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HighSchool:Math2TakeHomePacket
Student’sName:______________Teacher’sName:_____________
School:_____________
This Photo by Unknown Author is licensed under CC BY-SA-NC
TableofContentsforMath2
Day1………………………….….TransformationsReview
Day2………………TheGraphofaQuadraticFunction
Day3…………………………………….ReviewofFactoring
Day4…………………SolvetheQuadraticbyFactoring
Day5………….SolvebyusingtheQuadraticFormula
Day6……...……………SolvebyCompletingtheSquare
Day7……......…CompletetheSquaretowriteinVertexForm
Day8……..…ApplicationswithQuadratics(WordProblems)
Day9……..TransformationsofQuadraticFunctions
Day10…………....SolvingSystemsofEquations(LinearandQuadratic)
StudentreferenceVideos:https://virtualnerd.com/common-core/hsf-geometry/HSG-CO-congruence/A/5/ORscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos.
Day1:GeometricTransformationsName:_______________________________
____ 1.TheverticesofatriangleareP(–7,5),Q(–4,–8),andR(–7,4).Nametheverticesoftheimagereflected
acrossthex-axis.A. C. B. D.
_____2.TheverticesofatriangleareP(8,–7),Q(–7,–5),andR(7,6).Nametheverticesoftheimagereflectedacross
they-axis.A. C. B. D.
_____3. Describeinwordsthetranslationrepresentedbytherule
(x,y)®(x-2,y+1)A. 2unitstotherightand1unitdownB. 1unittotherightand2unitsdownC. 2unitstotheleftand1unitdownD. 2unitstotheleftand1unitup
_____4. If isreflectedinthey-axisfindthecoordinatesofDA’B’C’.
____5.WhichtransformationcorrectlymapsfigureAtofigureBshownbelow?
_____6.G(-9,15)isrotated90degreescounterclockwise.WhatarethecoordinatesofG’? A.(15,-9) C.(-9,-15) B.(15,9) D.(-15,-9)
A. rotationB. dilationC. reflectionD. translation
A. A’(1, –2), B’(1, –4), C’(4, –4) C. A’(2, –1), B’(2, –4), C’(4, –4) B. A’(1, 2), B’(1, 4), C’(4, 4) D. A’(–1, 2), B’(–1, 4), C’(–4, 4)
y
x
A
B
_____7.Whichofthefollowingrulesshowsamappingfora180°clockwiserotation?
A. (x,y)(-x,y) C.(x,y)(y,-x)B. (x,y)(-x,-y) D.(x,y)(-y,x)
_____8.Writearuletodescribethefollowingtransformation?
_____9.Namethelineofreflectionforthefollowingreflection.
_____10.Whichgraphshowsarotation?
A. (x, y) ® (x - 7, y +1) B. (x, y) ® (x + 7, y - 1) C. (x, y) ® (-x, y) D. (x, y) ® (-y, x)
A. x- axis B. y -axis C. x = 1 D. y = 1
2 4–2–4 x
2
4
–2
–4
y
2 4–2–4 x
2
4
–2
–4
y
2 4–2–4 x
2
4
–2
–4
y
2 4–2–4 x
2
4
–2
–4
y
A.
B.
C.
D.
11. DescribethetransformationthatoccurredtomovepointAontopointA’.Whatinformationdidyouusetocometothisconclusion?
12. Doesthefollowingpictureshowanexampleofareflection,atranslation,orneither?Explainhow
youcametothisconclusion.13. a)Describethetransformationthatmapsfootballhelmet1ontofootballhelmet2.
b)Describethetransformationthatmapsfootballhelmet2ontofootballhelmet3.c)Describeatransformationthatwouldmapfootballhelmet1directlyontofootballhelmet3.
Studentreferencevideos:https://virtualnerd.com/common-core/hsf-functions/HSF-IF-interpreting-functions/C/7/7a/how-do-you-graph-a-quadratic-functionORscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos.DAY2:THEGRAPHOFAQUADRATICFUNCTION
• TheQuadraticEquationiswrittenas:𝑦 = 𝑎𝑥% + 𝑏𝑥 + 𝑐,thisequationhas
adegreeof2.
§ Wherea,bandcareintegercoefficients(wherea 0)• Thegraphofthisequationiscalledaparabola.• ParabolasarefunctionsbecausetheypasstheVerticalLineTest.
