circles & other conics - mathematics vision project · it’s like a different form of the...

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2017 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 8

Circles & Other Conics

SECONDARY

MATH TWO

An Integrated Approach

SECONDARY MATH 2 // MODULE 8

CIRCLES AND OTHER CONICS

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 8 - TABLE OF CONTENTS

CIRCLES AND OTHER CONICS

Classroom Task: 8.1 Circling Triangles (or Triangulating Circles) – A Develop Understanding

Task

Deriving the equation of a circle using the Pythagorean Theorem (G.GPE.1)

Ready, Set, Go Homework: Circles and Other Conics 8.1

Classroom Task: 8.2 Getting Centered – A Solidify Understanding Task

Complete the square to find the center and radius of a circle given by an equation (G.GPE.1)


Classroom Task: 8.3 Circle Challenges – A Practice Understanding Task

Writing the equation of a circle given various information (G.GPE.1)


Classroom Task: 8.4 Directing Our Focus– A Develop Understanding Task

Derive the equation of a parabola given a focus and directrix (G.GPE.2)


Classroom Task: 8.5 Functioning with Parabolas – A Solidify Understanding Task

Connecting the equations of parabolas to prior work with quadratic functions (G.GPE.2)


Classroom Task: 8.6 Turn It Around – A Solidify Understanding Task

Writing the equation of a parabola with a vertical directrix, and constructing an argument that all

parabolas are similar (G.GPE.2)


SECONDARY MATH II // MODULE 8

CIRCLES & OTHER CONICS - 8.1

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

8.1 Circling Triangles (Or

Triangulating Circles)

A Develop Understanding Task

Usingthecornerofapieceofcoloredpaperandaruler,cutarighttrianglewitha6”hypotenuse,

likeso:

Usethistriangleasapatterntocutthreemorejustlikeit,sothatyouhaveatotaloffourcongruent

triangles.

1. Chooseoneofthelegsofthefirsttriangleandlabelitxandlabeltheotherlegy.Whatisthe

relationshipbetweenthethreesidesofthetriangle?

2. Whenyouaretoldtodoso,takeyourtrianglesuptotheboardandplaceeachofthemonthecoordinateaxislikethis:

Markthepointattheendofeachhypotenusewithapin.

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3. Whatshapeisformedbythepinsaftertheclasshaspostedalloftheirtriangles?Whywould

thisconstructioncreatethisshape?

4. Whatarethecoordinatesofthepinthatyouplacedin:a. thefirstquadrant?b. thesecondquadrant?c. thethirdquadrant?d. thefourthquadrant?

5. Nowthatthetriangleshavebeenplacedonthecoordinateplane,someofyourtriangleshavesidesthatareoflength–xor–y.Istherelationship!! + !! = 6!stilltrueforthesetriangles?Whyorwhynot?

6. Whatwouldbetheequationofthegraphthatisthesetonallpointsthatare6”awayfromtheorigin?

7. Isthepoint(0,-6)onthegraph?Howaboutthepoint(3,5.193)?Howcanyoutell?

8.Ifthegraphistranslated3unitstotherightand2unitsup,whatwouldbetheequationof

thenewgraph?Explainhowyoufoundtheequation.

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CIRCLES AND OTHER CONICS – 8.1


Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

8.1

Needhelp?Visitwww.rsgsupport.org

READY

Topic:FactoringspecialproductsFactorthefollowingasthedifferenceof2squaresorasaperfectsquaretrinomial.Donotfactoriftheyareneither.1.!! − 49

2.!! − 2! + 1 3.!! + 10! + 25

4.!! − !!

5.!! − 2!" + !! 6.25!! − 49!!

7.36!! + 60!" + 25!!

8.81!2 − 16!2 9.144!! − 312!" + 169!!

SET

Topic:WritingtheequationsofcirclesWritetheequationofeachcirclecenteredattheorigin.10.

11.

12.

READY, SET, GO! Name PeriodDate

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8.1


13.

14.

15.

