MATHIV–Chapter9.2
PolarGraphs
NAME:____________________________
Directions:UseDesmosforallgraphsunlessotherwisestated.ToproperlyuseDesmos,clickthewrench
toolintheupperrighthandcornerandchangethegridoptionfromCartesiantoPolar(fromthesquare
gridtothecirculargrid).Whenwritingequations,usetheformr=(risfoundundertheABCoption),
sin/cos(foundunderfunctions)andtheta(foundabovethebackspaceinABC).Youmayfindithelpfulto
removethestaplefromthispacketwhileworking.Thepagesarenumberedtoavoidtroubleputtingthe
packetbacktogether.
PARTONE:CIRCLES
Thestandardformforacircleis:
𝑟 = 𝑎 sin 𝜃 𝑟 = 𝑎 cos 𝜃
Beginbygraphingthefunction𝑟 = sin 𝜃,thenattempttransformingthegraphbychangingtheavalue.
Forexample,graph𝑟 = 2 sin 𝜃and𝑟 = 3 sin 𝜃.Howdoesaaffectthegraph?
Nowgraphthefunction𝑟 = cos 𝜃.Howisthisdifferentfromthesinefunction?
Adjusttheavalue.Doesitbehavethesamewayforcosineasitdoesforsine?
Intheemptyspacebelowthestandardformequations,sketchagraphofeachfunction,labelingwherea
appearsinthecircle.
Then,maketwoflashcards,oneforeachequation.Writethestandardformononeside,andsketcha
graphontheotherside.
ClearyourDesmosgraphsbeforemovingon.
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PARTTWO:LIMACONS(LIM-a-son)
Standardformforalimaconis:
𝑟 = 𝑎 + 𝑏 sin 𝜃 𝑟 = 𝑎 − 𝑏 sin 𝜃
𝑟 = 𝑎 + 𝑏 cos 𝜃 𝑟 = 𝑎 − 𝑏 cos 𝜃
Limaconsarethepolargraphsthathavethemostvariation.Let’sbeginbyfirstdeterminingjustthe
locationofeachlimacongraph,andthendiscusshowaandbaffectthegraph.Limaconswilllooksimilar
tocircles,whichshouldmakesensegivenhowclosetheequationistotheequationofacircle,theonly
changeherebeingtheadditionofaconstantterm.Beginbygraphing𝑟 = 1 + sin 𝜃.Noticethatthisgraph
looksalmostlikeacircle,butnotquite.Whereisthemajorityofthegraphlocated?Overthepositivexaxis,thenegativex-axis,thepositivey-axis,orthenegativey-axis?
Nowgraph𝑟 = 1 − sin 𝜃.Whereisthemajorityofthisgraphlocated?
Whatabout𝑟 = 1 + cos 𝜃?
Whatabout𝑟 = 1 − cos 𝜃?
Usethisinformationtosketchagraphofabasiclimacon(alsoreferredtoasacardioid)beloweachofthe
standardformsabove.
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Nowlet’strytodeterminewhataandbcontrol.Todeterminethis,wewillonlygraphvariationsof
𝑟 = 𝑎 + 𝑏 sin 𝜃. Let’sbeginwiththegraph𝑟 = 2 + 2 sin 𝜃.Noticethattheshapelookssimilartothe
previouslimacongraphs,andinboththeseexamples,we’vekepta=b.Graph𝑟 = 3 + 3 sin 𝜃.Istheshape
thesameordifferent?
Inthisexampleandthepreviousexamples,howthelengthfromthepointoftheshapetotheouteredge
comparetotheaandbvalues?
Labelwhatyou’venoticedontheimagebelow.
a=b
Nowlet’smakeasmallerthanb.Graph𝑟 = 2 + 3 sin 𝜃.Howhastheshapechanged?
Nowgraph𝑟 = 1 + 3 sin 𝜃.Noticethatboththesizeoftheinnerandtheouterloophavechanged.These
sizesaredeterminedbyaandb.Trytodeterminethepattern.Considergraphing𝑟 = 1 + 4 sin 𝜃,
𝑟 = 2 + 4 sin 𝜃,and𝑟 = 3 + 4 sin 𝜃toclarifyorreinforceyourtheories.
Labelwhatyou’venoticedontheimagebelow.
a<b
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Nowlet’smakealargerthanb.Graph𝑟 = 5 + 4 sin 𝜃.Howhastheshapechangednow?
Continueadjustingthevalueofa,keepingitbiggerthanb.Noticenowthatwehavetwolengthstolook
at:thelengthoftheshapeabovethex-axisandthelengthoftheshapebelowthex-axis.Thisiscontrolled
byaandb.Trytodeterminethepattern.
Labelwhatyou’venoticedontheimagebelow.
a>b
Alloftheserulesregardingaandbaretrueregardlessofwhichstandardformwebeginwith.
Makefourflashcards,onewitheachstandardformsanda=bononeside,andgraphtheshapethatis
createdwhena=bontheotherside.Then,maketwomoreflashcardsthatillustrateanyofthestandard
formsandtheshapethatappearswhena<banda>b.
ClearyourDesmosgraphsbeforemovingon.
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PARTTHREE:ROSES
Thestandardformforaroseis:
𝑟 = 𝑎 sin(𝑛𝜃)
𝑟 = 𝑎 cos(𝑛𝜃)
wheren≥2andisaninteger
Beginbygraphing𝑟 = sin(2𝜃).Noticethatthegraphlooksabitlikeaflower,hencethename.Eachloop
ofthegraphiscalledapetal.Howmanypetalsdoesthisgraphhave?Howmanyareineachquadrant?
