Lesson15

M1ALGEBRAII

Lesson15:StructureinGraphsofPolynomialFunctions

StudentOutcomes

§ Studentsgraphpolynomialfunctionsanddescribeendbehaviorbaseduponthedegreeofthepolynomial.

LessonNotesSofarinthismodule,studentshavepracticedfactoringpolynomialsusingseveraltechniquesandexaminedhowtheycanusethefactoredformofthepolynomialtoidentifyinterestingcharacteristicsofthegraphsofthesefunctions.Inthislesson,studentscontinueexploringgraphsofpolynomialfunctionsinordertoidentifyhowthedegreeofthepolynomialinfluencestheendbehaviorofthesegraphs.Theyalsodiscusshowtoidentify𝑦-interceptsofthegraphsofpolynomialfunctionsandaregivenanopportunitytoconstructviableargumentsandcritiquethereasoningofothersintheOpeningExercise(MP.3).

OpeningExercise(8minutes)OpeningExercise

Sketchthegraphof𝒇 𝒙 = 𝒙𝟐.Whatwillthegraphof𝒈 𝒙 = 𝒙𝟒looklike?Sketchitonthesamecoordinateplane.Whatwillthegraphof𝒉 𝒙 = 𝒙𝟔looklike?

Havestudentsrecallandsketchthegraphof𝑓 𝑥 = 𝑥,.Discussthecharacteristicsofthegraph,wherethe𝑥-interceptis,andwhythegraphstaysabovethe𝑥-axisoneithersideofthe𝑥-intercept.

Inpairsoringroups,havethemdiscussorwritewhattheythinkthegraphof𝑔 𝑥 = 𝑥.willlooklikeandhowtheythinkitcomparestothegraphof𝑓 𝑥 = 𝑥,.Oncetheydoso,theyshouldsketchtheirideaofthegraphof𝑔ontopofthegraphof𝑓.Discusswithstudentswhattheyhavesketched,andemphasizethesimilaritiesbetweenthetwographs.

Since𝒈 𝒙 = 𝒙𝟐 𝟐,𝒈(𝒙)willincreasefasteras𝒙increasesthan𝒇 𝒙 does.Bothgraphspassthrough 𝟎, 𝟎 .Thebasicshapesarethesame,butneartheoriginthegraphof𝒈isflatterthanthegraphof𝒇.

Finally,inpairsoringroups,havestudentsdiscussorwritewhattheythinkthegraphofℎ 𝑥 = 𝑥5willlooklikeandhowtheythinkitwillcomparetographsof𝑓and𝑔.Oncetheydoso,studentsshouldsketchonthesamegraphtheprevioustwographs.Again,discussgraphswithstudents,andemphasizethesimilaritiesbetweengraphs.

Since𝒉 𝒙 = 𝒙𝟐 ⋅ 𝒙𝟐 ⋅ 𝒙𝟐,thegraphof𝒉againpassesthroughtheorigin.Sincewearesquaringandmultiplyingbysquares,thegraphof𝒉shouldlookaboutthesameasthegraphsof𝒇and𝒈butincreaseevenfasterandbeevenflatterneartheorigin.

150

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M1ALGEBRAII

§ Usingagraphingutility,havestudentsgraphallthreefunctionssimultaneouslytoconfirmtheirsketches.

Discussion(5minutes)

UsethegraphsfromtheOpeningExercisetoframethefollowingdiscussionaboutendbehavior.

Askstudentstocompareanddescribethebehaviorofthevalue𝑓(𝑥)astheabsolutevalueof𝑥increaseswithoutbound.Introducethetermendbehaviorasawaytotalkaboutthefunctionandwhathappenstoitsgraphbeyondtheboundedregionofthecoordinateplanethatisdrawnonpaper.Thatis,theendbehaviorisawaytodescribewhathappenstothefunctionas𝑥approachespositiveandnegativeinfinitywithouthavingtodrawthegraph.