GO

Topic:VerifyingPythagoreantriplesIdentifywhichsetsofnumberscouldbethesidesofarighttriangle.Showyourwork.16. 9,12,15{ } 17. 9,10, 19{ } 18. 1, 3, 2{ }

19. 2, 4, 6{ } 20. 3, 4, 5{ } 21. 10, 24, 26{ }

22. 2, 7,3{ } 23. 2 2, 5 3, 9{ } 24. 4ab3 10, 6ab3, 14ab3{ }

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8.2 Getting Centered

A Solidify Understanding Task

Malik’sfamilyhasdecidedtoputinanewsprinkling

systemintheiryard.Malikhasvolunteeredtolaythe

systemout.Sprinklersareavailableatthehardwarestoreinthefollowingsizes:

• Fullcircle,maximum15’radius

• Halfcircle,maximum15’radius

• Quartercircle,maximum15’radius

Allofthesprinklerscanbeadjustedsothattheysprayasmallerradius.Malikneedstobesure

thattheentireyardgetswatered,whichheknowswillrequirethatsomeofthecircularwater

patternswilloverlap.Hegetsoutapieceofgraphpaperandbeginswithascalediagramofthe

yard.Inthisdiagram,thelengthofthesideofeachsquarerepresents5feet.

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1. Ashebeginstothinkaboutlocatingsprinklersonthelawn,hisparentstellhimtotrytocoverthewholelawnwiththefewestnumberofsprinklerspossiblesothattheycansavesomemoney.TheequationofthefirstcirclethatMalikdrawstorepresenttheareawateredbythesprinkleris:

(! + 25)! + (! + 20)! = 225Drawthiscircleonthediagramusingacompass.

2. Layoutapossibleconfigurationforthesprinklingsystemthatincludesthefirstsprinklerpatternthatyoudrewin#1.

3. Findtheequationofeachofthefullcirclesthatyouhavedrawn.

Malikwrotetheequationofoneofthecirclesandjustbecausehelikesmessingwiththealgebra,hedidthis:

Originalequation: (! − 3)! + (! + 2)! = 225

!! − 6! + 9 + !! + 4! + 4 = 225

!! + !! − 6! + 4! − 212 = 0

Malikthought,“That’sprettycool.It’slikeadifferentformoftheequation.Iguessthattherecouldbedifferentformsoftheequationofacircleliketherearedifferentformsoftheequationofaparabolaortheequationofaline.”Heshowedhisequationtohissister,Sapana,andshethoughthewasnuts.Sapana,said,“That’sacrazyequation.Ican’teventellwhere

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thecenterisorthelengthoftheradiusanymore.”Maliksaid,“Nowit’slikeapuzzleforyou.I’llgiveyouanequationinthenewform.I’llbetyoucan’tfigureoutwherethecenteris.”

Sapanasaid,“Ofcourse,Ican.I’lljustdothesamethingyoudid,butworkbackwards.”

4. MalikgaveSapanathisequationofacircle:!! + !! − 4! + 10! + 20 = 0

HelpSapanafindthecenterandthelengthoftheradiusofthecircle.

5. Sapanasaid,“Ok.Imadeoneforyou.What’sthecenterandlengthoftheradiusforthiscircle?”

!! + !! + 6! − 14! − 42 = 0

6. Sapanasaid,“Istilldon’tknowwhythisformoftheequationmightbeuseful.Whenwehaddifferentformsforotherequationslikelinesandparabolas,eachofthevariousformshighlighteddifferentfeaturesoftherelationship.”Whymightthisformoftheequationofacirclebeuseful?

!! + !! + !" + !" + ! = 0

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8.2


READY Topic:MakingperfectsquaretrinomialsFillinthenumberthatcompletesthesquare.Thenwritethetrinomialinfactoredform.

1. 2.

3. 4.

Onthenextset,leavethenumberthatcompletesthesquareasafraction.Thenwritethetrinomialinfactoredform.5. 6. 7.

8. 9.

10.

SET Topic:Writingequationsofcircleswithcenter(h,k)andradiusr.Writetheequationofeachcircle.11. 12. 13.

x2 + 6x + _________ x2 −14x + _________

x2 − 50x + _________ x2 − 28x + _________

x2 −11x + _________ x2 + 7x + _________ x2 +15x + _________

x2 + 23x + _________ x2 − 1

5x + _________ x2 − 3

4x + _________


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8.2


Writetheequationofthecirclewiththegivencenterandradius.Thenwriteitinexpandedform.14.Center:(5,2)Radius:13 15.Center:(-6,-10)Radius:9

16.Center:(0,8)Radius:15 17.Center:(19,-13)Radius:1

18.Center:(-1,2)Radius:10 19.Center:(-3,-4)Radius:8

Go Topic:VerifyingifapointisasolutionIdentifywhichpointisasolutiontothegivenequation.Showyourwork.

