湖北省武汉六中高三(国际部)数学复习课件：41三角函数的图像

Unit4:TrigonometricFunctions

Lesson1:TheGraphsofsin,cosandtanUnit4:TrigonometricFunctionTrigonometryIngrade10youwereintroducedtotrigonometrybyapplyingittorighttrianglesIngrade11youusedtrigonometrytosolveobliquetriangles(triangleswithoutarightangle)Thisrequiredyoutousesin,cosandtanforanglesgreaterthan90˚Next,youcreatedgraphsofsinandcosKnownastrigonometricfunctionsIngrade12,wewillcreategraphsofsin,cosandtanforanglesbetween0and2πWenowlookatthetrigonometricfunctionsinradiansTrigonometryIngrade10youwGraphsofsinandcosThegraphsoff(x)=sinxandf(x)=cosxwhenxisindegreesare:Orifweextendthembeyond0and360:GraphsofsinandcosThegraphTerminologyThefunctionsf(x)=sinxandf(x)=cosxareperiodicTheyhavearepeatingpatternTheperiodisthehorizontallengthoftherepeatingpatternTheaxisofcurveisequationofthehorizontallinethatcutsthegraphinhalfTheamplitudeisverticaldistancefromtheaxisofcurvetothemaximum(orminimum)pointBecauseitisadistance,theamplitudeisalwayspositiveTerminologyThefunctionsf(x)PropertiesoftheGraphofsinOneperiodAxisofcurveAmplitudeMaximumMinimumPeriod:Axisofcurve:Maximum:Minimum:Amplitude:360˚Theliney=01-11PropertiesoftheGraphofsinPropertiesoftheGraphofcosOneperiodAxisofcurveAmplitudeMaximumMinimumPeriod:Axisofcurve:Maximum:Minimum:Amplitude:360˚Theliney=01-11PropertiesoftheGraphofcosExample1UseyourTI-83or“Graph”tocreateagraphoff(x)=sinxwherexisinradiansFromthegraph,determineTheperiodTheaxisofcurveThemaximumandminimumvaluesTheamplitudeExample1UseyourTI-83or“GrExample1:SolutionOneperiodAxisofcurveAmplitudeMaximumMinimumPeriod:Axisofcurve:Maximum:Minimum:Amplitude:2πTheliney=01-11Example1:SolutionOneperiodAExample1:NotesThegraphoff(x)=sinxhasthesameshapeandpropertieswhenxisinradiansasitdoeswhenxisindegrees:Theonlydifferenceistheperiod360˚forxindegrees2πforxinradiansThismakessensebecausetheonlythingthatchangedwastheunitsforxandtheperioddependsonx360˚Example1:NotesThegraphoffExample2UseyourTI-83or“Graph”tocreateagraphoff(x)=cosxwherexisinradiansFromthegraph,determineTheperiodTheaxisofcurveThemaximumandminimumvaluesTheamplitudeExample2UseyourTI-83or“GrExample2:SolutionOneperiodAxisofcurveAmplitudeMaximumMinimumPeriod:Axisofcurve:Maximum:Minimum:Amplitude:2πTheliney=01-11Example2:SolutionOneperiodAExample2:NotesThegraphoff(x)=cosxhasthesameshapeandpropertieswhenxisinradiansasitdoeswhenxisindegrees:Theonlydifferenceistheperiod360˚forxindegrees2πforxinradiansThisisexactlywhatwesawforf(x)=sinx360˚Example2:NotesThegraphoffExample3UseyourTI-83or“Graph”tocreateagraphoff(x)=tanxwherexisinradiansFromthegraph,determineTheperiodTheaxisofcurveThemaximumandminimumvaluesTheamplitudeExample3UseyourTI-83or“GrExample3:SolutionOneperiodAxisofcurvePeriod:Axisofcurve:Maximum:Minimum:Amplitude:πTheliney=0nonenonenoneThegraphoftanhasverticalasymptotes!Example3:SolutionOneperiodAExample3:NotesAlthoughitisperiodic(period=π),thegraphoff(x)=tanxlooksnothinglikef(x)=sinxorf(x)=cosxf(x)=tanxhasnoamplitudebecauseithasnomaximumorminimumvaluesf(x)=tanxhasasymptotesatoddmultiplesofi.e.Example3:NotesAlthoughitisWhyf(x)=tanxHasAsymptotesUsingthequotientidentity,wecanseethatf(x)=tanxisarationalfunction:So,f(x)=tanxwillhaveasymptoteswherevercosx=0Becausecosx=0when f(x)=tanxhasasymptoteswhenWhyf(x)=tanxHasAsymptoteExample4(a)Onthesameaxis,graph f(x)=sin(x) f(x)=sin(x)+2 f(x)=sin(x)–3Makesurexisinradians(b)DescribewhatishappeningExample4(a)Onthesameaxis,Example4:SolutionThegraphoff(x)=sinxismovingupanddownExample4:SolutionThegraphoExample4:NotesByaddingavaluec,tof(x)=sinx,wemovethefunction…Upwhenc>0Downwhenc<0ThisisknownasaverticaltranslationThevalueofcisaddedtothey-coordinateofeverypointonthegraphoff(x)=sinx