Notetoteacher:Itisimportanttonotethatendbehaviorcannotbegivenaprecisemathematicaldefinitionuntiltheconceptofalimitisintroducedincalculus.Togetaroundthisdifficulty,mosthighschooltextbooksdrawpicturesandstatethingslike,“As𝑥 → ∞,𝑓 𝑥 → ∞.”Wedothisalso,butitisimportanttocarefullydescribetostudentsthemeaningofthephrase,“As𝑥approachespositiveinfinity,”beforeusingthephrase(oritssymbolversion)todescribeendbehavior.Thatisbecausethephraseappearstomeanthatthesymbol𝑥isliterally“movingalongthenumberlinetotheright.”Nottrue!Recallthatavariableisjustaplaceholderforwhichanumbercanbesubstituted(thinkofablankorboxusedinGrade2equations)and,therefore,doesnotactuallymoveorvary.

Thephrase,“As𝑥 → ∞,”canbeprofitablydescribedasaprocessbywhichtheuserofthephrasethinksofrepeatedlysubstitutinglargerandlargerpositivenumbersinfor𝑥,eachtimeperformingwhatevercalculationisrequiredbytheproblemforthatnumber(whichinthislessonisfindingthevalueofthefunction).

Thisishowmathematiciansoftenusethephraseeventhoughtheprecisedefinitionoflimitremovesanyneedtothinkofalimitasaprocess.

§ ENDBEHAVIOR(description):Let𝑓beafunctionwhosedomainandrangearesubsetsoftherealnumbers.Theendbehaviorofafunction𝑓isadescriptionofwhathappenstothevaluesofthefunction

• as𝑥approachespositiveinfinity,and• as𝑥approachesnegativeinfinity.

Helpstudentsunderstandthedescriptionofendbehaviorusingthefollowingpicture.

Lesson15

M1ALGEBRAII

As𝑥 → −∞, As𝑥 → ∞,𝑓 𝑥 → −∞ 𝑓 𝑥 → ∞

Askstudentstomakeageneralizationabouttheendbehaviorofpolynomialsofevendegreeinwritingindividuallyorwithapartner.Theyshouldconcludethatanevendegreepolynomialfunctionhasthesameendbehavioras𝑥 → ∞andas𝑥 → −∞.Afterstudentshavegeneralizedtheendbehavior,havethemcreatetheirowngraphicorganizerlikethefollowing.

𝑥 → ∞

𝑥 → −∞

𝑓(𝑥) → −∞

𝑓(𝑥) → ∞

Graphof𝑦 = 𝑓(𝑥)

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M1ALGEBRAII

Ifstudentssuspectthatendbehaviorofapolynomialfunctionwithevendegreewillalwaysincrease,thensuggestexaminingthegraphsof𝑓 𝑥 = 1 − 𝑥,and𝑔 𝑥 = −𝑥..

Example1(8minutes)

StudentsarenowgoingtolookatanewsetoffunctionsbutasksimilarquestionstothoseaskedintheOpeningExercise.

Example1

Sketchthegraphof𝒇 𝒙 = 𝒙𝟑.Whatwillthegraphof𝒈 𝒙 = 𝒙𝟓looklike?Sketchthisonthesamecoordinateplane.Whatwillthegraphof𝒉 𝒙 = 𝒙𝟕looklike?Sketchthisonthesamecoordinateplane.

Havestudentsrecallandsketchthegraphof𝑓 𝑥 = 𝑥>.Discussthecharacteristicsofthegraph,wherethe𝑥-interceptis,andwhythegraphisabovethe𝑥-axisfor𝑥 > 0andbelowthe𝑥-axisfor𝑥 < 0.

Inpairsoringroups,havestudentsdiscussorwritewhattheythinkthegraphof𝑔 𝑥 = 𝑥Bwilllooklikeandhowitwillrelatetothegraphof𝑓 𝑥 = 𝑥>.Theyshouldsketchtheirresultsontopoftheoriginalgraphof𝑓.Discusswithstudentswhattheyhavesketched,andemphasizethesimilaritiestothegraphof𝑓 𝑥 = 𝑥>.

Finally,inpairsoringroups,havestudentsdiscussorwritewhattheythinkthegraphofℎ 𝑥 = 𝑥Cwilllooklikeandhowitwillrelatetothegraphsof𝑓and𝑔.Theyshouldsketchonthesamegraphtheyusedwiththeprevioustwographs.Again,discussthegraphswithstudents,andemphasizethesimilaritiesbetweengraphs.

Usingagraphingutility,havestudentsgraphallthreefunctionssimultaneouslytoconfirmtheirsketches.

