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IM2Sem1FinalReviewH 1
IM2Semester1FinalExamReviewH(StudyGuideQuestions43-45,49-51&55-57)
FeaturesofaQuadratic&FactoredFormZeros
Todetermineneededfeaturesfromaquadraticsituation(problems43–45):
Startbyvisualizingthesituationasaquadraticgraph.Then,identifythelocationonthegraphofanyneededinformation.
1.Apersonstandsatawindowthatismodeledbythepoint(0, 480)onthecoordinateplane.Hethenthrowsapaperairplanewhosepathismodeledbyaquadraticequation.Selectonepieceofinformationthathelpstodeterminethetimeittakestheairplanetotravelfromthewindowtoitsmaximumheight.
A. EndBehaviorB. PositivezeroonlyC. NegativezeroonlyD. ZerosE. Y-interceptF. Vertex
2.Apersonstandsatawindowthatismodeledbythepoint(0, 480)onthecoordinateplane.Hethenthrowsapaperairplanewhosepathismodeledbyaquadraticequation.Selectonepieceofinformationthathelpstodeterminethetimeittakestheairplanetotravelsfromthewindowtowhereitlands.
A. EndBehaviorB. PositivezeroonlyC. NegativezeroonlyD. ZerosE. Y-interceptF. Vertex
3.Apersonstandsatawindowthatismodeledbythepoint(0, 480)onthecoordinateplane.Hethenthrowsapaperairplanewhosepathismodeledbyaquadraticequation.Selectonepieceofinformationthathelpstodeterminetheinitialheightoftheairplanewhenitisthrownfromthewindow.
A. EndBehaviorB. PositivezeroonlyC. NegativezeroonlyD. ZerosE. Y-interceptF. Vertex
4.Apersonstandsatawindowthatismodeledbythepoint(0, 480)onthecoordinateplane.Hethenthrowsapaperairplanewhosepathismodeledbyaquadraticequation.Selectonepieceofinformationthathelpstodeterminetheheightoftheairplanewhenittravelsfromthewindowtoitsmaximumheight.
A. EndBehaviorB. PositivezeroonlyC. NegativezeroonlyD. ZerosE. Y-interceptF. Vertex
5.Acatapultisfiredoffofthegroundsothattheobjectisreleasedatapointthatismodeledby(0, 7)onthecoordinateplane.Thepathofthecatapultedobjectismodeledbyaquadraticequation.Selectonepieceofinformationthathelpstodeterminethetimeswhentheobjectisontheground.
A. EndBehaviorB. PositivezeroonlyC. NegativezeroonlyD. ZerosE. Y-interceptF. Vertex
6.Acatapultisfiredoffofthegroundsothattheobjectisreleasedatapointthatismodeledby(0, 7)onthecoordinateplane.Thepathofthecatapultedobjectismodeledbyaquadraticequation.Selectonepieceofinformationthathelpstodeterminetheheightoftheobjectwhenittravelsfromthereleasepointtoitsmaximumheight.
A. EndBehaviorB. PositivezeroonlyC. NegativezeroonlyD. ZerosE. Y-interceptF. Vertex
7.Acatapultisfiredoffofthegroundsothattheobjectisreleasedatapointthatismodeledby(0, 7)onthecoordinateplane.Thepathofthecatapultedobjectismodeledbyaquadraticequation.Selectonepieceofinformationthathelpstodeterminethenumberofsecondstheobjecttravelsthroughtheairfromthereleasepointuntilitreachesitsmaximumheight.
A. EndBehaviorB. PositivezeroonlyC. NegativezeroonlyD. ZerosE. Y-interceptF. Vertex
8.Acatapultisfiredoffofthegroundsothattheobjectisreleasedatapointthatismodeledby(0, 7)onthecoordinateplane.Thepathofthecatapultedobjectismodeledbyaquadraticequation.Selectonepieceofinformationthathelpstodeterminethetimeittakestheobjecttotravelfromthereleasepointtowhereitlandsontheground.
A. EndBehaviorB. PositivezeroonlyC. NegativezeroonlyD. ZerosE. Y-interceptF. Vertex
9.Acatapultisfiredoffofthegroundsothattheobjectisreleasedatapointthatismodeledby(0, 7)onthecoordinateplane.Thepathofthecatapultedobjectismodeledbyaquadraticequation.Selectonepieceofinformationthathelpstodeterminethenumberofsecondsittakesfortheobjecttotravelfromthecatapultonthegroundtothereleasepoint.
A. EndBehaviorB. PositivezeroonlyC. NegativezeroonlyD. ZerosE. Y-interceptF. Vertex
x
yMaximum Height
Landing Point
Release Point
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IM2Sem1FinalReviewH 2
Tocomparequadraticsfromtheirdescriptions(problems49–51):Foreachfunction,drawasketchbasedontheinformationyou’regiven. Vertex:Wherethegraphturns–allpointswillbemirroredontotheothersideofthevertexaswell.
