Inthisuni twedescribepolynomialfunctions andlookatsomeoftheirproperties Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideotutorialo nthist ID: 22800 Download Pdf
Algebra 2. Chapter 5. This Slideshow was developed to accompany the textbook. Larson Algebra 2. By Larson. , R., Boswell, L., . Kanold. , T. D., & Stiff, L. . 2011 . Holt . McDougal. Some examples and diagrams are taken from the textbook..
Defn. : . Polynomial function. In the form of: . .. . The coefficients are real numbers.. The exponents are non-negative integers.. The domain of the function is the set of all real numbers..
Standard 15. Graph and analyze polynomial and radical functions to determine:. Domain and range. X and y intercepts. Maximum and minimum values. Intervals of increasing and decreasing. End behavior. With the function: f(x) = .
Objectives:. To approximate . x. -intercepts of a polynomial function with a graphing utility. To locate and use relative . extrema. of polynomial functions. To sketch the graphs of polynomial functions.
Objectives: Identify Polynomial functions. Determine end behavior recognize characteristics of polynomial functions. Use factoring to find zeros of polynomial functions.. Polynomials of degree 2 or higher have graphs that are smooth and continuous. By smooth we mean the graphs have rounded curves with no sharp corners. By continuous we mean the graphs have no breaks and can be drawn without lifting your pencil from the rectangular coordinate system..
Definitions. Coefficient. : the numerical factor of each term.. Constant. : the term without a variable.. Term. : a number or a product of a number and variables raised . to a power.. Polynomial. : a finite sum of terms of the form .
Objective: . Recognize the shape of basic polynomial functions. Describe the graph of a polynomial function. Identify properties of general polynomial functions: Continuity, End Behaviour, Intercepts, Local .
Now, we have learned about several properties for polynomial functions. Finding y-intercepts. Finding x-intercepts (zeros). End behavior (leading coefficient, degree). Testing values for zeros/factors (synthetic division) .
. An order . differential equation has a . parameter family of solutions … or will it?. . 0. 1. 2. 3. 4. 0. 0. 1. 2. 3. 4. 1. 1. 2. 3. 4. 0. 2. 2. 3. 4. 0. 1. 3. 3. 4. 0. 1. 2. 4. 4. 0. 1. 2.
Section 2.4. Terms. Divisor: . Quotient: . Remainder:. Dividend: . PF. FF . . Long Division. Use long division to find . divided by . .. . Division Algorithm for Polynomials. Let . and . be polynomials with the degree of .
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Inthisuni twedescribepolynomialfunctions andlookatsomeoftheirproperties Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideotutorialo nthist
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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.Wecangiveageneraldentionofapolynomial,anddeneitsdegree.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:weneedtoknowonlythehighestpowerofxtondoutthedegree.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`undened'. 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?Onewaytondoutistosketchthegraphsofthefunctions. 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,butweshallllinonlythosevaluesneartheturningpointsofthefunctions. 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,sohowdoesthisdierfromthegraphofthefunctionsf(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+xcolumntondeverythingelse.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)Whena0,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)Whena0,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
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