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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