AxisofSymmetry:Thislineofsymmetryiscalledtheaxisofsymmetry.Itisalwaysaverticallinethatgoesthroughtheturningpointofthecurve.ToFindtheAxisofSymmetry(AOS):𝒙 = *𝒃
𝟐𝒂
TurningPoint:Isanothertermforthevertexoftheparabola.The“vertex”hasthecoordinatesof.
ToFindTurningPoint(VERTEX):Findtheaxisofsymmetry,thensubstitutethatvalueinplaceofxintheequation..*/
%0, 𝑓 3*/
%0, 45
Rootsoftheequationarethepointswheretheparabolacrossesthex–axis!HowtoGraphParabolas:
1.Findtheaxisofsymmetrybyusingtheformula.2.Substitutethexvaluebackintothe
equationtofindtheturningpointanddescribeitasamaxorminpt. 3.Makeatableofvalues. 4.Graphthepoints.
¹
( )yx,
When “a” is positive, the parabola opens: UP Where the curve reaches a MINIMUM.
When “a” is negative, the parabola opens: DOWN Where the curve reaches a MAXIMUM.
EX1:GRAPH:
EX2:GRAPH: EXPLORINGTHEGRAPHEDQUADRATICEQUATIONStudent reference videos: https://virtualnerd.com/common-core/hsa-algebra/HSA-REI-equations-inequalities-reasoning/B/4/4b/two-solutions-by-graphingORscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos. Thex–intercepts(whenx=0)oftheparabola arecalledtheSOLUTIONSorROOTSoftheequation( )Howmanyrootsarepossibletoobtainfromaquadraticequation?Drawapicturetoillustrateeachsituation
4xxy 2 -=
32xxy 2 +--=
cbxaxy ++= 2
02 =++ cbxax
x
y
x
y
x
y
EX1.Giventhefollowinggraphoftheequationy=x2–7x+10.Answerthefollowingquestions.
Whatistheaxisofsymmetry?__________Whatarethecoordinatesoftheturningpoint?________
IstheT.P.amaxorminimumpoint?________Howmanyrootsarethere?__________Whatarethesolutionsofthisequation?_____Whatarethesolutionscalled?________
EX2.GRAPH:
Whatistheaxisofsymmetry?__________
Whatarethecoordinatesoftheturningpoint?_______ IstheT.P.amaxorminimumpoint?________ Howmanyrootsarethere?__________ Whatarethesolutionsofthisequation?________ Whatdoyoucallthesesolutions?__________
56xxy 2 ---=
Day2Practice:PartsofaParabola Name:___________________________Thegraphofaquadraticiscalla________________.Usingthefollowinggraphandwordbank,fillintheboxesprovidedbelowandidentifythekeyparts!x-interceptsvertexy-interceptsx-axisy-axis
Identifythekeypartsofthegivenparabolas.
Studentreferencevideo:https://virtualnerd.com/common-core/hsa-algebra/HSA-SSE-expressions-seeing-structure/B/3/3a/ORscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos.Day3:FactoringandSolving
MultipleChoiceSection
1. If6x2y2werefactoredoutof24x2y3–18x3y2thebinomialleftwouldbe?
A4y–3x B4y2–18x C4x2y–3y D4x2y–6x
2. Whichisthegreatestcommonfactorof12x2+9x3–6x4?
A6x
B3x
C6x2
D3x2
3. Whichofthefollowingisequaltox2+4x+3? A(x+1)(x+3) B(x+2)(x+2) C(x+4)(x–1) D(x–4)(x–1)
4. Whicharethefactorsoftheexpression3x2+23x–8? A(3x+1)(x–8) B(3x–8)(x+1) C(3x–1)(x+8) D(3x–1)(x–8)
5. Whatisthefactoredformof6x2–16x+8? A(6x–2)(x–4)
B2(3x+2)(x+2)
C2(3x–2)(x–2)
D3(3x+2)(x–2)
6. Whichisafactorof2x2–11x–40? A2x+5 B2x–5 C2x–8 D2x+8
7. Thesumoftwicex2addedto7xis–3.Whichofthefollowingcouldbethevalueofx? A–3 B3 C2 D–2
8. Whichshows3x2–8x+4completelyfactored? A(3x–2)(x+2) B(3x+4)(x–1) C(x+4)(3x–1) D(3x–2)(x–2)
9. Whatarethesolutionsforthequadraticequation2x2+x=3?
A–1,
B1,
C1,–
D–1,–
10. Bobissolvingthisequationbyfactoring. 3x2+17x+10=0
Whichexpressioncouldbeoneofhiscorrectfactors?Ax+5
Bx–5 C3x–2 D3x–5
11. Iftheareaoftherectangleshownisgivenbytheexpression3x2+7x–6,andthewidthis(x+3),whichofthefollowingcouldrepresentthelength?