20. y = 45x − 2 a.(-15,-14) b.(10,10)

21. a.(-4,-12) b. − 5 , 3 5

22. y = x2 + 8 a. 7 , 15 b. 7,−1

23. y = −4x2 +120 a. 5 3 ,−180 b. 5 3, 40

24. x2 + y2 = 9 a. b. −2, 5

25. 4x2 − y2 = 16 a. −3 , 10 b. −2 2 , 4

y = 3 x

8,−1( )

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CIRCLES & OTHER CONICS -8.3



8.3 Circle Challenges

A Practice Understanding Task

OnceMalikandSapanastartedchallengingeachother

withcircleequations,theygotalittlemorecreative

withtheirideas.Seeifyoucanworkoutthe

challengesthattheygaveeachothertosolve.Besuretojustifyallofyouranswers.

1. Malik’schallenge:

Whatistheequationofthecirclewithcenter(-13,-16)andcontainingthepoint(-10,-16)on

thecircle?

2. Sapana’schallenge:

Thepoints(0,5)and(0,-5)aretheendpointsofthediameterofacircle.Thepoint(3,y)is

onthecircle.Whatisavaluefory?


Findtheequationofacirclewithcenterinthefirstquadrantandistangenttothelines

x=8,y=3,andx=14.

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Thepoints(4,-1)and(-6,7)aretheendpointsofthediameterofacircle.Whatisthe

equationofthecircle?


Isthepoint(5,1)inside,outside,oronthecircle!! − 6! + !! + 8! = 24?Howdoyouknow?


Thecircledefinedby(! − 1)! + (! + 4)! = 16istranslated5unitstotheleftand2unitsdown.Writetheequationoftheresultingcircle.

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Therearetwocircles,thefirstwithcenter(3,3)andradiusr1,andthesecondwithcenter

(3,1)andradiusr2.

a. Findvaluesr1andr2sothatthefirstcircleiscompletelyenclosedbythesecondcircle.

b. Findonevalueofr1andonevalueofr2sothatthetwocirclesintersectattwopoints.

c. Findonevalueofr1andonevalueofr2sothatthetwocirclesintersectatexactlyone

point.

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8.3


READY Topic:Findingthedistancebetweentwopoints

Simplify.Usethedistanceformulad = x2 − x1( )2 + y2 − y1( )2 tofindthedistancebetweenthe

givenpoints.Leaveyouranswerinsimplestradicalform.1. A 18,−12( ) B 10,4( ) 2.G −11,−9( ) H −3,7( ) 3. J 14,−20( ) K 5,5( )

4.M 1,3( ) P −2,7( ) 5.Q 8,2( ) R 3,7( )

6.S −11,2 2( ) T −5,−4 2( ) 7.W −12,−2 2( ) Z −7,−3 2( ) SET Topic:WritingequationsofcirclesUsetheinformationprovidedtowritetheequationofthecircleinstandardform,

(! − !)! + (! − !)! = !!8.Center −16,−5( ) andthecircumferenceis22π

9.Center 13,−27( ) andtheareais196π

10.Diametermeasures15unitsandthecenterisattheintersectionofy=x+7andy=2x–5

11.Liesinquadrant2Tangentto x = −12 and x = −4


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8.3


12.Center(-14,9)Pointoncircle(1,11)13.CenterliesontheyaxisTangenttoy=-2andy=-1714.Threepointsonthecircleare −8,5( ), 3,−6( ), 14,5( )

15.Iknowthreepointsonthecircleare(-7,6),(9,6),and(-4,13).Ithinkthattheequationofthecircleis x −1( )2 + y − 6( )2 = 64 .Isthisthecorrectequationforthecircle?Justifyyouranswer.

GO Topic:FindingthevalueofBinaquadraticintheformof Ax2 + Bx +C inordertocreateaperfectsquaretrinomial.FindthevalueofBthatwillmakeaperfectsquaretrinomial.Thenwritethetrinomialinfactoredform.16. x2 + _____ x + 36 17. x2 + _____ x +100 18. x2 + _____ x + 225

19.9x2 + _____ x + 225 20.16x2 + _____ x +169 21. x2 + _____ x + 5

22. x2 + _____ x + 254 23. x2 + _____ x + 9

4 24. x2 + _____ x + 49

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8.4 Directing Our Focus

A Develop

Understanding Task

Onaboardinyourclassroom,yourteacherhassetupapointandalinelikethis:

Focus(pointA)

directrix(line!)

