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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
WCPSS Math 2 Unit 4: MVP MODULE 2
Structures of Expressions
SECONDARY
MATH TWO
An Integrated Approach
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SECONDARY MATH 2 // MODULE 2
QUADRATIC FUNCTIONS
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MODULE 2 - TABLE OF CONTENTS
STRUCTURES OF EXPRESSIONS
2.1 Transformers: Shifty y’s – A Develop Understanding Task
Connecting transformations to quadratic functions and parabolas
(NC.M2.F-IF.7, NC.M2.F-BF.3)
READY, SET, GO Homework: Structure of Expressions 2.1
2.2 Transformers: More Than Meets the y’s – A Solidify Understanding Task
Working with vertex form of a quadratic, connecting the components to transformations
(NC.M2.F-IF.7, NC.M2.F-BF.3)
READY, SET, GO Homework: Structure of Expressions 2.2
2.3 Building the Perfect Square – A Develop Understanding Task
Visual and algebraic approaches to completing the square (NC.M2.F-IF.8)
READY, SET, GO Homework: Structure of Expressions 2.3
2.4 A Square Deal– A Solidify Understanding Task
Visual and algebraic approaches to completing the square (NC.M2.F-IF.8)
READY, SET, GO Homework: Structure of Expressions 2.4
2.5 Be There or Be Square– A Practice Understanding Task
Visual and algebraic approaches to completing the square (NC.M2.F-IF.8)
READY, SET, GO Homework: Structure of Expressions 2.5
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QUADRATIC FUNCTIONS
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2.8 The Wow Factor – A Solidify Understanding Task
Connecting the factored and expanded forms of a quadratic when a-value is not equal to one.
(NC.M2.F-IF.8, NC.M2.A-SSE.3)
READY, SET, GO Homework: Structure of Expressions 2.8
2.9 Lining Up Quadratics – A Solidify Understanding Task Focus on
the vertex and intercepts for quadratics (NC.M2.F-IF.7, NC.M2.F-BF.1)
READY, SET, GO Homework: Structure of Expressions 2.9
2.10 I’ve Got a Fill-in – A Practice Understanding Task
Building fluency in rewriting and connecting different forms of a quadratic
(NC.M2.F-IF.8, NC.M2.F-BF.1, NC.M2.A-SSE.3)
READY, SET, GO Homework: Structure of Expressions
3.4 Pascal's Pride – A Solidify Understanding Task
Building fluency in multiplying polynomials. (NC.M2.A-APR.1)
READY, SET, GO Homework: Polynomial Functions
3.3 It All Adds Up – A Develop Understanding Task
Building fluency in adding and subtratcting polynomials. (NC.MA-APR.1)
READY, SET, GO Homework: Polynomial Functions
2.6 Factor Fixin’ – A Solidify Understanding Task Connecting the factored and expanded forms of a quadratic(NC.M2.F-IF.8, NC.M2.F-BF.1, NC.M2.A-SSE.3) READY, SET, GO Homework: Structure of Expressions 2.6
2.7 The x Factor – A Solidify Understanding Task Connecting the factored and expanded or standard forms of a quadratic(NC.M2.F-IF.8, NC.M2.F-BF.1, NC.M2.A-SSE.3) READY, SET, GO Homework: Structure of Expressions 2.7
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STRUCTURES OF EXPRESSIONS – 2.1
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2.1 Transformers: Shifty y’s
A Develop Understanding Task
OptimaPrimeisdesigningarobotquiltforhernewgrandson.Sheplansfortherobottohavea
squareface.Theamountoffabricthatsheneedsforthefacewilldependontheareaoftheface,so
Optimadecidestomodeltheareaoftherobot’sfacemathematically.SheknowsthattheareaAofa
squarewithsidelengthxunits(whichcanbeinchesorcentimeters)ismodeledbythe
function, ! ! = !!squareunits.1. Whatisthedomainofthefunction! ! inthiscontext?
2. Matcheachstatementabouttheareatothefunctionthatmodelsit:
MatchingEquation
(A,B,C,orD)
Statement FunctionEquation
Thelengthofeachsideisincreasedby5units.
A) AB) !(!) = 5!!
Thelengthofeachsideismultipliedby5units.
C) BD) ! = (! + 5)!
Theareaofasquareisincreasedby5squareunits.
E) CF) ! = (5!)!
Theareaofasquareismultipliedby5. G) DH) ! = !! + 5
Optimastartedthinkingaboutthegraphof! = !!(inthedomainofallrealnumbers)andwonderingabouthowchangestotheequationofthefunctionlikeadding5ormultiplyingby5
affectthegraph.Shedecidedtomakepredictionsabouttheeffectsandthencheckthemout.
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3. Predicthowthegraphsofeachofthefollowingequationswillbethesameordifferentfrom
thegraphof! = !!.Similaritiestothegraphof
! = !!Differencesfromthegraphof
! = !!! = 5!!
! = (! + 5)!
! = (5!)!
! = !! + 5
4. Optimadecidedtotestherideasusingtechnology.Shethinksthatitisalwaysagoodidea
tostartsimple,soshedecidestogowith! = !! + 5.Shegraphsitalongwith! = !!inthesamewindow.Testityourselfanddescribewhatyoufind.
5. Knowingthatthingsmakealotmoresensewithmorerepresentations,Optimatriesafew
moreexampleslike! = !! + 2 and! = !! − 3,lookingatbothatableandagraphforeach.Whatconclusionwouldyoudrawabouttheeffectofaddingorsubtractinganumberto! = !!?Carefullyrecordthetablesandgraphsoftheseexamplesinyournotebookandexplainwhyyour
conclusionwouldbetrueforanyvalueofk,given,! = !! + !.
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6. Afterheramazingsuccesswithadditioninthelastproblem,Optimadecidedtolookatwhat
happenswithadditionandsubtractioninsidetheparentheses,orasshesaysit,“addingtothe!beforeitgetssquared”.Usingyourtechnology,decidetheeffectofhintheequations:! = (! + ℎ)!and! = (! − ℎ)!.(Choosesomespecificnumbersforh.)Recordafewexamples(bothtablesandgraphs)inyournotebookandexplainwhythiseffectonthegraphoccurs.
7. Optimathoughtthat#6wasverytrickyandhopedthatmultiplicationwasgoingtobemore
straightforward.Shedecidestostartsimpleandmultiplyby-1,soshebeginswith
! = −!!.Predictwhattheeffectisonthegraphandthentestit.Whydoesithavethiseffect?
8. Optimaisencouragedbecausethatonewaseasy.Shedecidestoendherinvestigationfor
thedaybydeterminingtheeffectofamultiplier,a,intheequation:! = !!!.Usingbothpositiveandnegativenumbers,fractionsandintegers,createatleast4tablesandmatchinggraphsto
determinetheeffectofamultiplier.
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.1 2.1
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READY Topic:Findingkeyfeaturesinthegraphofaquadraticequation
Makeapointonthevertexanddrawadottedlinefortheaxisofsymmetry.Labelthecoordinatesofthevertexandstatewhetherit’samaximumoraminimum.Writetheequationfortheaxisofsymmetry.
1. 2. 3.
4. 5. 6.
7. Whatconnectionexistsbetweenthecoordinatesofthevertexandtheequationoftheaxisofsymmetry?
8. Lookbackat#6.Trytofindawaytofindtheexactvalueofthecoordinatesofthevertex.Testyourmethodwitheachvertexin1-5.Explainyourconjecture.
9. Howmanyx-interceptscanaparabolahave?
10. Sketchaparabolathathasnox-intercepts,thenexplainwhathastohappenforaparabolatohavenox-intercepts.
READY, SET, GO! Name Period Date
8
6
4
2
–2
–5 5
8
6
4
2
5
8
6
4
2
–2
–5 5
6
4
2
–2
–4
–5
4
2
–2
–4
–6
5
6
4
2
–2
–4
5
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STRUCTURES OF EXPRESSIONS – 2.1 2.1
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SET
Topic:TransformationsonquadraticsMatching:Choosetheareamodelthatisthebestmatchfortheequation.
______11.x! + 4 ______12. x + 4 ! ______13. 4x ! _______14.4x!a. b.
c. d.
x -3 -2 -1 0 1 2 3f(x)=x2 9 4 1 0 1 4 9
15a)g(x)= b)x -3 -2 -1 0 1 2 3g(x) 2 -3 -6 -7 -6 -3 2
c) Inwhatwaydidthegraphmove?
d) Whatpartoftheequationindicatesthismove?
Atableofvaluesandthegraphforf(x)=x2isgiven.Comparethevaluesinthetableforg(x)tothoseforf(x).Identifywhatstaysthesameandwhatchanges.a)Usethisinformationtowritethevertexformoftheequationofg(x).b)Graphg(x).c)Describehowthegraphchangedfromthegraphoff(x).Usewordssuchasright,left,up,anddown.d)Answerthequestion.
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16a)g(x)= b)
x -3 -2 -1 0 1 2 3g(x) 11 6 3 2 3 6 11
c) Inwhatwaydidthegraphmove?d) Whatpartoftheequationindicatesthismove?
