Contents iii

Getting Started 1

About This Book 1 Eight Standards for Mathematical Practice 1Test-Taking Strategies 3

Chapter R: Review 4

R.1 Expressions, Equations, and Functions A-CED.1; A-CED.4; A-REI.3 5Writing and Evaluating Algebraic Expressions 5Solving Equations 9Literal Equations 13

R.2 Linear Functions and Rate of Change 14The Slope-Intercept Form of a Line 14Rate of Change 18

R.3 Functions A-REI.10; F-IF.1; F-IF.2 22Graphs of Functions 23

R.4 Solving Systems of Linear Equations and Inequalities A-CED.3; 25A-REI.5; A-REI.6; A-REI.11; A-REI.12

Solving Systems of Equations by Graphing 25Solving Systems by Elimination or Substitution 28Graphing Linear Inequalities 32Solving Systems of Linear Inequalities by Graphing 33

R.5 Polynomial Operations A-SSE.2; A-APR.1 36Power Rules 36Products and Quotients to a Power 37Zero and Negative Exponents 38Multiplying Polynomials 39

R.6 Parabolas A-APR.3; F-BF.3 41 Translating Parabolas in Vertex Form 42

Key to the icons:

The computer icon indicates Digital Activities that can be found at www.amscomath.com.

The globe icon indicates where Real-World Model Problems are found in the text.

Contents

iv Contents

Chapter 1: Themes in Algebra 2 44

1.1 Functions 45Properties of Functions 45Domain and Range of Functions 48Translating Function Graphs 49Scaling Function Graphs 52Odd and Even Functions 52

1.2 Models 54Spreadsheet and Graphing Calculator: Drawing a Scatter Plot 55Modeling Data with Trend Lines 56

Regression 57Spreadsheet and Graphing Calculator: Linear Models 58Multi-Part Problem Practice 63

1.3 Working with Models N-Q.2; A-SSE.1a; A-CED.3; F-BF.1a 64Linear Programming 65Multi-Part Problem Practice 70

1.4 Seeing Structure in Equations and ExpressionsA-SSE.1b; A-REI.6 71The Form of an Equation 71Structure and Factoring 71Systems of Equations with More than Two Variables 72

Chapter 1 Key Ideas 74Chapter 1 Review 75

Chapter 2: Quadratics 78

2.1 Algebra 1 Review: Factoring PolynomialsA-SSE.2 79Special Product Patterns 80Structure and Factoring 82Factoring by Grouping 84

2.2 Polynomial Patterns A-SSE.2 85 Factoring Sums and Differences of Cubes 85

Factoring Two-Variable Polynomials 86

2.3 Patterns and Equations A-SSE.2; A-APR.4 88Algebra 1 Review: The Square Root Principle 88Algebra 1 Review: The Zero-Product Property 89Using Structure in Expressions to Solve an Equation 92Factoring and Identities 93

Contents v

2.4 Algebra 1 Review: The Quadratic Formula A-REI.4a; A-REI.4b 94Completing the Square 94The Quadratic Formula 97Graphing Calculator: The Quadratic Formula 99Multi-Part Problem Practice 102

2.5 Imaginary and Complex NumbersN-CN.1; N-CN.2; N-CN.8 103Adding and Subtracting Complex Numbers 104Multiplying Complex Numbers 105Optional: Complex Conjugates 106Factoring Identities and Complex Numbers 107

2.6 Solutions of Quadratic EquationsN-CN.7; A-APR.3; A-REI.4b; F-IF.8a 108Graphs and the Number of Solutions to a Quadratic 108Graphing Calculator: Graphing Quadratic Equations 109The Discriminant 109Complex Solutions to Quadratic Equations 110

2.7 Modeling with Quadratic Functions A-CED.2; F-IF.4; S-ID.6a 114Spreadsheet and Graphing Calculator: Modeling with Quadratic Functions 114Multi-Part Problem Practice 118

2.8 Parabolas at the Origin F-IF.4; G-GPE.2 118GeometricDefinitionofaParabola 118Graphing a Parabola at the Origin 120

Chapter 2 Key Ideas 126Chapter 2 Review 127Cumulative Review for Chapters 1–2 129

Chapter 3: Polynomials 132

3.1 Multivariable PolynomialsA-SSE.2; A-APR.1 133Combining Like Terms in Multivariable Polynomials 134Evaluating Multivariable Polynomials 135Operations with Multivariable Polynomials 136