Even-DegreePo

sitiv

eLead

ingCo

efficient

NegativeLead

ingCo

efficient

𝑓(𝑥) = 𝑥,

𝑓(𝑥) = −𝑥,

As𝑥 → ∞,𝑓(𝑥) → ∞

As𝑥 → −∞,𝑓(𝑥) → ∞

As𝑥 → ∞,𝑓(𝑥) → −∞

As𝑥 → −∞,𝑓(𝑥) → −∞

Lesson15

M1ALGEBRAII

Askstudentstocompareanddescribethebehaviorofthevalueof𝑓(𝑥)astheabsolutevalueof𝑥increaseswithoutbound.Guidethemtousetheterminologyoftheendbehaviorofthefunction.

Askstudentstomakeageneralizationabouttheendbehaviorofpolynomialsofodddegreeindividuallyorwithapartner.Afterstudentshavegeneralizedtheendbehavior,havethemcreatetheirowngraphicorganizerlikethefollowing.

Ifstudentssuspectthatpolynomialfunctionswithodddegreealwayshavethevalueofthefunctionincreaseas𝑥increases,thensuggestexaminingafunctionwithanegativeleadingcoefficient,suchas𝑓 𝑥 = 4 − 𝑥or𝑔 𝑥 = −𝑥>.

§ HowdothesegraphsdifferfromthoseintheOpeningExercise?Whyaretheydifferent?ú StudentsmaytalkabouthowtheOpeningExercisegraphsstayabovethe𝑥-axiswhileinthisexamplethegraphscut

throughthe𝑥-axis.Guidestudentsasnecessarytoconcludingthattheendbehaviorofeven-degreepolynomialfunctionsisthatbothendsbothapproachpositiveinfinityorbothapproachnegativeinfinitywhiletheendbehaviorofodd-degreepolynomialfunctionsisthatthebehavioras𝑥 → ∞isoppositeofthebehavioras𝑥 → −∞.

𝑓(𝑥) = 𝑥>

𝑓(𝑥) = −𝑥>

Odd-Degree

Positiv

eLead

ingCo

efficient

NegativeLead

ingCo

efficient

As𝑥 → ∞,𝑓(𝑥) → ∞

As𝑥 → −∞,𝑓(𝑥) → −∞

As𝑥 → ∞,𝑓(𝑥) → −∞

As𝑥 → −∞,𝑓(𝑥) → ∞

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M1ALGEBRAII

Exercise1(8minutes)

Keepingtheresultsoftheexamplesaboveinmind,havestudentsworkwithpartnersoringroupstoanswerthefollowingquestions.

Exercise1

a. Considerthefollowingfunction,𝒇 𝒙 = 𝟐𝒙𝟒 + 𝒙𝟑 − 𝒙𝟐 + 𝟓𝒙 + 𝟑,withamixtureofoddandevendegreeterms.PredictwhetheritsendbehaviorwillbelikethefunctionsintheOpeningExerciseormorelikethefunctionsfromExample1.Graphthefunction𝒇usingagraphingutilitytocheckyourprediction.

Studentsseethatthisfunctionactsmoreliketheeven-degreemonomialfunctionsfromtheOpeningExercise.

b. Considerthefollowingfunction,𝒇 𝒙 = 𝟐𝒙𝟓 − 𝒙𝟒 − 𝟐𝒙𝟑 + 𝟒𝒙𝟐 + 𝒙 + 𝟑,withamixtureofoddandevendegreeterms.PredictwhetheritsendbehaviorwillbelikethefunctionsintheOpeningExerciseormorelikethefunctionsfromExample1.Graphthefunction𝒇usingagraphingutilitytocheckyourprediction.

Studentsseethatthisfunctionactsmorelikeodd-degreemonomialfunctionsfromExample1.Theycandrawaconclusionsuchasthatthefunctionbehaveslikethehighestdegreeterm.

c. Thinkingbacktoourdiscussionof𝒙-interceptsofgraphsofpolynomialfunctionsfromthepreviouslesson,sketchagraphofaneven-degreepolynomialfunctionthathasno𝒙-intercepts.

Studentsmaydrawthegraphofaquadraticfunctionthatstaysabovethe𝒙-axissuchasthegraphof𝒇 𝒙 = 𝒙𝟐 + 𝟏.

d. Similarly,canyousketchagraphofanodd-degreepolynomialfunctionwithno𝒙-intercepts?

