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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

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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 − !

!