Axisofsymmetry:Thex-valueofthevertex Y-intercept:Wherethegraphcrossesthey-axis(standingline) Direction:
Thetwopossiblefunctionequationsarevertexformandfactoredform:
Vertexform:𝑓 𝑥 = 𝑎 𝑥 − ℎ ! + 𝑘Vertexwillbeat(ℎ, 𝑘)
Factoredform:𝑓 𝑥 = 𝑎(𝑥 − 𝑟!)(𝑥 − 𝑟!)Therootswillbeat(𝑟!, 0)andat(𝑟!, 0).
10.Twodifferentquadraticfunctionsaredescribedbelow:
• FunctionA:Thisfunctionhasavertexof(−4, 3)andpassesthroughthepoint(−2,−1)
• FunctionB:Thisfunctionisrepresentedbytheequation𝑓 𝑥 = −(𝑥 + 1)(𝑥 + 7)
Whichstatementistrueaboutthesetwoquadraticfunctions?
a. Bothfunctionsopendownwards.
b. Bothfunctionshavethesamey-intercept.
c. Bothfunctionshavethesamevertex.
d. ThevertexoffunctionAisloweronthegraphthanthevertexoffunctionB.
11.Twodifferentquadraticfunctionsaredescribedbelow:
• FunctionA:Thisfunctionhasavertexof(−2, 3)andpassesthroughthepoint(−1, 5)
• FunctionB:Thisfunctionisrepresentedbytheequation𝑓 𝑥 = 𝑥 − 3 ! − 4
Whichstatementistrueaboutthesetwoquadraticfunctions?
a. Bothfunctionshavethesamey-intercept.
b. ThevertexoffunctionAishigheronthegraphthanthevertexoffunctionB.
c. Bothfunctionshavethesameaxisofsymmetry.
d. Bothfunctionshavethesamevertex.
12.Twodifferentquadraticfunctionsaredescribedbelow:
• FunctionA:Thisfunctionhasavertexof(1, 1)andpassesthroughthepoint(0, 4)
• FunctionB:Thisfunctionisrepresentedbytheequation𝑓 𝑥 = − 𝑥 + 1 ! + 5
Whichstatementistrueaboutthesetwoquadraticfunctions?
a. Bothfunctionsopenupwards.
b. Bothfunctionshavethesamey-intercept.
c. Bothfunctionshavetworealsolutions.
d. Bothfunctionshavethesameaxisofsymmetry.
Opensupwardswhenaispositive
Opensdownwardswhenaisnegative
Name:_________________________________________________
IM2Sem1FinalReviewH 3
13.Twodifferentquadraticfunctionsaredescribedbelow:
• FunctionA:Thisfunctionhasavertexof(5, 2)andpassesthroughthepoint(6, 1)
• FunctionB:Thisfunctionisrepresentedbytheequation𝑓 𝑥 = 2 𝑥 − 3 ! + 2
Whichstatementistrueaboutthesetwoquadraticfunctions?
a. Bothfunctionsopenupwards.
b. Bothfunctionshavethesamey-intercept.
c. ThevertexoffunctionAisloweronthegraphthanthevertexoffunctionB.
d. FunctionAhastworealsolutionsandfunctionBhastwoimaginarysolutions.
14.Twodifferentquadraticfunctionsaredescribedbelow:
• FunctionA:Thisfunctionhasavertexof(3,−6)andpassesthroughthepoint(0, 3)
• FunctionB:Thisfunctionisrepresentedbytheequation𝑓 𝑥 = (𝑥 − 3)(𝑥 − 1)
Whichstatementistrueaboutthesetwoquadraticfunctions?
a. Bothfunctionshavethesamey-intercept.
b. ThevertexoffunctionBisloweronthegraphthanthevertexoffunctionA.
c. FunctionAhastworealsolutionsandfunctionBhastwoimaginarysolutions.
d. Bothfunctionshavethesameaxisofsymmetry.
15.Twodifferentquadraticfunctionsaredescribedbelow:
• FunctionA:Thisfunctionhasavertexat(−3,−4)andpassesthroughthepoint(−2,−3)
• FunctionB:Thisfunctionisrepresentedbytheequation𝑓 𝑥 = (𝑥 − 3)(𝑥 + 3)
Whichstatementistrueaboutthesetwoquadraticfunctions?
a. Bothfunctionsopenupwards.
b. Bothfunctionshavethesamey-intercept.
c. ThevertexoffunctionAisloweronthegraphthanthevertexoffunctionB.
d. FunctionAhastworealsolutionsandfunctionBhastwoimaginarysolutions.