Changestheaxisofcurvetotheliney=cExample4:NotesByaddingavaExample5(a)Onthesameaxis,graph f(x)=sin(x) f(x)=2sin(x) f(x)=0.5sin(x) f(x)=-3sin(x)Makesurexisinradians(b)DescribewhatishappeningExample5(a)Onthesameaxis,Example5:SolutionThegraphoff(x)=sinxisbeingstretchedandcompressed.Thegraphoff(x)=sinxis“flipped”overthex-axisandstretchedExample5:SolutionThegraphoExample5:NotesBymultiplyingf(x)=sinxbyavalueawe…Stretchthefunctionwhena>1Compressthefunctionwhen0<a<1ThisisknownasaverticaldilationWealsoreflectthefunctionoverthex-axiswhena<0ThisisknownasaverticalreflectionInbothcases,They-coordinateofeverypointonthegraphoff(x)=sinx

ismultipliedbyaTheamplitudeischangedto|a|Means“absolutevalue”andyouignorethenegativeExample5:NotesBymultiplyingExample6(a)Onthesameaxis,graph

Makesurexisinradians(b)DescribewhatishappeningExample6(a)Onthesameaxis,Example6:SolutionThegraphoff(x)=sinxismovingrightandleftExample6:SolutionThegraphoExample6:NotesBysubtractingavaluedintheargumentoff(x)=sinx,wemovethefunction…Leftwhend<0Rightwhend>0ThisisknownasahorizontaltranslationThevalueofdisaddedtothex-coordinateofeverypointonthegraphoff(x)=sinx

Dealingwithhorizontaltranslationsiscounter-intuitiveWhend<0thefunctionlookslike:f(x)=sin(x+d)andwemoveitleftWhend>0thefunctionlookslike:f(x)=sin(x–d)andwemoveitrightCommonlyreferredtoasaphaseshiftExample6:NotesBysubtractingExample7(a)Onthesameaxis,graph

Makesurexisinradians(b)DescribewhatishappeningExample7(a)Onthesameaxis,Example7:SolutionThegraphoff(x)=sinxisbeingstretchedorcompressedhorizontallyExample7:SolutionThegraphoExample7:NotesBymultiplyingtheargumentoff(x)=sinxbyavaluekwe…Compressthefunctionhorizontallywhenk>1Stretchthefunctionhorizontallywhen0<k<1ThisisknownasahorizontaldilationThex-coordinateofeverypointonthegraphoff(x)=sinx

