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IM2Sem1FinalExamStudyGuide 1

IntegratedMath2–Semester1FinalExamStudyGuide

1.Solvethefollowingequation:

40 − 2𝑥 = 4(𝑥 − 5)2.Solvethefollowingequation:

10 + 5𝑥 = 2(𝑥 + 1)3.Solvethefollowingequation:

20 + 5𝑥 = 7𝑥 + 4

4.Solvethequadraticequationbyfactoring,completingthesquareorbyusingtheQuadraticFormula.Roundtothenearesttenthifnecessary.2𝑥! + 𝑥 − 5 = 0

5.Solvethequadraticequationbyfactoring,completingthesquareorbyusingtheQuadraticFormula.Roundtothenearesttenthifnecessary.3𝑥! − 5𝑥 = 2

6.Solvethequadraticequationbyfactoring,completingthesquareorbyusingtheQuadraticFormula.Roundtothenearesttenthifnecessary.4𝑥! + 4𝑥 = −1

7.Findtheequationoftheaxisofsymmetryandthecoordinatesofthevertexofthegraphof𝑦 = −𝑥! + 2𝑥 + 5.

8.Findtheequationoftheaxisofsymmetryandthecoordinatesofthevertexofthegraphof𝑦 = 𝑥! + 4𝑥 − 5.

9.Findtheequationoftheaxisofsymmetryandthecoordinatesofthevertexofthegraphof𝑦 = −2𝑥! + 4𝑥 + 6.

10.Whichequationcorrespondstothegraphshown?

a. 𝑦 = 𝑥 − 2 !b. 𝑦 = 𝑥 + 2 !c. 𝑦 = 𝑥! − 2d. 𝑦 = 𝑥! + 2

11.Whichequationcorrespondstothegraphshown?

a. 𝑦 = 𝑥 + 1 ! − 3b. 𝑦 = 𝑥 − 1 ! + 3c. 𝑦 = 𝑥! + 3d. 𝑦 = 𝑥 + 3 ! − 1

12.Whichequationcorrespondstothegraphshown?

a. 𝑦 = − 𝑥 − 4 ! + 5b. 𝑦 = − 𝑥 + 4 ! + 5c. 𝑦 = − 𝑥 − 4 ! − 5d. 𝑦 = −𝑥! + 5

13.Find(2ℎ + 5)(3ℎ − 2). 14.Find(5𝑟 − 2)(−2𝑟 + 3). 15.Find(𝑏 + 4)(2𝑏 − 5).

16.Find(𝑦 + 3𝑧)(2𝑦 − 𝑧). 17.Find(3𝑦 + 5𝑧)(3𝑦 − 7𝑧). 18.Find(−2𝑦 + 1)(2𝑦 + 4).

19.Findthedifferencebycombiningliketerms.6𝑥! − 3𝑥 − 3 − (−4𝑥! − 4𝑥 + 10)

20.Findthedifferencebycombiningliketerms.5𝑥! − 4𝑥 − 2 − (−3𝑥! + 2𝑥 − 5)

21.Findthedifferencebycombiningliketerms.−7𝑥! + 6𝑥 − 8 − (2𝑥! − 5𝑥 + 3)

22.Distributebymultiplication.2𝑥(−6𝑥 + 5)

23.Distributebymultiplication.−3𝑥(7𝑥 − 2)

24.Distributebymultiplication.2𝑥(4𝑥 + 3)

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IM2Sem1FinalExamStudyGuide 2

25.Find5𝑥! + 3𝑥 − 2 + (2𝑥! − 7𝑥 − 1).

26.Find−9𝑥! − 6𝑥 − 6 + (3𝑥! + 6𝑥 − 4).

27.Find7𝑥! − 5𝑥 − 8 + (−𝑥! − 5𝑥 + 6).

28.Willthesumof −64and−3bereal,orcomplex?Ifitisreal,isitrationalorirrational?

29.Willthesumof 36and−5bereal,orcomplex?Ifitisreal,isitrationalorirrational?

30.Willthesumof 12and−4bereal,orcomplex?Ifitisreal,isitrationalorirrational?

31.Whatisthesimplifiedformoftheexpression −81?

32.Whatisthesimplifiedformoftheexpression −45?

33.Whatisthesimplifiedformoftheexpression− −36?

