Lesson 14: Graphing Factored Polynomials...8 9 10 MP.5 MP.3 Scaffolding: §Consider beginning the...
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Lesson 14 ALGEBRA II Lesson 14: Graphing Factored Polynomials Student Outcomes § Students use the factored forms of polynomials to find zeros of a function. § Students use the factored forms of polynomials to sketch the components of graphs between zeros. Lesson Notes In this lesson, students use the factored form of polynomials to identify important aspects of the graphs of polynomial functions and, therefore, important aspects of the situations they model. Using the factored form, students identify zeros of the polynomial (and thus -intercepts of the graph of the polynomial function) and see how to sketch a graph of the polynomial functions by examining what happens between the -intercepts. They are also introduced to the concepts of relative minima and maxima and determining the possible degree of the polynomial by noting the number of relative extrema by looking at the graph of a function. A relative maximum (or minimum) is a property of a function that is visible in its graph. A relative maximum occurs at an -value, , in the domain of the function, and the relative maximum value is the corresponding function value at . If a relative maximum of a function occurs at , then , is a relative maximum point. As an example, if (10,300) is a relative maximum point of a function , then the relative maximum value of is 300 and occurs at 10. When speaking about relative extrema, however, relative maximum is often used informally to refer to either a relative maximum at , a relative maximum value, or a relative maximum point when the context is clear. Definitions of relevant vocabulary are included at the end of the lesson The use of a graphing utility is recommended for some examples in this lesson to encourage students to focus on understanding the structure of the polynomials without the tedium of repeated graphing by hand. Opening Exercise (10 minutes) Prompt students to answer part (a) of the Opening Exercise independently or in pairs before continuing with the scaffolded questions. Opening Exercise An engineer is designing a roller coaster for younger children and has tried some functions to model the height of the roller coaster during the first yards. She came up with the following function to describe what she believes would make a fun start to the ride: = − + − + , where () is the height of the roller coaster (in yards) when the roller coaster is yards from the beginning of the ride. Answer the following questions to help determine at which distances from the beginning of the ride the roller coaster is at its lowest height. a. Does this function describe a roller coaster that would be fun to ride? Explain. Yes, the roller coaster quickly goes to the top and then drops you down. This looks like a fun ride. No, I don’t like roller coasters that climb steeply, and this one goes nearly straight up. b. Can you see any obvious -values from the equation where the roller coaster is at height ? The height is when is because, at that value, each term is equal to . c. Using a graphing utility, graph the function on the interval ≤≤, and identify e 1 2 3 4 5 6 7 8 9 P.5 P.3 Scaffolding: § Consider beginning the class by reviewing graphs of simpler functions modeling simple roller coasters, such as () = − 8 + 4. § A more visual approach may be taken by first describing and analyzing the graph of before connecting each concept to the algebra associated with the function. Pose questions such as When is the roller coaster going up? Going down? How many times does the roller coaster touch the bottom? P.5 & P.7
Transcript of Lesson 14: Graphing Factored Polynomials...8 9 10 MP.5 MP.3 Scaffolding: §Consider beginning the...
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
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Scaffolding:§ Considerbeginningthe
classbyreviewinggraphsofsimplerfunctionsmodelingsimplerollercoasters,suchas𝐺(𝑥) = −𝑥8 + 4𝑥.
§ Amorevisualapproachmaybetakenbyfirstdescribingandanalyzingthegraphof𝐻beforeconnectingeachconcepttothealgebraassociatedwiththefunction.PosequestionssuchasWhenistherollercoastergoingup?Goingdown?Howmanytimesdoestherollercoastertouchthebottom?
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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.
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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.
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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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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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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.
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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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§ 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.
