3.3Function and graphs A-level考试纯数学P1课件.ppt

FunctionandgraphsChapter3

ContentsThepointofintersectionoftwographsUsingfactorstosketchgraphsPredictingfunctionsfromtheirgraphsThemodulusofanumberTheshapesofgraphsoftheformy=ax2+bx+c

Usingfactorstosketchgraphsf(x)=x2–6x+5g(x)=12x–4x2-1y123450321-1x-2

EnhancingVisualEffectsYoucandrawthegraphofanyfunctionofthistypewhichcanbefactorisedasa(x–r)(x–s),Byfirstnotingthatitcutsthex-axisatthepoints(r,0)and(s,0).Thesignoftheconstantatellsyouwhetherit‘bendsupwards’(likey=x2)or‘bensdownwards’.

ExampleSketchthegraphoff(x)=3x2–2x–1yx

Youcanextendthefactormethodtodrawinggraphsoffunctionswithmorethantwofactors.forexamplef(x)=a(x–r)(x–s)(x–t)

ExampleSketchthegraphoff(x)=2x(x–1)(x–4)yx

ExampleSketchthegraphoff(x)=–(x+2)(x–1)2yx

PredictingfunctionsfromtheirgraphsYoucanalsousethefactorformtopredicttheequationofafunctionofthetypef(x)=ax2+bx+c,ifyouknowthepointswhereitsgraphcrossesthex-axisandthecoordinatesofoneotherpointonthegraph.

ExampleFindtheequationofthegraphofthetypey=ax2+bx+cwhichcrossesthex-axisatthepoints(1,0)and(4,0)andalsopassedthroughthepoint(3,-4)

Theshapesofgraphsoftheformy=ax2+bx+cAllthegraphshavethesamegeneralshape,whichiscalledaparabola.Theseparabolashaveaverticalaxisofsymmetry.Thepointwhereaparabolameetsitsaxisofsymmetryiscalledthevertex.Changingcmovesthegraphupanddowninthey-direction.Changingbmovestheaxisofsymmetryofthegraphinthex-direction.Ifaandbhavethesamesigntheaxisofsymmetryistotheleftofthey-axis;ifaandbhaveoppositesingstheaxisofsymmetryistotherightofthey-axis.Ifaispositivethevertexisatthelowestpointofthegraphofthegraph;ifaisnegativethevertexisatthehighestpoint.Thelargerthesi