17a)g(x)= b)
x -4 -3 -2 -1 0 1 2g(x) 9 4 1 0 1 4 9
c) Inwhatwaydidthegraphmove?d) Whatpartoftheequationindicatesthismove?
18a)g(x)= b)
x 0 1 2 3 4 5 6g(x) 9 4 1 0 1 4 9
c) Inwhatwaydidthegraphmove?d) Whatpartoftheequationindicatesthismove?
GO Topic:FindingSquareRootsSimplifythefollowingexpressions
19. 49a!b! 20. x + 13 ! 21. x − 16 !
22. 36x + 25 ! 23. 11x − 7 ! 24. 9!! 2!! − ! !
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.2
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2.2 Transformers: More Than
Meets the y’s
A Solidify Understanding Task
Writetheequationforeachproblembelow.Useasecond
representationtocheckyourequation.
1. Theareaofasquarewithsidelengthx,wheretheside
lengthisdecreasedby3,theareaismultipliedby2andthen4squareunitsareaddedtothearea.
2.
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3.
4.
x f(x)
-4 7
-3 2
-2 -1
-1 -2
0 -1
1 2
2 7
3 14
4 23
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Grapheachequationwithoutusingtechnology.Besuretohavetheexactvertexandatleasttwocorrectpointsoneithersideofthelineofsymmetry.
5. ! ! = −!! + 3
6. ! ! = (! + 2)! − 5
7. ℎ ! = 3(! − 1)! + 2
8. Given:! ! = !(! − ℎ)! + !a. Whatpointisthevertexoftheparabola?
b. Whatistheequationofthelineofsymmetry?
c. Howcanyoutelliftheparabolaopensupordown?
d. Howdoyouidentifythedilation?
9. Doesitmatterinwhichorderthetransformationsaredone?Explainwhyorwhynot.
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STRUCTURES OF EXPRESSIONS – 2.2 2.2
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READY Topic:Standardformofquadraticequations
Thestandardformofaquadraticequationisdefinedas ! = !!! + !" + !, (! ≠ !).Identifya,b,andcinthefollowingequations.
Example:Given!!! + !" − !,a=4,b=7,andc=-61. y = 5x! + 3x + 6 2. y = x! − 7x + 3 3. y = −2x! + 3x
a=____________ a=____________ a=____________b=____________ b=____________ b=____________c=____________ c=____________ c=____________
4. y = 6x! − 5 5. y = −3x! + 4x 6. y = 8x! − 5x − 2
a=____________ a=____________ a=____________b=____________ b=____________ b=____________c=____________ c=____________ c=____________
Multiplyandwriteeachproductintheformy=ax2+bx+c.Thenidentifya,b,andc.7. y = x x − 4 8. y = x − 1 2x − 1 9. y = 3x − 2 3x + 2
a=____________ a=____________ a=____________b=____________ b=____________ b=____________c=____________ c=____________ c=____________
10. y = x + 6 x + 6 11. y = x − 3 ! 12. y = − x + 5 !
a=____________ a=____________ a=____________b=____________ b=____________ b=____________c=____________ c=____________ c=____________
READY, SET, GO! Name Period Date
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STRUCTURES OF EXPRESSIONS – 2.2 2.2
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SET
Topic:Graphingastandardy=x2parabola13. Graphtheequationy = x!.Includeatleast3accuratepointsoneachsideoftheaxisofsymmetry.
a. Statethevertexoftheparabola.
b. Completethetableofvaluesfory = x!.! ! !-3
-2
-1
0
1
2
3
Topic:Writingtheequationofatransformedparabolainvertexform.Findavaluefor!suchthatthegraphwillhavethespecifiednumberofx-intercepts.14. ! = !! +! 15. ! = !! +! 16. ! = !! +!
2(x-intercepts) 1(x-intercept) no(x-intercepts)
17. y = −x! + ω 18. y = −x! + ω 19. y = −x! + ω2(x-intercepts) 1(x-intercept) no(x-intercepts)
Graphthefollowingequations.Statethevertex.(Beaccuratewithyourkeypointsandshape!)20. y = x − 1 ! 21. y = x − 1 ! + 1 22. y = 2 x − 1 ! + 1
Vertex?_____________________ Vertex?_____________________ Vertex?_____________________
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23. y = x + 3 ! 24. y = − x + 3 ! − 4 25. y = −0.5 x + 1 ! + 4
Vertex?_____________________Vertex?_____________________ Vertex?_____________________
GO Topic:FeaturesofParabolasUsethetabletoidentifythevertex,theequationfortheaxisofsymmetry(AoS),andstatethe
numberofx-intercept(s)theparabolawillhave,ifany.Statewhetherthevertexwillbea
minimumoramaximum.
26. x y-4-3-2-1012
103-2-5-6-5-2
27. x y-2-101234
49281341413
28. x y-7-6-5-4-3-2-1
-9373-9-29-57
29. x y-8-7-6-5-4-3-2
-9-8-9-12-17-24-33
a.
b.
c.
d.
Vertex:_________
AoS:____________
x-int(s):________
MINorMAX
a.
b.
c.
d.
Vertex:_________
AoS:____________
x-int(s):________
MINorMAX
a.
b.
c.
d.
Vertex:_________
AoS:____________
x-int(s):________
MINorMAX
a.
b.
c.
d.
Vertex:_________
AoS:____________
x-int(s):________
MINorMAX
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2.3 Building the Perfect Square
A Develop Understanding Task
QuadraticQuilts
Optimahasaquiltshopwhereshesellsmanycolorful
quiltblocksforpeoplewhowanttomaketheirown
quilts.Shehasquiltdesignsthataremadesothattheycanbesizedtofitanybed.She
basesherdesignsonquiltsquaresthatcanvaryinsize,soshecallsthelengthofthe
sideforthebasicsquarex,andtheareaofthebasicsquareisthefunction! ! = !!.Inthisway,shecancustomizethedesignsbymakingbiggersquaresorsmallersquares.
1. IfOptimaadds3inchestothesideofthesquare,whatistheareaofthesquare?
WhenOptimadrawsapatternforthesquareinproblem#1,itlookslikethis:
2. Useboththediagramandtheequation,! ! = (! + 3)!toexplainwhytheareaofthequiltblocksquare,! ! ,isalsoequalto!! + 6! + 9.
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ThecustomerservicerepresentativesatOptima’sshopworkwithcustomerordersandwrite
uptheordersbasedontheareaofthefabricneededfortheorder.Asyoucanseefrom
problem#2therearetwowaysthatcustomerscancallinanddescribetheareaofthequilt
block.Onewaydescribesthelengthofthesidesoftheblockandtheotherwaydescribesthe
areasofeachofthefoursectionsoftheblock.
Foreachofthefollowingquiltblocks,drawthediagramoftheblockandwritetwoequivalent
equationsfortheareaoftheblock.
3. Blockwithsidelength:! + 2.
4. Blockwithsidelength:! + 1.
5. Whatpatternsdoyounoticewhenyourelatethediagramstothetwoexpressionsfor
thearea?
6. Optimalikestohaveherlittledog,Clementine,aroundtheshop.Onedaythedoggota
littlehungryandstartedtochewuptheorders.WhenOptimafoundtheorders,oneof
themwassochewedupthattherewereonlypartialexpressionsforthearea
remaining.HelpOptimabycompletingeachofthefollowingexpressionsforthearea
sothattheydescribeaperfectsquare.Then,writethetwoequivalentequationsforthe
areaofthesquare.
a. !! + 4!
b. !! + 6!
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c. !! + 8!
d. !! + 12!
7. If !! + !" + !isaperfectsquare,whatistherelationshipbetweenbandc?Howdoyouusebtofindc,likeinproblem6?
Willthisstrategyworkifbisnegative?Whyorwhynot?
Willthestrategyworkifbisanoddnumber?Whathappenstocifbisodd?
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READY Topic:Graphinglinesusingtheintercepts
Findthex-interceptandthey-intercept.Thengraphtheequation.
1. 3! + 2! = 12 2. 8! − 12! = −24 3. 3! − 7! = 21a. x-intercept: a. x-intercept: a. x-intercept:b. y-intercept: b. y-intercept: b. y-intercept:
4. 5! − 10! = 20 5. 2! = 6! − 18 6. ! = −6! + 6a. x-intercept: a. x-intercept: a. x-intercept:b. y-intercept: b. y-intercept: b. y-intercept:
READY, SET, GO! Name Period Date
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SET
Topic:CompletingthesquarebypayingattentiontothepartsMultiply.Showeachstep.Circlethepairofliketermsbeforeyousimplifytoatrinomial.
7. ! + 5 ! + 5 8. 3! + 7 3! + 7 9. 9! + 1 ! 10. 4! + 11 !
11. Writearuleforfindingthecoefficient“B”ofthex-term(themiddleterm)whenmultiplyingandsimplifying ax + q !.
Inproblems12–17, (a)Fillinthenumberthatcompletesthesquare.(b)Thenwritethetrinomialastheproductoftwofactors.
