Inthisuni twedescribepolynomialfunctions andlookatsomeoftheirproperties Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideotutorialo nthist ID: 22800
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.
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+an 1xn 1+:::+a2x2+a1x+a0wherethea'sarerealnumbers(sometimescalledthecoecientsofthepolynomial).Althoughthisgeneralformulamightlookquitecomplicated,particularexamplesaremuchsimpler.Forexample,f(x)=4x3 3x2+2isapolynomialofdegree3,as3isthehighestpowerofxintheformula.Thisiscalledacubicpolynomial,orjustacubic.Andf(x)=x7 4x5+1isapolynomialofdegree7,as7isthehighestpowerofx.Noticeherethatwedon'tneedeverypowerofxupto7:weneedtoknowonlythehighestpowerofxto ndoutthedegree.Anexampleofakindyoumaybefamiliarwithisf(x)=4x2 2x 4whichisapolynomialofdegree2,as2isthehighestpowerofx.Thisiscalledaquadratic.Functionscontainingotheroperations,suchassquareroots,arenotpolynomials.Forexample,f(x)=4x3+p x 1isnotapolynomialasitcontainsasquareroot.Andf(x)=5x4 2x2+3=xisnotapolynomialasitcontainsa`dividebyx'. KeyPointApolynomialisafunctionoftheformf(x)=anxn+an 1xn 1+:::+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.Thiswillalwayshappenforfunctionsofanydegreeiftheyaremultipliedby 1.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 x2 x 6 2 0 0 2 6 x2 4x 0 3 4 3 0 x2 6x 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+x 4?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+x 4 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(n 1)turningpoints.Forinstance,aquadratichasonlyoneturningpoint. Acubiccouldhaveuptotwoturningpoints,andsowouldlooksomethinglikethis. However,somecubicshavefewerturningpoints:forexamplef(x)=x3.Butnocubichasmorethantwoturningpoints. www.mathcentre.ac.uk6c\rmathcentre2009 Inthesameway,aquarticcouldhaveuptothreeturningturningpoints,andsowouldlooksome-thinglikethis. Again,somequarticshavefewerturningpoints,butnonehasmore. KeyPointApolynomialofdegreencanhaveupto(n 1)turningpoints. 5.RootsofpolynomialfunctionsYoumayrecallthatwhen(x a)(x b)=0,weknowthataandbarerootsofthefunctionf(x)=(x a)(x b).Nowwecanusetheconverseofthis,andsaythatifaandbareroots,thenthepolynomialfunctionwiththeserootsmustbef(x)=(x a)(x b),oramultipleofthis.Forexample,ifaquadratichasrootsx=3andx= 2,thenthefunctionmustbef(x)=(x 3)(x+2),oraconstantmultipleofthis.Thiscanbeextendedtopolynomialsofanydegree.Forexample,iftherootsofapolynomialarex=1,x=2,x=3,x=4,thenthefunctionmustbef(x)=(x 1)(x 2)(x 3)(x 4);oraconstantmultipleofthis.Letusalsothinkaboutthefunctionf(x)=(x 2)2.Wecanseestraightawaythatx 2=0,sothatx=2.Forthisfunctionwehaveonlyoneroot.Thisiswhatwecallarepeatedroot,andarootcanberepeatedanynumberoftimes.Forexample,f(x)=(x 2)3(x+4)4hasarepeatedrootx=2,andanotherrepeatedrootx= 4.Wesaythattherootx=2hasmultiplicity3,andthattherootx= 4hasmultiplicity4.Theusefulthingaboutknowingthemultiplicityofarootisthatithelpsuswithsketchingthegraphofthefunction.Ifthemultiplicityofarootisoddthenthegraphcutsthroughthex-axisatthepoint(x;0).Butifthemultiplicityiseventhenthegraphjusttouchesthex-axisatthepoint(x;0).Forexample,takethefunctionf(x)=(x 3)2(x+1)5(x 2)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)=(x 2)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(x a)isafactoroff(x).Ifaisarootoff(x),andif(x a)misafactoroff(x)but(x a)m+1isnotafactor,thenwesaythattheroothasmultiplicitym.Atarootofoddmultiplicitythegraphofthefunctioncrossesthex-axis,whereasatarootofevenmultiplicitythegraphtouchesthex-axis. Exercises1.Whatisapolynomialfunction?2.Whichofthefollowingfunctionsarepolynomialfunctions?(a)f(x)=4x2+2(b)f(x)=3x3 2x+p x(c)f(x)=12 4x5+3x2(d)f(x)=sinx+1(e)f(x)=3x12 2=x(f)f(x)=3x11 2x12 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)=(x 2)3(x+4)4(b)f(x)=(x 1)(x+2)2(x 4)39.Sketchthefollowingfunctions:(a)f(x)=(x 2)2(x+1)(b)f(x)=(x 1)2(x+3)Answers1.Apolynomialfunctionisafunctionthatcanbewrittenintheformf(x)=anxn+an 1xn 1+an 2xn 2+:::+a2x2+a1x+a0;whereeacha0,a1,etc.representsarealnumber,andwherenisanaturalnumber(including0).2.(a)f(x)=4x2+2isapolynomial(b)f(x)=3x3 2x+p xisnotapolynomial,becauseofp x(c)f(x)=12 4x5+3x2isapolynomial(d)f(x)=sinx+1isnotapolynomial,becauseofsinx(e)f(x)=3x12 2=xisnotapolynomial,becauseof2=x(f)f(x)=3x11 2x12isapolynomial3.(a)Thehighestpowerofxmustbe3,soexamplesmightbef(x)=x3 2x+1orf(x)=x3 2.(b)Thehighestpowerofxmustbe1,soexamplesmightbef(x)=xorf(x)=6x 5.(c)Thehighestpowerofxmustbe4,soexamplesmightbef(x)=x4 3x3+2xorf(x)=x4 x 5.(d)Thehighestpowerofxmustbe2,soexamplesmightbef(x)=x2orf(x)=x2 5.4. 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)(n 1)turningpoints.7.(a)f(x)=(x 1)(x 2)(x 3)(x 4)oramultiple(b)f(x)=(x 2)(x+4)oramultiple(c)f(x)=(x 12)(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