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.

1

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

2

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

3

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

5

PRACTICE

Graph1Aand1Bandidentifythefollowing.

6

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

7

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.

8

ActivitySet1AnswerKey

9

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