12.a)x! + 8x + _____ 13.a)x! + 10x + _____ 14.a) x! + 16x + _____
b) b) b)
15.a)x! + 6x + _____ 16.a)x! + 22x + _____ 17.a) x! + 18x + _____
b) b) b)
Inproblems18–26, (a)Findthevalueof“B,”thatwillmakeaperfectsquaretrinomial.(b)Thenwritethetrinomialasaproductoftwofactors.
18. x! + Bx + 16a)b)
19. x! + Bx + 121a)b)
20. x! + Bx + 625a)b)
21. x! + Bx + 225a)b)
22. x! + Bx + 49a)b)
23. x! + Bx + 169a)b)
24. x! + Bx + !"!
a)b)
25. x! + Bx + !!
a)b)
26. x! + Bx + !"!
a)b)
GO Topic:FeaturesofhorizontalandverticallinesFindtheinterceptsofthegraphofeachequation.Statewhetherit’sanx-ory-intercept.
27. y=-4.5 28. x=9.5 29. x=-8.2 30. y=112
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2.4 A Square Deal
A Solidify Understanding Task
QuadraticQuilts,Revisited
RememberOptima’squiltshop?Shebasesherdesignsonquiltsquaresthatcanvaryinsize,soshecallsthelengthofthesideforthebasicsquarex,andtheareaofthebasicsquareisthefunctionA ! = !!.Inthisway,shecancustomizethedesignsbymakingbiggersquaresorsmallersquares.
1. Sometimesacustomerordersmorethanonequiltblockofagivensize.Forinstance,whenacustomerorders4blocksofthebasicsize,thecustomerservicerepresentativeswriteupanorderforA ! = 4!!.Modelthisareaexpressionwithadiagram.
2. Oneofthecustomerservicerepresentativesfindsanenvelopethatcontainstheblockspicturedbelow.Writetheorderthatshowstwoequivalentequationsfortheareaoftheblocks.
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3. Whatequationsfortheareacouldcustomerservicewriteiftheyreceivedanorderfor2blocksthataresquaresandhavebothdimensionsincreasedby1inchincomparisontothebasicblock?Writetheareaequationsintwoequivalentforms.Verifyyouralgebrausingadiagram.
4. Ifcustomerservicereceivesanorderfor3blocksthatareeachsquareswithbothdimensionsincreasedby2inchesincomparisontothebasicblock?Again,show2differentequationsfortheareaandverifyyourworkwithamodel.
5. Clementineisatitagain!Whenisthatdoggoingtolearnnottochewuptheorders?(ShealsochewsOptima’sshoes,butthat’sastoryforanotherday.)Herearesomeoftheordersthathavebeenchewedupsotheyaremissingthelastterm.HelpOptimabycompletingeachofthefollowingexpressionsfortheareasothattheydescribeaperfectsquare.Then,writethetwoequivalentequationsfortheareaofthesquare.
2!! + 8!
3x! + 24!
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Sometimesthequiltshopgetsanorderthatturnsoutnottobemoreorlessthanaperfectsquare.Customerservicealwaystriestofillorderswithperfectsquares,oratleast,theystartthereandthenadjustasneeded.
6. Nowhere’sarealmess!CustomerservicereceivedanorderforanareaA x = 2!! + 12! + 13.Helpthemtofigureoutanequivalentexpressionfortheareausingthediagram.
7. Optimareallyneedstogetorganized.Here’sanotherscrambleddiagram.Writetwoequivalentequationsfortheareaofthisdiagram:
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8. Optimarealizedthatnoteveryoneisinneedofperfectsquaresandnotallordersarecominginasexpressionsthatareperfectsquares.Determinewhetherornoteachexpressionbelowisaperfectsquare.Explainwhytheexpressionisorisnotaperfectsquare.Ifitisnotaperfectsquare,findtheperfectsquarethatseems“closest”tothegivenexpressionandshowhowtheperfectsquarecanbeadjustedtobethegivenexpression.
A. ! ! = !! + 6! + 13 B. ! ! = !! − 8! + 16
C. ! ! = !! − 10! − 3 D. ! ! = 2!! + 8! + 14
E. ! ! = 3!! − 30! + 75 F. ! ! = 2!! − 22! + 11
9. Nowlet’sgeneralize.Givenanexpressionintheform!"! + !" + ! (! ≠ 0),describeastep-by-stepprocessforcompletingthesquare.
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READY
Topic:Findy-interceptsinparabolas
Statethey-interceptforeachofthegraphs.Thenmatchthegraphwithitsequation.
1. 2. 3.
4. 5. 6.
a. ! ! = −!! + 2! − 1 b. ! ! = −. 25!! − 2! + 5 c. ! ! = !! + 3! − 5
d. ! ! =. 5!! + ! − 7 e. ! ! = −.25!! + 3! + 1 f. ! ! = !! − 4! + 4
READY, SET, GO! Name Period Date
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SET
Topic:Completingthesquarewhena>1.Fillinthemissingconstantsothateachexpressionrepresents5perfectsquares.Thenstatethedimensionsofthesquaresineachproblem.
7. 5!! + 30! + _______ 8. 5!! − 50! + _______ 9. 5!! + 10! + _______
10. Giventhescrambleddiagrambelow,writetwoequivalentequationsforthearea.
Determineifeachexpressionbelowisaperfectsquareornot.Ifitisnotaperfectsquare,find
theperfectsquarethatseems“closest”tothegivenexpressionandshowhowtheperfectsquare
canbeadjustedtobethegivenexpression.
11. ! ! = !! + 10! + 14 12. ! ! = 2!! + 16! + 613.! ! = 3!! + 18! − 12
GO Topic:Evaluatingfunctions.
Findtheindicatedfunctionvaluewhen! ! = !!! − !! − !" !"# ! ! = −!!! + ! − !.14. ! 1 15.! 5 16. ! 10 17. ! −5 18. ! 0 + ! 0
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2.5 Be There or Be Square
A Practice Understanding Task
QuiltsandQuadraticGraphs
Optima’sniece,Jennyworksintheshop,takingordersanddrawingquiltdiagrams.Whenthe
shopisn’ttoobusy,Jennypullsouthermathhomeworkandworksonit.Oneday,sheis
workingongraphingparabolasandnoticesthattheequationssheisworkingwithlooksalot
likeanorderforaquiltblock.Forinstance, Jennyissupposedtographtheequation:! = (! − 3)! + 4 . Shethinks,“That’sfunny.Thiswouldbeanorderwherethelengthofthestandardsquareisreducedby3andthenweaddalittlepieceoffabricthathasasareaof4.
Wedon’tusuallygetorderslikethat,butitstillmakessense.Ibettergetbacktothinking
aboutparabolas.Hmmm…”
1. FullydescribetheparabolathatJennyhasbeenassignedtograph.
2. Jennyreturnstoherhomework,whichisaboutgraphingquadraticfunctions.Muchtoher
dismay,shefindsthatshehasbeengiven:! = !! − 6! + 9. “Ohdear”,thinksJenny.“Ican’ttellwherethevertexisoridentifyanyofthetransformationsoftheparabolainthis
form.NowwhatamIsupposedtodo?”
“Waitaminute—isthistheareaofaperfectsquare?”UseyourworkfromBuildingthePerfectSquaretoanswerJenny’squestionandjustifyyouranswer.
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3. Jennysays,“IthinkI’vefiguredouthowtochangetheformofmyquadraticequationso
thatIcangraphtheparabola.I’llchecktoseeifIcanmakemyequationaperfectsquare.”
Jenny’sequationis:! = !! − 6! + 9.
Seeifyoucanchangetheformoftheequation,findthevertex,andgraphtheparabola.
a. ! = !! − 6! + 9 Newformoftheequation:_________________________________
b. Vertexoftheparabola:______________
c. Graph(withatleast3accuratepointsoneachsideofthelineofsymmetry):
4. ThenextquadratictographonJenny’shomeworkis! = !! + 4! + 2.Doesthisexpressionfitthepatternforaperfectsquare?Whyorwhynot?
a. Useanareamodeltofigureouthowtocompletethesquaresothattheequation
canbewritteninvertexform,! = ! ! − ℎ ! + !.
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b. Istheequationyouhavewrittenequivalenttotheoriginalequation?Ifnot,what
adjustmentsneedtobemade?Why?
c. Identifythevertexandgraphtheparabolawiththreeaccuratepointsonbothsides
ofthelineofsymmetry.
5. Jennyhopedthatshewasn’tgoingtoneedtofigureouthowtocompletethesquareonan
equationwherebisanoddnumber.Ofcourse,thatwasthenextproblem.HelpJennytofindthevertexoftheparabolaforthisquadraticfunction:
! ! = !! + 7! + 10
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6. Don’tworryifyouhadtothinkhardabout#5.Jennyhastodoacouplemore:
a. ! ! = !! − 5! + 3 b. ! ! = !! − ! − 5
7. Itjustgetsbetter!HelpJennyfindthevertexandgraphtheparabolaforthequadratic
function:ℎ ! = 2!! − 12! + 17
8. Thisoneisjusttoocute—you’vegottotryit!Findthevertexanddescribetheparabola
thatisthegraphof:! ! = !! !