Havestudentsworkinpairsorgroupsanddiscoverthatbecauseofthe“cutthrough”natureofgraphsofodd-degreepolynomialfunctionthereisalwaysan𝒙-intercept.

Conclusion:Graphsofodd-poweredpolynomialfunctionsalwayshavean𝒙-intercept,whichmeansthatodd-degreepolynomialfunctionsalwayshaveatleastonezero(orroot)andthatpolynomialfunctionsofodd-degreealwayshaveoppositeendbehaviorsas𝒙 → ∞and𝒙 → −∞.

Havestudentsconcludethatthegraphsofodd-degreepolynomialfunctionsalwayshaveatleastone𝑥-interceptandsothefunctionsalwayshaveatleastonezero.Thegraphsofeven-degreepolynomialfunctionsmayormaynothave𝑥-intercepts.

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M1ALGEBRAII

Exercise2(8minutes)

Inthisexercise,studentsusewhattheylearnedtodayaboutendbehaviortodeterminewhetherornotthepolynomialfunctionusedtomodelthedatahasanevenorodddegree.

Exercise2

TheCenterforTransportationAnalysis(CTA)studiesallaspectsoftransportationintheUnitedStates,fromenergyandenvironmentalconcernstosafetyandsecuritychallenges.A1997studycompiledthefollowingdataofthefueleconomyinmilespergallon(mpg)ofacarorlighttruckatvariousspeedsmeasuredinmilesperhour(mph).Thedataarecompiledinthetablebelow.

FuelEconomybySpeed

Speed(mph) FuelEconomy(mpg)𝟏𝟓 𝟐𝟒. 𝟒𝟐𝟎 𝟐𝟕. 𝟗𝟐𝟓 𝟑𝟎. 𝟓𝟑𝟎 𝟑𝟏. 𝟕𝟑𝟓 𝟑𝟏. 𝟐𝟒𝟎 𝟑𝟏. 𝟎𝟒𝟓 𝟑𝟏. 𝟔𝟓𝟎 𝟑𝟐. 𝟒𝟓𝟓 𝟑𝟐. 𝟒𝟔𝟎 𝟑𝟏. 𝟒𝟔𝟓 𝟐𝟗. 𝟐𝟕𝟎 𝟐𝟔. 𝟖𝟕𝟓 𝟐𝟒. 𝟖

Source:TransportationEnergyDataBook,Table4.28.http://cta.ornl.gov/data/chapter4.shtml

a. Plotthedatausingagraphingutility.Whichvariableistheindependentvariable?

Speedistheindependentvariable.

b. Thisdatacanbemodeledbyapolynomialfunction.Determineifthefunctionthatmodelsthedatawouldhaveanevenorodddegree.

Itseemswecouldmodelthisdatabyaneven-degreepolynomialfunction.

c. Istheleadingcoefficientofthepolynomialthatcanbeusedtomodelthisdatapositiveornegative?

Theleadingcoefficientwouldbenegativesincetheendbehaviorofthisfunctionistoapproachnegativeinfinityonbothsides.

d. Listtwopossiblereasonsthedatamighthavetheshapethatitdoes.

Possibleresponses:Fueleconomyimprovesuptoacertainspeed,butthenwindresistanceathigherspeedsreducesfueleconomy;theincreasedgasneededtogohigherspeedsreducesfueleconomy.

Closing(3minutes)

§ Inthislesson,studentsexploredthecharacteristicsofthegraphsofpolynomialfunctionsofevenandodd-degree.Graphsofeven-degreepolynomialsdemonstratethesameendbehavioras𝑥 → ∞asitdoesas𝑥 → −∞,whilegraphsofodd-degreepolynomialsdemonstrateoppositeendbehavioras𝑥 → ∞asitdoesas𝑥 → −∞.Becauseofthisfact,graphsofodd-degreepolynomialfunctionsalwaysintersectthe𝑥-axis;therefore,odd-degreepolynomialfunctionshaveatleastonezeroorroot.