3.2 Dividing Polynomials A-APR.6 138Long Division of Polynomials 138Long Division of Polynomials with Remainder 141Synthetic Division 142Dividing Expressions Using a Computer Algebra System 146

3.3 Remainder and Factor Theorems A-APR.2 147The Remainder Theorem 147The Factor Theorem 148

vi Contents

3.4 Solving Polynomial Equations AlgebraicallyA-SSE.1a; A-APR.3 152Solving Cubic Equations 152Solving Quartic Equations 153

3.5 Finding Zeros of Polynomial Functions N-CN.9; A-APR.3; F-IF.7c 154Graphing Polynomial Functions by Plotting Points 155Graphing Calculator: Graphing Polynomial Functions 157Graphing Polynomial Functions Using Zeros 159Fundamental Theorem of Algebra 160Linear Factorization Theorem 160Summary of Finding Zeros of Polynomial Functions 161

3.6 Optional: Descartes’ Rule of Signs 1653.7 Transformations of Polynomial Functions F-BF.3 1683.8 Modeling with Polynomial FunctionsA-CED.2; A-CED.3; F-IF.4; F-IF.6; 170

F-IF.7c; F-IF.9

Spreadsheet and Graphing Calculator: Modeling Polynomial Functions 170Multi-Part Problem Practice 174

3.9 Solving Systems of Polynomial EquationsA-REI.7; A-REI.11 175Solving Polynomial Systems Graphically 175Graphing Calculator: Solving a System of Polynomial Equations 176Solving Polynomial Systems Algebraically 177Multi-Part Problem Practice 178

Chapter 3 Key Ideas 179Chapter 3 Review 180Cumulative Review for Chapters 1–3 182

Chapter 4: Rational Expressions 184

4.1 Multiplying and Dividing Rational Expressions A-SSE.1a; A-SSE.1b; 185A-SSE.2; A-APR.6; A-APR.7

Simplifying Rational Expressions 185Multiplying Rational Expressions 187Dividing Rational Expressions 189

4.2 Adding and Subtracting Rational ExpressionsA-SSE.1a; A-SSE.1b; A-SSE.2; 192A-APR.6; A-APR.7

Adding and Subtracting Rational Expressions with Common Denominators 192AddingandSubtractingRationalExpressionswithDifferentDenominators 194Least Common Multiple 195

Contents vii

4.3 Rational Equations A-CED.1; A-REI.1; A-REI.2 197Algebra 1 Review: Evaluating Rational Expressions and Equations 197Rational Equations 200Extraneous Solutions 201Multi-Part Problem Practice 204

4.4 Graphing Rational Functions A-REI.11; F-IF.4; F-BF.3 205TranslatingandReflectingRationalFunctions 206Solving a System of Rational Equations by Graphing 212Multi-Part Problem Practice 216

Chapter 4 Key Ideas 216Chapter 4 Review 218Cumulative Review for Chapters 1–4 220

Chapter 5: Powers and Radicals 222

5.1 Radical OperationsN-RN.2 223Review: Roots 223Simplifying Radical Expressions 225Product Rule for Radicals 227Quotient Rule for Radicals 228Multiplying Square Root Radicals with Negative Radicands 231

5.2 More Operations with Radicals N-RN.2 232Adding and Subtracting Like Radicals 232Multiplying Monomial and Binomial Radical Expressions 234Rationalizing the Denominator 235Summary: Simplifying Radical Expressions 237Multi-Part Problem Practice 238

5.3 Exponent Notation N-RN.1; N-RN.2; A-SSE.2 239Exponent Notation for Roots 239Derivation of Roots as Powers 241Rules of Exponents and Fractional Exponents 242MultiplyingandDividingRadicalswithDifferentIndicesbutSameRadicand 242

5.4 Radical Equations A-REI.1; A-REI.2 245Squaring Principle 245Extraneous Solutions 246Power Principle 248Multi-Part Problem Practice 250

viii Contents

5.5 Radical Function Graphs F-IF.7b; F-BF.3 251

Graphing a Square Root Function 251Graphing a Cube Root Function 253

Chapter 5 Key Ideas 256Chapter 5 Review 257Cumulative Review for Chapters 1–5 258

Chapter 6: Exponential Functions 260

6.1 Exponential Function Graphs F-IF.6; F-IF.8b; F-BF.3 261Exponential Functions 261Exponential Function Graphs 262Translating Exponential Function Graphs 264Rate of Change in Exponential Functions 265