! + 2! − 3
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Topic:RecognizingQuadraticEquationsIdentifywhetherornoteachequationrepresentsaquadraticfunction.Explainhowyouknewitwasaquadratic.
1. x! + 13x − 4 = 0 2. 3x! + x = 3x! − 2 3. x 4x − 5 = 0Quadraticorno?
Justification:
Quadraticorno?
Justification:
Quadraticorno?
Justification:
4. 2x − 7 + 6x = 10 5. 2! + 6 = 0 6. 32 = 4x!
Quadraticorno?
Justification:
Quadraticorno?
Justification:
Quadraticorno?
Justification:
SET
Topic:Changingfromstandardformofaquadratictovertexform.Changetheformofeachequationtovertexform:! = ! ! − ! ! + !.Statethevertexandgraphtheparabola.Showatleast3accuratepointsoneachsideofthelineofsymmetry.
7. y = x! − 4x + 1
vertex:
8. y = x! + 2x + 5
vertex:
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9. y = x! + 3x + !"!
vertex:
10. y = !! x
! − x + 5
vertex:
11. Oneoftheparabolasinproblems9–10shouldlook“wider”thantheothers.Identifytheparabola.Explainwhythisparabolalooksdifferent.
Fillintheblankbycompletingthesquare.Leavethenumberthatcompletesthesquareasanimproperfraction.Thenwritethetrinomialinfactoredform.
12. 13. 14.
15. 16. 17.
x2 −11x + _________ x2 + 7x + _________ x2 +15x + _________
x2 + 23x + _________ x2 − 1
5x + _________ x2 − 3
4x + _________
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GO Topic:Writingrecursiveequationsforquadraticfunctions.
Identifywhetherthetablerepresentsalinearorquadraticfunction.Ifthefunctionislinear,writeboththeexplicitandrecursiveequations.Ifthefunctionisquadratic,writeonlytherecursiveequation.
18. 19.
Typeoffunction:
Equation(s):
! ! !1 02 33 64 95 12
Typeoffunction:
Equation(s):
! ! !1 72 103 164 255 37
20. 21.
Typeoffunction:
Equation(s):
! ! !1 82 103 124 145 16
Typeoffunction:
Equation(s):
! ! !1 282 403 544 705 88
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2.6 Factor Fixin’
A Develop Understanding Task
Atfirst,Optima’sQuiltsonlymadesquareblocksforquiltersandOptimaspenthertimemaking
perfectsquares.Customerservicerepresentativesweretrainedtoaskforthelengthoftheside
oftheblock,x,thatwasbeingordered,andtheywouldletthecustomerknowtheareaoftheblock
tobequiltedusingtheformula! ! = !!.Optimafoundthatmanycustomersthatcameintothestoreweremakingdesignsthatrequireda
combinationofsquaresandrectangles.So,Optima’sQuiltshasdecidedtoproduceseveralnew
linesofrectangularquiltblocks.Eachnewlineisdescribedintermsofhowtherectangularblock
hasbeenmodifiedfromtheoriginalsquareblock.Forexample,onelineofquiltblocksconsistsof
startingwithasquareblockandextendingonesidelengthby5inchesandtheothersidelengthby
2inchestoformanewrectangularblock.Thedesigndepartmentknowsthattheareaofthisnew
blockcanberepresentedbytheexpression:A(x)=(x+5)(x+2),buttheydonotfeelthatthis
expressiongivesthecustomerarealsenseofhowmuchbiggerthisnewblockis(e.g.,howmuch
moreareaithas)whencomparedtotheoriginalsquareblocks.
1. Canyoufindadifferentexpressiontorepresenttheareaofthisnewrectangularblock?You
willneedtoconvinceyourcustomersthatyourformulaiscorrectusingadiagram.
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HerearesomeadditionalnewlinesofblocksthatOptima’sQuiltshasintroduced.Findtwodifferent
algebraicexpressionstorepresenteachrectangle,andillustratewithadiagramwhyyour
representationsarecorrect.
2. Theoriginalsquareblockwasextended3inchesononesideand4inchesontheother.
3. Theoriginalsquareblockwasextended4inchesononlyoneside.
4. Theoriginalsquareblockwasextended5inchesoneachside.
5. Theoriginalsquareblockwasextended2inchesononesideand6inchesontheother.
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Customersstartorderingcustom-madeblockdesignsbyrequestinghowmuchadditionalareathey
wantbeyondtheoriginalareaof!!.Onceanorderistakenforacertaintypeofblock,customerserviceneedstohavespecificinstructionsonhowtomakethenewdesignforthemanufacturing
team.Theinstructionsneedtoexplainhowtoextendthesidesofasquareblocktocreatethenew
lineofrectangularblocks.
Thecustomerservicedepartmenthasplacedthefollowingordersonyourdesk.Foreach,describe
howtomakethenewblocksbyextendingthesidesofasquareblockwithaninitialsidelengthof
!.Yourinstructionsshouldincludediagrams,writtendescriptionsandalgebraicdescriptionsoftheareaoftherectanglesinusingexpressionsrepresentingthelengthsofthesides.
6. !! + 5! + 3! + 15
7. !! + 4! + 6! + 24
8. !! + 9! + 2! + 18
9. !! + 5! + ! + 5
Someoftheordersarewritteninanevenmoresimplifiedalgebraiccode.Figureoutwhatthese
entriesmeanbyfindingthesidesoftherectanglesthathavethisarea.Usethesidesoftherectangle
towriteequivalentexpressionsforthearea.
10. !! + 11! + 10
11. !! + 7! + 10
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12. !! + 9! + 8
13. !! + 6! + 8
14. !! + 8! + 12
15. !! + 7! + 12
16. !! + 13! + 12
17. Whatrelationshipsorpatternsdoyounoticewhenyoufindthesidesoftherectanglesfora
givenareaofthistype?
18. Acustomercalledandaskedforarectanglewithareagivenby:!! + 7! + 9.Thecustomerservicerepresentativesaidthattheshopcouldn’tmakethatrectangle.Doyouagreeordisagree?
Howcanyoutellifarectanglecanbeconstructedfromagivenarea?
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Topic:CreatingBinomialQuadraticsMultiply.(Usethedistributiveproperty,writeinstandardform.)
1. x 4x − 7 2. 5x 3x + 8 3. 3x 3x − 2
4. Theanswerstoproblems1,2,&3arequadraticsthatcanberepresentedinstandardformax2+bx+c.Whichcoefficient,a,b,orcequals0foralloftheexercisesabove?
Factorthefollowing.(Writetheexpressionsastheproductoftwolinearfactors.)
5. !! + 4! 6. 7!! − 21! 7. 12!! + 60! 8. 8!! + 20!
Multiply9. ! + 9 ! − 9 10. ! + 2 ! − 2 11. 6! + 5 6! − 5 12. 7! + 1 7! − 1
13. Theanswerstoproblems9,10,11,&12arequadraticsthatcanberepresentedinstandardformax2+bx+c.Whichcoefficient,a,b,orcequals0foralloftheexercisesabove?
SET
Topic:FactoringTrinomialsFactorthefollowingquadraticexpressionsintotwobinomials.14. !! + 14! + 45 15. !! + 18! + 45 16. !! + 46! + 45
17. !! + 11! + 24 18. !! + 10! + 24 19. !! + 14! + 24
20. !! + 12! + 36 21. !! + 13! + 36 22. !! + 20! + 36
23. !! − 15! − 100 24. !! + 20! + 100 25. !! + 29! + 100
READY, SET, GO! Name Period Date
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26. Lookbackateach“row”offactoredexpressionsinproblems14to25above.Explainhowitispossiblethatthecoefficient(b)ofthemiddletermcanbedifferentnumbersineachproblemwhenthe“outside”coefficients(a)and(c)arethesame.(Recallthestandardformofaquadraticis!!! + !" + !.)
GO Topic:Takingthesquarerootofperfectsquares.
Onlysomeoftheexpressionsinsidetheradicalsignareperfectsquares.Identifywhichonesareperfectsquaresandtakethesquareroot.Leavetheonesthatarenotperfectsquaresundertheradicalsign.Donotattempttosimplifythem.(Hint:Checkyouranswersbysquaringthem.Youshouldbeabletogetwhatyoustartedwith,ifyouareright.)
27. 17xyz ! 28. 3x − 7 ! 29. 121a!b!
30. x! + 8x + 16 31. x! + 14x + 49 32. x! + 14x − 49
33. x! + 10x + 100 34. x! + 20x + 100 35. x! − 20x + 100
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2.7 The x Factor
A Solidify Understanding Task
NowthatOptima’sQuiltsisacceptingordersforrectangularblocks,theirbusinessingrowingby
leapsandbounds.Manycustomerwantrectangularblocksthatarebiggerthanthestandard
squareblockononeside.Sometimestheywantonesideoftheblocktobethestandardlength,!,withtheothersideoftheblock2inchesbigger.
