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Activity 1 knoxschools.org/kcsathome
Algebra 1
KCSatHomeAlgebra1SummerPacket
ActivitySet1A. QuadraticFunctionsObjective:Thestudentwillbeableto:• Graphquadraticsembeddedinareal-worldsituation.• Factoraquadraticfunctiontorevealthezerosofthefunction.• Determinetheminimumandmaximumofaquadraticbycompletingthesquare.• Knowandapplythequadraticformula.
ActivitySet2B. LinearFunctionsandEquationsObjective:Thestudentwillbeableto:• Calculateandwritetheequationfortheslopeofaline.• Rewriteanequationinstandardformtoslopeinterceptform• Modelandcomparelinearfunctionsusingmultiplerepresentations.• Representandsolvesystemsoflinearfunctions• Solvemulti-stepequationsusingpropertiesofequalityandnumberproperties.
ActivitySet3C. ExponentLawsandExponentialFunctionsObjective:Thestudentwillbeableto:• Usepropertiesofexponentstorewriteexponentialexpressions.• Evaluatepowersthathavezerosornegativeexponents.• Writetheexplicitformulaforgeometricsequencesinfunctionform.• Createanexponentialfunctiongivenagraph.• Representexponentialgrowthanddecayfunctions.
ActivitySet4D. PolynomialsExpressionsObjective:Thestudentwillbeableto:• Simplypolynomialsbyaddingandsubtracting.• Multiplymonomialsandpolynomialsusingmodelsandstrategies.• Usearithmeticoperationstosimplyexpressions.• Identifythegreatestcommonfactorofthetermsofapolynomialexpression.• Factorpolynomialsusingstrategiessuchasgroupingordifferenceofsquares.
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ActivitySet1
A. QuadraticFunctionsI.Featuresofaquadraticfunction.Vocabularyinred.
II.GraphingQuadraticsinStandardForm
• Standardform:f(x)=ax2+bx+c=0 • Findtheaxisofsymmetryusing,= #$
%&
• Findtheminimumbysubstituting2inforXintheequation.Thisisyouryvalue.
• Findtherootsbyfactoring(orbestmethod)• Theyinterceptisalsoyourcvalue.• Thegraphopensupwardbecausethe‘a’value
Ispositive.
X2-4x+3
#(#()
*(+)=2(vertex)
22-4(2)+3=-1
(x–3)(x–1)x–3=0x–1=0x=3x=1
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Example:(SEEKCSVideo)
III.GraphingQuadraticsinVertexForm
• Vertexform:f(x)=a(x-h)2+k• Settheinsideofthefunctionto0• Theoutsideofthefunctionisthe
y-intercept.
• Tofindtherootsyoucansolvetheequationandfollowsteps
abovefromstandardform
• Oryoucancreateatableofvaluesbysubstituting
• Withoneoftherootsandthevertexyoucandeterminetheotherroot.(set
y=0andfindallsolutions)
Let’susethesameequation:X2-4x+3vertexform:(x-2)2–1
(x-2)=0x=2y=-1(minimum)
x y
0 3
1 0
3 0
ßYinterceptßOneoftherootsßSecondroot
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IV.RealWorldProblemswithQuadraticFunctions(SeeKCSvideo)
• Basedonthisinformation,theyinterceptis0feetbecausetherocketistakingofffromtheground.
• Findtheaxisofsymmetryusing, = #$%&
• Substitute4ast.y=256(maximum)• Sinceoneoftherootshastobe0,andthevertexisat(4,256),theotherrootmustbearound
8.
• Substitute8infort(Itshouldbecloseto0ifcorrect).Itequals0exactlysotheotherrootwouldbe(8,0)
4
V.Solvequadraticsbycompletingthesquare
• Determinetherootsoftheequationx2+10x+16• Isolatex2+10x.Youcancompletethesquareand
Rewritethisasaperfectsquaretrinomial.
• Determinetheconstanttermthatwouldcompletethesquare.Addthistermtobothsidesoftheequation.
• Factortheleftsideoftheequation.• Determinethesquarerootofeachsideoftheequation.• Setthefactoroftheperfectsquaretrinomialequalto
Eachsquarerootoftheconstant.
• Solveforx.Therootsarex=-2andx=-8
VI.QuadraticFormula
- =#.± .0#(12
*1alsowrittenas
#.
*1±
.0#(12
*1
• Thefirstterm#$
%&representtheaxisofsymmetry.
• Thesignsofthesecondareopposites±becausethesolutionsliethesamedistanceawayfromtheintersectionoftheaxisofsymmetryandtheliney=0,butinoppositedirections.
• Solvingforthediscriminantb2-4acwilldeterminehowmanysolutions.• Ifitismorethan0,therearetworealsolutions,lessthan0therearenosolutions,and
equalto0thereisonesolution.
Ex.Ashleighdeterminestherootsforthe
quadraticequations2x2+3x–9=-10.
WhatdidAshleighdoincorrectly?
X2+10x+16-16=0-16X2+10x=-16X2+10x+___=-16+___X2+10x+25=-16+25X2+10x+25=9(x+5)2=9
3(- + 5)2=±√9x+5=±√9x+5=+3andx+5=-3x=-5+3 x=-5-3x=-2 x=-8
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PRACTICE
Graph1Aand1Bandidentifythefollowing.
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2.Aswimteammemberperformsadivefroma14-foothighspringboard.Theparabolabelow
showsthepathofherdive.
A) Whatistheaxisofsymmetryandwhatdoesitmean?_____________________________________________
_____________________________________________B) Findf(6)___________
C) Whatdoesthesolution8feetmean?______________
_____________________________________________
3.Thefunctionrepresentingaparabolawithavertexat(-3,-2)andpassingthroughthepoint
(-1,10)iswhichofthefollowing?
A) ( ) 232 2 +-= xy B) ( ) 232 2 -+= xy C) ( ) 23
21 2 +-= xy D) ( )23 3 2y x= + -
4.Calculatethezerosofeachquadraticfunction.
A) f(x)=2x2+x-36 B.f(x)=x2-10x+24 C.y=2(x+3)2-2
5.Determinetherootsoftheequationx
2+18x–40bycompletingthesquare.
6.Determinethenumberofrealzeros,theaxisofsymmetry,andthevertexforthefunction
A.f(x)=4x2+2x+8 B.f(x)=-2x2+4x+6
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7.ThePerrisPandasbaseballteamhasanewpromotionalactivitytoencouragefanstoattend
games:launchingfreeT-shirts!TheycanlaunchaT-shirtintheairwithaninitialvelocityof91
feetpersecondfrom5½feetofftheground(theheightoftheteammascot).
AT-shirt’sheightcanbemodeledwiththequadraticfunctionh(t)=-16t2+91t+5.5,wheretisthetimeinsecondsandh(t)istheheightofthelaunchedT-shirtinfeet.TheywanttoknowhowlongitwilltakeforaT-shirttolandbackonthegroundafterbeinglaunched(ifnofans
grabitbeforethen!)Usethequadraticformulatosolve.
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ActivitySet1AnswerKey
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3.D.
4A.
4B.x=6orx=4
4C.x=-4orx=-2
5.
7.
6.A.f(x)=4x2+2x+8
6B.f(x)=-2x2+4x+6
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