Lesson14

ALGEBRAII

Lesson14:GraphingFactoredPolynomials

StudentOutcomes

§ Studentsusethefactoredformsofpolynomialstofindzerosofafunction.§ Studentsusethefactoredformsofpolynomialstosketchthecomponentsofgraphsbetweenzeros.

LessonNotesInthislesson,studentsusethefactoredformofpolynomialstoidentifyimportantaspectsofthegraphsofpolynomialfunctionsand,therefore,importantaspectsofthesituationstheymodel.Usingthefactoredform,studentsidentifyzerosofthepolynomial(andthus𝑥-interceptsofthegraphofthepolynomialfunction)andseehowtosketchagraphofthepolynomialfunctionsbyexaminingwhathappensbetweenthe𝑥-intercepts.Theyarealsointroducedtotheconceptsofrelativeminimaandmaximaanddeterminingthepossibledegreeofthepolynomialbynotingthenumberofrelativeextremabylookingatthegraphofafunction.Arelativemaximum(orminimum)isapropertyofafunctionthatisvisibleinitsgraph.Arelativemaximumoccursatan𝑥-value,𝑐,inthedomainofthefunction,andtherelativemaximumvalueisthecorrespondingfunctionvalueat𝑐.Ifarelativemaximumofafunction𝑓occursat𝑐,then 𝑐, 𝑓 𝑐 isarelativemaximumpoint.Asanexample,if(10,300)isarelativemaximumpointofafunction𝑓,thentherelativemaximumvalueof𝑓is300andoccursat10.Whenspeakingaboutrelativeextrema,however,relativemaximumisoftenusedinformallytorefertoeitherarelativemaximumat𝑐,arelativemaximumvalue,orarelativemaximumpointwhenthecontextisclear.Definitionsofrelevantvocabularyareincludedattheendofthelesson

Theuseofagraphingutilityisrecommendedforsomeexamplesinthislessontoencouragestudentstofocusonunderstandingthestructureofthepolynomialswithoutthetediumofrepeatedgraphingbyhand.

OpeningExercise(10minutes)

Promptstudentstoanswerpart(a)oftheOpeningExerciseindependentlyorinpairsbeforecontinuingwiththescaffoldedquestions.

OpeningExercise

Anengineerisdesigningarollercoasterforyoungerchildrenandhastriedsomefunctionstomodeltheheightoftherollercoasterduringthefirst𝟑𝟎𝟎yards.Shecameupwiththefollowingfunctiontodescribewhatshebelieveswouldmakeafunstarttotheride:

𝑯 𝒙 = −𝟑𝒙𝟒 + 𝟐𝟏𝒙𝟑 − 𝟒𝟖𝒙𝟐 + 𝟑𝟔𝒙,

where𝑯(𝒙)istheheightoftherollercoaster(inyards)whentherollercoasteris𝟏𝟎𝟎𝒙yardsfromthebeginningoftheride.Answerthefollowingquestionstohelpdetermineatwhichdistancesfromthebeginningoftheridetherollercoasterisatitslowestheight.

a. Doesthisfunctiondescribearollercoasterthatwouldbefuntoride?Explain.

Yes,therollercoasterquicklygoestothetopandthendropsyoudown.Thislookslikeafunride.

No,Idon’tlikerollercoastersthatclimbsteeply,andthisonegoesnearlystraightup.

b. Canyouseeanyobvious𝒙-valuesfromtheequationwheretherollercoasterisatheight𝟎?

Theheightis𝟎when𝒙is𝟎because,atthatvalue,eachtermisequalto𝟎.

c. Usingagraphingutility,graphthefunction𝑯ontheinterval𝟎 ≤ 𝒙 ≤ 𝟑,and identify

e

-1 0 1 2 3 4

-1

1

2

3

4

5

6

7

8

9

10

MP.5

MP.3

Scaffolding:§ Considerbeginningthe

classbyreviewinggraphsofsimplerfunctionsmodelingsimplerollercoasters,suchas𝐺(𝑥) = −𝑥8 + 4𝑥.

§ Amorevisualapproachmaybetakenbyfirstdescribingandanalyzingthegraphof𝐻beforeconnectingeachconcepttothealgebraassociatedwiththefunction.PosequestionssuchasWhenistherollercoastergoingup?Goingdown?Howmanytimesdoestherollercoastertouchthebottom?

