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Coach is the leader in standards-based, state-customized instruction for grades K–12 in all assessed subjects. Our student texts deliver everything you need to meet your state standards and prepare your class for grade-level success! Virginia End-of-Course Coach, Algebra II Your Complete Test Preparation Program! Coach lessons have just what you’re looking for: ✔✔ Easy-to-follow,✔predictable✔lesson✔plans ✔✔ Focused✔instruction✔with✔modeled✔examples ✔✔ Guided✔practice✔with✔hints✔and✔support ✔✔ Higher-level✔thinking✔activities✔ PLUS additional review and practice that target assessed skills Used by more students in the U.S. than any other state-customized series, Coach books are proven effective. Triumph Learning has been a trusted name in educational publishing for more than 40 years, and we continue to work with teachers and administrators to keep our books up to date— improving test scores and maximizing student learning. Please visit our Web site for detailed product descriptions of all our instructional materials, including sample pages and more. ® www.triumphlearning.com This book is printed on paper containing a minimum of 10% post-consumer waste. 230VA Phone: (800) 221-9372 • Fax: (866) 805-5723 • E-mail: [email protected] Developed in Consultation with Virginia Educators Table of Contents Virginia Standards of Learning Correlation Chart . . . . . . . . . . . . . . 6 Virginia Standards of Learning Algebra II Chapter 1 Expressions and Operations . . . . . . . . . . . . . . . . . . . . 9 Lesson 1 Properties of Exponents . . . . . . . . . . . . . . . . . . . . . . . . 10 AII.1.a Lesson 2 Use Patterns to Factor Polynomials . . . . . . . . . . . . . . . 14 AII.1.d Lesson 3 Operations on Rational Expressions. . . . . . . . . . . . . . . 19 AII.1.a Lesson 4 Expressions and Operations with Radicals . . . . . . . . . 24 AII.1.b Lesson 5 Convert Between Radical and Exponential Forms of Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . 29 AII.1.c Lesson 6 Introduction to Complex Numbers . . . . . . . . . . . . . . . . 33 AII.3 Lesson 7 Add and Subtract Complex Numbers . . . . . . . . . . . . . 38 AII.3 Lesson 8 Multiply Complex Numbers . . . . . . . . . . . . . . . . . . . . . 42 AII.3 Chapter 2 Equations and Inequalities. . . . . . . . . . . . . . . . . . . . . 49 Lesson 9 Solve Absolute Value Equations . . . . . . . . . . . . . . . . . . 50 AII.4.a Lesson 10 Solve Absolute Value Inequalities. . . . . . . . . . . . . . . . . 56 AII.4.a Lesson 11 Solve Quadratic Equations Graphically . . . . . . . . . . . . 62 AII.4.b Lesson 12 Solve Quadratic Equations Algebraically . . . . . . . . . . . 68 AII.4.b Lesson 13 Solve Rational Equations . . . . . . . . . . . . . . . . . . . . . . . 74 AII.4.c Lesson 14 Solve Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . 80 AII.4.d Lesson 15 Use Equations and Inequalities to Solve Real-World Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 86 AII.4.a, AII.4.b, AII.4.c, AII.4.d Solve Nonlinear Systems of Equations. . . . . . . . . . . . . 92 AII.5 Lesson 16 Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Chapter 3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Lesson 17 Arithmetic Sequences and Series. . . . . . . . . . . . . . . . 104 AII.2 Lesson 18 Graph Transformations of Functions . . . . . . . . . . . . . 111 AII.6 Lesson 19 Characteristics of Functions . . . . . . . . . . . . . . . . . . . . 119 AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.e Lesson 20 Graph Transformations of Quadratic Functions . . . . . 127 AII.6 Lesson 21 Characteristics of Quadratic Functions . . . . . . . . . . . 134 AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.f Lesson 22 Graph Transformations of Cubic Functions . . . . . . . . 141 AII.6 Lesson 23 Graph Transformations of Polynomial Functions of Higher Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 AII.6 Duplicating any part of this book is prohibited by law. Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4 230VA_Alg_II_SE_Final.indd 4 7/28/11 8:50 AM Virginia Standards of Learning Algebra II Lesson 24 Characteristics of Polynomial Functions . . . . . . . . . . 156 AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.f Lesson 25 Graph Transformations of Exponential Functions . . . 163 AII.6 Lesson 26 Characteristics of Exponential Functions . . . . . . . . . . 169 AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.e, AII.7.f Lesson 27 Geometric Sequences and Series . . . . . . . . . . . . . . . 175 AII.2 Lesson 28 Graph Transformations of Logarithmic Functions . . . 182 AII.6 Lesson 29 Characteristics of Logarithmic Functions . . . . . . . . . . 189 AII.7.a, AII.7.b, AII.7.c, AII.7.d, AII.7.e Lesson 30 Functions and Their Inverses . . . . . . . . . . . . . . . . . . . 195 AII.7.g Lesson 31 Characteristics of Functions and Their Inverses . . . . 203 AII.7.a, AII.7.e, AII.7.g Lesson 32 Composition of Functions. . . . . . . . . . . . . . . . . . . . . . 