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M1ALGEBRAII

As𝑥 → ∞,𝑓(𝑥) → ∞As𝑥 → −∞,𝑓(𝑥) → −∞

As𝑥 → ∞,𝑓(𝑥) → −∞As𝑥 → −∞,𝑓(𝑥) → ∞

As𝑥 → ∞,𝑓(𝑥) → ∞As𝑥 → −∞,𝑓(𝑥) → ∞

As𝑥 → ∞,𝑓(𝑥) → −∞As𝑥 → −∞,𝑓(𝑥) → −∞

Odd-DegreeEven-DegreePo

sitiveLead

ingCo

efficient

NegativeLead

ingCo

efficient

§ Studentsalsolearnedthatitisthehighestdegreetermofthepolynomialthatdeterminesifthegraphexhibitsodd-degreeendbehaviororeven-degreeendbehavior.Thismakessensebecausethehighestdegreetermofapolynomialdeterminesthedegreeofthepolynomial.

Havestudentssummarizethelessoneitherwithagraphicorganizerorawrittensummary.Agraphicorganizerisincludedbelow.

RelevantVocabulary

EVENFUNCTION:Let𝒇bea functionwhosedomainandrangeisasubset oftherealnumbers.Thefunction𝒇iscalledeveniftheequation𝒇(𝒙) = 𝒇(−𝒙)istrueforeverynumber𝒙inthedomain.

Even-degreepolynomialfunctionsaresometimesevenfunctions,like𝒇(𝒙) = 𝒙𝟏𝟎,andsometimesnot,like𝒈(𝒙) = 𝒙𝟐 − 𝒙.

ODDFUNCTION:Let𝒇beafunctionwhosedomainandrangeisasubsetoftherealnumbers.Thefunction𝒇iscalledoddiftheequation𝒇 −𝒙 =−𝒇(𝒙)istrueforeverynumber𝒙inthedomain.

Odd-degreepolynomialfunctionsaresometimesoddfunctions,like𝒇(𝒙) = 𝒙𝟏𝟏,andsometimesnot,like𝒉(𝒙) = 𝒙𝟑 − 𝒙𝟐.

ExitTicket(5minutes)

Lesson15

M1ALGEBRAII

Name Date

Lesson15:StructureinGraphsofPolynomialFunctions

ExitTicketWithoutusingagraphingutility,matcheachgraphbelowincolumn1withthefunctionincolumn2thatitrepresents.

a.

1. 𝑦 = 3𝑥>

b.

2. 𝑦 = 12 𝑥

,

c.

3. 𝑦 = 𝑥> − 8

d.

4. 𝑦 = 𝑥. − 𝑥> + 4𝑥 + 2

e.

5. 𝑦 = 3𝑥B − 𝑥> + 4𝑥 + 2

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M1ALGEBRAII

ExitTicketSampleSolutions

Withoutusingagraphingutility,matcheachgraphbelowincolumn1withthefunctionincolumn2thatitrepresents.

a.

1. 𝒚 = 𝟑𝒙𝟑

b.

2. 𝒚 = 𝟏𝟐 𝒙

𝟐

c.

3. 𝒚 = 𝒙𝟑 − 𝟖

d.

4. 𝒚 = 𝒙𝟒 − 𝒙𝟑 + 𝟒𝒙 + 𝟐

e.

5. 𝒚 = 𝟑𝒙𝟓 − 𝒙𝟑 + 𝟒𝒙 + 𝟐

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M1ALGEBRAII

ProblemSet

1. GraphthefunctionsfromtheOpeningExercisesimultaneouslyusingagraphingutilityandzoominattheorigin.

a. At𝒙 = 𝟎. 𝟓,orderthevaluesofthefunctionsfromleasttogreatest.

b. At𝒙 = 𝟐. 𝟓,orderthevaluesofthefunctionsfromleasttogreatest.

c. Identifythe𝒙-value(s)wheretheorderreverses.Writeabriefsentenceonwhyyouthinkthisswitchoccurs.

2. TheNationalAgriculturalStatisticsService(NASS)isanagencywithintheUSDAthatcollectsandanalyzesdatacoveringvirtuallyeveryaspectofagricultureintheUnitedStates.Thefollowingtablecontainsinformationontheamount(intons)ofthefollowingvegetablesproducedintheU.S.from1988–1994forprocessingintocanned,frozen,andpackagedfoods:limabeans,snapbeans,beets,cabbage,sweetcorn,cucumbers,greenpeas,spinach,andtomatoes.