1. Drawandlabelthisblock.Writetwodifferentexpressionsfortheareaoftheblock.
Sometimestheywantblockswithonesidethatisthestandardlength,x,andonesidethatis2
incheslessthanthestandardsize.
2. Drawandlabelthisblock.Writetwodifferentexpressionsfortheareaoftheblock.Useyour
diagramandverifyalgebraicallythatthetwoexpressionsareequivalent.
Therearemanyothersizeblocksrequested,withthesidelengthsallbasedonthestandardlength,
!.Drawandlabeleachofthefollowingblocks.Useyourdiagramstowritetwoequivalentexpressionsforthearea.Verifyalgebraicallythattheexpressionsareequal.
3. Onesideis1”lessthanthestandardsizeandtheothersideis2”morethanthestandardsize.
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4. Onesideis2”lessthanthestandardsizeandtheothersideis3”morethanthestandardsize.
5. Onesideis2”morethanthestandardsizeandtheothersideis3”lessthanthestandardsize.
6. Onesideis3”morethanthestandardsizeandtheothersideis4”lessthanthestandardsize.
7. Onesideis4”morethanthestandardsizeandtheothersideis3”lessthanthestandardsize.
8. Anexpressionthathas3termsintheform:!!! + !" + !iscalledatrinomial.Lookbackatthetrinomialsyouwroteinquestions3-7.Howcanyoutellifthemiddleterm(!")isgoingtobepositiveornegative?
9. Onecustomerhadanunusualrequest.Shewantedablockthatisextended2inchesononeside
anddecreasedby2inchesontheother.Oneoftheemployeesthinksthatthisrectanglewillhave
thesameareaastheoriginalsquaresinceonesidewasdecreasedbythesameamountastheother
sidewasincreased.Whatdoyouthink?Useadiagramtofindtwoexpressionsfortheareaofthis
block.
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10. Theresultoftheunusualrequestmadetheemployeecurious.Isthereapatternorawayto
predictthetwoexpressionsforareawhenonesideisincreasedandtheothersideisdecreasedby
thesamenumber?Trymodelingthesetwoproblems,lookatyouranswerto#8,andseeifyoucan
findapatternintheresult.
a. (! + 1)(! − 1)
b. (! + 3)(! − 3)
11. Whatpatterndidyounotice?Whatistheresultof(! + !)(! − !)?
12. Somecustomerswantbothsidesoftheblockreduced.Drawthediagramforthefollowing
blocksandfindatrinomialexpressionfortheareaofeachblock.Usealgebratoverifythetrinomial
expressionthatyoufoundfromthediagram.
a. (! − 2)(! − 3)
b. (! − 1)(! − 4)
13. Lookbackoveralltheequivalentexpressionsthatyouhavewrittensofar,andexplainhowto
tellifthethirdterminthetrinomialexpression!!! + !" + !willbepositiveornegative.
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14. Optima’squiltshophasreceivedanumberofordersthataregivenasrectangularareasusinga
trinomialexpression.Findtheequivalentexpressionthatshowsthelengthsofthetwosidesofthe
rectangles.
a. !! + 9! + 18
b. !! + 3x − 18
c. !! − 3! − 18
d. !! − 9! + 18
e. !! − 5! + 4
f. !! − 3! − 4
g. !! + 2! − 15
15. Writeanexplanationofhowtofactoratrinomialintheform:!! + !" + !.
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Topic:Exploringthedensityofthenumberline.Findthreenumbersthatarebetweenthetwogivennumbers.
1. 5 !! !"# 6 !! 2. −2 !! !"# − 1!! 3. !! !"# !! 4. 3 !"# 5
5. 4 !"# 23 6. −9 !! !"# − 8.5 7. !! !"# !
!8. 13 !"# 14
SET
Topic:FactoringQuadraticsTheareaofarectangleisgivenintheformofatrinomialexpression.Findtheequivalentexpressionthatshowsthelengthsofthetwosidesoftherectangle.
9. x! + 9x + 8 10. x! − 6x + 8 11. x! − 2x − 8 12. x! + 7x − 8
13. x! − 11x + 24 14. x! − 14x + 24 15. x! − 25x + 24 16. x! − 10x + 24
17. x! − 2x − 24 18. x! − 5x − 24 19. x! + 5x − 24 20. x! − 10x + 25
21. x! − 25 22. x! − 2x − 15 23. x! + 10x − 75 24. x! − 20x + 51
25. x! + 14x − 32 26. x! − 1 27. x! − 2x + 1 28. x! + 12x − 45
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GO Topic:GraphingParabolas
Grapheachparabola.Includethevertexandatleast3accuratepointsoneachsideoftheaxisofsymmetry.Thendescribethetransformationinwords.
29. ! ! = !!
Description:
30. ! ! = !! − 3
Description:
31. ℎ ! = ! − 2 !
Description:
32. ! ! = − ! + 1 ! + 4
Description:
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2.8 The Wow Factor
A Solidify Understanding Task
Optima’sQuiltssometimesgetsordersforblocksthataremultiplesofagivenblock.Forinstance,
Optimagotanorderforablockthatwasexactlytwiceasbigastherectangularblockthathasaside
thatis1”longerthanthebasicsize,!, andonesidethatis3”longerthanthebasicsize.1. Drawandlabelthisblock.Writetwoequivalentexpressionsfortheareaoftheblock.
2. Ohdear!Thisorderwasscrambledandthepiecesareshownhere.Putthepiecestogetherto
makearectangularblockandwritetwoequivalentexpressionsfortheareaoftheblock.
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3. Whatdoyounoticewhenyoucomparethetwoequivalentexpressionsinproblems#1and#2?
4. Optimahasalotofneworders.Usediagramstohelpyoufindequivalentexpressionsforeachof
thefollowing:
a. 5!! + 10!
b. 3!! + 21! + 36
c. 2!! + 2! − 4
d. 2!! − 10! + 12
e. 3!! − 27
Becausesheisagreatbusinessmanager,Optimaoffershercustomerslotsofoptions.Oneoptionis
tohaverectanglesthathavesidelengthsthataremorethanone!.Forinstance,Optimamadethiscoolblock:
5. Writetwoequivalentexpressionsforthisblock.Usethedistributivepropertytoverifythat
youransweriscorrect.
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6. Herewehavesomepartialorders.Wehaveoneoftheexpressionsfortheareaoftheblock
andweknowthelengthofoneofthesides.Useadiagramtofindthelengthoftheothersideand
writeasecondexpressionfortheareaoftheblock.Verifyyourtwoexpressionsfortheareaofthe
blockareequivalentusingalgebra.
a. Area: 2!! + 7! + 3 Side: (! + 3)
Equivalentexpressionforarea:
b. Area: 5!! + 8! + 3 Side: (! + 1)
Equivalentexpressionforarea:
c. Area: 2!! + 7! + 3 Side: (2! + 1)
Equivalentexpressionforarea:
7. Whataresomepatternsyouseeinthetwoequivalentexpressionsforareathatmighthelp
youtofactor?
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8. Businessisbooming!Moreandmoreordersarecomingin!Usediagramsornumber
patterns(orboth)towriteeachofthefollowingordersinfactoredform:
a. 3!! + 16! + 5
b. 2!! − 13! + 15
c. 3!! + ! − 10
d. 2!! + 9! − 5
9. InThe !Factor,youwrotesomerulesfordecidingaboutthesignsinsidethefactors.Dothoserulesstillworkinfactoringthesetypesofexpressions?Explainyouranswer.
10. ExplainhowOptimacantelliftheblockisamultipleofanotherblockorifonesidehasa
multipleof!inthesidelength.
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11. There’sonemoretwistonthekindofblocksthatOptimamakes.Thesearethetrickiestof
allbecausetheyhavemorethanone!inthelengthofbothsidesoftherectangle!Here’sanexample:
Writetwoequivalentexpressionsforthisblock.Usethedistributivepropertytoverifythatyour
answeriscorrect.
12. Allright,let’strythetrickyones.Theymaytakealittlemessingaroundtogetthefactored
expressiontomatchthegivenexpression.Makesureyoucheckyouranswerstobesurethat
you’vegotthemright.Factoreachofthefollowing:
a. 6!! + 7! + 2
b. 10!! + 17! + 3
c. 4!! − 8! + 3
d. 4!! + 4! − 3
e. 9!! − 9! − 10
12. Writea“recipe”forhowtofactortrinomialsintheform,!"! + !" + !.
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Topic:ComparingarithmeticandgeometricsequencesThefirstandfifthtermsofeachsequencearegiven.Fillinthemissingnumbers.
1.
Arithmetic 3 1875
Geometric 3 1875
2.
Arithmetic -1458 -18
Geometric -1458 -18
3.
Arithmetic 1024 4
Geometric 1024 4
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SET
Topic:Writinganareamodelasaquadraticexpression
Writetwoequivalentexpressionsfortheareaofeachblock.Letxbethesidelengthofeachofthelargesquares.