MP.5&

MP.7

Lesson14

ALGEBRAII

whentherollercoasteris𝟎yardsofftheground.Thelowestpointsofthegraphon𝟎 ≤ 𝒙 ≤ 𝟑arewhenthe𝒙-valuesatisfies𝑯 𝒙 = 𝟎,whichoccurswhen𝒙is𝟎,𝟐,and𝟑.

d. Whatdothe𝒙-valuesyoufoundinpart(c)meanintermsofdistancefromthebeginningoftheride?

Thedistancesrepresent𝟎yards,𝟐𝟎𝟎yards,and𝟑𝟎𝟎yards,respectively.

e. Whydorollercoastersalwaysstartwiththelargesthillfirst?

Sotheycanbuildupspeedfromgravitytohelppropelthecarsthroughtherestofthetrack.

f. Verifyyouranswerstopart(c)byfactoringthepolynomialfunction𝑯.

Somestudentsmayneedsomehintsorguidancewithfactoring.

𝑯 𝒙 = −𝟑𝒙𝟒 + 𝟐𝟏𝒙𝟑 − 𝟒𝟖𝒙𝟐 + 𝟑𝟔𝒙= −𝟑𝒙(𝒙𝟑 − 𝟕𝒙𝟐 + 𝟏𝟔𝒙 − 𝟏𝟐)

Fromthegraph,wesuspectthat(𝒙 − 𝟑)isafactor;usinglongdivision,weobtain

𝑯 𝒙 = −𝟑𝒙 𝒙 − 𝟑 𝒙𝟐 − 𝟒𝒙 + 𝟒 = −𝟑𝒙 𝒙 − 𝟑 𝒙 − 𝟐 𝒙 − 𝟐 = −𝟑𝒙 𝒙 − 𝟑 𝒙 − 𝟐 𝟐.

Thesolutionstotheequation𝑯 𝒙 = 𝟎are𝟎,𝟐,and𝟑.Therefore,therollercoasterisatthebottomat𝟎yards,𝟐𝟎𝟎yards,and𝟑𝟎𝟎yardsfromthestartoftheride.

g. Howdoyouthinktheengineercameupwiththefunctionforthismodel?

Letstudentsdiscussthisquestioningroupsorasawholeclass.Thefollowingconclusionshouldbemade:Tostartatheight𝟎yardsandend𝟑𝟎𝟎yardslateratheight𝟎yards,shemultiplied𝒙by𝒙 − 𝟑(tocreatezerosat𝟎and𝟑).Tocreatethebottomofthehillat𝟐𝟎𝟎yards,shemultipliedthisfunctionby 𝒙 − 𝟐 𝟐.Sheneededtomultiplyby−𝟑toguaranteetherollercoastershapeandtoadjusttheoverallheightoftherollercoaster.

h. Whatiswrongwiththisrollercoastermodelatdistance𝟎yardsand𝟑𝟎𝟎yards?Whymightthisnotinitiallybothertheengineerwhensheisfirstdesigningthetrack?

Themodelappearstoabruptlystartat𝟎yardsandabruptlyendat𝟑𝟎𝟎yards.Infact,therollercoasterlooksasifitwillcrashintothegroundat𝟑𝟎𝟎yards!Theengineermaybeplanningto“smooth”outthetracklaterat𝟎yardsand𝟑𝟎𝟎yardsaftershehasselectedtheoverallshapeoftherollercoaster.

MP.3

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ALGEBRAII

Discussion(4minutes)

Bymanipulatingapolynomialfunctionintoitsfactoredform,wecanidentifythezerosofthefunctionaswellasidentifythegeneralshapeofthegraph.ThinkingabouttheOpeningExercise,whatelsecanwesayaboutthepolynomialfunctionanditsgraph?

§ Thedegreeofthepolynomialfunction𝐻is4.Howcanyoufindthedegreeofthefunctionfromitsfactoredform?

ú Addthehighestdegreetermfromeachfactor:§ −3isadegree0factor§ 𝑥isdegree1factor§ 𝑥 − 3isdegree1factor§ 𝑥 − 2 8isadegree2factor,since 𝑥 − 2 8 = (𝑥 − 2)(𝑥 − 2).

ú Thus,0 + 1 + 1 + 2 = 4.§ Howmany𝑥-interceptsdoesthegraphofthepolynomialfunctionhave?

ú Forthisgraph,therearethree: 0,0 ,(2,0),and(3,0).

Youmaywanttoincludeadiscussionthatthezerosofafunctioncorrespondtothe𝑥-interceptsofthegraphofthefunction.

