Chapter1-Polynomial

Functions

LessonPackage

MHF4U

Chapter1OutlineUnitGoal:Bytheendofthisunit,youwillbeabletoidentifyanddescribesomekeyfeaturesofpolynomialfunctions,andmakeconnectionsbetweenthenumeric,graphical,andalgebraicrepresentationsofpolynomialfunctions.

Section Subject LearningGoals CurriculumExpectations

L1 PowerFunctions-describekeyfeaturesofgraphsofpowerfunctions-learnintervalnotation-beabletodescribeendbehaviour

C1.1,1.2,1.3

L2 CharacteristicsofPolynomialFunctions

-describecharacteristicsofequationsandgraphsofpolynomialfunctions-learnhowdegreerelatedtoturningpointsand𝑥-intercepts

C1.1,1.2,1.3,1.4

L3 FactoredFormPolynomialFunctions

-connecthowfactoredformequationrelatedto𝑥-interceptsofgraphofpolynomialfunction-givengraph,determineequationinfactoredform

C1.5,1.7,1.8

L4 TransformationsofPolynomialFunctions

-understandhowtheparameters𝑎, 𝑘, 𝑑,and𝑐transformpowerfunctions

C1.6

L5 SymmetryinPolynomialFunctions

-understandthepropertiesofevenandoddpolynomialfunctions C1.9

Assessments F/A/O MinistryCode P/O/C KTACNoteCompletion A P PracticeWorksheetCompletion F/A P

Quiz–PropertiesofPolynomialFunctions F P

PreTestReview F/A P Test-Functions O C1.1,1.2,1.3,1.4,1.5,1.6,1.7,

1.8,1.9 P K(21%),T(34%),A(10%),C(34%)

L1–1.1–PowerFunctionsLessonMHF4UJensenThingstoRememberAboutFunctions

• Arelationisafunctionifforevery𝑥-valuethereisonly1corresponding𝑦-value.Thegraphofarelationrepresentsafunctionifitpassesthe________________________________________,thatis,ifaverticallinedrawnanywherealongthegraphintersectsthatgraphatnomorethanonepoint.

• The________________ofafunctionisthecompletesetofallpossiblevaluesoftheindependentvariable(𝑥)

o Setofallpossible𝑥-valesthatwilloutputreal𝑦-values

• The________________ofafunctionisthecompletesetofallpossibleresultingvaluesofthedependentvariable(𝑦)

o Setofallpossible𝑦-valueswegetaftersubstitutingallpossible𝑥-values

• Forthefunction𝑓 𝑥 = (𝑥 − 1)) + 3

• Thedegreeofafunctionisthehighestexponentintheexpressiono 𝑓 𝑥 = 6𝑥- − 3𝑥) + 4𝑥 − 9hasadegreeof_____.

• An__________________________isalinethatacurveapproachesmoreandmorecloselybutnevertouches.

Thefunction𝒚 = 𝟏

𝒙3𝟑hastwoasymptotes:

VerticalAsymptote:Divisionbyzeroisundefined.Thereforetheexpressioninthedenominatorofthefunctioncannotbezero.Thereforex≠-3.Thisiswhytheverticallinex=-3isanasymptoteforthisfunction.HorizontalAsymptote:Fortherange,therecanneverbeasituationwheretheresultofthedivisioniszero.Thereforetheliney=0isahorizontalasymptote.Forallfunctionswherethedenominatorisahigherdegreethanthenumerator,therewillbyahorizontalasymptoteaty=0.

PolynomialFunctionsApolynomialfunctionhastheform

𝑓 𝑥 = 𝑎6𝑥6 + 𝑎678𝑥678 + 𝑎67)𝑥67) + ⋯+ 𝑎)𝑥) + 𝑎8𝑥8 + 𝑎:

• 𝑛Isawholenumber• 𝑥Isavariable• the_________________________𝑎:, 𝑎8, … , 𝑎6arerealnumbers• the___________________ofthefunctionis𝑛,theexponentofthegreatestpowerof𝑥• 𝑎6,thecoefficientofthegreatestpowerof𝑥,isthe________________________________• 𝑎:,thetermwithoutavariable,isthe___________________________• Thedomainofapolynomialfunctionisthesetofrealnumbers__________________• Therangeofapolynomialfunctionmaybeallrealnumbers,oritmayhavealowerboundoran

upperbound(butnotboth)• Thegraphofpolynomialfunctionsdonothavehorizontalorverticalasymptotes• Thegraphsofpolynomialfunctionsofdegree0are__________________________.Theshapesofother

graphsdependsonthedegreeofthefunction.Fivetypicalshapesareshownforvariousdegrees:A___________________________isthesimplesttypeofpolynomialfunctionandhastheform:

𝑓 𝑥 = 𝑎𝑥6• 𝑎isarealnumber• 𝑥isavariable• 𝑛isawholenumber

Example1:Determinewhichfunctionsarepolynomials.Statethedegreeandtheleadingcoefficientofeachpolynomialfunction.a)𝑔 𝑥 = sin 𝑥b)𝑓 𝑥 = 2𝑥Cc)𝑦 = 𝑥- − 5𝑥) + 6𝑥 − 8d)𝑔 𝑥 = 3F

IntervalNotationInthiscourse,youwilloftendescribethefeaturesofthegraphsofavarietyoftypesoffunctionsinrelationtoreal-numbervalues.Setsofrealnumbersmaybedescribedinavarietyofways:1)asaninequality−3 < 𝑥 ≤ 52)interval(orbracket)notation(−3, 5]3)graphicallyonanumberlineNote:

• Intervalsthatareinfiniteareexpressedusing_____________________or_____________________________• ________________________indicatethattheendvalueisincludedintheinterval• ________________________indicatethattheendvalueisNOTincludedintheinterval• A________________bracketisalwaysusedatinfinityandnegativeinfinity

Example2:Belowarethegraphsofcommonpowerfunctions.Usethegraphtocompletethetable.

PowerFunction

SpecialName Graph Domain Range

EndBehaviouras𝒙 →−∞

EndBehaviouras

𝒙 → ∞

𝒚 = 𝒙 Linear

𝒚 = 𝒙𝟐 Quadratic

𝒚 = 𝒙𝟑 Cubic

PowerFunction

SpecialName Graph Domain Range

EndBehaviouras𝒙 →−∞

EndBehaviouras

𝒙 → ∞

𝒚 = 𝒙𝟒 Quartic

𝒚 = 𝒙𝟓 Quintic

𝒚 = 𝒙𝟔 Sextic

KeyFeaturesofEVENDegreePowerFunctionsWhentheleadingcoefficient(a)ispositive Whentheleadingcoefficient(a)isnegative

Endbehaviour

Endbehaviour

Domain

Domain

Range Range

Example:

𝑓 𝑥 = 2𝑥C

Example:𝑓 𝑥 = −3𝑥)

LineSymmetryAgraphhaslinesymmetryifthereisaverticalline𝑥 = 𝑎thatdividesthegraphintotwopartssuchthateachpartisareflectionoftheother.Note:

KeyFeaturesofODDDegreePowerFunctionsWhentheleadingcoefficient(a)ispositive Whentheleadingcoefficient(a)isnegative

Endbehaviour

Endbehaviour

Domain

Domain

Range Range

Example:

𝑓 𝑥 = 3𝑥Q

Example:𝑓 𝑥 = −2𝑥-

PointSymmetryAgraphhaspointpointsymmetryaboutapoint(𝑎, 𝑏)ifeachpartofthegraphononesideof(𝑎, 𝑏)canberotated180°tocoincidewithpartofthegraphontheothersideof(𝑎, 𝑏).Note:

Example3:Writeeachfunctionintheappropriaterowofthesecondcolumnofthetable.Givereasonsforyourchoices.𝑦 = 2𝑥 𝑦 = 5𝑥U 𝑦 = −3𝑥) 𝑦 = 𝑥V𝑦 = − )

Q𝑥W 𝑦 = −4𝑥Q 𝑦 = 𝑥8: 𝑦 = −0.5𝑥Y

EndBehaviour Functions Reasons

Q3toQ1

Q2toQ4

Q2toQ1

Q3toQ4

Example4:Foreachofthefollowingfunctionsi)Statethedomainandrangeii)Describetheendbehavioriii)Identifyanysymmetry

a)b)c)