34.Whichpolynomialdoesthegraphrepresent?

a. 𝑦 = (𝑥 − 2)(𝑥 − 4)b. 𝑦 = (𝑥 − 2)(𝑥 + 4)c. 𝑦 = 𝑥 + 2 𝑥 − 4 d. 𝑦 = (𝑥 + 2)(𝑥 + 4)

35.Whichpolynomialdoesthegraphrepresent?

a. 𝑦 = (𝑥 − 1)(𝑥 − 3)b. 𝑦 = (𝑥 − 1)(𝑥 + 3)c. 𝑦 = 𝑥 + 1 𝑥 − 3 d. 𝑦 = (𝑥 + 1)(𝑥 + 3)

36.Whichpolynomialdoesthegraphrepresent?

a. 𝑦 = (𝑥 − 1)(𝑥 − 4)b. 𝑦 = (𝑥 − 1)(𝑥 + 4)c. 𝑦 = 𝑥 + 1 𝑥 − 4 d. 𝑦 = (𝑥 + 1)(𝑥 + 4)

37.Whichequationrepresentsaparabolawithavertexat(−5, 4)?

a. 𝑓 𝑥 = 2 𝑥 + 5 ! + 4b. 𝑓 𝑥 = −3(𝑥 + 5)(𝑥 + 4)c. 𝑓 𝑥 = 4 𝑥 + 5 𝑥 − 4 d. 𝑓 𝑥 = −6 𝑥 − 5 ! + 4

38.Whichequationrepresentsaparabolawithavertexat(−6,−7)?

a. 𝑓 𝑥 = 5(𝑥 − 6)(𝑥 − 7)b. 𝑓 𝑥 = 4(𝑥 + 6)(𝑥 − 7)c. 𝑓 𝑥 = −8 𝑥 + 6 ! − 7d. 𝑓 𝑥 = 3 𝑥 − 6 ! − 7

39.Whichequationrepresentsaparabolawithavertexat(8, 1)?

a. 𝑓 𝑥 = −6 𝑥 + 8 ! + 1b. 𝑓 𝑥 = 3 𝑥 − 8 ! + 1c. 𝑓 𝑥 = −5(𝑥 − 8)(𝑥 − 1)d. 𝑓 𝑥 = 4(𝑥 + 8)(𝑥 + 1)

40.Solveforxandextractthesquarerootsofanyperfectsquares:

48 = 𝑥!

41.Solveforxandextractthesquarerootsofanyperfectsquares:

72 = 𝑥!

42.Solveforxandextractthesquarerootsofanyperfectsquares:

40 = 𝑥!

43.Abaseballplayerstandsatapointthatismodeledby(0, 0)onthecoordinateplane.Hethenthrowsabaseballthatismodeledbyaquadraticequation.Whichpieceofinformation(quadraticproperty)helpsdeterminewhentheballlandsontheground?

44.Abaseballplayerstandsatapointthatismodeledby(0, 0)onthecoordinateplane.Hethenthrowsabaseballthatismodeledbyaquadraticequation.Whichpieceofinformation(quadraticproperty)helpsdeterminetheinitialheightoftheball(atthemomenthethrowsit)?

45.Abaseballplayerstandsatapointthatismodeledby(0, 0)onthecoordinateplane.Hethenthrowsabaseballthatismodeledbyaquadraticequation.Whichpieceofinformation(quadraticproperty)helpsdeterminewhentheballstartsfalling?

46.Usethequadraticformulatosolve.

2𝑥! + 5𝑥 + 4 = 0

47.Usethequadraticformulatosolve.

𝑥! − 4𝑥 − 2 = 0

48.Usethequadraticformulatosolve.

−5𝑥! + 2𝑥 − 2 = 0

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IM2Sem1FinalExamStudyGuide 3

49.Twodifferentquadraticfunctionsaredescribedbelow:

• FunctionA:Thisfunctionhasrootsat2and6,andhasay-interceptat-13.

• FunctionB:Thisfunctionisrepresentedbytheequation𝑓 𝑥 = 2 𝑥 − 4 ! + 5

Whichstatementistrueaboutthesetwoquadraticfunctions?

a. Bothfunctionsopendownwards.

b. Bothfunctionshavetworealsolutions.

c. Bothfunctionshavethesameaxisofsymmetry.

d. ThevertexofbothfunctionsareinQuadrantII.

50.Twodifferentquadraticfunctionsaredescribedbelow:

• FunctionA:Thisfunctionhasavertexat(2, 3)andpassesthrough(4,−1).

• FunctionB:Thisfunctionisrepresentedbytheequation𝑓 𝑥 = − 𝑥 + 1 !

Whichstatementistrueaboutthesetwoquadraticfunctions?

a. Bothfunctionsopenupward.b. Bothfunctionshavethe

samey-intercept.c. FunctionAhastworeal

solutionsandfunctionBhastwoimaginarysolutions.

d. Bothfunctionshavethesameaxisofsymmetry.

51.Twodifferentquadraticfunctionsaredescribedbelow:

• FunctionA:Thisfunctionhasrootsat−4&−2andpassesthrough(1, 8)

• FunctionB:Thisfunctionisrepresentedbytheequation𝑓 𝑥 = −(𝑥 − 2)(𝑥 − 4)

Whichstatementistrueaboutthesetwoquadraticfunctions?

a. ThevertexoffunctionAisloweronthegraphthanthevertexoffunctionB.

b. Bothfunctionshavethesameroots.

c. Bothfunctionshavethesamey-intercept.

d. ThevertexofbothfunctionsareinQuadrantIII.

52.Thecost,y,indollars,ofparkingacarforxhoursataparkinglotduringthedayisshownusingthefunctionbelow.