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IM2Sem1FinalReviewH 4
Todeterminezerosfromfactorsandtoanalyzeerrors(problems55-57):Eachquadraticiscorrectlyfactored,butnotnecessarilycompletelyfactored–watchoutforthea-value. Ifa()hasanumberinfrontofx,dividebothpartsinthat()byit,andwritethenumberinfront. Donotdivideitoutoftheother()! Forexample: −7𝑥 + 2 𝑥 + 1 → −7 !!!
!!+ !
!!𝑥 + 1 → −7 𝑥 − !
!(𝑥 + 1)
Rememberthatyoumustswitchthesignsoftherootswhenyoutakethemoutoftheparentheses. Forexample:thezerosof−7 𝑥 − !
!𝑥 + 1 wouldbe 𝑥 = + !
! 𝑎𝑛𝑑 𝑥 = −1
16.Emilycorrectlyfactored2𝑥! + 12𝑥 + 18as2(𝑥 + 3)(𝑥 + 3).Shethenclaimedthatthezerosofthatquadraticfunction𝑓 𝑥 = 2𝑥! +12𝑥 + 18arelocated𝑥 = 2at𝑥 = −3and.
A. ExplainEmily’smistake.
B. Determinethecorrectzeros.
17.Megancorrectlyfactored5𝑥! + 𝑥 − 18as(5𝑥 − 9)(𝑥 + 2).Shethenclaimedthatthezerosofthatquadraticfunction𝑓 𝑥 = 5𝑥! + 𝑥 −18arelocated𝑥 = 9at𝑥 = −2and.
A. ExplainMegan’smistake.
B. Determinethecorrectzeros.
18.Jeremycorrectlyfactored6𝑥! − 10𝑥 − 4as(3𝑥 + 1)(2𝑥 − 4).Hethenclaimedthatthezerosofthatquadraticfunction𝑓 𝑥 = 6𝑥! −10𝑥 − 4arelocated𝑥 = !
!at𝑥 = −2
and.A. ExplainJeremy’smistake.
B. Determinethecorrectzeros.
19.Stephencorrectlyfactored– 𝑥! + 𝑥 + 12as– (𝑥 + 3)(𝑥 − 4).Hethenclaimedthatthezerosofthatquadraticfunction𝑓 𝑥 = −𝑥! + 𝑥 +12arelocated𝑥 = 3at𝑥 = −4and.
A. ExplainStephen’smistake.
B. Determinethecorrectzeros.
20.Margaretcorrectlyfactored2𝑥! + 14𝑥 + 24as2(𝑥 + 4)(𝑥 + 3).Shethenclaimedthatthezerosofthatquadraticfunction𝑓 𝑥 = 2𝑥! +14𝑥 + 24arelocatedatthepoint(−4,−3).
A. ExplainMargaret’smistake.
B. Determinethecorrectzeros.
21.Jorgecorrectlyfactored8𝑥! + 14𝑥 − 15as8 𝑥 − !
!𝑥 + !
!.
Hethenclaimedthatthezerosofthatquadraticfunction𝑓 𝑥 = 8𝑥! +14𝑥 − 15arelocated𝑥 = !
!at𝑥 = !
!
and.A. ExplainJorge’smistake.
B. Determinethecorrectzeros.
Answers1.𝐹 2.𝐵 3.𝐸 4.𝐹 5.𝐷 6.𝐹 7.𝐹 8.𝐵 9.𝐶 10.𝐴 11.𝐵 12.𝐵 13.𝐷 14.𝐴 15.𝐴16.a.Sincebothfactorswerethesame,itseemslikeEmilylookedforasecondroot(whenthereisonlyone),anddecidedtouseaasaroot.b.𝑥 = −3
17.a.Margaretforgottofactorouta.
5𝑥 − 9 𝑥 + 2 = 5 𝑥 −95
𝑥 + 2
b.𝑥 = !!
𝑎𝑛𝑑 𝑥 = −2
18.a.Jeremyforgottoswitchthesignsoftheroots,whichmeansthatpluggingintherootswillnotmakezero.b.𝑥 = − !
! 𝑎𝑛𝑑 𝑥 = 2
19.a.Stephenforgottoswitchthesignsoftheroots.b.𝑥 = −3 𝑎𝑛𝑑 𝑥 = 4
20.a.Margaretputthetworootstogetherasonepoint,buttheyarenot.Theyshouldbetwoseparatex’s,twoseparatepoints.b.𝑥 = −4 𝑎𝑛𝑑 𝑥 = −3𝑜𝑟 −4, 0 𝑎𝑛𝑑(−3, 0)
21.a.Jorgeonlychangedthesignsononeoftheroots.Heneededtoswitchboth.b.𝑥 = !
! 𝑎𝑛𝑑 − !
!
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