𝒚 = −𝒙

𝒚 = 𝟎. 𝟓𝒙𝟐

𝒚 = 𝟒𝒙𝟑

i)Domain: Range:ii)As_____________________andas______________________Thegraphextendsfromquadrant_____to_____iii)

i)Domain: Range:ii)As_____________________andas______________________Thegraphextendsfromquadrant_____to_____iii)

i)Domain: Range:ii)As_____________________andas______________________Thegraphextendsfromquadrant_____to_____iii)

L2–1.2–CharacteristicsofPolynomialFunctionsLessonMHF4UJensenInsection1.1welookedatpowerfunctions,whicharesingle-termpolynomialfunctions.Manypolynomialfunctionsaremadeupoftwoormoreterms.Inthissectionwewilllookatthecharacteristicsofthegraphsandequationsofpolynomialfunctions.NewTerminology–LocalMin/Maxvs.AbsoluteMin/MaxLocalMinorMaxPoint–Pointsthatareminimumormaximumpointsonsomeintervalaroundthatpoint.AbsoluteMaxorMin–Thegreatest/leastvalueattainedbyafunctionforALLvaluesinitsdomain.Investigation:GraphsofPolynomialFunctionsThedegreeandtheleadingcoefficientintheequationofapolynomialfunctionindicatetheendbehavioursofthegraph.Thedegreeofapolynomialfunctionprovidesinformationabouttheshape,turningpoints(localmin/max),andzeros(x-intercepts)ofthegraph.Completethefollowingtableusingtheequationandgraphsgiven:

Inthisgraph,(-1,4)isa___________________and(1,-4)isa___________________.Thesearenotabsoluteminandmaxpointsbecausethereareotherpointsonthegraphofthefunctionthataresmallerandgreater.Sometimeslocalminandmaxpointsarecalled______________________________.

Onthegraphofthisfunction…Thereare____localmin/maxpoints.____arelocalminand____isalocalmax.Oneofthelocalminpointsisalsoanabsolutemin(itislabeled).

EquationandGraph DegreeEvenorOdd

Degree?

LeadingCoefficient EndBehaviour

Numberofturningpoints

Numberofx-intercepts

𝑓 𝑥 = 𝑥$ + 4𝑥 − 5

𝑓 𝑥 = 3𝑥* − 4𝑥+ − 4𝑥$ + 5𝑥 + 5

𝑓 𝑥 = 𝑥+ − 2𝑥

𝑓 𝑥 = −𝑥* − 2𝑥+ + 𝑥$ + 2𝑥

𝑓 𝑥 = 2𝑥- − 12𝑥* + 18𝑥$ + 𝑥 − 10

𝑓 𝑥 = 2𝑥1 + 7𝑥* − 3𝑥+ − 18𝑥$ + 5

SummaryofFindings:

• Apolynomialfunctionofdegree𝑛hasatmost_____________localmax/minpoints(turningpoints)• Apolynomialfunctionofdegree𝑛mayhaveupto_____distinctzeros(x-intercepts)• Ifapolynomialfunctionis_________degree,itmusthaveatleastonex-intercept,andanevennumberof

turningpoints• Ifapolynomialfunctionis________degree,itmayhavenox-intercepts,andanoddnumberofturningpoints• Anodddegreepolynomialfunctionextendsfrom…

o _____quadrantto_____quadrantifithasapositiveleadingcoefficiento _____quadrantto_____quadrantifithasanegativeleadingcoefficient

• Anevendegreepolynomialfunctionextendsfrom…o _____quadrantto_____quadrantifithasapositiveleadingcoefficiento _____quadrantto_____quadrantifithasanegativeleadingcoefficient

EquationandGraph DegreeEvenorOdd

Degree?

LeadingCoefficient EndBehaviour

Numberofturningpoints

Numberofx-intercepts

𝑓 𝑥 = 5𝑥1 + 5𝑥* − 2𝑥+ + 4𝑥$ − 3𝑥

𝑓 𝑥 = −2𝑥+ + 4𝑥$ − 3𝑥 − 1

𝑓 𝑥 = 𝑥* + 2𝑥+ − 3𝑥 − 1

Note:OdddegreepolynomialshaveOPPOSITEendbehaviours

Note:EvendegreepolynomialshaveTHESAMEendbehaviour.