We’regoingtocallthelineadirectrixandthepointafocus.They’vebeenlabeledonthedrawing.

Similartothecirclestask,theclassisgoingtoconstructageometricfigureusingthefocus(pointA)

anddirectrix(line!).

1. Cuttwopiecesofstringwiththesamelength.

2. Markthemidpointofeachpieceofstringwithamarker.

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3. Positionthestringontheboardsothatthemidpointisequidistantfromthefocus(pointA)

andthedirectrix(line !),which means that it must be perpendicular to the directrix.Whileholdingthestringinthisposition,putapinthroughthemidpoint.Dependingonthesizeof

yourstring,itwilllooksomethinglikethis:

4. Usingyoursecondstring,usethesameproceduretopostapinontheothersideofthe

focus.

5. Asyourclassmatesposttheirstrings,whatgeometricfiguredoyoupredictwillbemadeby

thetacks(thecollectionofallpointslike(!, !)showinthefigureabove)?Why?

6. Whereisthevertexofthefigurelocated?Howdoyouknow?

7. Whereisthelineofsymmetrylocated?Howdoyouknow?

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8. ConsiderthefollowingconstructionwithfocuspointAandthe!-axisasthedirectrix.Usearulertocompletetheconstructionoftheparabolainthesamewaythattheclass

constructedtheparabolawithstring.

9. Youhavejustconstructedaparabolabaseduponthedefinition:Aparabolaisthesetofall

points(!, !)equidistantfromalinel(thedirectrix)andapointnotontheline(thefocus).Usethisdefinitiontowritetheequationoftheparabolaabove,usingthepoint(!, !)torepresentanypointontheparabola.

10. Howwouldtheparabolachangeifthefocuswasmovedup,awayfromthedirectrix?

11. Howwouldtheparabolachangeifthefocusweretobemoveddown,towardthedirectrix?

12. Howwouldtheparabolachangeifthefocusweretobemoveddown,belowthedirectrix?

(!, !)

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8.4


READY Topic:GraphingQuadraticsGrapheachsetoffunctionsonthesamecoordinateaxes.Describeinwhatwaythegraphsarethesameandinwhatwaytheyaredifferent.

1. 2.

3. 4.

y = x2, y = 2x2, y = 4x2 y = 14x2, y = −x2, y = −4x2

y = 14x2, y = x2 − 2, y = 1

4x2 − 2, y = 4x2 − 2 y = x2, y = −x2, y = x2 + 2, y = −x2 + 2


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8.4


SET Topic:Sketchingaparabolausingtheconicdefinition.UsetheconicdefinitionofaparabolatosketchaparaboladefinedbythegivenfocusFandtheequationofthedirectrix.Beginbygraphingthefocus,thedirectrix,andpointP1.UsethedistanceformulatofindFP1andfindthe

verticaldistancebetweenP1andthedirectrixbyidentifyingpointHonthedirectrixandcountingthe

distance.LocatethepointP2,(thepointontheparabolathatisareflectionofP1acrosstheaxisof

symmetry.)LocatethevertexV.Sincethevertexisapointontheparabola,itmustlieequidistantbetween

thefocusandthedirectrix.Sketchtheparabola.Hint:theparabolaalways“hugs”thefocus.

Example:F(4,3),P1(8,6),y=1 FP1= P1H=5P2islocatedat(0,6)Vislocatedat(4,2)

5.F(1,-1),P1(3,-1) y=-3 6.F(-5,3),P1(-1,3)y=77.F(2,1),P1(-2,1)y=-3 8.F(1,-1),P1(-9,-1)y=9

4 − 8( )2 + 3− 6( )2 = 16 + 9 = 25 = 5

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8.4


9.Findasquarepieceofpaper(apost-itnotewillwork).Foldthesquareinhalfverticallyandputadotanywhereonthefold.Lettheedgeofthepaperbethedirectrixandthedotbethefocus.Foldtheedgeofthepaper(thedirectrix)uptothedotrepeatedlyfromdifferentpointsalongtheedge.Thefoldlinesbetweenthefocusandtheedgeshouldmakeaparabola.Experimentwithanewpaperandmovethefocus.Useyourexperimentstoanswerthefollowingquestions.10.Howwouldtheparabolachangeifthefocusweremovedup,awayfromthedirectrix?11.Howwouldtheparabolachangeifthefocusweremoveddown,towardthedirectrix?12.Howwouldtheparabolachangeifthefocusweremoveddown,belowthedirectrix?