§ Notethattherearefourfactors,butonlythree𝑥-intercepts.Whyisthat?ú Twoofthefactorsarethesame.

Remindstudentsthatthe𝑥-interceptsofthegraphof𝑦 = 𝑓 𝑥 aresolutionstotheequation𝑓 𝑥 = 0.Valuesof𝑟thatsatisfy𝑓 𝑟 = 0arecalledzeros(orroots)ofthefunction.Someofthesezerosmayberepeated.

§ Canyoumakeonechangetothepolynomialfunctionsuchthatthenewgraphwouldhavefour𝑥-intercepts?

ú Changeoneofthe(𝑥 − 2)factorsto 𝑥 − 1 ,forexample.

Example1(10minutes)

Studentsarenowgoingtoexamineafewpolynomialfunctionsinfactoredformandcomparethezerosofthefunctiontothegraphofthefunctiononthecalculator.Helpstudentswithpart(a),andaskthemtodopart(b)ontheirown.

Scaffolding:Encouragestrugglinglearnerstographtheoriginalandthefactoredformsusingagraphingutilitytoconfirmthattheyarethesame.

Scaffolding:§ Foradvancedlearners,

considerchallengingstudentstoconstructavarietyoffunctionstomeetdifferentcriteriasuchasthreefactorsandno𝑥-interceptsorfourfactorswithtwo𝑥-intercepts.

§ Studentsmayenjoychallengingeachotherbytryingtoguesstheequationthatgoeswiththegraphoftheirclassmates.

Lesson14

ALGEBRAII

Example1

Grapheachofthefollowingpolynomialfunctions.Whatarethefunction’szeros(countingmultiplicities)?Whatarethesolutionsto𝒇 𝒙 = 𝟎?Whatarethe𝒙-interceptstothegraphofthefunction?Howdoesthedegreeofthepolynomialfunctioncomparetothe𝒙-interceptsofthegraphofthefunction?

a. 𝒇 𝒙 = 𝒙(𝒙 − 𝟏)(𝒙 + 𝟏)

Zeros: −𝟏, 𝟎, 𝟏

Solutionsto𝒇 𝒙 = 𝟎: −𝟏, 𝟎, 𝟏

𝒙-intercepts: −𝟏, 𝟎, 𝟏

Thedegreeis𝟑,whichisthesameasthenumberof𝒙-intercepts.

Beforegraphingthenextequation,askstudentswheretheythinkthegraphof𝑓willcrossthe𝑥-axisandhowtherepeatedfactorwillaffectthegraph.Aftergraphing,studentsmayneedtotracenear𝑥 = −3dependingonthegraphingwindowtoobtainaclearpictureofthe𝑥-intercept.

b. 𝒇 𝒙 = (𝒙 + 𝟑)(𝒙 + 𝟑)(𝒙 + 𝟑)(𝒙 + 𝟑)

Zeros: −𝟑,−𝟑,−𝟑,−𝟑(repeatedzero)

Solutionsto𝒇 𝒙 = 𝟎: −𝟑

𝒙-intercept: −𝟑

Thedegreeis𝟒,whichisgreaterthanthenumberof𝒙-intercepts.

Bynow,studentsshouldhaveanideaofwhattoexpectinpart(c).Itmaybeworthnotingthedifferencesintheendbehaviorofthegraphs,whichwillbeexploredfurtherinLesson15.Discussthedegreeofeachpolynomial.

c. 𝒇 𝒙 = (𝒙 − 𝟏)(𝒙 − 𝟐)(𝒙 + 𝟑)(𝒙 + 𝟒)(𝒙 + 𝟒)

Zeros: −𝟒,−𝟒,−𝟑, 𝟏, 𝟐

Solutionsto𝒇 𝒙 = 𝟎: −𝟒,−𝟑, 𝟏, 𝟐

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ALGEBRAII

𝒙-intercepts: −𝟒,−𝟑, 𝟏, 𝟐

Thedegreeis𝟓,whichisgreaterthanthenumberof𝒙-intercepts.

d. 𝒇 𝒙 = (𝒙𝟐 + 𝟏)(𝒙 − 𝟐)(𝒙 − 𝟑)

Zeros: 𝟐, 𝟑

Solutionsto𝒇 𝒙 = 𝟎: 𝟐, 𝟑

𝒙-intercepts: 𝟐, 𝟑

Thedegreeis𝟒,whichisgreaterthanthenumberof𝒙-intercepts.

§ Whyisthefactor𝑥8 + 1neverzeroandhowdoesthisaffectthegraphof𝑓?