Example1:Describetheendbehavioursofeachfunction,thepossiblenumberofturningpoints,andthepossiblenumberofzeros.Usethesecharacteristicstosketchpossiblegraphsofthefunctiona)𝑓 𝑥 = −3𝑥1 + 4𝑥+ − 8𝑥$ + 7𝑥 − 5Possiblegraphsof5thdegreepolynomialfunctionswithanegativeleadingcoefficient:b)𝑔 𝑥 = 2𝑥* + 𝑥$ + 2Possiblegraphsof4thdegreepolynomialfunctionswithapositiveleadingcoefficient:Example2:Filloutthefollowingchart

Degree Possible#of𝒙-intercepts Possible#ofturningpoints1 2 3 4 5

Note:Odddegreefunctionsmusthaveanevennumberofturningpoints.

Note:Evendegreefunctionsmusthaveanoddnumberofturningpoints.

Example3:Determinethekeyfeaturesofthegraphofeachpolynomialfunction.Usethesefeaturestomatcheachfunctionwithitsgraph.Statethenumberof𝑥-intercepts,thenumberoflocalmax/minpoints,andthenumberofabsolutemax/minpointsforthegraphofeachfunction.Howarethesefeaturesrelatedtothedegreeofeachfunction?a)𝑓 𝑥 = 2𝑥+ − 4𝑥$ + 𝑥 + 1b)𝑔 𝑥 = −𝑥* + 10𝑥$ + 5𝑥 − 4c)ℎ 𝑥 = −2𝑥1 + 5𝑥+ − 𝑥d)𝑝 𝑥 = 𝑥- − 16𝑥$ + 3a)b)c)d)FiniteDifferencesForapolynomialfunctionofdegree𝑛,where𝑛isapositiveinteger,the𝑛9:differences…

• areequal• havethesamesignastheleadingcoefficient• areequalto𝑎 ∙ 𝑛!,where𝑎istheleadingcoefficient

Note:𝑛!isreadas𝑛factorial.𝑛! = 𝑛×(𝑛 − 1)×(𝑛 − 2)×…×2×15! = 5×4×3×2×1 = 120

Example4:Thetableofvaluesrepresentsapolynomialfunction.Usefinitedifferencestodetermine

a) thedegreeofthepolynomialfunctionb) thesignoftheleadingcoefficientc) thevalueoftheleadingcoefficient

a)b)c) Example5:Forthefunction2𝑥* − 4𝑥$ + 𝑥 + 1whatisthevalueoftheconstantfinitedifferences?𝐹𝑖𝑛𝑖𝑡𝑒𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠 =

𝒙 𝒚 Firstdifferences Seconddifferences Thirddifferences

-3 -36 -2 -12 -1 -2 0 0 1 0 2 4 3 18 4 48

L3–1.3–FactoredFormPolynomialFunctionsLessonMHF4UJensenInthissection,youwillinvestigatetherelationshipbetweenthefactoredformofapolynomialfunctionandthe𝑥-interceptsofthecorrespondinggraph,andyouwillexaminetheeffectofrepeatedfactoronthegraphofapolynomialfunction.

FactoredFormInvestigation

Ifwewanttographthepolynomialfunction𝑓 𝑥 = 𝑥$ + 3𝑥' + 𝑥( − 3𝑥 − 2accurately,itwouldbemostusefultolookatthefactoredformversionofthefunction:𝑓 𝑥 = 𝑥 + 1 ( 𝑥 + 2 𝑥 − 1

Let’sstartbylookingatthegraphofthefunctionandmakingconnectionstothefactoredformequation.

Graphof𝑓(𝑥): Fromthegraph,answerthefollowingquestions…

a)Whatisthedegreeofthefunction? b)Whatisthesignoftheleadingcoefficient? c)Whatarethe𝑥-intercepts? d)Whatisthe𝑦-intercept?e)The𝑥-interceptsdividethegraphintointofourintervals.Writetheintervalsinthefirstrowofthetable.Inthesecondrow,chooseatestpointwithintheinterval.Inthethirdrow,indicatewhetherthefunctionispositive(abovethe𝑥-axis)ornegative(belowthe𝑦-axis).

Interval

TestPoint

Signof𝒇(𝒙)

f)Whathappenstothesignoftheof𝑓(𝑥)neareach𝑥-intercept?

