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file:///E|/scan/contents.txt Algebra and Trigonometry, 3rd Ed Author: Judith A Beecher, Judith A Penna, Marvin L Bittinger Publisher: Addison Wesley; (February 2, 2007) Format: PDF ISBN-10: 0321466209 ISBN-13: 9780321466204 Pages: 1058 Size: 22.3 Mb Home Page: http://www.aw-bc.com/catalog/academic/product/0,1144,0321466209,00.html A book to really help students visualize mathematics for better comprehension By creating algebraic visual side-by-sides to solve various problems in the examples, the authors show students the relationship of the algebraic solution with the visual, often graphical, solution In addition, the authors have added a variety of new tools to help students better use the book for maximum effectiveness to not only pass the course, but truly understand the material Table of Contents Chapter R Basic Concepts of Algebra R.1 The Real-Number System R.2 Integer Exponents, Scientific Notation, and Order of Operations R.3 Addition, Subtraction, and Multiplication of Polynomials R.4 Factoring R.5 Rational Expressions R.6 Radical Notation and Rational Exponents R.7 The Basics of Equation Solving Chapter Graphs, Functions, and Models 1.1 Introduction to Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling 1.5 More on Functions 1.6 The Algebra of Functions 1.7 Symmetry and Transformations Chapter Functions, Equations, and Inequalities 2.1 Linear Equations, Functions, and Models file:///E|/scan/contents.txt (1 of 3) [15.12.2007 17:03:31] file:///E|/scan/contents.txt 2.2 2.3 2.4 2.5 2.6 The Complex Numbers Quadratic Equations, Functions, and Models Analyzing Graphs of Quadratic Functions More Equation Solving Solving Linear Inequalities Chapter Polynomial And Rational Functions 3.1 Polynomial Functions and Models 3.2 Graphing Polynomial Functions 3.3 Polynomial Division; The Remainder and Factor Theorems 3.4 Theorems about Zeros of Polynomial Functions 3.5 Rational Functions 3.6 Polynomial and Rational Inequalities 3.7 Variation and Applications Chapter Exponential and Logarithmic Functions 4.1 Inverse Functions 4.2 Exponential Functions and Graphs 4.3 Logarithmic Functions and Graphs 4.4 Properties of Logarithmic Functions 4.5 Solving Exponential and Logarithmic Equations 4.6 Applications and Models: Growth and Decay, and Compound Interest Chapter The Trigonometric Functions 5.1 Trigonometric Functions of Acute Angles 5.2 Applications of Right Triangles 5.3 Trigonometric Functions of Any Angle 5.4 Radians, Arc Length, and Angular Speed 5.5 Circular Functions: Graphs and Properties 5.6 Graphs of Transformed Sine and Cosine Functions Chapter Trigonometric Identities, Inverse Functions, and Equations 6.1 Identities: Pythagorean and Sum and Difference 6.2 Identities: Cofunction, Double-Angle, and Half-Angle 6.3 Proving Trigonometric Identities 6.4 Inverses of the Trigonometric Functions 6.5 Solving Trigonometric Equations Chapter Applications of Trigonometry 7.1 The Law of Sines 7.2 The Law of Cosines 7.3 Complex Numbers: Trigonometric Form 7.4 Polar Coordinates and Graphs file:///E|/scan/contents.txt (2 of 3) [15.12.2007 17:03:31] file:///E|/scan/contents.txt 7.5 Vectors and Applications 7.6 Vector Operations Chapter Systems of Equations and Matrices 8.1 Systems of Equations in Two Variables 8.2 Systems of Equations in Three Variables 8.3 Matrices and Systems of Equations 8.4 Matrix Operations 8.5 Inverses of Matrices 8.6 Determinants and Cramer's Rule 8.7 Systems of Inequalities and Linear Programming 8.8 Partial Fractions Chapter Analytic Geometry Topics 9.1 The Parabola 9.2 The Circle and the Eclipse 9.3 The Hyperbola 9.4 Nonlinear Systems of Equations and Inequalities 9.5 Rotation of Axes 9.6 Polar Equations of Conics 9.7 Parametric Equations Chapter 10 Sequences, Series, and Combinatorics 10.1 Sequences and Series 10.2 Arithmetic Sequences and Series 10.3 Geometric Sequences and Series 10.4 Mathematical Induction 10.5 Combinatorics: Permutations 10.6 Combinatorics: Combinations 10.7 The Binomial Theorem 10.8 Probability file:///E|/scan/contents.txt (3 of 3) [15.12.2007 17:03:31] BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page Basic Concepts of Algebra R.1 R.2 R.3 R.4 R.5 R.6 R.7 The Real-Number System Integer Exponents, Scientific Notation, and