Polynomial functions - Math Centre

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f (x)f (x) = x 2 + x + 5f (x) = x 2 + x + 1f (x) = x 2 + xf (x) = x 2 + x − 4xAswecanseestraightaway,varyingtheconstanttermtranslatesthe x 2 + xcurvevertically.Furthermore,thevalueoftheconstantisthepointatwhichthegraphcrossesthe f(x)axis.4. Turning points of polynomial <strong>functions</strong>Aturningpointofafunctionisapointwherethegraphofthefunctionchangesfromslopingdownwardstoslopingupwards,orviceversa.Sothegradientchangesfromnegativetopositive,orfrompositivetonegative. Generallyspeaking,curvesofdegree ncanhaveupto (n − 1)turningpoints.For instance, a quadratic has only one turningpoint.Acubiccouldhaveuptotwoturningpoints,andsowouldlooksomethinglikethis.However,somecubicshavefewerturningpoints: forexample f(x) = x 3 . Butnocubichasmorethantwoturningpoints.www.mathcentre.ac.uk 6 c○mathcentre2009
Inthesameway,aquarticcouldhaveuptothreeturningturningpoints,andsowouldlooksomethinglikethis.Again,somequarticshavefewerturningpoints,butnonehasmore.Key PointApolynomialofdegree ncanhaveupto (n − 1)turningpoints.5. Roots of polynomial <strong>functions</strong>Youmayrecallthatwhen (x − a)(x − b) = 0,weknowthat aand barerootsofthefunctionf(x) = (x − a)(x − b). Nowwecanusetheconverseofthis,andsaythatif aand bareroots,thenthepolynomialfunctionwiththeserootsmustbe f(x) = (x − a)(x − b),oramultipleofthis.Forexample,ifaquadratichasroots x = 3and x = −2,thenthefunctionmustbe f(x) =(x−3)(x+2),oraconstantmultipleofthis.Thiscanbeextendedtopolynomialsofanydegree.Forexample,iftherootsofapolynomialare x = 1, x = 2, x = 3, x = 4,thenthefunctionmustbef(x) = (x − 1)(x − 2)(x − 3)(x − 4),oraconstantmultipleofthis.Letusalsothinkaboutthefunction f(x) = (x − 2) 2 .Wecanseestraightawaythat x − 2 = 0,sothat x = 2. Forthisfunctionwehaveonlyoneroot. Thisiswhatwecallarepeatedroot,andarootcanberepeatedanynumberoftimes. Forexample, f(x) = (x − 2) 3 (x + 4) 4 hasarepeatedroot x = 2,andanotherrepeatedroot x = −4. Wesaythattheroot x = 2hasmultiplicity3,andthattheroot x = −4hasmultiplicity4.Theusefulthingaboutknowingthemultiplicityofarootisthatithelpsuswithsketchingthegraphofthefunction.Ifthemultiplicityofarootisoddthenthegraphcutsthroughthe x-axisatthepoint (x, 0). Butifthemultiplicityiseventhenthegraphjusttouchesthe x-axisatthepoint (x, 0).Forexample,takethefunctionf(x) = (x − 3) 2 (x + 1) 5 (x − 2) 3 (x + 2) 4 .•Theroot x = 3hasmultiplicity2,sothegraphtouchesthe x-axisat (3, 0).www.mathcentre.ac.uk 7 c○mathcentre2009
Page 1 and 2: Polynomial functionsmc-TY-polynomia
Page 3 and 4: 3. Graphs of polynomial functionsWe
Page 5: Whathappensifthecoefficientof xisne
Page 10 and 11: f (x)f (x) = 4x 2f (x) = x 2xf (x)

Polynomial functions - Math Centre

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