209 AII.7.g, AII.7.h Lesson 33 Solutions, Zeros, x-Intercepts, and Factors of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 AII.8 Chapter 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Chapter 4 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Lesson 34 Curves of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 AII.9 Lesson 35 Direct, Inverse, and Joint Variation. . . . . . . . . . . . . . . 237 AII.10 Lesson 36 Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 242 AII.11 Lesson 37 Permutations and Combinations . . . . . . . . . . . . . . . . 252 AII.12 Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Practice Test 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Duplicating any part of this book is prohibited by law. Practice Test 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Algebra II Formula Sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Z-score Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 5 230VA_Alg_II_SE_Final.indd 5 7/28/11 8:50 AM Chapter 1 • Lesson 1 Properties of Exponents SOL: AII.1.a You can apply the properties of exponents to simplify expressions in exponential form. In the examples below, the variables a and b are real numbers and the variables m and n are integers. Product of Powers: To multiply powers with the same base, add the exponents. m n mn a a a Power of a Power: To raise a number in exponential form to a power, multiply the exponents. m n mn (a ) a Power of a Product: To find a power of a product, find the power of each factor and multiply. (ab)m ambm Quotient of Powers: To divide powers with the same base, subtract the exponents. am __ amn if a 0 an Power of a Quotient: To raise a quotient to a power, raise both the numerator and the denominator to that power. ( __ba ) m m a __ m b Power of Zero: Any nonzero number raised to the power of zero is 1. 0 a 1 if a 0 n a 1 if a 0 __ n a 1 an if a 0 ___ a n Duplicating any part of this book is prohibited by law. Negative Exponents: Any nonzero number raised to a negative exponent is equal to the reciprocal of that number with a positive exponent. 10 • Chapter 1: Expressions and Operations 230VA_Alg_II_SE_Final.indd 10 7/28/11 8:50 AM Example 1 4 73 Simplify and then evaluate the expression: 7_____ 5 7 Strategy Step 1 Use the product of powers and quotient of powers to simplify. Use the product of powers to rewrite the expression in the numerator. 4 3 43 7 7 7 7 7 4 3 7 7 __ 7 So, 7_____ 5 5 7 Step 2 7 Use the quotient of powers to rewrite the fraction. 77 775 72 __ 75 Step 3 Evaluate the expression. 2 7 49 Solution 4 73 in simplified form is 72 and has a value of 49. The expression 7_____ 5 7 You can also apply properties of exponents to evaluate an algebraic expression. An algebraic expression is a combination of numbers and variables that are connected by one or more operations. Example 2 Simplify the expression: (x3y2z0)3 Strategy Step 1 Apply the properties of exponents that relate to multiplication. Use the power of a product to rewrite the expression. (x3y2z0)3 (x3)3 (y2)3 (z0)3 Step 2 Find the power of each power. Duplicating any part of this book is prohibited by law. 3 3 33 9 (x ) x x (y2)3 y 2 3 y6 (z0)3 z0 3 z0 1 Note: You apply the power of zero to determine that z0 1. Step 3 Rewrite the expression in simplest form. (x3y2z0)3 x9 y6 1 x9y6 Solution The simplified form of (x3y2z0)3 is x9y6. 11 230VA_Alg_II_SE_Final.indd 11 7/28/11 8:50 AM Example 3 3 6 What is the value of the expression ___ 1 ? 2 Strategy Step 1 Apply the properties of negative exponents. Write an equivalent expression with positive exponents. 6 3 1 __ 3 6 1 21 ___ 1 2 1 6 3 2 __ So, ___ 1 3 2 6 Step 2 Evaluate the numerator and the denominator. Then simplify. 1 2 ___ 2 __ 3 216 6 2 is a common factor in both the numerator and the denominator, so: 1 2 ______ 1 2 ___ 1 ___ 216 2 108 108 1 Solution 3 6 1 . ___ The value of the expression ___ 1 is 108 2 Coached Example 3 What is the result when a4b is divided by a 2b2? 3 Write an expression that represents a4b 2 a b divided by a 2b : ______ 2 ____ 4 ____ a b Simplify the expression using the quotient of powers property. b___ Add and subtract the exponents. a 4 (______) b___ a___ b___ Rewrite the expression so that all the exponents are positive. ___ a ___ b___ 3 When a4b is divided by a 2b2, the quotient is ____________. Duplicating any part of this book is prohibited by law. 4 (2) a 12 • Chapter 1: Expressions and Operations 230VA_Alg_II_SE_Final.indd 12 7/28/11 8:50 AM Lesson 1: Properties of Exponents Lesson Practice Choose the correct answer. 1. What is d 6 d 4 in simplified form? A. d 2 B. d 10 5. 25s A. ___ 6 t 25s 2 ___ B. t5 10s 2 C. ___ 5 t 2 25s D. ___ 6 t D. d 64 Which is equivalent to x5 x 8? 1 A. __ 40 B. (t ) 2 C. d 24 2. 5s 2 ? Which is equivalent to ___ 3 6. x 1 __ x3 yz 4 2 81x y A. _____ z C. x3 z B. ____ 4 2 40 D. x Evaluate: (80)2 3z D. ___ 4 2 ( ) x y A. 64 Duplicating any part of this book is prohibited by law. 4. ( 3x y ) 3x y ( ____ z ) 4 2 C. 3. 4 3x ? Which is equivalent to _____ 2 1 B. 8 C. 1 D. 0 7. ( ) Simplify: (2jk)3 _13 j 2k 5 2 A. _89 j 7k 13 B. _23 j 7k 13 3 (5) Evaluate: _3 C. _23 j 7k 13 15 A. __ 9 27 B. ___ 125 125 C. ___ 27 27 D. ___ 125 D. _89 j 12k 30 8. 2 ab Which is equivalent to ________ 3 2 3 ? (4a b ) A. 64a11b5 11 a B. ____ 5 64b a11 C. ____ 5 64b 11 a D. 64__ 5 b 13 230VA_Alg_II_SE_Final.indd 13 7/28/11 8:50 AM