6.2 Modeling with Exponential FunctionsA-SSE.1b; A-SSE.3c; F-IF.4; 268F-IF.7e; F-LE.5; S-ID.6a

Spreadsheet and Graphing Calculator: Modeling Exponential Functions 268Multi-Part Problem Practice 277

6.3 Combining Functions F-BF.1b 278Evaluating a Combined Function 280

6.4 Inverse and Composite Functions F-BF.4a 281Inverse Functions 281The Graph of a Function and Its Inverse 283Optional: When Does a Function Have an Inverse Function? 284Composite Functions 286Optional: Domain Restrictions of Composite Functions 287Multi-Part Problem Practice 289

Chapter 6 Key Ideas 290Chapter 6 Review 290Cumulative Review for Chapters 1–6 292

Chapter 7: Logarithmic Functions 294

7.1 LogarithmsF-LE.4 295Logarithmic Functions 295Common Logarithms 297Solving Logarithmic Equations 297

7.2 Logarithmic Function Graphs A-REI.11; F-BF.3 300Translating Logarithmic Function Graphs 302

Contents ix

7.3 Natural Logarithms and eF-LE.4 3057.4 Laws of LogarithmsF-LE.4 308

Logarithmic Identities 308Logarithmic Equations 309Logarithms of Products 311Logarithms of Quotients 312Logarithms of Powers 313Change-of-Base Formula 313Summary of Logarithm Rules 314Solving Exponential Equations 316

7.5 Modeling with LogarithmsA-CED.1; F-IF.4; F-IF.6; F-IF.7e; F-IF.9 317Spreadsheet and Graphing Calculator: Modeling Logarithmic Functions 317Multi-Part Problem Practice 323

7.6 More Logarithmic Operations A-SSE.2; F-LE.4 324Breaking Up and Combining Logarithmic Expressions 324Derivations of the Rules of Logarithms 325Multi-Part Problem Practice 327

Chapter 7 Key Ideas 328Chapter 7 Review 329Cumulative Review for Chapters 1–7 330

Chapter 8: Sequences and Series 332

8.1 Arithmetic Sequences F-IF.3; F-BF.1a; F-BF.2; F-LE.2 333Recursive Formula for Arithmetic Sequences 334Explicit Formula for the General Term 335

8.2 Optional: Arithmetic Series 342Sigma Notation 343PartialSumofanInfiniteArithmeticSeries 344Derivation of Formula for Arithmetic Series 344Arithmetic Series in History 347Multi-Part Problem Practice 349

8.3 Geometric Sequences F-IF.3; F-BF.1a; F-BF.2; F-LE.2 350Recursive Formula for Geometric Sequences 350Explicit Formula for the General Term 351What Type of Sequence? 355

x Contents

8.4 Geometric Series A-SSE.4 361Geometric Series and Partial Sums 361Derivation of Formula for Geometric Series 361InfiniteGeometricSeries 364DerivationofFormulaforInfiniteGeometricSeries 365Multi-Part Problem Practice 370

8.5 Binomial TheoremA-APR.5 370Binomial Expansion 370Factorial Notation 374BinomialCoefficients 374Binomial Theorem 377

Chapter 8 Key Ideas 380Chapter 8 Review 382Cumulative Review for Chapters 1–8 384

Chapter 9: Trigonometry 386

9.1 Geometry Review: Right TrianglesG-SRT.8 387Right Triangle Basics 387Special Right Triangles 389

9.2 Geometry Review: Trigonometric Functions G-SRT.6; G-SRT.7; 394G-SRT.8

Trigonometric Ratios 394Sine, Cosine, and Complementary Angles 397Sine, Cosine, and Tangent for Special Triangles 401

9.3 Angles of Rotation and Trigonometric FunctionsF-TF.1 406Reference Angles and Trigonometric Functions 408Angles of Rotation and Trigonometric Functions 408Radian Measure of Angles 414

9.4 Trigonometric Functions and the Unit CircleF-TF.2; F-TF.8 417The Unit Circle 417Trigonometric Identities 419Trigonometric Functions and the Unit Circle 421Multi-Part Problem Practice 423

Contents xi

9.5 Trigonometric Function Graphs F-IF.4; F-IF.7e; F-BF.3; F-TF.5 424Properties of Trigonometric Function Graphs 424Graphs Using the Unit Circle 426Scaling Trigonometric Function Graphs 427Translating Trigonometric Function Graphs 431Sine and Cosine Identities 435Graph of the Tangent Function 436Multi-Part Problem Practice 441

