Graphs of Cubic Functions.docx
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Graphs of Cubic Functions.docx
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GraphsofCubicFunctions
GraphsofCubicFunctions
1.Introduction
Inmathematics,theformofacubicfunctionisf(x)=ax3+bx2+cx+d,thataisnonzero;orafunctionthatisdefinedbyapolynomialofdegreethree.Usually,thecoefficientsa,b,canddarerealnumbers,settingf(x)=0andthethreerootswillbeobtained.Thisinvestigationexplorestangentstocubicfunctionsandtheirrelationshipwiththerootsofthefunction.Thetangentgetsthroughthemidpointofthetworootsofthethreeroots.Theinvestigationwillbecarriedoutinthreesections:
introduction,mathematicalinvestigationandanalysis,conclusion.Infirstsection,themaincontentisbackground,thepurposeofthisinvestigationandsoon.Insecondsection,inordertofindtherelationshipbetweenthetangentandthecubicfunctionandtoverifythecorrectnessoftherelationship,thecontentisdividedintothefollowingthreeparts:
tangenttoacubicfunctionwith3distinctrealroots,tangentstoothercubicfunctions,tangentstoageneralcubicfunction.Infinalsection,thecontentmainlyfocusedonsummaryofallmainfindingsandtheevaluationoftheresultsandthemethodused.
2.MathematicalInvestigationandAnalysis
2.1Tangenttoacubicfunctionwith3distinctrealroots
Considerthefunctionf(x)=x3–3x2–10x+24,drawagraphforthiscubicfunctionwithonlinegraphingpackageDesmos.Fromthegraphofthecubicfunction,itiseasytofindthatthecubicfunctionhasthreedistinctroots.Inordertoverifythecorrectnessofthethreedistinctroots,thefunctionf(x)=x3–3x2–10x+24cantransformasy=(x+3)(x–2)(x–4),thethreedistinctrootscaneasilybeobtainedthatarex=-3,x=2,x=4,whichconsistentwiththefollowinggraph.
Then,findtheequationofthetangentatthemidpointofthefirsttworoots.Thefirsttworootsarex=-3,x=2,andthemidpointofthetworootsisx=(-3+2)/2=-0.5.Substitutex=-0.5intothefunctionandgetthevaluey=28.13,so,thecoordinateofthepointsis(-0.5,28.13)asshowninthefollowinggraph.
Thederivationofthecubicfunctionis
=3x2-6x-10,substitutex=-0.5intothederivationfunctionandgetthevalueof
=-6.25.Forthereasonthederivationofthefunctiony=f(x)atx=
isthegradientofthetangentlineatpoint
thatis
=k.Inwhich,kreferstothegradientofthetangentline.So,thefunctionoftangentlineisy-
inwhich,
.Here,
=-0.5,
=28.13,k=-6.25,thetangentlineisy-28.13=-6.25(x+0.5),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Fromtheaboveanalysisandgraph,itiseasyfindingthatthetangentlineexactlygetsthroughthethirdrootorthetangentmeetsthecurvejustatthepointofthethirdroot.
Liketheaboveprocess,forthemidpointofthesecondandthirdroots,ifthereanyruleliketheaboveanalysis?
Thefollowingstepscanmakeitsure.
First,themidpointofsecondandthirdrootsisx=3,substitutex=3intothefunctionandgetthevaluey=-6,so,thecoordinateofthepointis(3,-6).Thederivationofthecubicfunctionis
=3x2-6x-10,substitutex=3intothederivationfunctionandgetthevalueof
=-1.Here,
=3,
=-6,k=-1,thetangentlineisy+6=-(x–3),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Fromtheaboveanalysisandgraph,itiseasytofindthatthetangentlineexactlygetsthroughthefirstrootorthetangentmeetsthecurvejustatthepointofthefirstroot.
2.2Tangentstoothercubicfunctions
Howdoestheconditionsofothercubicfunctions?
Arethereanyruleslikethefunctionf(x)=x3–3x2–10x+24?
Inthefollowingparts,5differentcubicfunctionswillbeanalyzedandaconjectureaboutcubicfunctionsandtheirtangentswillbestated.
