CK-12 Algebra II with Trigonometry Lori Jordan Kate Dirga SayThankstotheAuthors Clickhttp://www.ck12.org/saythanks (Nosigninrequired) www.ck12.org AUTHORS LoriJordan To access a customizable version of this book, as well as other KateDirga interactivecontent,visitwww.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web- basedcollaborativemodeltermedtheFlexBook®textbook,CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through theFlexBookPlatform®. 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Printed: February28,2015 iii Contents www.ck12.org Contents 1 EquationsandInequalities 1 1.1 RealNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 SimplifyingAlgebraicExpressionsandEquations . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 SolvingLinearEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 SolvingLinearInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.5 SolvingAbsoluteValueEquationsandInequalities . . . . . . . . . . . . . . . . . . . . . . . . 43 1.6 InterpretingWordProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2 LinearEquationsandFunctions 56 2.1 FindingtheSlopeandEquationofaLine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2 StandardFormofaLine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3 GraphingLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.4 RelationsandFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.5 GraphingLinearInequalitiesinTwoVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.6 GraphingAbsoluteValueFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.7 AnalyzingScatterplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3 SystemsofLinearEquationsandInequalities 137 3.1 SolvingLinearSystemsbyGraphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.2 SolvingLinearSystemsbySubstitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.3 SolvingLinearSystemsbyLinearCombinations(Elimination) . . . . . . . . . . . . . . . . . . 183 3.4 GraphingandSolvingLinearInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 3.5 SolvingLinearSystemsinThreeVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 4 Matrices 235 4.1 OperationsonMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 4.2 MultiplyingMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 4.3 DeterminantsandCramer’sRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 4.4 IdentityandInverseMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 4.5 SolvingLinearSystemsUsingInverseMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 281 5 QuadraticFunctions 293 5.1 SolvingQuadraticsbyFactoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 5.2 SolvingQuadraticsbyUsingSquareRoots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 5.3 ComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 5.4 CompletingtheSquare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 5.5 TheQuadraticFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 5.6 AnalyzingtheGraphofaQuadraticFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 6 PolynomialFunctions 367 6.1 PropertiesofExponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 6.2 Adding,SubtractingandMultiplyingPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . 377 iv www.ck12.org Contents 6.3 FactoringandSolvingPolynomialEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 6.4 DividingPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 6.5 FindingallSolutionsofPolynomialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 6.6 AnalyzingtheGraphofPolynomialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 7 Roots,RadicalsandFunctionOperations 428 7.1 UsingRationalExponentsandnthRoots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 7.2 GraphingSquareRootandCubedRootFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 437 7.3 SolvingRadicalEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 7.4 FunctionOperationsandtheInverseofaFunction . . . . . . . . . . . . . . . . . . . . . . . . . 466 8 ExponentialandLogarithmicFunctions 478 8.1 ExponentialGrowthandDecay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 8.2 LogarithmicFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 8.3 PropertiesofLogarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 8.4 SolvingExponentialandLogarithmicEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 516 9 RationalFunctions 522 9.1 Direct,Inverse,andJointVariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 9.2 GraphingRationalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 9.3 Simplifying,Multiplying,andDividingRationalExpressions . . . . . . . . . . . . . . . . . . . 553 9.4 Adding&SubtractingRationalExpressionsandComplexFractions . . . . . . . . . . . . . . . 559 9.5 SolvingRationalEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 10 ConicSections 577 10.1 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 10.2 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 10.3 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 10.4 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 10.5 GeneralConicEquationsandSolvingNon-LinearSystems . . . . . . . . . . . . . . . . . . . . 