湖北省武汉六中高三(国际部)数学复习课件：42三角函数图像的变换

Unit4:TrigonometricFunctions

Lesson2:SinusoidalFunctionsUnit4:TrigonometricFunctionSinusoidagraphwhoseshapelookslikethegraphofsin(x)sin,cosandtransformationsofthesefunctionsareallsinusoids(AKAsinusoidalfunctions) SinusoidagraphwhoseshapeloTransformationsInLesson1,welookedatthedifferentfunctiontransformationsWecansummarizethesewiththegeneralfunctiontransformation

Ifweapplythistothesineandcosinefunctions,wegetTransformationsInLesson1,weTransformationsForsimplicity,considerthetransformedsinefunction:verticalstretches,compressions&reflectionsGivesyoutheamplitudeHorizontalstretches,compressions&reflectionsGivesyoutheperiod:ShiftsupordownGivesyoutheaxisofcurveShiftsleftorrightCalledthephaseshiftTransformationsForsimplicity,Example1Whatistheamplitude,period,phaseshiftandaxisofcurveofExample1WhatistheamplitudeExample1:SolutionIngeneral:Wehave:a=3c=–1k=2theamplitudeis3theaxisofcurveisy=–1d=thephaseshiftistheperiodisExample1:SolutionIngeneral:Example2Whatistheamplitude,period,phaseshiftandverticalshiftofExample2WhatistheamplitudeExample2:SolutionIngeneral:Wehave:FACTOR!a=–2c=6theamplitudeis2theaxisofcurveisy=6k=0.5IgnorethenegativeExample2:SolutionIngeneral:Example2:NotesThetipoftheday:Alwaysfactorthecoefficientonthex-termtocorrectlyidentifythetransformationsIfyoudon’t,yourphaseshiftwillbeincorrectReminder:Theamplitudeistheverticaldistancefromtheaxisofcurvetothemaximumvalue.So,althoughthevalueofacanbenegative,theamplitudeisalwayspositiveExample2:NotesThetipoftheExample3Whatistheequationofthecosinefunctionafterithasbeenstretchedsothatitsperiodis,itsamplitudeis8andithasexperiencedaphaseshiftoftotheleftandhasbeenshiftedup1unitExample3WhatistheequationExample3:Solutionamplitudeis8phaseshiftis(left)ShifteduponeunitTheperiodisa=8d=c=1Example3:SolutionamplitudeiExample4Whatistheequationofthesinefunctionafterithasbeenstretchedsothatitsperiodis4,ithasexperiencedaphaseshiftoftotherightandithasbeenreflectedinthex-axisExample4WhatistheequationExample4:SolutionamplitudeisunchangedFunctionisreflectedinthex-axisphaseshiftis(right)ThereisnoverticalshiftTheperiodis4a=1d=c=0a=-1Example4:SolutionamplitudeiExample4:NotesItispossiblefortheperiodtonotbeamultipleofπIfthisisthecase,thek-valuewillbeintermsofπTheperiodisrarelyamultipleofπinreal-worldapplicationsEx.motionofapendulum,theheightofthetides,voltageinanACcircuitExample4:NotesItispossibleTheGraphsofsin&cosInordertounderstandtransformationsofsinandcos,it’scrucialfortoknowwhatf(x)=sinxandf(x)=cosxlooklikeTohelpyouwiththis,alwaysthinkofthefive“keypoints”ofoneperiodforeachfunction:sinhasthreezeroes,onemaxandonemin.Eachperiodstartsontheaxisofcurvecoshastwozeroes,twomaxandonemin.EachperiodstartsatthemaximumBecausethesefunctionsareperiodicifyouneedmorethanoneperiod,simplyrepeatthepatternTheGraphsofsin&cosInordeHowtoDeterminetheEquationFromaGraphDrawahorizontallinethatdividesthefunctioninhalf(axisofcurve)Locatethestartingpointofasine/cosineperiodandmarkthispointwithanA.Locatetheendingpointofasine/cosineperiodandmarkthispointwithaB.Determinethehorizontaldistancebetweenthesetwopoints(theperiod).Determinetheverticaldistancefromtheaxisofcurvetothemaximumvalue(amplitude)Becausethex-coordinateofthestartingpointforasine/cosineperiodiszero,thex-coordinateofAgivesyouthephaseshiftHowtoDeterminetheEquationExample5Useasinecurvetodeterminetheequationofthefunctiongivenbelow-12Example5UseasinecurvetodExample5:SolutionAB12-12Axisofcurveis-9PeriodisπAmplitudeis3PhaseshiftisExample5:SolutionAB12-12AxisExample5:Solutionamplitudeis3phaseshiftis(right)Axisofcurveis-9Theperiodisa=3d=c=-9Example5:SolutionamplitudeiExample6RepeatExample5,butthistimeuseacosinecurvetodeterminetheequationExample6RepeatExample5,butExample6:SolutionAB12-12Axisofcurveis-9PeriodisπAmplitudeis3PhaseshiftisExample6:SolutionAB12-12AxisExample6:Solutionamplitudeis3phaseshiftis(left)Axisofcurveis-9Theperiodisa=3d=c=-9Example6:SolutionamplitudeiExample6:NotesWhenweusedacosinetomodelthefunctioninExample5,theonlychangewastothephaseshift.Thereasonforthis,isthatsineandcosinearethesamefunction–theonlydifferenceisaphaseshift:f(x)=sinxf(x)=cosxorExample6:NotesWhenweusedSummaryAsinusoidalfunctionisatransformedsineorcosinefunctionand,ingeneral,hastheequation:YoucandeterminetheequationofasinusoidusingasineorcosinefunctiongivenalistofpropertiesoragrapharelatestotheamplitudeamplitudeMUSTbepositivekrelatestotheperiodPerioddoesnotneedtobeintermsofπdisthephaseshiftcistheverticalshift(axisofcurve)Don’tforgettofactoroutwhat’sinfrontofx!f(x)=sinxandf(x)=cosxdifferonlybyaphaseshiftofSummaryAsinusoidalfunctioniPracticeProblemsP.275-277#1-6,8-13Note:Anygraphs/sketchescanbedoneusingyourTI-83ortheprogram“Graph”PracticeProblemsP.275-277#1Unit4:TrigonometricFunctions