ismultipliedby1/kTheperiodischangedfrom2πto:i.e.ifk=2,thetransformedfunctionwillhavetwoperiodsin2πExample7:NotesBymultiplyingSummary,Part1Thegraphsofsin,cosandtanhavethefollowingproperties:f(x)=sinxf(x)=cosxf(x)=tanxPeriod2π2ππMaxValue11N/AMinValue-1-1N/AAmplitude11N/AAxisofcurveY=0Y=0Y=0Asymptotesn/an/aGraphSummary,Part1ThegraphsofsSummary,Part2Wecantransformthegraphsoff(x)=sinxandf(x)=cosxinthefollowingways:Verticaltranslationf(x)=sin(x)+corf(x)=cos(x)+cMovetheaxisofcurvetoy=cHorizontaltranslationf(x)=sin(x–d)orf(x)=cos(x–d)AphaseshiftofdVerticalDilationf(x)=asin(x)orf(x)=acos(x)Changeamplitudeto|a|(nonegatives!)HorizontalDilationf(x)=sin(kx)orf(x)=cos(kx)ChangetheperiodtoSummary,Part2WecantransforPracticeProblemsP.258-260#1-11,15,19(notf)Note:Anygraphs/sketchescanbedoneusingyourTI-83ortheprogram“Graph”PracticeProblemsP.258-260#1Unit4:TrigonometricFunctions

Lesson1:TheGraphsofsin,cosandtanUnit4:TrigonometricFunctionTrigonometryIngrade10youwereintroducedtotrigonometrybyapplyingittorighttrianglesIngrade11youusedtrigonometrytosolveobliquetriangles(triangleswithoutarightangle)Thisrequiredyoutousesin,cosandtanforanglesgreaterthan90˚Next,youcreatedgraphsofsinandcosKnownastrigonometricfunctionsIngrade12,wewillcreategraphsofsin,cosandtanforanglesbetween0and2πWenowlookatthetrigonometricfunctionsinradiansTrigonometryIngrade10youwGraphsofsinandcosThegraphsoff(x)=sinxandf(x)=cosxwhenxisindegreesare:Orifweextendthembeyond0and360:GraphsofsinandcosThegraphTerminologyThefunctionsf(x)=sinxandf(x)=cosxareperiodicTheyhavearepeatingpatternTheperiodisthehorizontallengthoftherepeatingpatternTheaxisofcurveisequationofthehorizontallinethatcutsthegraphinhalfTheamplitudeisverticaldistancefromtheaxisofcurvetothemaximum(orminimum)pointBecauseitisadistance,theamplitudeisalwayspositiveTerminologyThefunctionsf(x)PropertiesoftheGraphofsinOneperiodAxisofcurveAmplitudeMaximumMinimumPeriod:Axisofcurve:Maximum:Minimum:Amplitude:360˚Theliney=01-11PropertiesoftheGraphofsinPropertiesoftheGraphofcosOneperiodAxisofcurveAmplitudeMaximumMinimumPeriod:Axisofcurve:Maximum:Minimum:Amplitude:360˚Theliney=01-11PropertiesoftheGraphofcosExample1UseyourTI-83or“Graph”tocreateagraphoff(x)=sinxwherexisinradiansFromthegraph,determineTheperiodTheaxisofcurveThemaximumandminimumvaluesTheamplitudeExample1UseyourTI-83or“GrExample1:SolutionOneperiodAxisofcurveAmplitudeMaximumMinimumPeriod:Axisofcurve:Maximum:Minimum:Amplitude:2πTheliney=01-11Example1:SolutionOneperiodAExample1:NotesThegraphoff(x)=sinxhasthesameshapeandpropertieswhenxisinradiansasitdoeswhenxisindegrees:Theonlydifferenceistheperiod360˚forxindegrees2πforxinradiansThismakessensebecausetheonlythingthatchangedwastheunitsforxandtheperioddependsonx360˚Example1:NotesThegraphoffExample2UseyourTI-83or“Graph”tocreateagraphoff(x)=cosxwherexisinradiansFromthegraph,determineTheperiodTheaxisofcurveThemaximumandminimumvaluesTheamplitudeExample2UseyourTI-83or“GrExample2:SolutionOneperiodAxisofcurveAmplitudeMaximumMinimumPeriod:Axisofcurve:Maximum:Minimum:Amplitude:2πTheliney=01-11Example2:SolutionOneperiodAExample2:NotesThegraphoff(x)=cosxhasthesameshapeandpropertieswhenxisinradiansasitdoeswhenxisindegrees:Theonlydifferenceistheperiod360˚forxindegrees2πforxinradiansThisisexactlywhatwesawforf(x)=sinx360˚Example2:NotesThegraphoffExample3UseyourTI-83or“Graph”tocreateagraphoff(x)=tanxwherexisinradiansFromthegraph,determineTheperiodTheaxisofcurveThemaximumandminimumvaluesTheamplitudeExample3UseyourTI-83or“GrExample3:SolutionOneperiodAxisofcurvePeriod:Axisofcurve:Maximum:Minimum:Amplitude:πTheliney=0nonenonenoneThegraphoftanhasverticalasymptotes!Example3:SolutionOneperiodAExample3:NotesAlthoughitisperiodic(period=π),thegraphoff(x)=tanxlooksnothinglikef(x)=sinxorf(x)=cosxf(x)=tanxhasnoamplitudebecauseithasnomaximumorminimumvaluesf(x)=tanxhasasymptotesatoddmultiplesofi.e.Example3:NotesAlthoughitisWhyf(x)=tanxHasAsymptotesUsingthequotientidentity,wecanseethatf(x)=tanxisarationalfunction:So,f(x)=tanxwillhaveasymptoteswherevercosx=0Becausecosx=0when f(x)=tanxhasasymptoteswhenWhyf(x)=tanxHasAsymptoteExample4(a)Onthesameaxis,graph f(x)=sin(x) f(x)=sin(x)+2 f(x)=sin(x)–3Makesurexisinradians(b)DescribewhatishappeningExample4(a)Onthesameaxis,Example4:SolutionThegraphoff(x)=sinxismovingupanddownExample4:SolutionThegraphoExample4:NotesByaddingavaluec,tof(x)=sinx,wemovethefunction…Upwhenc>0Downwhenc<0ThisisknownasaverticaltranslationThevalueofcisaddedtothey-coordinateofeverypointonthegraphoff(x)=sinx