622 11 SequencesandSeries 640 11.1 GeneralSequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 11.2 ArithmeticSequencesandSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 11.3 GeometricSequencesandSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 11.4 InfiniteSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 12 IntroductiontoProbability 678 12.1 TheFundamentalCountingPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 12.2 PermutationsandCombinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 12.3 TheBinomialTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 12.4 IntroductiontoProbability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 12.5 VennDiagramsandIndependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 13 TrigonometricRatios 727 13.1 RightTriangleTrigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 13.2 TheUnitCircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 13.3 IntroductiontoPolarCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 13.4 TheLawofSines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 13.5 TheLawofCosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789 14 TrigonometricFunctionsandIdentities 800 14.1 GraphingTrigonometricFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 v Contents www.ck12.org 14.2 UsingTrigonometricIdentities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827 14.3 SolvingTrigonometricEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838 14.4 SumandDifferenceFormulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844 14.5 DoubleandHalfAngleFormulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 vi www.ck12.org Chapter1. EquationsandInequalities C 1 HAPTER Equations and Inequalities Chapter Outline 1.1 REAL NUMBERS 1.2 SIMPLIFYING ALGEBRAIC EXPRESSIONS AND EQUATIONS 1.3 SOLVING LINEAR EQUATIONS 1.4 SOLVING LINEAR INEQUALITIES 1.5 SOLVING ABSOLUTE VALUE EQUATIONS AND INEQUALITIES 1.6 INTERPRETING WORD PROBLEMS AlgebraIIwithTrigonometry AlgebraIIisconsideredthethirdyearofmathintheon-levelmathematicshigh-schoolsequence. Dependingonthe level of the students in your class, the option of Trigonometry can be added at the end of the coursework. There is alotofmaterialtobecoveredinthiscourse,including: systemsofequations,quadratics,higherlevelpolynomials, exponential and logarithmic functions, modeling, rational functions, conics, sequences and series, probability, and trigonometry. EquationsandInequalities ThischapterofAlgebraIIisareviewofmuchofthetopicscoveredinAlgebraI.Studentswillreviewrealnumbers, solvingequations,inequalities,absolutevalueequations,andabsolutevalueinequalities. 1 1.1. RealNumbers www.ck12.org 1.1 Real Numbers Objective Tofamiliarizestudentswiththesubsetsofrealnumbersandreviewtheirproperties. ReviewQueue 1. Writedownoneexampleofeachofthefollowing: afraction,adecimal,aninteger,andasquareroot. 2. DoyouremembertheOrderofOperationsfromanothermathclass? Whatisit? 3. Whichfractionislarger? a) 5 or 1 6 2 b) 1 or 1 4 3 c) 6 or 7 7 9 Subsets of Real Numbers Objective Identifythesubsetsofrealnumbers. WatchThis MEDIA ClickimagetotheleftorusetheURLbelow. URL:http://www.ck12.org/flx/render/embeddedobject/57616 JamesSousa: IdentifyingSetsofRealNumbers Guidance Thereareseveraltypesofrealnumbers. Youareprobablyfamiliarwithfractions,decimals,integers,wholenumbers and even square roots. All of these types of numbers are real numbers. There are two main types of numbers: real andcomplex. Wewilladdresscomplex(imaginary)numbersintheQuadraticFunctionschapter. TABLE 1.1: RealNumbers Any number that can be plotted on Examples: 8,4.67,−1,π 3 anumberline. Symbol: R RationalNumbers Any number that can be written as Examples: −5,1,1.3,16 9 8 4 afraction,includingrepeatingdeci- mals. Symbol: Q √ √ IrrationalNumbers Real numbers that are not rational. Example: e,π,− 2, 3 5 When written as a decimal, these numbersdonotendnorrepeat. 2 www.ck12.org Chapter1. EquationsandInequalities TABLE 1.1: (continued) RealNumbers Any number that can be plotted on Examples: 8,4.67,−1,π 3 anumberline. Symbol: R Integers All positive and negative “count- Example: -4,6,23,-10 ing”numbersandzero. Symbol: Z WholeNumbers All positive “counting” numbers Example: 0,1,2,3,... andzero. NaturalNumbers All positive “counting” numbers. Example: 1,2,3,... Symbol: N A“counting”numberisanynumberthatcanbecountedonyourfingers. Therealnumberscanbegroupedtogetherasfollows: ExampleA Whatisthemostspecificsubsetoftherealnumbersthat-7isapartof? Solution: -7isaninteger. ExampleB Listallthesubsetsthat1.3liesin. Solution: 1.3isaterminatingdecimal. Therefore,itisconsideredarationalnumber. Itwouldalsobearealnumber. GuidedPractice √ 1. Whattypeofrealnumberis 5? 2. Listallthesubsetsthat-8isapartof. √ 3. TrueorFalse: − 9isanirrationalnumber. Answers √ 1. 5isanirrationalnumberbecause,whenconvertedtoadecimal,itdoesnotendnordoesitrepeat. 2. -8isanegativeinteger. Therefore,itisalsoarationalnumberandarealnumber. √ 3. − 9=−3,whichisaninteger. Thestatementisfalse. Vocabulary Subset Asetofnumbersthatiscontainedinalargergroupofnumbers. RealNumbers Anynumberthatcanbeplottedonanumberline. 3 1.1. RealNumbers www.ck12.org RationalNumbers Anynumberthatcanbewrittenasafraction,includingrepeatingdecimals. IrrationalNumbers Realnumbersthatarenotrational. Whenwrittenasadecimal,thesenumbersdonotendnorrepeat. Integers Allpositiveandnegative“counting”numbersandzero. WholeNumbers Allpositive“counting”numbersandzero. NaturalNumbersorCountingNumbers Numbersthancanbecountedonyourfingers;1,2,3,4,... TerminatingDecimal Whenadecimalnumberends. RepeatingDecimal Whenadecimalnumberrepeatsitselfinapattern. 1.666...,0.98989898... areexamplesofrepeatingdecimals. ProblemSet Whatisthemostspecificsubsetofrealnumbersthatthefollowingnumbersbelongin? 1. 5.67 √ 2. − 6 3. 9 5 4. 0 5. -75 √ 6. 16 ListALLthesubsetsthatthefollowingnumbersareapartof. 7. 4 8. 6 9 9. π Determineifthefollowingstatementsaretrueorfalse. 10. Integersarerationalnumbers. 11. Everywholenumberisarealnumber. 12. Integersareirrationalnumbers. 13. Anaturalnumberisarationalnumber. 14. Anirrationalnumberisarealnumber. 15. Zeroisanaturalnumber. 4