Conclusionsfrominvestigation:The𝑥-interceptsofthegraphofthefunctioncorrespondtotheroots(zeros)ofthecorrespondingequation.Forexample,thefunction𝑓 𝑥 = (𝑥 − 2)(𝑥 + 1)has𝑥-interceptsat___and___.Thesearetherootsoftheequation 𝑥 − 2 𝑥 + 1 = 0.Ifapolynomialfunctionhasafactor(𝑥 − 𝑎)thatisrepeated𝑛times,then𝑥 = 𝑎isazeroof_________𝑛.Order–theexponenttowhicheachfactorinanalgebraicexpressionisraised.Forexample,thefunction𝑓 𝑥 = 𝑥 − 3 ((𝑥 − 1)hasazerooforder______at𝑥 = 3andazerooforder______at𝑥 = 1.Thegraphofapolynomialfunctionchangessignatzerosof_________orderbutdoesnotchangesignatzerosof_________order.Shapesbasedonorderofzero:

𝑓 𝑥 = 0.01(𝑥 − 1) 𝑥 + 2 ((𝑥 − 4)'

ORDER2(-2,0)isan𝑥-interceptoforder2.Therefore,itdoesn’tchangesign.“Bouncesoff”𝑥-axis.Parabolicshape.

ORDER1(1,0)isan𝑥-interceptoforder1.Therefore,itchangessign.“Goesstraightthrough”𝑥-axis.LinearShape

ORDER3(4,0)isan𝑥-interceptoforder3.Therefore,itchangessign.“S-shape”through𝑥-axis.Cubicshape.

Example1:AnalyzingGraphsofPolynomialFunctionsForeachgraph,

i) theleastpossibledegreeandthesignoftheleadingcoefficientii) the𝑥-interceptsandthefactorsofthefunctioniii) theintervalswherethefunctionispositive/negative

a) i) ii) iii)b) i) ii) iii)

Example2:AnalyzeFactoredFormEquationstoSketchGraphs

Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-interceptTheexponenton𝑥whenallfactorsof𝑥aremultipliedtogether

OR

Addtheexponentsonthefactorsthatincludean𝑥.

Theproductofallthe𝑥coefficients

Usedegreeandsignofleadingcoefficienttodeterminethis

Seteachfactorequaltozeroandsolvefor𝑥

Set𝑥 = 0andsolvefor𝑦

Sketchagraphofeachpolynomialfunction:a)𝑓 𝑥 = (𝑥 − 1)(𝑥 + 2)(𝑥 + 3)

Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-intercept

b)𝑔 𝑥 = −2 𝑥 − 1 ((𝑥 + 2)

Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-intercept

c)ℎ 𝑥 = − 2𝑥 + 1 '(𝑥 − 3)

Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-intercept

d)𝑗 𝑥 = 𝑥$ − 4𝑥' + 3𝑥(

Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-intercept

Example3:RepresentingtheGraphofaPolynomialFunctionwithitsEquationa)Writetheequationofthefunctionshownbelow:

Note:mustputintofactoredformtofind𝑥-intercepts

Steps:1)Writetheequationofthefamilyofpolynomialsusingfactorscreatedfrom𝑥-intercepts2)Substitutethecoordinatesofanotherpoint(𝑥, 𝑦)intotheequation.3)Solvefor𝑎4)Writetheequationinfactoredform

b)Findtheequationofapolynomialfunctionthatisdegree4withzeros−1(order3)and1,andwitha𝑦-interceptof−2.

L4–1.4–TransformationsLessonMHF4UJensenInthissection,youwillinvestigatetherolesoftheparameters𝑎, 𝑘, 𝑑,and𝑐inpolynomialfunctionsoftheform𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄.Youwillapplytransformationstothegraphsofbasicpowerfunctionstosketchthegraphofitstransformedfunction.Part1:TransformationsInvestigationInthisinvestigation,youwillbelookingattransformationsofthepowerfunction𝑦 = 𝑥4.Completethefollowingtableusinggraphingtechnologytohelp.Thegraphof𝑦 = 𝑥4isgivenoneachsetofaxes;sketchthegraphofthetransformedfunctiononthesamesetofaxes.Thencommentonhowthevalueoftheparameter𝑎, 𝑘, 𝑑,or𝑐transformstheparentfunction.Effectsof𝑐on𝑦 = 𝑥4 + 𝑐TransformedFunction