A(3x+2)
B(x+2) C(3x–3) D(3x–2)
32323232
A = 3x2 + 7x – 6
?
(x + 3)
12. Thevolumeofthecubeshownisx3–5x2+6xifoneofthedimensionsisx,whichrepresentsthetwomissingdimensions?
A(x+2)(x–3) B(x+2)(x+3) C(x–2)(x–3) D(x–2)(x+3)
ConstructedResponse.13. Factor.6x3+9x2–12xy
14. Factoroutthegreatestcommonfactor.4a4b–6a2b2+12a3b
15. Factoroutthegreatestcommonfactor.5ab2+10ab
16. Whatarethefactorsofx2+x–6?
x
?
?
17. Writeinfactoredform.6y2–y–2
18. Whatarethefactorsofx2+9x+20?
19. Factor.8b2–10b+3
20. Writeinfactoredform.x2–9
21. Factorcompletely.4x2–16.
22. Whatarethefactorsof9x2–30x+25?
23. Whatvaluesofsmaketheequations2+7s+10=0true?
X=_______________
24. Whatarethesolutionsoftheequationx2–x=6?X=_______________
25. Findx.2x2=11x+6X=_______________
UsetheStudentReferencevideosfromDay3.Day4:SOLVINGQUADRATICEQUATIONSbyFactoringQUADRATICEQUATION: STANDARDFORMofaquadraticequation:( )HOWTOSOLVEQUADRATICEQUATIONS:
Step1: WriteequationinStandardForm. Step2: FactorthequadraticequationStep3: Seteach()=to0andsolveforthevariable.Step5: CheckeachoftherootsintheORIGINALquadraticequation.Substituteyoursolution
inplaceofxandcheckthaty=0.EXAMPLES:Solveeachequation.Checktheroots.1. Findtheroots: 2. Solvefory: 3. Findtheroots: 4. Solvefory:
cbxaxy ++= 2
02 =++ cbxax
035r12r 2 =+-
024y11y2 =++
06x5x 2 =--
28y3y2 =-
5. Findtheroots:
6. Findtheroots: SolvebyfactoringSPECIALQUADRATICS:ex. and ex. Solvethefollowingequations.Checkalltheroots:7. 8. 9. 10. 11. 12.
30xx 2 +=
3x7x3x3 2 -=-0bxax 2 =+ 0cax2 =+
04z2 =- 036a2 =- x4x 2 -=
0x8x 2 =+ 012x3 2 =- 45y5 2 =
StudentReferenceVideo:https://virtualnerd.com/algebra-2/quadratics/formula-discriminant/quadratic-formula/solve-by-quadratic-formula ORscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos.Day5:SolveQuadraticsbyusingtheQuadraticFormula
StudentReferenceVideo:https://virtualnerd.com/algebra-2/quadratics/solve-by-completing-square-roots/complete-square/ ORscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos. Day6:SolvingQuadraticsbyCompletingtheSquare
StudentReferenceVideo:https://virtualnerd.com/algebra-2/quadratics/solve-by-completing-square-roots/complete-square/completing-square-standard-to-vertex-leading-coefficient-not-1 OrscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos.Day7:WRITINGQUADRATICSINVERTEXFORM
BYCOMPLETINGTHESQUAREStandardForm:𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 VertexForm:𝒚 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌,where(𝒉, 𝒌)isthevertexoftheparabola.
StudentReferenceVideo: https://www.loom.com/share/15dd7e465d7243a5a7d2835dd52dec1aOrscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos.DAY8:APPLICATIONSWITHQUADRATICFUNCTIONSUsingagraph!1. Usingthegraphattheright,Itshowstheheighthinfeetofasmallrockettsecondsafteritislaunched.Thepathoftherocketisgivenbytheequation:h=-16t2+128t.Howlongistherocketintheair?_________Whatisthegreatestheighttherocketreaches?____Abouthowhighistherocketafter1second?_______After2seconds, a.abouthowhighistherocket?_________ b.istherocketgoinguporgoingdown?________After6seconds, a.abouthowhighistherocket?_______ b.istherocketgoinguporgoingdown?________ Doyouthinktherocketistravelingfasterfrom0to1secondorfrom3to4seconds?Explainyouranswer.