(Atthispointinthemodule,allpolynomialfunctionsaredefinedfromtherealnumberstotherealnumbers;hence,thefunctionscanhaveonlyrealnumberzeros.Wewillextendpolynomialfunctionstothedomainofcomplexnumberslater,andthenitwillbepossibletoconsidercomplexsolutionstoapolynomialequation.)

ú Forrealnumbers𝑥,thevalueof𝑥8isalwaysgreaterthanorequaltozero,so𝑥8 + 1willalwaysbestrictlygreaterthanzero.Thus,𝑥8 + 1 ≠ 0forallrealnumbers𝑥.Sincetherecanbeno𝑥-interceptfromthisfactor,thegraphof𝑓canhaveatmosttwo𝑥-intercepts.

Ifthereistime,considergraphingthefunctionsforparts(e)–(h)ontheboardandaskingstudentstomatchthefunctionstothegraphs.Encouragestudentstouseagraphingutilitytographtheirguesses,talkaboutthedifferencesbetweenguessesandtheactualgraph,andwhatmaycausethemineachcase.

e. 𝒇 𝒙 = 𝒙 − 𝟐 𝟐

Zeros: 𝟐, 𝟐

Solutionsto𝒇 𝒙 = 𝟎: 𝟐

𝒙-intercepts: 𝟐

Thedegreeis𝟐,whichisgreaterthanthenumberof𝒙-intercepts.

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ALGEBRAII

f. 𝒇 𝒙 = (𝒙 − 𝟏)(𝒙 + 𝟏)(𝒙 − 𝟐)(𝒙 + 𝟐)(𝒙 − 𝟑)(𝒙 + 𝟑)(𝒙 − 𝟒)

Zeros: 𝟏,−𝟏, 𝟐, −𝟐, 𝟑, −𝟑, 𝟒

Solutionsto𝒇 𝒙 = 𝟎: 𝟏,−𝟏, 𝟐, −𝟐, 𝟑, −𝟑, 𝟒

𝒙-intercepts: 𝟏,−𝟏, 𝟐, −𝟐, 𝟑, −𝟑, 𝟒

Thedegreeis𝟕,whichisequaltothenumberof𝒙-intercepts.

g. 𝒇 𝒙 = 𝒙𝟐 + 𝟐 𝟐

Zeros: None

Solutionsto𝒇 𝒙 = 𝟎: Nosolutions

𝒙-intercepts: No𝒙-intercepts

Thedegreeis𝟒,whichisgreaterthanthenumberof𝒙-intercepts.

h. 𝒇 𝒙 = 𝒙 + 𝟏 𝟐 𝒙 − 𝟏 𝟐𝒙

Zeros: −𝟏,−𝟏, 𝟏, 𝟏, 𝟎

Solutionsto𝒇 𝒙 = 𝟎: −𝟏, 𝟎, 𝟏

𝒙-intercepts: −𝟏, 𝟎, 𝟏

Thedegreeis𝟓,whichisgreaterthanthenumberof𝒙-intercepts.

Discussion(1minutes)

Askstudentstosummarizewhattheyhavelearnedsofar,eitherinwritingorwithapartner.Checkforunderstandingoftheconcepts,andhelpstudentsreachthefollowingconclusionsiftheydonotdosoontheirown.

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ALGEBRAII

§ The𝑥-interceptsinthegraphofafunctioncorrespondtothesolutionstotheequation𝑓 𝑥 = 0andcorrespondtothenumberofdistinctzerosofthefunction(butthe𝑥-interceptsdonothelpustodeterminethemultiplicityofagivenzero).

§ Thegraphofapolynomialfunctionofdegree𝑛hasatmost𝑛𝑥-interceptsbutmayhavefewer.§ Apolynomialfunctionwhosegraphhas𝑚𝑥-interceptsisatleastadegree𝑚polynomial.

Example2(8minutes)

Leadstudentsthroughthequestionsinordertoarriveatasketchofthefinalgraph.Themainpointofthisexerciseisthatifstudentsknowthe𝑥-interceptsofapolynomialfunction,thentheycansketchafairlyaccurategraphofthefunctionbyjustcheckingtoseeifthefunctionispositiveornegativeatafewpoints.Theyarenotgraphingbyplottingpointsandconnectingthedotsbutbyapplyingpropertiesofpolynomialfunctions.