9.6 Optional: Reciprocal Trigonometric Functions 441Cosecant, Secant, and Cotangent 441Reciprocal Trigonometric Function Graphs 444

9.7 Modeling with FunctionsS-ID.6a 448Chapter 9 Key Ideas 456Chapter 9 Review 458Cumulative Review for Chapters 1–9 460

Chapter 10: Probability 462

10.1 Introduction to Probability S-CP.1; S-MD.6 463Experimental Probability 463Theoretical Probability and Sample Spaces 465

10.2 Independent Events and the Multiplication Rule S-CP.1; 471S-CP.2; S-MD.6

Compound Events 474

10.3 Addition and Subtraction RulesS-CP.1; S-CP.7; S-MD.7 477“Or” and the Addition Rule 477Mutually Exclusive Events and the Addition Rule 479The Subtraction Rule 479The Origin of Probability Studies 481Multi-Part Problem Practice 484

10.4 Conditional Probability S-CP.3; S-CP.4; S-CP.5; S-CP.6 485Conditional Probability and Independent Events 485Conditional Probability and Frequency Tables 488Optional: Bayes’ Theorem 490

10.5 Normal Distribution S-ID.4 496Spreadsheet and Graphing Calculator: Generating a Normal Distribution Curve 500Approximating the Area Under a Normal Curve 501 Spreadsheet and Graphing Calculator: Approximating the Area Under a Normal 501 Curve

xii Contents

10.6 Surveys and Samples S-IC.1; S-IC.2; S-IC.3 505Multi-Part Problem Practice 508

10.7 Observational Studies S-IC.1; S-IC.2; S-IC.3; S-IC.4; S-IC.5; 509S-IC.6; S-CP.4; S-MD.7

ConfidenceIntervalandMarginofError 511Computing the Margin of Error 512

Chapter 10 Key Ideas 517Chapter 10 Review 518Cumulative Review for Chapters 1–10 522

Glossary 524

Digital Activities and Real-World Model Problems 532

Index 533

Getting Started 1

AboutThisBookNew York Algebra 2 is a full-year course, written to give students a strong understanding of the concepts of algebra as well as prepare them for the Algebra 2 (Common Core) Regents Examination. All instruction, model problems, and practice items were developed to support the Common Core Learning Standards (CCLS) and modules and lessons of engageny. Each chapter opens with lesson-by-lesson alignment with the standards. The eight Mathematical Practice Standards are imbedded throughout the text in selected Model Problems, extensive practice problem sets, and the comprehensive Chapter and Cumulative Reviews. Complete correlations of the lessons in New York Algebra 2 to engageny lessons are available in the teacher manual.

In New York Algebra 2, students will explore quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions and apply their knowledge to contextual problems. New York Algebra 2 builds on the themes of New York Algebra 1. Students see structure in expressions, transform functions, and use regressions as a method to analyze and model data. Fin ally, students will expand their understanding of probability by building on concepts introduced in earlier years. Throughout the text, prior learning is accessed to build a strong foundation for learning new concepts.

Each chapter incorporates multiple performance tasks that measure the ability of students to think critically and apply their knowledge in real-world situations. In addition, students and teachers have access to a companion Web site (www.amscomath.com) with activities and simulations linked directly to lessons in New York Algebra 2. Teachers also have the option to include a full range of digital simulations, electronic whiteboard lessons, videos, and interactive problems to stimulate conceptual understanding. Through the online MathX program, available separately, students and teachers have access to a comprehensive suite of instructional videos, adaptive practice exercises, quizzes, and tests with automated grading and reporting.Carefulandconsistentuseofthistextandthesupportingmaterialswillgivestudentsafirm

grasp of Algebra 2, prepare them for the Algebra 2 Regents Examination, and give them the tools they need to be college and career ready.

EightStandardsforMathematicalPracticeThe mathematical practices are a common thread for students to think about and understand math as they progress from Kindergarten through high school. Students should use the mathematical practices as a method to break down concepts and solve problems, including representing problems logically, justifying conclusions, applying mathematics to practical situations, explaining the mathematics accurately to other students, or deviating from a known proceduretofindashortcut.