4. 5.
6.
7. Problems4,5,and6allcontainthesamenumberofsquaresmeasuring!!and1!.A.Whatisdifferentaboutthem?
B.Howdoesthisdifferenceaffectthequadraticexpressionthatrepresentsthem?
C.Describehowthearrangementofthesquaresandrectanglesaffectsthefactoredform.
Topic:Factoringquadraticexpressionswhen! > !Factorthefollowingquadraticexpressions.
8. 4x! + 7x − 2 9. 2x! − 7x − 15 10. 6x! + 7x − 3 11. 4x! − x − 3
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.8 2.8
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12. 4x! + 19x − 5 13. 3x! − 10x + 8 14. 6x! + x − 2 15. 3x! − 14x − 24
16. 2x! + 9x + 10 17. 5x! + 31x + 6 18. 5x! + 7x − 6 19. 4x! + 8x − 5
20. 3x! − 75 21. 3x! + 7x + 2 22. 4x! + 8x − 5 23. 2x! + x − 6
GO
Topic:Findingtheequationofthelineofsymmetryofaparabola
Giventhex-interceptsofaparabola,writetheequationofthelineofsymmetry.
24. x-intercepts:(-3,0)and(3,0) 25. x-intercepts:(-4,0)and(16,0)
26. x-intercepts:(-2,0)and(5,0) 27. x-intercepts:(-14,0)and(-3,0)
28. x-intercepts:(17,0)and(33,0) 29. x-intercepts: −0.75, 0 and 2.25, 0
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Whenever we’re thinking about algebra and working
with variables, it is useful to consider how it relates to
the number system and operations on numbers. Right
now, polynomials are on our minds, so let’s see if we
can make some useful comparisons between whole
numbers and polynomials.
Let’s start by looking at the structure of numbers and polynomials. Consider the number 132. The
way we write numbers is really a shortcut because:
132 = 100 + 30 + 2
1. Compare 132 to the polynomial 𝑥2 + 3𝑥 + 2. How are they alike? How are they different?
2. Write a polynomial that is analogous to the number 2,675.
When two numbers are to be added together, many people use a procedure like this:
132
+ 451
583
3. Write an analogous addition problem for polynomials and find the sum of the two polynomials.
4. How does adding polynomials compare to adding whole numbers?
CC
BY
Anders
San
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5. Use the polynomials below to find the specified sums in a-f.
𝑓(𝑥) = 𝑥3 + 3𝑥2 − 2𝑥 + 10 𝑔(𝑥) = 2𝑥 − 1 ℎ(𝑥) = 2𝑥2 + 5𝑥 − 12 𝑘(𝑥) = −𝑥2 − 3𝑥 + 4
a) ℎ(𝑥) + 𝑘(𝑥) b) 𝑔(𝑥) + 𝑓(𝑥) c) 𝑓(𝑥) + 𝑘(𝑥)
______________________________ _______________________________ ______________________________
d) 𝑙(𝑥) + 𝑚(𝑥) e) 𝑚(𝑥) + 𝑛(𝑥) f) 𝑙(𝑥) + 𝑝(𝑥)
𝑝(𝑥) = x2 - 7xy + 4𝑛(𝑥) = 4xy + 2x2𝑙(𝑥) = 4x2 - 3y2 + 5xy 𝑚(𝑥) = 8xy + 3y2
______________________________ ______________________________ ______________________________
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6. What patterns do you see when polynomials are added?
Subtraction of whole numbers works similarly to addition. Some people line up subtraction
vertically and subtract the bottom number from the top, like this:
368
-157
211
7. Write the analogous polynomials and subtract them.
8. Is your answer to #7 analogous to the whole number answer? If not, why not?
9. Subtracting polynomials can easily lead to errors if you don’t carefully keep track of your
positive and negative signs. One way that people avoid this problem is to simply change all the
signs of the polynomial being subtracted and then add the two polynomials together. There are two
common ways of writing this:
(𝑥3 + 𝑥2 − 3𝑥 − 5) − (2𝑥3 − 𝑥2 + 6𝑥 + 8)
Step 1: = (𝑥3 + 𝑥2 − 3𝑥 − 5) + (−2𝑥3 + 𝑥2 − 6𝑥 − 8)
Step 2: = (−𝑥3 + 2𝑥2 − 9𝑥 − 13)
Or, you can line up the polynomials vertically so that Step 1 looks like this:
Step 1: 𝑥3 + 𝑥2 − 3𝑥 − 5
+(−2𝑥3 + 𝑥2 − 6𝑥 − 8)
Step 2: −𝑥3 + 2𝑥2 − 9𝑥 − 13
The question for you is: Is it correct to change all the signs and add when subtracting? What
mathematical property or relationship can justify this action?
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10. Use the given polynomials to find the specified differences in a-d.
𝑓(𝑥) = 𝑥3 + 2𝑥2 − 7𝑥 − 8 𝑔(𝑥) = −4𝑥 − 7 ℎ(𝑥) = 4𝑥2 − 𝑥 − 15 𝑘(𝑥) = −𝑥2 + 7𝑥 + 4
a) ℎ(𝑥) − 𝑘(𝑥) b) 𝑓(𝑥) − ℎ(𝑥) c) 𝑓(𝑥) − 𝑔(𝑥)
______________________________ _______________________________ ______________________________
d) 𝑘(𝑥) − 𝑓(𝑥) e) 𝑙(𝑥) − 𝑚(𝑥)
______________________________
11. List three important things to remember when subtracting polynomials.
𝑙(𝑥) = 5x2 - 7y2 + 4xy 𝑚(𝑥) = -10x2 + 9y2 -12xy + 4
______________________________
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POLYNOMIAL FUNCTIONS –
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READY Topic:Usingthedistributiveproperty
Multiply.
1. 2𝑥(5𝑥% + 7) 2. 9𝑥(−𝑥% − 3) 3. 5𝑥%(𝑥, + 6𝑥.)
4. −𝑥(𝑥% − 𝑥 + 1) 5. −3𝑥.(−2𝑥% + 𝑥 − 1) 6. −1(𝑥% − 4𝑥 + 8)
SET Topic:Addingandsubtractingpolynomials
Add.Writeyouranswersindescendingorderoftheexponents.(Standardform)
7. (3𝑥, + 5𝑥% − 1) + (2𝑥. + 𝑥) 8. (4𝑥% + 7𝑥 − 4) + (𝑥% − 7𝑥 + 14)
9. (2𝑥. + 6𝑥% − 5𝑥) + (𝑥4 + 3𝑥% + 8𝑥 + 4) 10. (−6𝑥4 − 2𝑥 + 13) + (4𝑥4 + 3𝑥% + 𝑥 − 9)
Subtract.Writeyouranswersindescendingorderoftheexponents.(Standardform)
11. (5𝑥% + 7𝑥 + 2) − (3𝑥% + 6𝑥 + 2) 12. (10𝑥, + 2𝑥% + 1) − (3𝑥, + 3𝑥 + 11)
13. (7𝑥. − 3𝑥 + 7) − (4𝑥% − 3𝑥 − 11) 14. (𝑥, − 1) − (𝑥, + 1)
READY, SET, GO! Name PeriodDate
GO Topic:Usingexponentrulestocombineexpressions
Simplify.
19. 20. 21.x-5 ∙ x4 ∙ x-2 x3 ∙ x-7 ∙ x-2 x4 ∙ x4 ∙ x-1
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Multiplying polynomials can require a bit of skill in the
algebra department, but since polynomials are
structured like numbers, multiplication works very
similarly. When you learned to multiply numbers, you may have learned to use an area
model. To multiply 12 × 15 the area model and the related procedure probably looked like this:
You may have used this same idea with quadratic expressions. Area models help us think about
multiplying, factoring, and completing the square to find equivalent expressions. We modeled
(𝑥 + 2)(𝑥 + 5) = 𝑥2 + 7𝑥 + 10 as the area of a rectangle with sides of length 𝑥 + 2 and 𝑥 + 5. The
various parts of the rectangle are shown in the diagram below:
CC
BY
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1
𝑥 + 5
𝑥2 𝑥 𝑥 𝑥 𝑥 𝑥
1
𝑥 1
𝑥
1 1 1 1
1 1 1 1 𝑥+
2
12
15
10
50
+ 20
100
180
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Some people like to shortcut the area a model a little bit to just have sections of area that
correspond to the lengths of the sides. In this case, they might draw the following.