Lesson2:SinusoidalFunctionsUnit4:TrigonometricFunctionSinusoidagraphwhoseshapelookslikethegraphofsin(x)sin,cosandtransformationsofthesefunctionsareallsinusoids(AKAsinusoidalfunctions) SinusoidagraphwhoseshapeloTransformationsInLesson1,welookedatthedifferentfunctiontransformationsWecansummarizethesewiththegeneralfunctiontransformation

Ifweapplythistothesineandcosinefunctions,wegetTransformationsInLesson1,weTransformationsForsimplicity,considerthetransformedsinefunction:verticalstretches,compressions&reflectionsGivesyoutheamplitudeHorizontalstretches,compressions&reflectionsGivesyoutheperiod:ShiftsupordownGivesyoutheaxisofcurveShiftsleftorrightCalledthephaseshiftTransformationsForsimplicity,Example1Whatistheamplitude,period,phaseshiftandaxisofcurveofExample1WhatistheamplitudeExample1:SolutionIngeneral:Wehave:a=3c=–1k=2theamplitudeis3theaxisofcurveisy=–1d=thephaseshiftistheperiodisExample1:SolutionIngeneral:Example2Whatistheamplitude,period,phaseshiftandverticalshiftofExample2WhatistheamplitudeExample2:SolutionIngeneral:Wehave:FACTOR!a=–2c=6theamplitudeis2theaxisofcurveisy=6k=0.5IgnorethenegativeExample2:SolutionIngeneral:Example2:NotesThetipoftheday:Alwaysfactorthecoefficientonthex-termtocorrectlyidentifythetransformationsIfyoudon’t,yourphaseshiftwillbeincorrectReminder:Theamplitudeistheverticaldistancefromtheaxisofcurvetothemaximumvalue.So,althoughthevalueofacanbenegative,theamplitudeisalwayspositiveExample2:NotesThetipoftheExample3Whatistheequationofthecosinefunctionafterithasbeenstretchedsothatitsperiodis,itsamplitudeis8andithasexperiencedaphaseshiftoftotheleftandhasbeenshiftedup1unitExample3WhatistheequationExample3:Solutionamplitudeis8phaseshiftis(left)ShifteduponeunitTheperiodisa=8d=c=1Example3:SolutionamplitudeiExample4Whatistheequationofthesinefunctionafterithasbeenstretchedsothatitsperiodis4,ithasexperiencedaphaseshiftoftotherightandithasbeenreflectedinthex-axisExample4WhatistheequationExample4:SolutionamplitudeisunchangedFunctionisreflectedinthex-axisphaseshiftis(right)ThereisnoverticalshiftTheperiodis4a=1d=c=0a=-1Example4:SolutionamplitudeiExample4:NotesItispossiblefortheperiodtonotbeamultipleofπIfthisisthecase,thek-valuewillbeintermsofπTheperiodisrarelyamultipleofπinreal-worldapplicationsEx.motionofapendulum,theheightofthetides,voltageinanACcircuitExample4:NotesItispossibleTheGraphsofsin&cosInordertounderstandtransformationsofsinandcos,it’scrucialfortoknowwhatf(x)=sinxandf(x)=cosxlooklikeTohelpyouwiththis,alwaysthinkofthefive“keypoints”ofoneperiodforeachfunction:sinhasthreezeroes,onemaxandonemin.Eachperiodstartsontheaxisofcurvecoshastwozeroes,twomaxandonemin.EachperiodstartsatthemaximumBecausethesefunctionsareperiodicifyouneedmorethanoneperiod,simplyrepeatthepatternTheGraphsofsin&cosInordeHowtoDeterminetheEquationFromaGraphDrawahorizontallinethatdividesthefunctioninhalf(axisofcurve)Locatethestartingpointofasine/cosineperiodandmarkthispointwithanA.Locatetheendingpointofasine/cosineperiodandmarkthispointwithaB.Determinethehorizontaldistancebetweenthesetwopoints(theperiod).Determinetheverticaldistancefromtheaxisofcurvetothemaximumvalue(amplitude)Becausethex-coordinateofthestartingpointforasine/cosineperiodiszero,thex-coordinateofAgivesyouthephaseshiftHowtoDeterminetheEquationExample5Useasinecurvetodeterminetheequationofthefunctiongivenbelow-12Example5UseasinecurvetodExample5:SolutionAB12-12Axisofcurveis-9PeriodisπAmplitudeis3PhaseshiftisExample5:SolutionAB12-12AxisExample5:Solutionamplitudeis3phaseshiftis(right)Axisofcurveis-9Theperiodisa=3d=c=-9Example5:SolutionamplitudeiExample6RepeatExample5,butthistimeuseacosinecurvetodeterminetheequationExample6RepeatExample5,butExample6:SolutionAB12-12Axisofcurveis-9PeriodisπAmplitudeis3Pha