Nowgraph𝑟 = sin(4𝜃).Howmanypetalsdoesthisgraphhave?Howmanyareineachquadrant?
Consider𝑟 = sin(6𝜃).Howmanypetalsdoyouthinkthisgraphhas?Howmanywillbeineachquadrant?
Graphtoconfirm.
Nowlookattheplacementofallthepetalsonanyofthesegraphs.Dothesepetalseveroverlapthex-axis
ory-axis?
Nowgraph𝑟 = sin(3𝜃).Howmanypetalsdoesthisgraphhave?Identifythegraphthatoverlapsanaxis.
Wewillcallthisthe“startingpetal.”Doesitoverlapthepositivex-axis,thenegativex-axis,thepositiveyaxis,orthenegativey-axis?
Nowgraph𝑟 = sin(5𝜃).Howmanypetalsdoesthisgraphhave?Whereisthestartingpetal?
Considerthegraphof𝑟 = sin(7𝜃).Howmanypetalsdoyouthinkthisgraphhas?Wheredoyouthinkthe
startingpetalis?Graphtoconfirm.
Graphafewmoreoddfunctions,notingthelocationofthestartingpetal.Doyounoticeapattern?
Describeit.
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Nowgraph𝑟 = cos(2𝜃).Howmanypetalsdoesthisgraphhave?Doesnbehavethesamewayincosineas
itdoesinsine?
Noticewherethepetalsareplacedinthisgraph.Howisthisdifferentthanthesinefunction?Consider
graphing𝑟 = sin(2𝜃)atthesametimetocompare.
Nowgraph𝑟 = cos(4𝜃)toconfirmyourideas.
Nowlet’strytheoddcosinegraphs.Graph𝑟 = cos(3𝜃),𝑟 = cos(5𝜃),and𝑟 = cos(7𝜃).Identifythe
locationofthestartingpetalforthesegraphs.Doesitoverlapthepositivex-axis,thenegativex-axis,the
positivey-axis,orthenegativey-axis?Doesitchangefromgraphtographlikesine?
Ifniseven,whatistrueaboutthenumberofpetals?
Ifnisodd,whatistrueaboutthenumberofpetals?
Nowadjusttheavalue.Considergraphing𝑟 = 5 sin(3𝜃).Graphvariationsofthisfunctionuntilyoucan
determinewhatadoes.Howdoesaaffectthegraph?
Intheemptyspacebelowthestandardformequations,sketchtwographsofeachfunction(oneeven,one
odd),labelingwhereaappearsintherose.
Then,makefourflashcards,twoforeachequation(oneeven,oneodd).Writethestandardformandifn
isevenoroddononeside,andsketchagraphontheotherside.
ClearyourDesmosgraphsbeforemovingon.
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PARTFOUR:SPIRALSOFARCHIMEDES
StandardformforaspiralofArchimedesis:
𝑟 = 𝑎𝜃 + 𝑏
Beginbygraphing𝑟 = 𝜃.Noticethatthisgraphcreatesaspiral.Whatdirectionisthespiralgoing?
Clockwiseorcounterclockwise?
Let’sbeginbyidentifyingwhataffectahas.Keepingapositive,adjustthevalue.Makeitlarger.Makeit
smaller.Zoomouttogetabetterpicture.Howdoesincreasingaaffectthespiral?(Thisismuchmore
generalthanyourpreviousanswers.)
Nowlet’sidentifywhataffectbhas.Graph𝑟 = 𝜃 + 1.Howhasyourgraphchanged?
Nowgraph𝑟 = 𝜃 + 2.Thencontinuetoadjustbuntilyoucandeterminewhatbchanges.Whatdoesb
change?
Intheemptyspacebelowthestandardformequations,sketchagraphofaspiralofArchimedes,labeling
wherebappears.
Then,makeaflashcard.Writethestandardformononesideandsketchthegraphontheother.
ClearyourDesmosgraphsbeforemovingon.
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PARTFIVE:LEMNISCATES(LEM-nis-keyts)
Standardformforalemniscateis:
𝑟 ! = 𝑎! sin 2𝜃 𝑟 ! = 𝑎! cos (2𝜃)
InsteadofusingDesmos,we’llbegraphingtheseinWolframAlpha.Gotowolframalpha.com.Entering
equationsintoWolframAlphaisnotasuserfriendlyasDesmos,sobecarefultocopymynotationwhile
graphing.Sometimes,Wolframcanbeparticular,andyoumayhavetorefreshtogetittograph.
Beginbygraphing𝑟 ! = 4 sin 2𝜃 .We’lltypethisintothesearchbaras“graphr^2=4sin(2theta)”.What
symboldoesthisshapelooklike?Whattwoquadrantsdoesthisgraphappearin?
Adjusttheavalue.Trygraphing𝑟 ! = 9 sin 2𝜃 ,then𝑟 ! = 16 sin 2𝜃 .Howisthevalueofachangingour
graph?(Ifyou’rehavingahardtimeseeingithere,revisitthisquestionafterworkingwiththecosine
graphs.)
Nowgraph𝑟 ! = 4cos (2𝜃).Howisthisgraphdifferentthanthesinegraph?Adjusttheavalue.Howdoes
aaffectthegraph?Isthisthesameordifferentthanhowitaffectedthesinegraph?
Intheemptyspacebelowthestandardformequations,sketchagraphofeachfunction,labelingwherea
appearsinthelemniscate.
Then,maketwoflashcards,oneforeachequation.Writethestandardformononesideandifnisevenor
odd,andsketchagraphontheotherside.
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