71K - views

Polynomial functions mcTYpolynomial Manycommonfunctionsarepolynomialfunctions

Inthisuni twedescribepolynomialfunctions andlookatsomeoftheirproperties Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideotutorialo nthist

Embed :
Pdf Download Link

Download Pdf - The PPT/PDF document "Polynomial functions mcTYpolynomial Many..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Polynomial functions mcTYpolynomial Manycommonfunctionsarepolynomialfunctions






Presentation on theme: "Polynomial functions mcTYpolynomial Manycommonfunctionsarepolynomialfunctions"— Presentation transcript:

Polynomialfunctions mc-TY-polynomial-2009-1 Manycommonfunctionsarepolynomialfunctions.Inthisunitwedescribepolynomialfunctionsandlookatsomeoftheirproperties.Inordertomasterthetechniquesexplainedhereitisvitalthatyouundertakeplentyofpracticeexercisessothattheybecomesecondnature.Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto:recognisewhenaruledescribesapolynomialfunction,andwritedownthedegreeofthepolynomial,recognizethetypicalshapesofthegraphsofpolynomials,ofdegreeupto4,understandwhatismeantbythemultiplicityofarootofapolynomial,sketchthegraphofapolynomial,givenitsexpressionasaproductoflinearfactors.Contents1.Introduction22.Whatisapolynomial?23.Graphsofpolynomialfunctions34.Turningpointsofpolynomialfunctions65.Rootsofpolynomialfunctions7 www.mathcentre.ac.uk1c\rmathcentre2009 1.IntroductionApolynomialfunctionisafunctionsuchasaquadratic,acubic,aquartic,andsoon,involvingonlynon-negativeintegerpowersofx.Wecangiveageneralde ntionofapolynomial,andde neitsdegree.2.Whatisapolynomial?Apolynomialofdegreenisafunctionoftheformf(x)=anxn+an1xn1+:::+a2x2+a1x+a0wherethea'sarerealnumbers(sometimescalledthecoecientsofthepolynomial).Althoughthisgeneralformulamightlookquitecomplicated,particularexamplesaremuchsimpler.Forexample,f(x)=4x33x2+2isapolynomialofdegree3,as3isthehighestpowerofxintheformula.Thisiscalledacubicpolynomial,orjustacubic.Andf(x)=x74x5+1isapolynomialofdegree7,as7isthehighestpowerofx.Noticeherethatwedon'tneedeverypowerofxupto7:weneedtoknowonlythehighestpowerofxto ndoutthedegree.Anexampleofakindyoumaybefamiliarwithisf(x)=4x22x4whichisapolynomialofdegree2,as2isthehighestpowerofx.Thisiscalledaquadratic.Functionscontainingotheroperations,suchassquareroots,arenotpolynomials.Forexample,f(x)=4x3+p x1isnotapolynomialasitcontainsasquareroot.Andf(x)=5x42x2+3=xisnotapolynomialasitcontainsa`dividebyx'. KeyPointApolynomialisafunctionoftheformf(x)=anxn+an1xn1+:::+a2x2+a1x+a0:Thedegreeofapolynomialisthehighestpowerofxinitsexpression.Constant(non-zero)polynomials,linearpolynomials,quadratics,cubicsandquarticsarepolynomialsofdegree0,1,2,3and4respectively.Thefunctionf(x)=0isalsoapolynomial,butwesaythatitsdegreeis`unde ned'. www.mathcentre.ac.uk2c\rmathcentre2009 3.GraphsofpolynomialfunctionsWehavemetsomeofthebasicpolynomialsalready.Forexample,f(x)=2isaconstantfunctionandf(x)=2x+1isalinearfunction. f(x)x 1 2 ) = 2) = 2 + 1 Itisimportanttonoticethatthegraphsofconstantfunctionsandlinearfunctionsarealwaysstraightlines.Wehavealreadysaidthataquadraticfunctionisapolynomialofdegree2.Herearesomeexamplesofquadraticfunctions:f(x)=x2;f(x)=2x2;f(x)=5x2:Whatistheimpactofchangingthecoecientofx2aswehavedoneintheseexamples?Onewayto ndoutistosketchthegraphsofthefunctions. f(x)x f(x) = x2 ) = 2 f(x) = 5x2 Youcanseefromthegraphthat,asthecoecientofx2isincreased,thegraphisstretchedvertically(thatis,intheydirection).Whatwillhappenifthecoecientisnegative?Thiswillmeanthatallofthepositivef(x)valueswillnowbecomenegative.Sowhatwillthegraphsofthefunctionslooklike?Thefunctionsarenowf(x)=x2;f(x)=2x2;f(x)=5x2: www.mathcentre.ac.uk3c\rmathcentre2009 f(x)x f(x) = -x2 f(x) = -2x2 f(x) = -5x2 Noticeherethatallofthesegraphshaveactuallybeenre\rectedinthex-axis.Thiswillalwayshappenforfunctionsofanydegreeiftheyaremultipliedby1.Nowletuslookatsomeotherquadraticfunctionstoseewhathappenswhenwevarythecoecientofx,ratherthanthecoecientofx2.Weshalluseatableofvaluesinordertoplotthegraphs,butweshall llinonlythosevaluesneartheturningpointsofthefunctions. x 5 4 3 2 1 0 1 2 x2+x 6 2 0 0 2 6 x2+4x 0 3 4 3 0 x2+6x 5 8 9 8 5 Youcanseethesymmetryineachrowofthetable,demonstratingthatwehaveconcentratedontheregionaroundtheturningpointofeachfunction.Wecannowusethesevaluestoplotthegraphs. f(x)x + 4 f(x) = x2 + 6x + Asyoucansee,increasingthepositivecoecientofxinthispolynomialmovesthegraphdownandtotheleft. www.mathcentre.ac.uk4c\rmathcentre2009 Whathappensifthecoecientofxisnegative? x 2 1 0 1 2 3 4 5 x2x 6 2 0 0 2 6 x24x 0 3 4 3 0 x26x 5 8 9 8 5 Againwecanusethesetablesofvaluestoplotthegraphsofthefunctions. f(x)x f(x) = x2 - 4x f(x) = x2 - 6x f(x) = x2 - x Asyoucansee,increasingthenegativecoecientofx(inabsoluteterms)movesthegraphdownandtotheright.Sonowweknowwhathappenswhenwevarythex2coecient,andwhathappenswhenwevarythexcoecient.Butwhathappenswhenwevarytheconstanttermattheendofourpolynomial?Wealreadyknowwhatthegraphofthefunctionf(x)=x2+xlookslike,sohowdoesthisdi erfromthegraphofthefunctionsf(x)=x2+x+1,orf(x)=x2+x+5,orf(x)=x2+x4?Asusual,atableofvaluesisagoodplacetostart. x 2 1 0 1 2 x2+x 2 0 0 2 6 x2+x+1 3 1 1 3 7 x2+x+5 7 5 5 7 11 x2+x4 2 4 4 2 2 Ourtableofvaluesisparticularlyeasytocompletesincewecanuseouranswersfromthex2+xcolumnto ndeverythingelse.Wecanusethesetablesofvaluestoplotthegraphsofthefunctions. www.mathcentre.ac.uk5c\rmathcentre2009 f(x)x + + x + + Aswecanseestraightaway,varyingtheconstanttermtranslatesthex2+xcurvevertically.Furthermore,thevalueoftheconstantisthepointatwhichthegraphcrossesthef(x)axis.4.TurningpointsofpolynomialfunctionsAturningpointofafunctionisapointwherethegraphofthefunctionchangesfromslopingdownwardstoslopingupwards,orviceversa.Sothegradientchangesfromnegativetopositive,orfrompositivetonegative.Generallyspeaking,curvesofdegreencanhaveupto(n1)turningpoints.Forinstance,aquadratichasonlyoneturningpoint. Acubiccouldhaveuptotwoturningpoints,andsowouldlooksomethinglikethis. However,somecubicshavefewerturningpoints:forexamplef(x)=x3.Butnocubichasmorethantwoturningpoints. www.mathcentre.ac.uk6c\rmathcentre2009 Inthesameway,aquarticcouldhaveuptothreeturningturningpoints,andsowouldlooksome-thinglikethis. Again,somequarticshavefewerturningpoints,butnonehasmore. KeyPointApolynomialofdegreencanhaveupto(n1)turningpoints. 5.RootsofpolynomialfunctionsYoumayrecallthatwhen(xa)(xb)=0,weknowthataandbarerootsofthefunctionf(x)=(xa)(xb).Nowwecanusetheconverseofthis,andsaythatifaandbareroots,thenthepolynomialfunctionwiththeserootsmustbef(x)=(xa)(xb),oramultipleofthis.Forexample,ifaquadratichasrootsx=3andx=2,thenthefunctionmustbef(x)=(x3)(x+2),oraconstantmultipleofthis.Thiscanbeextendedtopolynomialsofanydegree.Forexample,iftherootsofapolynomialarex=1,x=2,x=3,x=4,thenthefunctionmustbef(x)=(x1)(x2)(x3)(x4);oraconstantmultipleofthis.Letusalsothinkaboutthefunctionf(x)=(x2)2.Wecanseestraightawaythatx2=0,sothatx=2.Forthisfunctionwehaveonlyoneroot.Thisiswhatwecallarepeatedroot,andarootcanberepeatedanynumberoftimes.Forexample,f(x)=(x2)3(x+4)4hasarepeatedrootx=2,andanotherrepeatedrootx=4.Wesaythattherootx=2hasmultiplicity3,andthattherootx=4hasmultiplicity4.Theusefulthingaboutknowingthemultiplicityofarootisthatithelpsuswithsketchingthegraphofthefunction.Ifthemultiplicityofarootisoddthenthegraphcutsthroughthex-axisatthepoint(x;0).Butifthemultiplicityiseventhenthegraphjusttouchesthex-axisatthepoint(x;0).Forexample,takethefunctionf(x)=(x3)2(x+1)5(x2)3(x+2)4:Therootx=3hasmultiplicity2,sothegraphtouchesthex-axisat(3;0). www.mathcentre.ac.uk7c\rmathcentre2009 