Givetimeforstudentstoworkthroughparts(a)and(b)inpairsorsmallgroupsbeforecontinuingwiththediscussioninparts(c)-(i).Whensketchingthegraphinpart(j),itisimportanttoletstudentsknowthatwecannotpinpointexactlythehighandlowpointsonthegraph—therelativemaximumandrelativeminimumpoints.Forthisreason,omitascaleonthe𝑦-axisinthesketch.

Example2

Considerthefunction𝒇 𝒙 = 𝒙𝟑 − 𝟏𝟑𝒙𝟐 + 𝟒𝟒𝒙 − 𝟑𝟐.

a. Usethefactthat𝒙 − 𝟒isafactorof𝒇tofactorthispolynomial.

Usingpolynomialdivisionandthenfactoring,𝒇 𝒙 = 𝒙 − 𝟒 𝒙𝟐 − 𝟗𝒙 + 𝟖 = 𝒙 − 𝟒 𝒙 − 𝟖 𝒙 − 𝟏 .

b. Findthe𝒙-interceptsforthegraphof𝒇.

The𝒙-interceptsare𝟏,𝟒,and𝟖.

c. Atwhich𝒙-valuescanthefunctionchangefrombeingpositivetonegativeorfromnegativetopositive?

Onlyatthe𝒙-intercepts𝟏,𝟒,and𝟖.

d. Tosketchagraphof𝒇,weneedtoconsiderwhetherthefunctionispositiveornegativeonthefourintervals𝒙 < 𝟏,𝟏 <𝒙 < 𝟒,𝟒 < 𝒙 < 𝟖,and𝒙 > 𝟖.Whyisthat?

Thefunctioncanonlychangesignatthe𝒙-intercepts;therefore,oneachofthoseintervals,thegraphwillalwaysbeaboveoralwaysbebelowtheaxis.

e. Howcanwetellifthefunctionispositiveornegativeonanintervalbetween𝒙-intercepts?

Evaluatethefunctionatasinglepointinthatinterval.Sincethefunctioniseitheralwayspositiveoralwaysnegativebetween𝒙-intercepts,checkingasinglepointwillindicatebehaviorontheentireinterval.

f. For𝒙 < 𝟏,isthegraphaboveorbelowthe𝒙-axis?Howcanyoutell?

Since𝒇 𝟎 = −𝟑𝟐isnegative,thegraphisbelowthe𝒙-axisfor𝒙 < 𝟏.

g. For𝟏 < 𝒙 < 𝟒,isthegraphaboveorbelowthe𝒙-axis?Howcanyoutell?

Since𝒇(𝟐) = 𝟏𝟐ispositive,thegraphisabovethe𝒙-axisfor𝟏 < 𝒙 < 𝟒.

h. For𝟒 < 𝒙 < 𝟖,isthegraphaboveorbelowthe𝒙-axis?Howcanyoutell?

Since𝒇(𝟓) = −𝟏𝟐isnegative,thegraphisbelowthe𝒙-axisfor𝟒 < 𝒙 < 𝟖.

i. For𝒙 > 𝟖,isthegraphaboveorbelowthe𝒙-axis?Howcanyoutell?

Since𝒇 𝟏𝟎 = 𝟏𝟎𝟖ispositive,thegraphisabovethe𝒙-axisfor𝒙 > 𝟖.

Lesson14

ALGEBRAII

Scaffolding:ForEnglishlanguagelearners,thetermrelativemayneedsomeadditionalinstructionandpracticetohelpdifferentiateitfromotherusesofthisword.

Itmayhelptothinkoftheotherpointsintheintervalcontainingtherelativemaximumasallbeingrelated,andofalltherelativespresent,𝑐isthevaluethatgivesthehighestfunctionvalue.

j. Usetheinformationgeneratedinparts(f)–(i)tosketchagraphof𝒇.

k. Graph𝒚 = 𝒇(𝒙)ontheintervalfrom[𝟎, 𝟗]usingagraphingutility,andcompareyoursketchwiththegraphgeneratedbythegraphingutility.

Discussion(6minutes)

§ Let’sexaminethegraphof𝑓 𝑥 = 𝑥M − 13𝑥8 + 44𝑥 − 32for1 ≤ 𝑥 ≤ 4.Isthereanumber𝑐inthatintervalwherethevalue𝑓(𝑐)isgreaterthanorequaltoanyothervalueofthefunctiononthatinterval?Doweknowexactlywherethatis?

ú Thereisavalueof𝑐suchthat𝑓(𝑐)thatisgreaterthanorequaltotheothervalues.Itseemsthat2 < 𝑐 < 2.5,butwedonotknowitsexactvalue.

