Polynomial functions - Math Centre

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f (x)xf (x) = −x 2f (x) = −2x 2f (x) = −5x 2Noticeherethatallofthesegraphshaveactuallybeenreflectedinthe x-axis. Thiswillalwayshappenforfunctionsofanydegreeiftheyaremultipliedby −1.Nowletuslookatsomeotherquadratic functions toseewhathappens whenwevarythecoefficientof x,ratherthanthecoefficientof x 2 .Weshalluseatableofvaluesinordertoplotthegraphs,butweshallfillinonlythosevaluesneartheturningpointsofthefunctions.x −5 −4 −3 −2 −1 0 1 2x 2 + x 6 2 0 0 2 6x 2 + 4x 0 −3 −4 −3 0x 2 + 6x −5 −8 −9 −8 −5Youcanseethesymmetryineachrowofthetable,demonstratingthatwehaveconcentratedontheregionaroundtheturningpointofeachfunction. Wecannowusethesevaluestoplotthegraphs.f (x)f (x) = x 2 + xf (x) = x 2 + 4xxf (x) = x 2 + 6xAsyoucansee,increasingthepositivecoefficientof xinthispolynomialmovesthegraphdownandtotheleft.www.mathcentre.ac.uk 4 c○mathcentre2009
Whathappensifthecoefficientof xisnegative?x −2 −1 0 1 2 3 4 5x 2 − x 6 2 0 0 2 6x 2 − 4x 0 −3 −4 −3 0x 2 − 6x −5 −8 −9 −8 −5Againwecanusethesetablesofvaluestoplotthegraphsofthefunctions.f (x)f (x) = x 2 − xf (x) = x 2 − 4xxf (x) = x 2 − 6xAsyoucansee,increasingthenegativecoefficientof x(inabsoluteterms)movesthegraphdownandtotheright.Sonowweknowwhathappenswhenwevarythe x 2 coefficient,andwhathappenswhenwevarythe xcoefficient. Butwhathappenswhenwevarytheconstanttermattheendofourpolynomial? Wealreadyknowwhatthegraphofthefunction f(x) = x 2 + xlookslike,sohowdoesthisdifferfromthegraphofthefunctions f(x) = x 2 + x + 1,or f(x) = x 2 + x + 5,orf(x) = x 2 + x − 4?Asusual,atableofvaluesisagoodplacetostart.x −2 −1 0 1 2x 2 + x 2 0 0 2 6x 2 + x + 1 3 1 1 3 7x 2 + x + 5 7 5 5 7 11x 2 + x − 4 −2 −4 −4 −2 2Ourtableofvaluesisparticularlyeasytocompletesincewecanuseouranswersfromthe x 2 +xcolumntofindeverythingelse. Wecanusethesetablesofvaluestoplotthegraphsofthefunctions.www.mathcentre.ac.uk 5 c○mathcentre2009
Page 1 and 2: Polynomial functionsmc-TY-polynomia
Page 3: 3. Graphs of polynomial functionsWe
Page 7 and 8: Inthesameway,aquarticcouldhaveuptot
Page 10 and 11: f (x)f (x) = 4x 2f (x) = x 2xf (x)

Polynomial functions - Math Centre

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