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
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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)
Ready, Set, Go Homework: Circles and Other Conics 8.2
Classroom Task: 8.3 Circle Challenges – A Practice Understanding Task
Writing the equation of a circle given various information (G.GPE.1)
Ready, Set, Go Homework: Circles and Other Conics 8.3
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)
Ready, Set, Go Homework: Circles and Other Conics 8.4
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)
Ready, Set, Go Homework: Circles and Other Conics 8.5
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)
Ready, Set, Go Homework: Circles and Other Conics 8.6
SECONDARY MATH II // MODULE 8
CIRCLES & OTHER CONICS - 8.1
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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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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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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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CIRCLES & OTHER CONICS - 8.2
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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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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 + _________
READY, SET, GO! Name PeriodDate
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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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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?
3. Malik’schallenge:
Findtheequationofacirclewithcenterinthefirstquadrantandistangenttothelines
x=8,y=3,andx=14.
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4. Sapana’schallenge:
Thepoints(4,-1)and(-6,7)aretheendpointsofthediameterofacircle.Whatisthe
equationofthecircle?
5. Malik’schallenge:
Isthepoint(5,1)inside,outside,oronthecircle!! − 6! + !! + 8! = 24?Howdoyouknow?
6. Sapana’schallenge:
Thecircledefinedby(! − 1)! + (! + 4)! = 16istranslated5unitstotheleftand2unitsdown.Writetheequationoftheresultingcircle.
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7. Malik’schallenge:
Therearetwocircles,thefirstwithcenter(3,3)andradiusr1,andthesecondwithcenter
(3,1)andradiusr2.
a. Findvaluesr1andr2sothatthefirstcircleiscompletelyenclosedbythesecondcircle.
b. Findonevalueofr1andonevalueofr2sothatthetwocirclesintersectattwopoints.
c. Findonevalueofr1andonevalueofr2sothatthetwocirclesintersectatexactlyone
point.
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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
READY, SET, GO! Name PeriodDate
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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
4
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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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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
READY, SET, GO! Name PeriodDate
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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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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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8.5 Functioning With
Parabolas
A Solidify Understanding Task
Sketchthegraphofeachparabola(accurately),findthevertexandusethegeometricdefinitionofa
parabolatofindtheequationoftheseparabolas.
1. Directrix! = −4,FocusA(2,-2)
Vertex________________
Equation:
2. Directrix! = 2,FocusA(-1,0)
Vertex________________
Equation:
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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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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
READY, SET, GO! Name PeriodDate
Figure1
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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
26
SECONDARY MATH II // MODULE 8
CIRCLES AND OTHER CONICS – 8.5
Mathematics Vision Project
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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______________________________________
27
SECONDARY MATH II // MODULE 8
CIRCLES AND OTHER CONICS – 8.5
Mathematics Vision Project
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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
28
SECONDARY MATH II // MODULE 8
CIRCLES & OTHER CONICS-8.6
Mathematics Vision Project
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8.6 Turn It Around
A Solidify Understanding Task
Annikaisthinkingmoreaboutthegeometricviewof
parabolasthatshehasbeenworkingoninmathclass.
Shethinks,“NowIseehowalltheparabolasthatcome
fromgraphingquadraticfunctionscouldalsocomefromagivenfocusanddirectrix.Inoticethat
alltheparabolashaveopenedupordownwhenthedirectrixishorizontal.Iwonderwhatwould
happenifIrotatedthefocusanddirectrix90degreessothatthedirectrixisvertical.Howwould
thatlook?Whatwouldtheequationbe?Hmmm….”SoAnnikastartstryingtoconstructaparabola
withaverticaldirectrix.Here’sthebeginningofherdrawing.UsearulertocompleteAnnika’s
drawing.
1. UsethedefinitionofaparabolatowritetheequationofAnnika’sparabola.
CC
BY
Mri
dul T
antia
http
s://f
lic.k
r/p/
77zq
Gh
29
SECONDARY MATH II // MODULE 8
CIRCLES & OTHER CONICS-8.6
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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?
30
SECONDARY MATH II // MODULE 8
CIRCLES AND OTHER CONICS – 8.6
Mathematics Vision Project
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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
READY, SET, GO! Name PeriodDate
31
SECONDARY MATH II // MODULE 8
CIRCLES AND OTHER CONICS – 8.6
Mathematics Vision Project
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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
⎛⎝⎜
⎞⎠⎟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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