Itcouldbementionedthattheexactvalueof𝑐canbefoundexactlyusingcalculus,butthisisatopicforanotherclass.Fornow,pointoutthattherelativemaximumorrelativeminimumpointofaquadraticfunctioncanalwaysbefound—theonlyoneisthevertexoftheparabola.

§ Ifsuchanumber𝑐exists,thenthefunctionhasarelativemaximumat𝑐.Therelativemaximumvalue,𝑓(𝑐),maynotbethegreatestoverallvalueofthefunction,butthereisanopenintervalaround𝑐sothatforevery𝑥inthatinterval,𝑓 𝑥 ≤ 𝑓(𝑐).Thatis,forvaluesof𝑥near𝑐(where“near”isarelativeterm),thepoint 𝑥, 𝑓 𝑥 onthegraphof𝑓isnothigherthan 𝑐, 𝑓 𝑐 .

§ Similarly,afunction𝑓hasarelativeminimumat𝑑ifthereisanopenintervalaround𝑑sothatforevery𝑥inthatinterval,𝑓 𝑥 ≥ 𝑓 𝑑 .Thatis,forvaluesof𝑥near𝑑,thepoint(𝑥, 𝑓 𝑥 )onthegraphof𝑓isnotlowerthanthepoint 𝑑, 𝑓 𝑑 .Inthiscase,therelativeminimumvalueis𝑓(𝑑).

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ALGEBRAII

§ Showtherelativemaximaandrelativeminimaonthegraph.Theimagebelowclarifiesthedistinctionbetweentherelativemaximumpointandtherelativeminimumvalue.Pointoutthattherearevaluesofthefunctionthatarelargerthan𝑓(𝑐),suchas𝑓(9),butthat𝑓(𝑐)isthehighestvalueamongthe“neighbors”of𝑐.

Theprecisedefinitionsofrelativemaximaandrelativeminimaarelistedintheglossaryoftermsforthislesson.Thesedefinitionsarenewtostudents,soitisworthgoingoverthemattheendofthelesson.Reiteratetostudentsthatifarelativemaximumoccursatavalue𝑐,thenthattherelativemaximumpointisthepoint(𝑐, 𝑓(𝑐))onthegraph,andtherelativemaximumvalueisthe𝑦-valueofthefunctionatthatpoint,𝑓(𝑐).Analogousdefinitionsholdforrelativeminimum,relativeminimumvalue,andrelativeminimumpoint.

Discussion

Foranyparticularpolynomial,canwedeterminehowmanyrelativemaximaorminimathereare?Considerthefollowingpolynomialfunctionsinfactoredformandtheirgraphs.

𝒇 𝒙 = (𝒙 + 𝟏)(𝒙 − 𝟑)

𝒈 𝒙 = 𝒙 + 𝟑 𝒙 − 𝟏 𝒙 − 𝟒

𝒉 𝒙 = (𝒙)(𝒙 + 𝟒)(𝒙 − 𝟐)(𝒙 − 𝟓)

Degreeofeachpolynomial:

𝟐 𝟑 𝟒

Numberof𝒙-interceptsineachgraph:

𝟐 𝟑 𝟒

Numberofrelativemaximumandminimumpointsshownineachgraph:

𝟏 𝟐 𝟑

Whatobservationscanwemakefromthisinformation?Thenumberofrelativemaximumandminimumpointsisonelessthanthedegreeandonelessthanthenumberof𝒙-intercepts.

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ALGEBRAII

Isthistrueforeverypolynomial?Considertheexamplesbelow.

𝒓 𝒙 = 𝒙𝟐 + 𝟏

𝒔 𝒙 = (𝒙𝟐 + 𝟐)(𝒙 − 𝟏)

𝒕 𝒙 = (𝒙 + 𝟑)(𝒙 − 𝟏)(𝒙 − 𝟏)(𝒙 − 𝟏)

Degreeofeachpolynomial:

𝟐 𝟑 𝟒

Numberof𝒙-interceptsineachgraph:

𝟎 𝟏 𝟐

Numberofrelativemaximumandminimumpointsshownineachgraph:

𝟏 𝟎 𝟏

Whatobservationscanwemakefromthisinformation?