GO Topic:Findingthecenterandradiusofacircle.Writeeachequationsothatitshowsthecenter(h,k)andradiusrofthecircle.Thiscalledthestandardformofacircle. ! − ! ! + ! − ! ! = !!13. 14. 15. 16. 17. 18.

19. 20. 21. 22.

x2 + y2 + 4y −12 = 0 x2 + y2 − 6x − 3= 0

x2 + y2 + 8x + 4y − 5 = 0 x2 + y2 − 6x −10y − 2 = 0

x2 + y2 − 6y − 7 = 0 x2 + y2 − 4x + 8y + 6 = 0

x2 + y2 − 4x + 6y − 72 = 0 x2 + y2 +12x + 6y − 59 = 0

x2 + y2 − 2x +10y + 21= 0 4x2 + 4y2 + 4x − 4y −1= 0

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CIRCLES & OTHER CONICS-8.5



8.5 Functioning With

Parabolas


Sketchthegraphofeachparabola(accurately),findthevertexandusethegeometricdefinitionofa

parabolatofindtheequationoftheseparabolas.

1. Directrix! = −4,FocusA(2,-2)

Vertex________________

Equation:

2. Directrix! = 2,FocusA(-1,0)

Vertex________________

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3. Directrix! = 3,FocusA(1,7)

Vertex________________

Equation:

4. Directrix! = 3,FocusA(2,-1)

Vertex________________

Equation:

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5. Giventhefocusanddirectrix,howcanyoufindthevertexoftheparabola?

6. Giventhefocusanddirectrix,howcanyoutelliftheparabolaopensupordown?

7. Howdoyouseethedistancebetweenthefocusandthevertex(orthevertexandthe

directrix)showingupintheequationsthatyouhavewritten?

8. Describeapatternforwritingtheequationofaparabolagiventhefocusanddirectrix.

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9. Annikawonderswhywearesuddenlythinkingaboutparabolasinacompletely

differentwaythanwhenwedidquadraticfunctions.Shewondershowthesedifferent

waysofthinkingmatchup.Forinstance,whenwetalkedaboutquadraticfunctions

earlierwestartedwith! = !!.“Hmmmm.….Iwonderwherethefocusanddirectrixwouldbeonthisfunction,”shethought.HelpAnnikafindthefocusanddirectrixfor

! = !!.

10. Annikathinks,“Ok,Icanseethatyoucanfindthefocusanddirectrixforaquadratic

function,butwhataboutthesenewparabolas.Aretheyquadraticfunctions?Whenwe

workwithfamiliesoffunctions,theyaredefinedbytheirratesofchange.Forinstance,

wecantellalinearfunctionbecauseithasaconstantrateofchange.”Howwouldyou

answerAnnika?Arethesenewparabolasquadraticfunctions?Justifyyouranswer

usingseveralrepresentationsandtheparabolasinproblems1-4asexamples.

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8.5


READY Topic:Standardformofaquadratic.Verifythatthegivenpointliesonthegraphoftheparaboladescribedbytheequation.(Showyourwork.)

1. 2.

3. 4.

SET Topic:Equationofparabolabasedonthegeometricdefinition

5.Verifythat y −1( ) = 14x2 istheequation

oftheparabolainFigure1byplugginginthe3pointsV(0,1),C(4,5)andE(2,2).Showyourworkforeachpoint!

6.Ifyoudidn’tknowthat(0,1)wasthevertexoftheparabola,couldyouhavefounditbyjustlookingattheequation?Explain.

6,0( ) y = 2x2 − 9x −18 −2,49( ) y = 25x2 + 30x + 9

5,53( ) y = 3x2 − 4x − 2 8,2( ) y = 14x2 − x − 6


Figure1

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8.5


7.UsethediagraminFigure2toderivethegeneralequationofaparabolabasedonthegeometricdefinitionofaparabola.RememberthatthedefinitionstatesthatMF=MQ.

8.Recalltheequationin#5, y −1( ) = 14x2 ,whatisthevalueof!?

9.Ingeneral,whatisthevalueof!foranyparabola?

10.InFigure3,thepointMisthesameheightasthefocusand FM ≅ MR .Howdothecoordinatesofthispointcomparewiththecoordinatesofthefocus?FillinthemissingcoordinatesforMandRinthediagram.