= 𝑥2 + 7𝑥 + 10
1. What is the property that all of these models are based upon?
2. Now that you’ve been reminded of the happy past, you are ready to use the strategy of your
choice to find equivalent expressions for each of the following:
a) (𝑥 + 3)(𝑥 + 4) b) (𝑥 + 7)(𝑥 − 2)
Maybe now you remember some of the different forms for quadratic expressions—factored form
and standard form. These forms exist for all polynomials, although as the powers get higher, the
algebra may get a little trickier. In standard form polynomials are written so that the terms are in
order with the highest-powered term first, and then the lower-powered terms. Some examples:
Quadratic: 𝑥2 − 3𝑥 + 8 or 𝑥2 − 9
Cubic: 2𝑥3 + 𝑥2 − 7𝑥 − 10 or 𝑥3 − 2𝑥2 + 15
Quartic: 𝑥4 + 𝑥3 + 3𝑥2 − 5𝑥 + 4
Hopefully, you also remember that you need to be sure that each term in the first factor is
multiplied by each term in the second factor and the like terms are combined to get to standard
form. You can use area models or boxes to help you organize, or you can just check every time to
be sure that you’ve got all the combinations. It can get more challenging with higher-powered
polynomials, but the principal is the same because it is based upon the mighty Distributive
Property.
𝑥 +5
𝑥 𝑥2 5𝑥
+2 2𝑥 10
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3. Tia’s favorite strategy for multiplying polynomials is to make a box that fits the two factors. She
sets it up like this: (𝑥 + 2)(𝑥2 − 3𝑥 + 5)
Try using Tia’s box method to multiply these two factors together and then combining like terms to
get a polynomial in standard form.
4. Try checking your answer by graphing the original factored polynomial, (𝑥 + 2)(𝑥2 − 3𝑥 + 5)
and then graphing the polynomial that is your answer. If the graphs are the same, you are right
because the two expressions are equivalent! If they are not the same, go back and check your work
to make the corrections.
5. Tehani’s favorite strategy is to connect the terms he needs to multiply in order like this:
(𝑥 − 3)(𝑥2 + 4𝑥 − 2)
Try multiplying using Tehani’s strategy and then check your work by graphing. Make any
corrections you need and figure out why they are needed so that you won’t make the same mistake
twice!
6. Use the strategy of your choice to multiply each of the following expressions. Check your work
by graphing and make any needed corrections.
a) (𝑥 + 5)(𝑥2 − 𝑥 − 3) b) (𝑥 − 2)(2𝑥2 + 6𝑥 + 1)
c) (𝑥 + 2)(𝑥 − 2)(𝑥 + 3)
𝑥2 −3𝑥 +5
𝑥
+2
d) (x + xy - 2y)(𝑥 − y)
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When graphing, it is often useful to have a perfect square quadratic or a perfect cube. Sometimes it
is also useful to have these functions written in standard form. Let’s try re-writing some related
expressions to see if we can see some useful patterns.
7H. Multiply and simplify both of the following expressions using the strategy of your choice:
a) 𝑓(𝑥) = (𝑥 + 1)2 b) 𝑓 (𝑥) = (𝑥 + 1)3
Check your work by graphing and make any corrections needed.
8H.Some enterprising young mathematician noticed a connection between the coefficients of the
in the polynomial and the number pattern known as Pascal’s Triangle. Put your answers from
problem 5 into the table. Compare your answers to the numbers in Pascal’s Triangle below and
describe the relationship you see.
(𝑥 + 1)0 1 1
(𝑥 + 1)1 𝑥 + 1 1 1
(𝑥 + 1)2 1 2 1
(𝑥 + 1)3 1 3 3 1
(𝑥 + 1)4
9H. It could save some time on multiplying the higher power polynomials if we could use Pascal’sTriangle to get the coefficients. First, we would need to be able to construct our own Pascal’s Triangle and add rows when we need to. Look at Pascal’s Triangle and see if you can figure out how to get the next row using the terms from the previous row. Use your method to find the terms in the next row of the table above.
10H. Now you can check your Pascal’s Triangle by multiplying out (𝑥 + 1)4 and comparing thecoefficients. Hint: You might want to make your job easier by using your answers from #5 in some way. Put your answer in the table above.
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11H. Make sure that the answer you get from multiplying (𝑥 + 1)4 and the numbers in Pascal’sTriangle match, so that you’re sure you’ve got both answers right. Then describe how to get the next row in Pascal’s Triangle using the terms in the previous row.
12H. Complete the next row of Pascal’s Triangle and use it to find the standard form of (𝑥 + 1)5.Write your answers in the table on #6.
13H. Pascal’s Triangle wouldn’t be very handy if it only worked to expand powers of 𝑥 + 1. Theremust be a way to use it for other expressions. The table below shows Pascal’s Triangle and the expansion of 𝑥 + 𝑎.
(𝑥 + 𝑎)0 1 1
(𝑥 + 𝑎)1 𝑥 + 𝑎 1 1
(𝑥 + 𝑎)2 𝑥2 + 2𝑎𝑥 + 𝑎2 1 2 1
(𝑥 + 𝑎)3 𝑥3 + 3𝑎𝑥2 + 3𝑎2𝑥 + 𝑎3 1 3 3 1
(𝑥 + 𝑎)4 𝑥4 + 4𝑎𝑥3 + 6𝑎2𝑥2 + 3𝑎3𝑥 + 𝑎4 1 4 6 4 1
What do you notice about what happens to the 𝑎 in each of the terms in a row?
14H. Use the Pascal’s Triangle method to find standard form for (𝑥 + 2)3. Check your answer bymultiplying.
15H. Use any method to write each of the following in standard form:
a) (𝑥 + 3)3 b) (𝑥 − 2)3 c) (𝑥 + 5)4
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READY Topic: Solve Quadratic Equations
1. 2. 3.
4. 5.
SET Topic: Multiplying polynomials
Multiply. Write your answers in standard form.
6. 𝑎 + 𝑏 𝑎 + 𝑏 7. 𝑥 − 3 𝑥! + 3𝑥 + 9
8. 𝑥 − 5 𝑥! + 5𝑥 + 25 9. 𝑥 + 1 𝑥! − 𝑥 + 1
10. 𝑥 + 7 𝑥! − 7𝑥 + 49 11. 𝑎 − 𝑏 𝑎! + 𝑎𝑏 + 𝑏!
READY, SET, GO! Name Period Date
Solve the following quadratic equations.
x2 = 121 x2 + 8x = -15(4x + 5)(x + 1) = 0
3x2 - 16r - 7 = 5 8x2 - 4x - 18 = 0
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( 𝑥 + 𝑎)! 1 1
(𝑥 + 𝑎)! 𝑥 + 𝑎 1 1
(𝑥 + 𝑎)! 𝑥! + 2𝑎𝑥 + 𝑎! 1 2 1
(𝑥 + 𝑎)! 𝑥! + 3𝑎𝑥! + 3𝑎!𝑥 + 𝑎! 1 3 3 1
(𝑥 + 𝑎)! 𝑥! + 4𝑎𝑥! + 6𝑎!𝑥! + 3𝑎!𝑥 + 𝑎! 1 4 6 4 1
Use the table above to write each of the following in standard form.
12. 𝑥 + 1 ! 13. 𝑥 − 5 ! 14. 𝑥 − 1 !
15, 𝑥 + 4 ! 16. 𝑥 + 2 ! 17. 3𝑥 + 1 !
GO Topic: Examining transformations on different types of functions
Graph the following functions.
18. 𝑔(𝑥) = 𝑥 + 2 19. ℎ(𝑥) = 𝑥! + 2 20. 𝑓(𝑥) = 2! + 2
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21. 𝑔(𝑥) = 3(𝑥 − 2) 22. ℎ(𝑥) = 3(𝑥 − 2)!
24. 𝑔 𝑥 = !! 𝑥 − 1 − 2 25. ℎ 𝑥 = !
! 𝑥 − 1 ! − 2
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.9
Mathematics Vision Project
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2.9 Lining Up Quadratics
A Practice Understanding Task
Grapheachfunctionandfindthevertex,they-interceptandthe!-intercepts.Besuretoproperlywritetheinterceptsaspoints.
1. ! = (! − 1)(! + 3)
LineofSymmetry_________
Vertex__________
!-intercepts_________ _________
y-intercept________
2. ! ! = 2(! − 2)(! − 6)
LineofSymmetry_________
Vertex__________
!-intercepts_________ _________
y-intercept________C
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.9
Mathematics Vision Project
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3. ! ! = −!(! + 4)
LineofSymmetry___________
Vertex__________
!-intercepts_________ _________
y-intercept________
4. Basedontheseexamples,howcanyouuseaquadraticfunctioninfactoredformto:a. Findthelineofsymmetryoftheparabola?
b. Findthevertexoftheparabola?
c. Findthe!-interceptsoftheparabola?
d. Findthey-interceptoftheparabola?
e. Findtheverticalstretch?
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.9
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5. Chooseanytwolinearfunctionsandwritethemintheform:!(!) = ! ! − ! , wheremistheslopeoftheline.Graphthetwofunctions.
Linearfunction1:
Linearfunction2:
6. Onthesamegraphas#5,graphthefunctionP(x)thatistheproductofthetwolinearfunctionsthatyouhavechosen.Whatshapeiscreated?
7. Describetherelationshipbetweenx-interceptsofthelinearfunctionsandthex-interceptsofthefunctionP(x).Whydoesthisrelationshipexist?