Order of Operations Addition, Subtraction, and Multiplication of Polynomials Factoring Rational Expressions Radical Notation and Rational Exponents The Basics of Equation Solving R SUMMARY AND REVIEW TEST A P P L I C A T I O N G ina wants to establish a college fund for her newborn daughter that will have accumulated $120,000 at the end of 18 yr If she can count on an interest rate of 6%, compounded monthly, how much should she deposit each month to accomplish this? This problem appears as Exercise 95 in Section R.2 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP Chapter R 12/2/04 2:42 PM Page • Basic Concepts of Algebra Identify various kinds of real numbers Use interval notation to write a set of numbers Identify the properties of real numbers Find the absolute value of a real number R.1 2.1 The Real-Number Polynomial Functions System and Modeling Real Numbers In applications of algebraic concepts, we use real numbers to represent quantities such as distance, time, speed, area, profit, loss, and temperature Some frequently used sets of real numbers and the relationships among them are shown below Natural numbers (positive integers): 1, 2, 3, … Whole numbers: 0, 1, 2, 3, … Integers: …, −3, −2, −1, 0, 1, 2, 3, … Rational numbers Rational numbers that are not integers: Real numbers Negative integers: −1, −2, −3, … 19 −7 −, − −, −−, −−, 8.3, −5 − Irrational numbers: Zero: 0.56, … √2, p, −√3, √27, −4.030030003…, … Numbers that can be expressed in the form p͞q, where p and q are integers and q 0, are rational numbers Decimal notation for rational numbers either terminates (ends) or repeats Each of the following is a rational number a) b) Ϫ7 0.25 d) Ϫ Ϫ0.45 11 c) for any nonzero integer a a ؊7 ؊7 ؍ , or ؊1 0؍ Terminating decimal Repeating decimal Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page Section R.1 • The Real-Number System The real numbers that are not rational are irrational numbers Decimal notation for irrational numbers neither terminates nor repeats Each of the following is an irrational number There is no repeating block of digits a) 3.1415926535 22 ͑ and 3.14 are rational approximations of the irrational number ͒ b) ͙2 1.414213562 c) Ϫ6.12122122212222 There is no repeating block of digits Although there is a pattern, there is no repeating block of digits The set of all rational numbers combined with the set of all irrational numbers gives us the set of real numbers The real numbers are modeled using a number line, as shown below Each point on the line represents a real number, and every real number is represented by a point on the line Ϫ2.9 ϪE Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 ͙3 * p The order of the real numbers can be determined from the number line If a number a is to the left of a number b, then a is less than b ͑a Ͻ b͒ Similarly, a is greater than b ͑a Ͼ b͒ if a is to the right of b on the number line For example, we see from the number line above that 17 Ϫ2.9 Ͻ Ϫ 35 , because Ϫ2.9 is to the left of Ϫ 35 Also, Ͼ ͙3, because 174 is to the right of ͙3 The statement a Յ b, read “a is less than or equal to b,” is true if either a Ͻ b is true or a b is true The symbol ʦ is used to indicate that a member, or element, belongs to a set Thus if we let ޑrepresent the set of rational numbers, we can see from the diagram on page that 0.56 ʦ ޑ We can also write ͙2 ޑto indicate that ͙2 is not an element of the set of rational numbers When all the elements of one set are elements of a second set, we say that the first set is a subset of the second set The symbol ʕ is used to denote this For instance, if we let ޒrepresent the set of real numbers, we can see from the diagram that ޑʕ ( ޒread “ ޑis a subset of )”ޒ Interval Notation Sets of real numbers can be expressed using interval notation For example, for real numbers a and b such that a Ͻ b, the open interval ͑a, b͒ is the set of real numbers between, but not including, a and b That is, ( a [ a ) (a, b) [a, ∞) b ͑a, b͒ ͕x ͉ a Ͻ x Ͻ b͖ The points a and b are endpoints of the interval The parentheses indicate that the endpoints are not included in the interval Some intervals extend without bound in one or both directions The interval ͓a, ϱ ͒, for example, begins at a and extends to the right without bound That is, ͓a, ϱ ͒ ͕x ͉ x Ն a͖ The bracket indicates that a is included in the interval Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP Chapter R 12/2/04 2:42 PM Page • Basic Concepts of Algebra The various types of intervals are listed below Intervals: Types, Notation, and Graphs TYPE INTERVAL NOTATION SET NOTATION Open ͑a, b͒ ͕x ͉ a Ͻ x Ͻ b͖ Closed Half-open Half-open Open ͓a, b͔ ͓a, b͒ ͑a, b͔ ͑a, ϱ ͒ GRAPH ͕x ͉ a Յ x Յ b͖ ͕x ͉ a Յ x Ͻ b͖ ͕x ͉ a Ͻ x Յ b͖ ͕x ͉ x Ͼ a͖ ( ) a b [ ] a b [ ) a b ( ] a b ( a Half-open ͓a, ϱ ͒ ͕x ͉ x Ն a͖ [ a Open ͑Ϫϱ, b͒ ͕x ͉ x Ͻ b͖ ) b Half-open ͑Ϫϱ, b͔ ͕x ͉ x Յ b͖ ] b The interval ͑Ϫϱ, ϱ ͒, graphed below, names the set of all real numbers, ޒ EXAMPLE Write interval notation for each set and graph the set a) ͕x ͉ Ϫ4 Ͻ x Ͻ 5͖ c) ͕x ͉ Ϫ5 Ͻ x Յ Ϫ2͖ b) ͕x ͉ x Ն 1.7͖ d) ͕ x ͉ x Ͻ ͙5 ͖ Solution a) ͕x ͉ Ϫ4 Ͻ x Ͻ 5͖ ͑Ϫ4, 5͒; Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 3 5 b) ͕x ͉ x Ն 1.7͖ ͓1.7, ϱ ͒; Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 c) ͕x ͉ Ϫ5 Ͻ x Յ Ϫ2͖ ͑Ϫ5, Ϫ2͔; Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page Section R.1 d) ͕ x ͉ x Ͻ ͙5 ͖ ͑ Ϫϱ, ͙5 ͒; • The Real-Number System Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 5 Properties of the Real Numbers The following properties can be used to manipulate algebraic expressions as well as real numbers Properties of the Real Numbers For any real numbers a, b, and c: a ϩ b b ϩ a and ab ba Commutative properties of addition and multiplication a ϩ ͑b ϩ c͒ ͑a ϩ b͒ ϩ c and a͑bc͒ ͑ab͒c Associative properties of addition and multiplication aϩ00ϩaa Additive identity property Ϫa ϩ a a ϩ ͑Ϫa͒ Additive inverse property aи11иaa Multiplicative identity property aи 1 и a ͑a a a a͑b ϩ c͒ ab ϩ ac 0͒ Multiplicative inverse property Distributive property Note that the distributive property is also true for subtraction since a͑b Ϫ c͒ a͓b ϩ ͑Ϫc͔͒ ab ϩ a͑Ϫc͒ ab Ϫ ac EXAMPLE State the property being illustrated in each sentence a) и и c) 14 ϩ ͑Ϫ14͒ e) 2͑a Ϫ b͒ 2a Ϫ 2b Solution SENTENCE b) ϩ ͑m ϩ n͒ ͑5 ϩ m͒ ϩ n d) и и PROPERTY a) и и b) c) d) e) Commutative property of multiplication: ab ba ϩ ͑m ϩ n͒ ͑5 ϩ m͒ ϩ n Associative property of addition: a ϩ ͑b ϩ c͒ ͑a ϩ b͒ ϩ c 14 ϩ ͑Ϫ14͒ Additive inverse property: a ϩ ͑Ϫa͒ 6и11и66 Multiplicative identity property: aи11иaa Distributive property: 2͑a Ϫ b͒ 2a Ϫ 2b a͑b ϩ c͒ ab ϩ ac Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP Chapter R 12/2/04 2:42 PM Page • Basic Concepts of Algebra Absolute Value The number line can be used to provide a geometric interpretation of absolute value The absolute value of a number a, denoted ͉a͉, is its distance from on the number line For example, ͉Ϫ5͉ 5, because the 3 distance of Ϫ5 from is Similarly, , because the distance of from is ԽԽ Absolute Value For any real number a, ͉a͉ ͭ a, if a Ն 0, Ϫa, if a Ͻ When a is nonnegative, the absolute value of a is a When a is negative, the absolute value of a is the opposite, or additive inverse, of a Thus, ͉a͉ is never negative; that is, for any real number a, ͉a͉ Ն Absolute value can be used to find the distance between two points on the number line a Distance Between Two Points on the Number Line For any real numbers a and b, the distance between a and b is ͉a Ϫ b͉, or equivalently, ͉b Ϫ a͉ b ͉a Ϫ b͉ ϭ ͉b Ϫ a͉ GCM EXAMPLE Solution Find the distance between Ϫ2 and The distance is ͉Ϫ2 Ϫ 3͉ ͉Ϫ5͉ 5, or equivalently, ͉3 Ϫ ͑Ϫ2͉͒ ͉3 ϩ 2͉ ͉5͉ We can also use the absolute-value operation on a graphing calculator to find the distance between two points On many graphing calculators, absolute value is denoted “abs” and is found in the MATH NUM menu and also in the CATALOG abs (Ϫ2Ϫ3) abs (3Ϫ(Ϫ2)) 5 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley ... Functions and Graphs 4.3 Logarithmic Functions and Graphs 4.4 Properties of Logarithmic Functions 4.5 Solving Exponential and Logarithmic Equations 4.6 Applications and Models: Growth and Decay, and. .. Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling 1.5 More on Functions 1.6 The Algebra of Functions 1.7 Symmetry and Transformations...file:///E|/scan/contents.txt Algebra and Trigonometry, 3rd Ed Author: Judith A Beecher, Judith A Penna, Marvin L Bittinger Publisher: Addison Wesley; (February 2, 2007) Format: PDF ISBN-10: 0321466209
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