Changestheaxisofcurvetotheliney=cExample4:NotesByaddingavaExample5(a)Onthesameaxis,graph f(x)=sin(x) f(x)=2sin(x) f(x)=0.5sin(x) f(x)=-3sin(x)Makesurexisinradians(b)DescribewhatishappeningExample5(a)Onthesameaxis,Example5:SolutionThegraphoff(x)=sinxisbeingstretchedandcompressed.Thegraphoff(x)=sinxis“flipped”overthex-axisandstretchedExample5:SolutionThegraphoExample5:NotesBymultiplyingf(x)=sinxbyavalueawe…Stretchthefunctionwhena>1Compressthefunctionwhen0<a<1ThisisknownasaverticaldilationWealsoreflectthefunctionoverthex-axiswhena<0ThisisknownasaverticalreflectionInbothcases,They-coordinateofeverypointonthegraphoff(x)=sinx

ismultipliedbyaTheamplitudeischangedto|a|Means“absolutevalue”andyouignorethenegativeExample5:NotesBymultiplyingExample6(a)Onthesameaxis,graph

Makesurexisinradians(b)DescribewhatishappeningExample6(a)Onthesameaxis,Example6:SolutionThegraphoff(x)=sinxismovingrightandleftExample6:SolutionThegraphoExample6:NotesBysubtractingavaluedintheargumentoff(x)=sinx,wemovethefunction…Leftwhend<0Rightwhend>0ThisisknownasahorizontaltranslationThevalueofdisaddedtothex-coordinateofeverypointonthegraphoff(x)=sinx

Dealingwithhorizontaltranslationsiscounter-intuitiveWhend<0thefunctionlookslike:f(x)=sin(x+d)andwemoveitleftWhend>0thefunctionlookslike:f(x)=sin(x–d)andwemoveitrightCommonlyreferredtoasaphaseshiftExample6:NotesBysubtractingExample7(a)Onthesameaxis,graph

Make