Therootx=1hasmultiplicity5,sothegraphcrossesthex-axisat(1;0).Therootx=2hasmultiplicity3,sothegraphcrossesthex-axisat(2;0).Therootx=2hasmultiplicity4,sothegraphtouchesthex-axisat(2;0).Totakeanotherexample,supposewehavethefunctionf(x)=(x2)2(x+1).Wecanseethatthelargestpowerofxis3,andsothefunctionisacubic.Weknowthepossiblegeneralshapesofacubic,andasthecoecientofx3ispositivethecurvemustgenerallyincreasetotherightanddecreasetotheleft.Wecanalsoseethattherootsofthefunctionarex=2andx=1.Therootx=2hasevenmultiplicityandsothecurvejusttouchesthex-axishere,whilstx=1hasoddmultiplicityandsoherethecurvecrossesthex-axis.Thismeanswecansketchthegraphasfollows. 2-1x f(x) KeyPointThenumberaisarootofthepolynomialfunctionf(x)iff(a)=0,andthisoccurswhen(xa)isafactoroff(x).Ifaisarootoff(x),andif(xa)misafactoroff(x)but(xa)m+1isnotafactor,thenwesaythattheroothasmultiplicitym.Atarootofoddmultiplicitythegraphofthefunctioncrossesthex-axis,whereasatarootofevenmultiplicitythegraphtouchesthex-axis. Exercises1.Whatisapolynomialfunction?2.Whichofthefollowingfunctionsarepolynomialfunctions?(a)f(x)=4x2+2(b)f(x)=3x32x+p x(c)f(x)=124x5+3x2(d)f(x)=sinx+1(e)f(x)=3x122=x(f)f(x)=3x112x12 www.mathcentre.ac.uk8c\rmathcentre2009 3.Writedownoneexampleofeachofthefollowingtypesofpolynomialfunction:(a)cubic(b)linear(c)quartic(d)quadratic4.Sketchthegraphsofthefollowingfunctionsonthesameaxes:(a)f(x)=x2(b)f(x)=4x2(c)f(x)=x2(d)f(x)=4x25.Considerafunctionoftheformf(x)=x2+ax,wherearepresentsarealnumber.Thegraphofthisfunctionisrepresentedbyaparabola.(a)Whena�0,whathappenstotheparabolaasaincreases?(b)Whena0,whathappenstotheparabolaasadecreases?6.Writedownthemaximumnumberofturningpointsonthegraphofapolynomialfunctionofdegree:(a)2(b)3(c)12(d)n7.Writedownapolynomialfunctionwithroots:(a)1;2;3;4(b)2;4(c)12;1;68.Writedowntherootsandidentifytheirmultiplicityforeachofthefollowingfunctions:(a)f(x)=(x2)3(x+4)4(b)f(x)=(x1)(x+2)2(x4)39.Sketchthefollowingfunctions:(a)f(x)=(x2)2(x+1)(b)f(x)=(x1)2(x+3)Answers1.Apolynomialfunctionisafunctionthatcanbewrittenintheformf(x)=anxn+an1xn1+an2xn2+:::+a2x2+a1x+a0;whereeacha0,a1,etc.representsarealnumber,andwherenisanaturalnumber(including0).2.(a)f(x)=4x2+2isapolynomial(b)f(x)=3x32x+p xisnotapolynomial,becauseofp x(c)f(x)=124x5+3x2isapolynomial(d)f(x)=sinx+1isnotapolynomial,becauseofsinx(e)f(x)=3x122=xisnotapolynomial,becauseof2=x(f)f(x)=3x112x12isapolynomial3.(a)Thehighestpowerofxmustbe3,soexamplesmightbef(x)=x32x+1orf(x)=x32.(b)Thehighestpowerofxmustbe1,soexamplesmightbef(x)=xorf(x)=6x5.(c)Thehighestpowerofxmustbe4,soexamplesmightbef(x)=x43x3+2xorf(x)=x4x5.(d)Thehighestpowerofxmustbe2,soexamplesmightbef(x)=x2orf(x)=x25.4. www.mathcentre.ac.uk9c\rmathcentre2009 f(x)x f(x) = 4x2f(x) = -4x2 f(x) = x2 f(x) = -x2 5.(a)Whena�0,theparabolamovesdownandtotheleftasaincreases.(b)Whena0,theparabolamovesdownandtotherightasadecreases.6.(a)1turningpoint(b)2turningpoints(c)11turningpoints(d)(n1)turningpoints.7.(a)f(x)=(x1)(x2)(x3)(x4)oramultiple(b)f(x)=(x2)(x+4)oramultiple(c)f(x)=(x12)(x+1)(x+6)oramultiple.8.(a)x=2oddmultiplicityx=4evenmultiplicity(b)x=1oddmultiplicityx=2evenmultiplicityx=4oddmultiplicity www.mathcentre.ac.uk10c\rmathcentre2009 9) f(x)x ) = ( ) = ( www.mathcentre.ac.uk11c\rmathcentre2009