Theobservationsmadeinthepreviousexamplesdonotholdfortheseexamples,soitisdifficulttodeterminefromthedegreeofthepolynomialfunctionthenumberofrelativemaximumandminimumpointsinthegraphofthefunction.Whatwecansayisthatforadegree𝒏polynomialfunction,thereareatmost𝒏 − 𝟏relativemaximaandminima.Youcanalsothinkabouttheinformationyoucangetfromagraph.Ifagraphofapolynomialfunctionhas𝒏relativemaximumandminimumpoints,youcansaythatthedegreeofthepolynomialisatleast𝒏 + 𝟏.

Closing(1minute)

§ Bylookingatthefactoredformofapolynomial,wecanidentifyimportantcharacteristicsofthegraphsuchas𝑥-interceptsanddegreeofthefunction,whichinturnallowustodevelopasketchofthegraph.

§ Apolynomialfunctionofdegree𝑛mayhaveupto𝑛𝑥-intercepts.§ Apolynomialfunctionofdegree𝑛mayhaveupto𝑛 − 1relativemaximaandminima.

RelevantVocabulary

INCREASING/DECREASING:Givenafunction𝒇whosedomainandrangearesubsetsoftherealnumbersand𝑰isanintervalcontainedwithinthedomain,thefunctioniscalledincreasingontheinterval𝑰if

𝒇 𝒙𝟏 < 𝒇(𝒙𝟐)whenever𝒙𝟏 < 𝒙𝟐in𝑰.

Itiscalleddecreasingontheinterval𝑰if

𝒇 𝒙𝟏 > 𝒇(𝒙𝟐)whenever𝒙𝟏 < 𝒙𝟐in𝑰.

RELATIVEMAXIMUM:Let𝒇beafunctionwhosedomainandrangearesubsetsoftherealnumbers.Thefunctionhasarelativemaximumat𝒄ifthereexistsanopeninterval𝑰ofthedomainthatcontains𝒄suchthat

𝒇 𝒙 ≤ 𝒇(𝒄)forall𝒙intheinterval𝑰.

If𝒇hasarelativemaximumat𝒄,thenthevalue𝒇(𝒄)iscalledtherelativemaximumvalue.

RELATIVEMINIMUM:Let𝒇beafunctionwhosedomainandrangearesubsetsoftherealnumbers.Thefunctionhasarelativeminimumat𝒄ifthereexistsanopeninterval𝑰ofthedomainthatcontains𝒄suchthat

𝒇 𝒙 ≥ 𝒇(𝒄)forall𝒙intheinterval𝑰.

If𝒇hasarelativeminimumat𝒄,thenthevalue𝒇(𝒄)iscalledtherelativeminimumvalue.

GRAPHOF𝒇:Givenafunction𝒇whosedomain𝑫andtherangearesubsetsoftherealnumbers,thegraphof𝒇isthesetoforderedpairsintheCartesianplanegivenby

𝒙, 𝒇 𝒙 𝒙 ∈ 𝑫}.

Lesson14

ALGEBRAII

GRAPHOF𝒚 = 𝒇 𝒙 :Givenafunction𝒇whosedomain𝑫andtherangearesubsetsoftherealnumbers,thegraphof𝒚 = 𝒇 𝒙 isthesetoforderedpairs 𝒙, 𝒚 intheCartesianplanegivenby

𝒙, 𝒚 𝒙 ∈ 𝑫and𝒚 = 𝒇(𝒙)}.

Lesson14

ALGEBRAII

LessonSummary

Apolynomialofdegree𝒏mayhaveupto𝒏𝒙-interceptsandupto𝒏 − 𝟏relativemaximum/minimumpoints.

Thefunction𝒇hasarelativemaximumat𝒄ifthereisanopenintervalaround𝒄sothatforall𝒙inthatinterval,𝒇(𝒙) ≤ 𝒇(𝒄).Thatis,lookingnearthepoint]𝒄, 𝒇(𝒄)^onthegraphof𝒇,thereisnopointhigherthan]𝒄, 𝒇(𝒄)^inthatregion.Thevalue𝒇(𝒄)isarelativemaximumvalue.

Thefunction𝒇hasarelativeminimumat𝒅ifthereisanopenintervalaround𝒅sothatforall𝒙inthatinterval,𝒇(𝒙) ≥ 𝒇(𝒅).Thatis,lookingnearthepoint]𝒅, 𝒇(𝒅)^onthegraphof𝒇,thereisnopointlowerthan]𝒅, 𝒇(𝒅)^inthatregion.Thevalue𝒇(𝒅)isarelativeminimumvalue.

Thepluralofmaximumismaxima,andthepluralofminimumisminima.