𝑦 = 2, 0 ≤ 𝑥 ≤ 3𝑥, 3 < 𝑥 ≤ 7

Createagraphthatmodelsthecostofparkingatthisparkinglot.

53.Thecost,y,indollars,ofparkingacarforxhoursataparkinglotduringthedayisshownusingthefunctionbelow.

𝑦 = 𝑥 + 2, 0 ≤ 𝑥 ≤ 5 9, 5 < 𝑥 ≤ 10

Createagraphthatmodelsthecostofparkingatthisparkinglot.

54.Thecost,y,indollars,ofparkingacarforxhoursataparkinglotduringthedayisshownusingthefunctionbelow.

𝑦 = 3𝑥, 0 ≤ 𝑥 < 2 7, 2 ≤ 𝑥 ≤ 6

Createagraphthatmodelsthecostofparkingatthisparkinglot.

55.Frankcorrectlyfactored6𝑥! − 17𝑥 − 3as (𝑥 − 3)(6𝑥 + 1).Hethenclaimedthatthezerosofthatquadraticfunction𝑓 𝑥 = 6𝑥! −17𝑥 − 3arelocatedat𝑥 = 3and𝑥 = −1.

A. ExplainFrank’smistake.B. Determinethecorrectzeros.

56.Eduardocorrectlyfactored5𝑥! + 32𝑥 − 21as (5𝑥 − 3)(𝑥 + 7).Hethenclaimedthatthezerosofthatquadraticfunction𝑓 𝑥 = 5𝑥! +32𝑥 − 21arelocatedat𝑥 = 7and𝑥 = − !

!.

A. ExplainEduardo’smistake.B. Determinethecorrectzeros.

57.Janicecorrectlyfactored3𝑥! − 19𝑥 + 20as (3𝑥 − 4)(𝑥 − 5).Shethenclaimedthatthezerosofthatquadraticfunction𝑓 𝑥 = 3𝑥! −19𝑥 + 20arelocatedat𝑥 = 5and𝑥 = 4.

A. ExplainJanice’smistake.B. Determinethecorrectzeros.

58.Arocketislaunchedfrom224feetabovethegroundattime𝑡 = 0.Thefunctionthatmodelsthissituationisgivenbyℎ = −16𝑡! +80𝑡 + 224,where𝑡ismeasuredinsecondsandℎisheightabovethegroundmeasuredinfeet.

A. Determinethemaximumheightobtainedbytherocket.Showallwork.

B. Determinethetimeatwhichtherockethitstheground.Showallwork.

59.Arocketislaunchedfrom192feetabovethegroundattime𝑡 = 0.Thefunctionthatmodelsthissituationisgivenbyℎ = −16𝑡! +64𝑡 + 192,where𝑡ismeasuredinsecondsandℎisheightabovethegroundmeasuredinfeet.

A. Determinethemaximumheightobtainedbytherocket.Showallwork.

B. Determinethetimeatwhichtherockethitstheground.Showallwork.

60.Arocketislaunchedfrom240feetabovethegroundattime𝑡 = 0.Thefunctionthatmodelsthissituationisgivenbyℎ = −16𝑡! +32𝑡 + 240,where𝑡ismeasuredinsecondsandℎisheightabovethegroundmeasuredinfeet.

A. Determinethemaximumheightobtainedbytherocket.Showallwork.

B. Determinethetimeatwhichtherockethitstheground.Showallwork.

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IM2Sem1FinalExamStudyGuide 4

61.Determinethesolutionstotheequation(usetheform𝑗 ± 𝑘,where𝑗and𝑘areintegers).

𝑥! + 13𝑥 = −20

62.Determinethesolutionstotheequation(usetheform𝑗 ± 𝑘,where𝑗and𝑘areintegers).

𝑥! − 15𝑥 = −40

63.Determinethesolutionstotheequation(usetheform𝑗 ± 𝑘,where𝑗and𝑘areintegers).

𝑥! + 17𝑥 = −35

64.Rewritethequadraticequationinstandardform.

𝑓 𝑥 = 2𝑥 + 5 ! − 4

65.Rewritethequadraticequationinstandardform.

𝑔 𝑥 = 4𝑥 − 7 ! + 5

66.Rewritethequadraticequationinstandardform.

ℎ 𝑥 = 5𝑥 − 3 ! + 7

67.Theequation𝑥! − 18𝑥 + 24 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.

Constant Value𝑝 𝑞

68.Theequation𝑥! + 24𝑥 − 13 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.

Constant Value𝑝 𝑞

69.Theequation𝑥! − 20𝑥 + 78 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.

Constant Value𝑝 𝑞

70.Graph.Labelthevertexandaxisofsymmetry.

𝑓 𝑥 = − 𝑥 + 3 ! + 4

71.Graph.Labelthevertexandaxisofsymmetry.

𝑔 𝑥 = 𝑥 + 2 ! − 9

72.Graph.Labelthevertexandaxisofsymmetry.

ℎ 𝑥 = 2 𝑥 − 2 ! − 8