MP1 Make sense of problems and persevere in solving them.Attack new problems by analyzing what students already know. Students should understand thatmanydifferentstrategiescanwork.Askleadingquestionstodirectthediscussion.Taketime to think. • explain the meaning of the problem• analyze given information, constraints, and relationships• plan a solution route• try simpler forms of the initial problem• use concrete objects to help conceptualize• monitor progress and change course, if needed• continually ask, “Does this make sense?”

GettingStarted

2 Getting Started

MP2 Reason abstractly and quantitatively.Represent problems with symbols and/or pictures. • make sense of quantities and their relationships• decontextualize—represent a situation symbolically and contextualize—consider what

given symbols represent• create a clear representation of the problem• consider the units involved• attend to the meaning of numbers and variables, not just how to compute them• use properties of operations and objects

MP3 Construct viable arguments and critique the reasoningof others.

Ask questions, defend answers, and/or make speculations using correct math vocabulary. • useassumptions,definitions,andpreviouslyestablishedresults• make conjectures and build a valid progression of statements• use counterexamples• justify conclusions and communicate them to others• determine whether the arguments of others seem right

MP4 Model with mathematics.Show the relevance of math by solving real-world problems. Look for opportunities to use math for current situations in and outside of school in all subject areas. • apply mathematics to solve everyday problems• analyzeandchartrelationshipsusingdiagrams,two-waytables,graphs,flowcharts,and

formulas to draw conclusions• apply knowledge to simplify a complicated situation• interpret results and consider whether answers make sense

MP5 Use appropriate tools strategically.Provide an assortment of tools for students and let them decide which ones to use. • choose appropriately from existing tools (pencil and paper, concrete models, ruler, protractor,

calculator, spreadsheet, dynamic geometry software, etc.) when solving mathematicalproblems

• detect possible errors by using estimation or other mathematical knowledge• use technology to explore and compare predictions and deepen understanding of concepts

MP6 Attend to precision.Use precise and detailed language in math. Instead of saying “I don’t get it,” students should be able to elaborate on where they lost the connection. Students should specify units in their answers and correctly label diagrams.• speak and write precisely using correct mathematical language• state the meaning of symbols and use them properly• specify units of measure and label axes appropriately• calculatepreciselyandefficiently• express answers with the proper degree of accuracy

MP7 Look for and make use of structure.Seepatternsandthesignificanceofgiveninformationandobjects.Usethesetosolvemorecomplex problems. • see the big picture• discern a pattern or structure

Getting Started 3

• recognizethesignificanceofgivenaspects• apply strategies to similar problems• step back for an overview and shift perspective• see complicated things as being composed of several objects

MP8 Look for and express regularity in repeated reasoning.Understand why a process works so students can apply it to new situations. • notice repeated calculations and look for both general methods and shortcuts• maintain oversight of the process while paying attention to the details• evaluate the reasonableness of intermediate results• create generalizations founded on observations

Test-TakingStrategiesGeneral Strategies

• Become familiar with the directions and format of the test ahead of time. Therewill be both multiple-choice and extended response questions where you mustshow the steps you used to solve a problem, including formulas, diagrams,graphs, charts, and so on, where appropriate.

• Pace yourself. Do not race to answer every question immediately. On the other hand,do not linger over any question too long. Keep in mind that you will need more time tocomplete the extended response questions than to complete the multiple-choice questions.

• Speed comes from practice. The more you practice, the faster you will becomeand the more comfortable you will be with the material. Practice as often as you can.

Specific Strategies

• Always scan the answer choices before beginning to work on a multiple-choicequestion. This will help you to focus on the kind of answer that is required. Areyou looking for fractions, decimals, percents, integers, squares, cubes, and so on? Eliminate choices that clearly do not answer the question asked.

• Do not assume that your answer is correct just because it appears among the choices.The wrong choices are usually there because they represent common student errors.Afteryoufindananswer,alwaysrereadtheproblemtomakesureyouhavechosentheanswer to the question that is asked, not the question you have in your mind.

• Sub-in. To sub-in means to substitute. You can sub-in friendly numbers for the variablestofindapatternanddeterminethesolutiontotheproblem.

• Backfill. If a problem is simple enough and you want to avoid doing the more complexalgebra, or if a problem presents a phrase such as x 5?,thenjustfillintheanswerchoicesthataregivenintheproblemuntilyoufindtheonethatworks.

• Do the math. This is the ultimate strategy. Don’t go wild searching in your mind fortricks, gimmicks, or math magic to solve every problem. Most of the time the best way toget the right answer is to do the math and solve the problem.