VegetableProductionbyYear

Year VegetableProduction(tons)1988 𝟏𝟏, 𝟑𝟗𝟑, 𝟑𝟐𝟎1989 𝟏𝟒, 𝟒𝟓𝟎, 𝟖𝟔𝟎1990 𝟏𝟓, 𝟒𝟒𝟒, 𝟗𝟕𝟎1991 𝟏𝟔, 𝟏𝟓𝟏, 𝟎𝟑𝟎1992 𝟏𝟒, 𝟐𝟑𝟔, 𝟑𝟐𝟎1993 𝟏𝟒, 𝟗𝟎𝟒, 𝟕𝟓𝟎1994 𝟏𝟖, 𝟑𝟏𝟑, 𝟏𝟓𝟎

Source:NASSStatisticsofVegetablesandMelons,1995,Table191.http://www.nass.usda.gov/Publications/Ag_Statistics/1995-1996/agr95_4.pdf

a. Plotthedatausingagraphingutility.

b. Determineifthedatadisplaythecharacteristicsofanodd-oreven-degreepolynomialfunction.

c. Listtwopossiblereasonsthedatamighthavesuchashape.

3. TheU.S.EnergyInformationAdministration(EIA)isresponsibleforcollectingandanalyzinginformationaboutenergyproductionanduseintheUnitedStatesandforinformingpolicymakersandthepublicaboutissuesofenergy,theeconomy,andtheenvironment.ThefollowingtablecontainsdatafromtheEIAaboutnaturalgasconsumptionfrom1950–2010,measuredinmillionsofcubicfeet.

U.S.NaturalGasConsumptionbyYear

Year U.S.naturalgastotalconsumption(millionsofcubicfeet)

1950 𝟓. 𝟕𝟕1955 𝟖. 𝟔𝟗1960 𝟏𝟏. 𝟗𝟕1965 𝟏𝟓. 𝟐𝟖1970 𝟐𝟏. 𝟏𝟒1975 𝟏𝟗. 𝟓𝟒1980 𝟏𝟗. 𝟖𝟖1985 𝟏𝟕. 𝟐𝟖1990 𝟏𝟗. 𝟏𝟕1995 𝟐𝟐. 𝟐𝟏

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M1ALGEBRAII

2000 𝟐𝟑. 𝟑𝟑2005 𝟐𝟐. 𝟎𝟏2010 𝟐𝟒. 𝟎𝟗

Source:U.S.EnergyInformationAdministration.http://www.eia.gov/dnav/ng/hist/n9140us2a.htm

a. Plotthedatausingagraphingutility.

b. Determineifthedatadisplaythecharacteristicsofanodd-oreven-degreepolynomialfunction.

c. Listtwopossiblereasonsthedatamighthavesuchashape.

4. Weusethetermevenfunctionwhenafunction𝒇satisfiestheequation𝒇 −𝒙 = 𝒇(𝒙)foreverynumber𝒙initsdomain.Considerthefunction𝒇 𝒙 = −𝟑𝒙𝟐 + 𝟕.Notethatthedegreeofthefunctioniseven,andeachtermisofanevendegree(theconstanttermisdegree𝟎).a. Graphthefunctionusingagraphingutility.

b. Doesthisgraphdisplayanysymmetry?

c. Evaluate𝒇 −𝒙 .

d. Is𝒇anevenfunction?Explainhowyouknow.

5. Weusethetermoddfunctionwhenafunction𝒇satisfiestheequation𝒇 −𝒙 = −𝒇(𝒙)foreverynumber𝒙initsdomain.Considerthefunction𝒇 𝒙 = 𝟑𝒙𝟑 − 𝟒𝒙.Thedegreeofthefunctionisodd,andeachtermisofanodddegree.

a. Graphthefunctionusingagraphingutility.

b. Doesthisgraphdisplayanysymmetry?

c. Evaluate𝒇(−𝒙).

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M1ALGEBRAII

d. Is𝒇anoddfunction?Explainhowyouknow.

6. Wehavetalkedabout𝒙-interceptsofthegraphofafunctioninboththislessonandthepreviousone.The𝒙-interceptscorrespondtothezerosofthefunction.Considerthefollowingexamplesofpolynomialfunctionsandtheirgraphstodetermineaneasywaytofindthe𝒚-interceptofthegraphofapolynomialfunction.

𝑓 𝑥 = 2𝑥, − 4𝑥 − 3 𝑓 𝑥 = 𝑥> + 3𝑥, − 𝑥 + 5 𝑓 𝑥 = 𝑥. − 2𝑥> − 𝑥, + 3𝑥 − 6