Considerthefunctiony=x3-5x2-x+5,drawagraphforthiscubicfunctionwithonlinegraphingpackageDesmos.Itiseasytofindthatthecubicfunctionhasthreedistinctroots,thefunctiony=x3-5x2-x+5canbetransformedasy=(x+1)(x-1)(x-5),thethreedistinctrootscaneasilybeobtainedthatarex=-1,x=1,x=5.First,themidpointofthefirsttworootsisx=0,substitutex=0intothefunctionandgetthevaluey=5,so,thecoordinateofthepointis(0,5).Thederivationofthecubicfunctionis
=3x2-10x-1,substitutex=0intothederivationfunctionandgetthevalueof
=-1.Here,
=0,
=5,k=-1,theequationoftangentlineisy–6=-(x–0),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Then,themidpointofsecondandthirdrootsisx=3,substitutex=3intothefunctionandgetthevaluey=-16,so,thecoordinateofthepointis(3,-16).Thederivationofthecubicfunctionis
=3x2-10x-1,substitutex=3intothederivationfunctionandgetthevalueof
=-4.Here,
=3,
=-16,k=-4,theequationoftangentlineisy+16=-4(x–3),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Considerthefunctiony=-2x3+10x2+2x-10,drawagraphforthiscubicfunctionwithonlinegraphingpackageDesmos.Itiseasytofindthatthecubicfunctionhasthreedistinctroots,thefunctiony=-2x3+10x2+2x–10canbetransformedasy=-2(x+1)(x–1)(x–5),thethreedistinctrootscaneasilybeobtainedthatarex=-1,x=1,x=5.First,themidpointofthefirsttworootsisx=0,substitutex=0intothefunctionandgetthevaluey=-10,so,thecoordinateofthepointis(0,5).Thederivationofthecubicfunctionis
=-6x2+20x+2,substitutex=0intothederivationfunctionandgetthevalueof
=2.Here,
=0,
=-10,k=2,thetangentlineisy+10=2(x–0),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Then,themidpointofsecondandthirdrootsisx=3,substitutex=3intothefunctionandgetthevaluey=32,so,thecoordinateofthepointis(3,32).Thederivationofthecubicfunctionis
=-6x2+20x+2,substitutex=3intothederivationfunctionandgetthevalueof
=8.Here,
=3,
=32,k=8,thetangentlineisy–32=8(x–3),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Forthepurposetofindthegeneralresults,arangeofcubicfunctionsareanalyzedusingtheabovemethod.Andtheresultsareasfollows:
Considerthefunctiony=3x3-33x2+93x-63,thethreedistinctrootscaneasilybeobtainedthatarex=1,x=3,x=7.Themidpointofthefirsttworootsisx=2andthecoordinateofthepointis(2,15).Theequationofthetangentlineisy–15=-3(x–2),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Then,themidpointofsecondandthirdrootsisx=5,thecoordinateofthepointis(5,-48).Thetangentlineisy+48=-12(x–5),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Considerthefunctiony=x3+13x2+47x+35,thethreedistinctrootscaneasilybeobtainedthatarex=-1,x=-5,x=-7.Themidpointofthefirsttworootsisx=-6andthecoordinateofthepointis(-6,5).Theequationoftangentlineisy–5=-(x+6),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Then,themidpointofsecondandthirdrootsisx=-3,thecoordinateofthepointis(-3,-16).Thetangentlineisy+16=-4(x+3),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Butthereissomespecialconditions,considerthefunctiony=x3-7x2+11x-5,thethreedistinctrootscaneasilybeobtainedthatarex=1,x=1,x=5.Thefunctionhastwosameroots.Themidpointofthefirsttworootsisx=1andthecoordinateofthepointis(1,0).Theequationoftangentlineisy=0,thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Then,themidpointofsecondandthirdrootsisx=3,thecoordinateofthepointis(3,-8).Thetangentlineisy+8=-4(x–3),thegraphofthetangentlineandthecubicfunctionisasthefollowinggraph:
Fromalltheaboveanalysis,itiseasytogiveouttheconjectureaboutthetangentsandthecubicfunctions.So,theconjectureisthatthetangentwhichgetsthroughthemidpointofthetworootswillmeetthecurveatthepointofthelastroot.
2.3TangentstoaGeneralCubicFunction
Forthepurposeoftestingthecorrectnessandtheuniversalityoftheconjecture,thefollowingpartmainlyfocusedonmathematicaldeductionforageneralcubicfunction,andverifyingtherelationshipbetweenthetangentandthecubicfunction.
Forthegeneralcubicfunctiony=ax3+bx2+cx+d,certainlya
0,assumedthatthecubicfunctionhasthreerealroots
(
),andthefunctioncanbetransformedastheformofy=ax3+bx2+cx+d=
.
Thederivationofthefunctioncanbewrittenasfollows:
So,when
/2,thegradientofthetangentis:
Substitute
/2intothecubicfunctionandgetthefollowingequation:
Andtheequationofthetangentisy-
wheny=0,thesolutionoftheequationisx=
.
If
/2,andstillcangetthesolutionoftheequationisx=
withthesamesteps.So,fromtheaboveanalysis,theconjectureisprovedtobetrue.
Fromtheaboveanalysis,forthegeneralcubicfunctiony=ax3+bx2+cx+d,theconjectureisthatthetangentwhichgetsthroughthemidpointofthetworootswillmeetthecurveatthepointofthelastroot.
3.Conclusion
Basedontheaboveanal
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