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STRUCTURES OF EXPRESSIONS – 2.9
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8. Describetherelationshipbetweeny-interceptsofthelinearfunctionsandthey-interceptsofthefunctionP(x).Whydoesthisrelationshipexist?
9. Giventheparabolatotheright,sketchtwolinesthatcouldrepresentitslinearfactors.
10. Writeanequationforeachofthesetwolines.
11. Howdidyouusethexandyinterceptsoftheparabolatoselectthetwolines?
12. Arethesetheonlytwolinesthatcouldrepresentthelinearfactorsoftheparabola?Ifso,explainwhy.Ifnot,describetheotherpossiblelines.
13. Useyourtwolinestowritetheequationoftheparabola.Isthistheonlypossibleequationoftheparabola?
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STRUCTURES OF EXPRESSIONS – 2.9 2.9
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READY
Topic:MultiplyingBinomialsUsingaTwo-WayTable
Multiplythefollowingbinomialsusingthegiventwo-waytabletoassistyou.Example:(2x+3)(5x–7)
=10x2+x–21
1. (3x–4)(7x–5) 2. (9x+2)(x+6) 3. (4x–3)(3x+11)
4. (7x+3)(7x–3) 5. (3x–10)(3x+10) 6. (11x+5)(11x–5)
7. (4x+5)2 8. (x+9)2 9. (10x–7)2
10. The“like-term”boxesin#’s7,8,and9revealaspecialpattern.Describetherelationshipbetween
themiddlecoefficient(b)andthecoefficients(a)and(c).
READY, SET, GO! Name Period Date
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SET
Topic:FactoredFormofaQuadraticFunction
Giventhefactoredformofaquadraticfunction,identifythevertex,intercepts,andverticalstretchoftheparabola.
11. ! = 4 ! − 2 ! + 6 12. ! = −3 ! + 2 ! − 6 13. ! = ! + 5 ! + 7a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________
b. x-inter(s)____________ b. x-inter(s)____________ b. x-inter(s)___________
c. y-inter________________ c. y-inter:_______________ c. y-inter_______________
d. Stretch_______________ d. Stretch_______________ d. Stretch_______________
14. ! = !! ! − 7 ! − 7 15. ! = − !
! ! − 8 ! + 4 16. ! = !! ! − 25 ! − 9
a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________
b. x-inter(s)____________ b. x-inter(s)____________ b. x-inter(s)___________
c. y-inter________________ c. y-inter:_______________ c. y-inter_______________
d. Stretch_______________ d. Stretch_______________ d. Stretch_______________
17. ! = !! ! − 3 ! + 3 18. ! = − ! − 5 ! + 5 19. ! = !
! ! + 10 ! + 10a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________
b. x-inter(s)____________ b. x-inter(s)____________ b. x-inter(s)___________
c. y-inter________________ c. y-inter:_______________ c. y-inter_______________
d. Stretch_______________ d. Stretch_______________ d. Stretch_______________
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STRUCTURES OF EXPRESSIONS – 2.9 2.9
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GO Topic:VertexFormofaQuadraticEquation
Giventhevertexformofaquadraticfunction,identifythevertex,intercepts,andverticalstretchoftheparabola.
20. ! = ! + 2 ! − 4 21. ! = −3 ! + 6 ! + 3 22. ! = 2 ! − 1 ! − 8a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________
b. x-inter(s)____________ b. x-inter(s)____________ b. x-inter(s)___________
c. y-inter________________ c. y-inter:_______________ c. y-inter_______________
d. Stretch_______________ d. Stretch_______________ d. Stretch_______________
23. ! = 4 ! + 2 ! − 64 24. ! = −3 ! − 2 ! + 48 25. ! = ! + 6 ! − 1a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________
b. x-inter(s)____________ b. x-inter(s)____________ b. x-inter(s)___________
c. y-inter________________ c. y-inter:_______________ c. y-inter_______________
d. Stretch_______________ d. Stretch_______________ d. Stretch_______________
26. Didyounoticethattheparabolasinproblems11,12,&13arethesameastheonesinproblems23,
24,&25respectively?Ifyoudidn’t,gobackandcomparetheanswersinproblems11,12,&13and
problems23,24,&25.
Provethat a. 4 ! − 2 ! + 6 = 4 ! + 2 ! − 64
b. −3 ! + 2 ! − 6 = −3 ! − 2 ! + 48
c. ! + 5 ! + 7 = ! + 6 ! − 1
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.10
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2.10 I’ve Got a Fill-in
A Practice Understanding Task
Foreachproblembelow,youaregivenapieceofinformationthattellsyoualot.Usewhatyou
knowaboutthatinformationtofillintherest.
1. Yougetthis: Fillinthis:
! = !! − ! − 12Factoredformoftheequation:
Graphoftheequation:
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Poi
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.10
Mathematics Vision Project
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2. Yougetthis: Fillinthis:
! = !! − 6! + 3Vertexformoftheequation:
Graphoftheequation:
3. Yougetthis: Fillinthis:Vertexformoftheequation:
Standardformoftheequation:
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.10
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
4. Yougetthis: Fillinthis:Factoredformoftheequation:
Standardformoftheequation:
5. Yougetthis: Fillinthis:
! = −!! − 6! + 16Eitherformoftheequationotherthanstandardform.
Vertexoftheparabola
x-interceptsandy-intercept
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.10
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6. Yougetthis: Fillinthis:
! = 2!! + 12! + 13Eitherformoftheequationotherthanstandardform.
Vertexoftheparabola
x-interceptsandy-intercept
7. Yougetthis: Fillinthis:
! = −2!! + 14! + 60Eitherformoftheequationotherthanstandardform.
Vertexoftheparabola
x-interceptsandy-intercept
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.10
2.10
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READY
Agolf-propracticeshisswingbydrivinggolfballsofftheedgeofacliffintoalake.Theheightoftheballabovethelake(measuredinmeters)asafunctionoftime(measuredinsecondsandrepresentedbythevariablet)fromthe
instantofimpactwiththegolfclubis
58.8 + 19.6t − 4.9t!.
Theexpressionsbelowareequivalent:
−4.9t! + 19.6t + 58.8 standardform
−4.9 t − 6 t + 2 factoredform
−4.9 t − 2 ! + 78.4 vertexform
1. Whichexpressionisthemostusefulforfindinghowmanysecondsittakesfortheballtohitthewater? Why?
2. Whichexpressionisthemostusefulforfindingthemaximumheightoftheball? Justifyyouranswer.
3. Ifyouwantedtoknowtheheightoftheballatexactly3.5seconds,whichexpressionwouldhelpthemosttofindtheanswer?Why?
4. Ifyouwantedtoknowtheheightofthecliffabovethelake,whichexpressionwouldyouuse?Why?
SET Topic:FindingmultiplerepresentationsofaquadraticOneformofaquadraticfunctionisgiven.Fill-inthemissingforms.5a.StandardForm b. VertexForm c. FactoredForm
! = ! + 5 ! − 3
d. Table(Includethevertexandatleast2pointsoneachsideofthevertex.)
x y
Showthefirstdifferencesandtheseconddifferences.
e. Graph
READY, SET, GO! Name Period Date
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.10
2.10
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6a.StandardForm b. VertexForm! = −3 ! − 1 ! + 3
c. FactoredForm
d. Table(Includethevertexandatleast2pointsoneachsideofthevertex.)
x y
Showthefirstdifferencesandtheseconddifferences.
e. Graph
7a.StandardForm! = −!! + 10! − 25
b. VertexForm c. FactoredForm
d. Table(Includethevertexandatleast2pointsoneachsideofthevertex.)
x y
Showthefirstdifferencesandtheseconddifferences.
e. Graph
8a.StandardForm b. VertexForm c. FactoredForm
d. Table(Includethevertexandatleast2pointsoneachsideofthevertex.)
x y
Showthefirstdifferencesandtheseconddifferences.
e. Graph
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SECONDARY MATH II // MODULE 2
STRUCTURES OF EXPRESSIONS – 2.10
2.10
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Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9a.StandardForm b. VertexForm c. FactoredFormSkipthisfornow
d. Tablex y0123456
122-4-6-4212
Showthefirstdifferencesandtheseconddifferences.
e. Graph
GO Topic:FactoringQuadratics
Verifyeachfactorizationbymultiplying.
10. x! + 12x − 64 = x + 16 x − 4 11. x! − 64 = x + 8 x − 8
12. x! + 20x + 64 = x + 16 x + 4 13. x! − 16x + 64 = x − 8 x − 8
Factorthefollowingquadraticexpressions,ifpossible.(Somewillnotfactor.)
14. x! − 5x + 6 15. x! − 7x + 6 16. x! − 5x − 36
17. m! + 16m + 63 18. s! − 3s − 1 19. x! + 7x + 2
20. x! + 14x + 49 21. x! − 9 22. c! + 11c + 3
23. Whichquadraticexpressionabovecouldrepresenttheareaofasquare?Explain.
24. WouldanyoftheexpressionsaboveNOTbetheside-lengthsforarectangle?Explain.
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