Valueof𝒄 Transformationsto𝒚 = 𝒙𝟒 Graphoftransformedfunctioncomparedto𝒚 = 𝒙𝟒

𝑦 = 𝑥4 + 1

𝑦 = 𝑥4 − 2

Effectsof𝑑on𝑦 = (𝑥 − 𝑑)4TransformedFunction

Valueof𝒅 Transformationsto𝒚 = 𝒙𝟒 Graphoftransformedfunctioncomparedto𝒚 = 𝒙𝟒

𝑦 = (𝑥 − 2)4

𝑦 = (𝑥 + 3)4

Effectsof𝑎on𝑦 = 𝑎𝑥4TransformedFunction

Valueof𝒂 Transformationsto𝒚 = 𝒙𝟒 Graphoftransformedfunctioncomparedto𝒚 = 𝒙𝟒

𝑦 = 2𝑥4

𝑦 =12𝑥

4

𝑦 = −2𝑥4

Effectsof𝑘on𝑦 = (𝑘𝑥)4TransformedFunction

Valueof𝒌 Transformationsto𝒚 = 𝒙𝟒 Graphoftransformedfunctioncomparedto𝒚 = 𝒙𝟒

𝑦 = (2𝑥)4

𝑦 =13𝑥

4

𝑦 = (−2𝑥)4

Summaryofeffectsof𝑎, 𝑘, 𝑑,and𝑐inpolynomialfunctionsoftheform𝑓(𝑥) = 𝑎 𝑘 𝑥 − 𝑑 ; + 𝑐

Valueof𝒄in𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄

𝑐 > 0

𝑐 < 0

Valueof𝒅in𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄

𝑑 > 0

𝑑 < 0

Valueof𝒂in𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄

𝑎 > 1or𝑎 < −1

−1 < 𝑎 < 1

𝑎 < 0

Valueof𝒌in𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄

𝑘 > 1or𝑘 < −1

−1 < 𝑘 < 1

𝑘 < 0

Note:𝑎and𝑐cause_________________transformationsandthereforeeffectthe𝑦-coordinatesofthefunction.𝑘and𝑑cause_________________transformationsandthereforeeffectthe𝑥-coordinatesofthefunction.Whenapplyingtransformationstoaparentfunction,makesuretoapplythetransformationsrepresentedby𝑎and𝑘BEFOREthetransformationsrepresentedby𝑑and𝑐.

Part2:DescribingTransformationsfromanEquationExample1:Describethetransformationsthatmustbeappliedtothegraphofeachpowerfunction,𝑓(𝑥),toobtainthetransformedfunction,𝑔(𝑥).Then,writethecorrespondingequationofthetransformedfunction.Then,statethedomainandrangeofthetransformedfunction.a)𝑓 𝑥 = 𝑥4,𝑔 𝑥 = 2𝑓 @

A𝑥 − 5

b)𝑓 𝑥 = 𝑥C,𝑔 𝑥 = @

4𝑓 −2 𝑥 − 3 + 4

Part3:ApplyingTransformationstoSketchaGraphExample2:Thegraphof𝑓(𝑥) = 𝑥Aistransformedtoobtainthegraphof𝑔(𝑥) = 3 −2 𝑥 + 1 A + 5.a)Statetheparametersanddescribethecorrespondingtransformationsb)Makeatableofvaluesfortheparentfunctionandthenusethetransformationsdescribedinparta)tomakeatableofvaluesforthetransformedfunction.

c)Graphtheparentfunctionandthetransformedfunctiononthesamegrid.

𝑓 𝑥 = 𝑥A𝒙 𝒚

Note:Whenchoosingkeypointsfortheparentfunction,alwayschoose𝑥-valuesbetween-2and2andcalculatethecorrespondingvaluesof𝑦.

Example3:Thegraphof𝑓(𝑥) = 𝑥4istransformedtoobtainthegraphof𝑔(𝑥) = − @A𝑥 + 2

4− 1.

a)Statetheparametersanddescribethecorrespondingtransformationsb)Makeatableofvaluesfortheparentfunctionandthenusethetransformationsdescribedinparta)tomakeatableofvaluesforthetransformedfunction.

c)Graphtheparentfunctionandthetransformedfunctiononthesamegrid.

