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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
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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
Lesson14
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
Lesson14
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.
Lesson14
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.
Lesson14
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).
Lesson14
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.
Lesson14
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.
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