Usingtheequation,findtheexactvalueoftheheightoftherocketat2seconds.2.Aballisthrownintheair.Thepathoftheballisrepresentedbytheequation
h=-t2+8t.Graphtheequationovertheinterval0£t£8ontheaccompanyinggrid.Whatisthemaximumheightoftheball?_______________Howlongistheballabove7meter?________________
50
1
100
150
200
250
2 3 4 5 6 7 8 time (seconds)
h (height (feet))
time (seconds)
height (meters)
StudentReferenceVideo: https://www.loom.com/share/891bd2e36409425aa7a6d9a9ddbe2d07OrscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos.SolvingAlgebraically
3. Aftertseconds,aballtossedintheairfromthegroundlevelreachesaheightofhfeetgivenbytheequationh=144t–16t2.
a. Whatistheheightoftheballafter3seconds?
b. Whatisthemaximumheighttheballwillreach?
c. Findthenumberofsecondstheballisintheairwhenitreachesaheightof224feet.
d. Afterhowmanysecondswilltheballhitthegroundbeforerebound?
4. Arocketcarryingfireworksislaunchedfromahill80feetabovealake.Therocketwillfallintolakeafterexplodingatitsmaximumheight.Therocket’sheightabovethesurfaceofthelakeisgivenbyh=-16t2+64t+80.
a. Whatistheheightoftherocketafter1.5second?
b. Whatisthemaximumheightreachedbytherocket?
c. Howlongwillittakefortherockettoreach128feet?
d. Afterhowmanysecondswilltherockethitthelake?
5. Arockisthrownfromthetopofatallbuilding.Thedistance,infeet,betweentherockandthegroundtsecondsafteritisthrownisgivenbyd=-16t2–4t+382.Howlongaftertherockisthrownisit370feetfromtheground?
StudentReferenceVideo:https://www.youtube.com/watch?v=hvyH-WJtMpcOrscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos.Day9:TransformationsofQuadraticFunctions
(StudentReference:seefirstproblem!)Day10:SOLVINGQUADACTIC–LINEARSYSTEMSSolveALGEBRAICALLYTwoequationswillbegiventoyouwiththedirectionstosolvethesystemalgebraically
• Oneequationwillbeaquadratic.• Thesecondequationwillbelinear.
Steps:
1. Solveforyinbothequationssotheyareiny=form.
2. Substitutethelinearequationintothe‘ypart’ofthequadraticequation.
3. Set=to0
4. SinceyournewequationisaquadraticyoumustFACTORTOSOLVEFORX.
YoushouldgetTWOsolutions!
5. Findybysubstitutingyour‘x’intooneoftheequationsandsolvefory.
6. Checksolutions
1.Solvethefollowingsystem:
2𝑥 = 𝑥% − 𝑥 + 2 Settheequationsequaltoeachotheroncetheyarebothiny=form.
−2𝑥 − 2𝑥 Set=to0bysubtracting2xfrombothsides.
0 = 𝑥% − 3𝑥 + 2 Now,factor.
0 = (𝑥 − 2)(𝑥 − 1) Seteach()=0andsolveforx.
2.Findthesolutionsof:
3.Solveforthesolutions:
x2y2xxy 2
=+-=
1yx3x4xy 2
=+-+-=
2yx13x7xy 2
=-+-=
StudentReferenceVideo:https://www.loom.com/share/a42989cfca234c13a4d52df8b162cfd8OrscanthisQRcodeonyoursmartphonetoviewthestudentreferencevideos.
SolveGRAPHICALLYExamples:4. Solvethefollowingsystemofequationsgraphicallyandcheck. 5.Solvethefollowingsystemofequationsgraphicallyandcheck.
3x4xy 2 -+-=1yx =+
4x4xy 2 ++=4x2y +-=
6. Thegraphsoftheequations and intersectin: (1) 1point (2) 2points (3) 3points (4) 4points7. Whichisasolutionorthefollowingsystemofequations? (1) (3,–9) (2) (0,0) (3) (5,5) (4) (6,0)8. Whenthegraphsoftheequations and aredrawnonthesamesetofaxes,
atwhichpointdothegraphsintersect?
(1) (4,2) (2) (5,1) (3) (3,3) (4) (2,4)
2xy = 2x =
15x2y -=
x6xy 2 -=
6x5xy 2 +-= 6yx =+