𝑓 𝑥 = 𝑥4𝒙 𝒚

Note:𝑘valuemustbefactoredoutintotheform[𝑘(𝑥 + 𝑑)]

Part4:DetermininganEquationGiventheGraphofaTransformedFunctionExample4:Transformationsareappliedtoeachpowerfunctiontoobtaintheresultinggraph.Determineanequationforthetransformedfunction.Thenstatethedomainandrangeofthetransformedfunction.a)b)

f(1)

f(-1)

f(1)f(-1)

𝑓(𝑥) = 2𝑥' + 3𝑥* − 2Notice:𝑓(1) =𝑓(−1) =∴

L5–1.3–SymmetryinPolynomialFunctionsMHF4UJensenInthissection,youwilllearnaboutthepropertiesofevenandoddpolynomialfunctions.SymmetryinPolynomialFunctions__________________–thereisaverticallineoverwhichthepolynomialremainsunchangedwhenreflected.___________________________________–thereisapointaboutwhichthepolynomialremainsunchangedwhenrotated180°Section1:PropertiesofEvenandOddFunctionsApolynomialfunctionofevenorodddegreeisNOTnecessarilyandevenoroddfunction.Thefollowingarepropertiesofallevenandoddfunctions:

EvenFunctions OddFunctionsAnevendegreepolynomialfunctionisanEVENFUNCTIONif:

• Linesymmetryoverthe__________• Theexponentofeachtermis_______• Mayhaveaconstantterm

AnodddegreepolynomialfunctionisanODDFUNCTIONif:

• Pointsymmetryaboutthe__________• Theexponentofeachtermis_____• Noconstantterm

Rule:

Rule:

Example:

Example:

𝑓(𝑥) = 2𝑥1 + 3𝑥Notice:𝑓(1) =𝑓(−1) =∴

Example1:Identifyeachfunctionasanevenfunction,oddfunction,orneither.Explainhowyoucantell.a)𝑦 = 𝑥1 − 4𝑥b)𝑦 = 𝑥1 − 4𝑥 + 2c)𝑦 = 𝑥' − 4𝑥* + 2

d)𝑦 = 3𝑥' + 𝑥1 − 4𝑥* + 2e)𝑦 = −3𝑥* − 6𝑥Example2:Chooseallthatapplyforeachfunctiona) b)

i)nosymmetry

ii)pointsymmetry

iii)linesymmetry

iv)oddfunction

v)evenfunction

i)nosymmetry

ii)pointsymmetry

iii)linesymmetry

iv)oddfunction

v)evenfunction

c)𝑃 𝑥 = 5𝑥1 + 3𝑥* + 2 d)𝑃 𝑥 = 𝑥8 + 𝑥* − 11

e) f)g)𝑃 𝑥 = 5𝑥9 − 4𝑥1 + 8𝑥Example3:Withoutgraphing,determineifeachpolynomialfunctionhaslinesymmetryaboutthey-axis,pointsymmetryabouttheorigin,orneither.Verifyyourresponsealgebraically.a)𝑓 𝑥 = 2𝑥' − 5𝑥* + 4

i)nosymmetry

ii)pointsymmetry

iii)linesymmetry

iv)oddfunction

v)evenfunction

i)nosymmetry

ii)pointsymmetry

iii)linesymmetry

iv)oddfunction

v)evenfunction

i)nosymmetry

ii)pointsymmetry

iii)linesymmetry

iv)oddfunction

v)evenfunction

i)nosymmetry

ii)pointsymmetry

iii)linesymmetry

iv)oddfunction

v)evenfunction

i)nosymmetry

ii)pointsymmetry

iii)linesymmetry

iv)oddfunction

v)evenfunction

Note:

(-1, 3)

b)𝑓 𝑥 = −3𝑥9 + 9𝑥1 + 2𝑥c)𝑥8 − 4𝑥1 + 6𝑥* − 4Section2:ConnectingfromthroughouttheunitExample4:Usethegivengraphtostate:a)𝑥-interceptsb)numberofturningpointsc)leastpossibledegreeb)anysymmetrypresentc)theintervalswhere𝑓 𝑥 < 0

d)Findtheequationinfactoredform