ExitTicket(5minutes)

Lesson14

ALGEBRAII

Name Date

Lesson14:GraphingFactoredPolynomials

ExitTicketSketchagraphofthefunction𝑓 𝑥 = 𝑥M + 𝑥8 − 4𝑥 − 4byfindingthezerosanddeterminingthesignofthefunctionbetweenzeros.Explainhowthestructureoftheequationhelpsguideyoursketch.

Lesson14

ALGEBRAII

ExitTicketSampleSolutions

Sketchagraphofthefunction𝒇 𝒙 = 𝒙𝟑 + 𝒙𝟐 − 𝟒𝒙 − 𝟒byfindingthezerosanddeterminingthesignofthefunctionbetweenzeros.Explainhowthestructureoftheequationhelpsguideyoursketch.

𝒇 𝒙 = (𝒙 + 𝟏)(𝒙 + 𝟐)(𝒙 − 𝟐)

Zeros: −𝟏,−𝟐,𝟐

For𝒙 < −𝟐: 𝒇 −𝟑 = −𝟏𝟎,sothegraphisbelowthe𝒙-axis onthisinterval.

For−𝟐 < 𝒙 < −𝟏: 𝒇 −𝟏. 𝟓 = 𝟎. 𝟖𝟕𝟓,sothegraphisabovethe 𝒙-axisonthisinterval.

For– 𝟏 < 𝒙 < 𝟐: 𝒇 𝟎 = −𝟒,sothegraphisbelowthe𝒙-axison thisinterval.

For𝒙 > 𝟐: 𝒇 𝟑 = 𝟐𝟎,sothegraphisabovethe𝒙-axison thisinterval.

ProblemSetSampleSolutions

1. Foreachfunctionbelow,identifythelargestpossiblenumberof𝒙-interceptsandthelargestpossiblenumberofrelativemaximaandminimabasedonthedegreeofthepolynomial.Thenuseacalculatororgraphingutilitytographthefunctionandfindtheactualnumberof𝒙-interceptsandrelativemaximaandminima.

a. 𝒇 𝒙 = 𝟒𝒙𝟑 − 𝟐𝒙 + 𝟏b. 𝒈 𝒙 = 𝒙𝟕 − 𝟒𝒙𝟓 − 𝒙𝟑 + 𝟒𝒙c. 𝒉 𝒙 = 𝒙𝟒 + 𝟒𝒙𝟑 + 𝟐𝒙𝟐 − 𝟒𝒙 + 𝟐

Function Largestnumberof𝒙-intercepts

Largestnumberofrelativemax/min

Actualnumberof𝒙-intercepts

Actualnumberofrelativemax/min

a.𝒇 𝟑 𝟐 𝟏 𝟐b.𝒈 𝟕 𝟔 𝟓 𝟒c.𝒉 𝟒 𝟑 𝟎 𝟑

2. Sketchagraphofthefunction𝒇 𝒙 = 𝟏𝟐 (𝒙 + 𝟓)(𝒙 + 𝟏)(𝒙 − 𝟐)byfindingthezerosanddeterminingthesignofthevaluesofthe

functionbetweenzeros.

3. Sketchagraphofthefunction𝒇 𝒙 = − 𝒙 + 𝟐 𝒙 − 𝟒 𝒙 − 𝟔 byfindingthezerosanddeterminingthesignofthevaluesofthefunctionbetweenzeros.

4. Sketchagraphofthefunction𝒇 𝒙 = 𝒙𝟑 − 𝟐𝒙𝟐 − 𝒙 + 𝟐byfindingthezerosanddeterminingthesignofthevaluesofthefunctionbetweenzeros.

5. Sketchagraphofthefunction𝒇 𝒙 = 𝒙𝟒 − 𝟒𝒙𝟑 + 𝟐𝒙𝟐 + 𝟒𝒙 − 𝟑bydeterminingthesignofthevaluesofthefunctionbetweenthezeros−𝟏,𝟏,and𝟑.

6. Afunction𝒇haszerosat−𝟏,𝟑,and𝟓.Weknowthat𝒇 −𝟐 and𝒇(𝟐)arenegative,while𝒇(𝟒)and𝒇(𝟔)arepositive.Sketchagraphof𝒇.

7. Thefunction𝒉(𝒕) = −𝟏𝟔𝒕𝟐 + 𝟑𝟑𝒕 + 𝟒𝟓representstheheightofaballtossedupwardfromtheroofofabuilding𝟒𝟓feetintheairafter𝒕seconds.Withoutgraphing,determinewhentheballwillhittheground.