Quick Review Solutions

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.

Example Solving by Finding x-Intercepts

Example Solving by Finding x-Intercepts

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

Quadratic Equation in x A quadratic equation in x is one that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers with a ≠ 0.

Completing the Square

Quadratic Equation

Example Solving Using the Quadratic Formula

Solving Quadratic Equations Algebraically These are four basic ways to solve quadratic equations algebraically. Factoring Extracting Square Roots Completing the Square Using the Quadratic Formula

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.

Example Solving by Finding Intersections

Example Solving by Finding Intersections

P.6 Complex Numbers

Quick Review

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.

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

Complex Numbers 10x – x2 = 40 x2 – 10x = -40 x2 – 10x + 25 = -40 +25

Complex Numbers

Complex Numbers The imaginary number i is the square root of –1.

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

If a and b are real numbers, then a + bi is a complex number. Complex Numbers 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 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.

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

Complex Numbers

* i -1 -i 1 i -1 -i 1 i -1 -i 1 i -1 -i 1 i -1 -i i -1 -i 1 1 Complex Numbers * i -1 -i 1 i -1 -i 1 i -1 -i 1 i -1 -i 1 i -1 -i i -1 -i 1 1

Complex Numbers Evaluate:

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.

Example Multiplying Complex Numbers

Example Multiplying Complex Numbers

Complex Conjugate

Discriminant of a Quadratic Equation

Example Solving a Quadratic Equation

Example Solving a Quadratic Equation

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

Complex Numbers 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 Show: (3 + 2i) (3 – 2i) = 3 2 + 2 2. = 3 2 – 2 2(-1) (3 + 2i) (3 – 2i) = 3.3 + 3(-2i) + 2i .3 + 2i (-2i) = 3 2 – 6i + 6i – 2 2i 2 = 3 2 – 2 2(-1) = 3 2 + 2 2 = 9 + 4 = 13

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

Complex Numbers

Complex Numbers

Solving Inequalities Algebraically and Graphically

Quick Review

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.

Solving Absolute Value Inequalities

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

Solving Absolute Value Inequalities 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 < -2 or x > 4 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4

Solving Absolute Value Inequalities |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 or 3x – 1 > 2 3x < -1 or 3x > 3 x < -1/3 or x > 1 ] [ -1 0 1 2 3 4 5

Example Solving an Absolute Value Inequality

Example Solving a Quadratic Inequality +++0--------0+++ -2 -1

Example Solving a Quadratic Inequality Solve x2 – x – 20 < 0 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 +++0-------0+++ -4 5 Solution is (-4,5)

Example Solving a Quadratic Inequality Solve x2 – 3x > 0 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 +++0------0+++ 0 3 Solution is (-∞,0) or (3, ∞)

Example Solving a Quadratic Inequality Solve x3 – 6x2 + 8x < 0 Find critical numbers x(x2 – 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 -----0++0----0+++ 0 2 4 Solution is (-∞,0] U [2,4]

Projectile Motion Suppose an object is launched vertically from a point so feet above the ground with an initial velocity of vo feet per second. The vertical position s (in feet) of the object t seconds after it is launched is s = -16t2 + vot + so.

Chapter Test

Chapter Test