Figure2

Figure3

M

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8.5


SketchthegraphbyfindingthevertexandthepointMandR(thereflectionofM)asdefinedinthediagramabove.Usethegeometricdefinitionofaparabolatofindtheequationoftheseparabolas.11.Directrixy=9,FocusF(-3,7) 12.Directrixy=-6,FocusF(2,-2)Vertex_____________ Vertex_____________Equation______________________________________ Equation______________________________________

13.Directrixy=5,FocusF(-4,-1) 14.Directrixy=-1,FocusF(4,-3)Vertex_____________ Vertex_____________Equation______________________________________ Equation______________________________________

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8.5


GO Topic:FindingminimumandminimumvaluesforquadraticsFindthemaximumorminimumvalueofthequadratic.Indicatewhichitis.

15. y = x2 + 6x − 5 16. y = 3x2 −12x + 8

17. y = − 12x2 +10x +13 18. y = −5x2 + 20x −11

19. y = 72x2 − 21x − 3 20. y = − 3

2x2 + 9x + 25

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8.6 Turn It Around


Annikaisthinkingmoreaboutthegeometricviewof

parabolasthatshehasbeenworkingoninmathclass.

Shethinks,“NowIseehowalltheparabolasthatcome

fromgraphingquadraticfunctionscouldalsocomefromagivenfocusanddirectrix.Inoticethat

alltheparabolashaveopenedupordownwhenthedirectrixishorizontal.Iwonderwhatwould

happenifIrotatedthefocusanddirectrix90degreessothatthedirectrixisvertical.Howwould

thatlook?Whatwouldtheequationbe?Hmmm….”SoAnnikastartstryingtoconstructaparabola

withaverticaldirectrix.Here’sthebeginningofherdrawing.UsearulertocompleteAnnika’s

drawing.

1. UsethedefinitionofaparabolatowritetheequationofAnnika’sparabola.

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2. Whatsimilaritiesdoyouseetotheequationsofparabolasthatopenupordown?Whatdifferencesdoyousee?

3. Tryanotherone:Writetheequationoftheparabolawithdirectrixx=4andfocus(0,3).

4. Onemoreforgoodmeasure:Writetheequationoftheparabolawithdirectrixx=-3andfocus(-2,-5).

5. Howcanyoupredictifaparabolawillopenleft,right,up,ordown?

6. Howcanyoutellhowwideornarrowaparabolais?

7. Annikahastwobigquestionsleft.Writeandexplainyouranswerstothesequestions.a. Areallparabolasfunctions?

b. Areallparabolassimilar?

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8.6


READY Topic:ReviewofCircles

Usethegiveninformationtowritetheequationofthecircleinstandardform.

1. Center:(-5,-8),Radius:11

2. Endpointsofthediameter:(6,0)and(2,-4)

3. Center(-5,4):Pointonthecircle(-9,1)

4. Equationofthecircleinthediagramtotheright.

SET Topic:WritingequationsofhorizontalparabolasUsethefocusF,pointM,apointontheparabola,andtheequationofthedirectrixtosketchtheparabola(labelyourpoints)andwritetheequation.Putyourequationintheform! = !

!" (! − !)! + !where“p”isthedistancefromthefocustothevertex.

5.F(1,0),M(1,4)x=-3 6.F(3,1),M(3,-5)x=9


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8.6


7.F(7,-5),M(4,-1)x=9 8.F(-1,2),M(6,-9)x=-7

GO Topic:Identifyingkeyfeaturesofaquadraticwritteninvertexform

State(a)thecoordinatesofthevertex,(b)theequationoftheaxisofsymmetry,(c)thedomain,and(d)therangeforeachofthefollowingfunctions.

9. f x( ) = x − 3( )2 + 5 10. f x( ) = x +1( )2 − 2 11. f x( ) = − x − 3( )2 − 7

12. f x( ) = −3 x − 34

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⎞⎠⎟2

+ 45 13. f x( ) = 1

2x − 4( )2 +1 14. f x( ) = 1

4x + 2( )2 − 4

15.Comparethevertexformofaquadratictothegeometricdefinitionofaparabolabasedonthefocus

anddirectrix.Describehowtheyaresimilarandhowtheyaredifferent.

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circles & other conics - mathematics vision project · it’s like a different form of the...

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