63. IM2 Semester 1 Final Exam Review Gmathbygrosvenor.weebly.com/.../4/8/...1_final_exam.pdf · IM2...
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Name: _________________________________________________ IM2 Sem1 Final Review G 1 IM2 Semester 1 Final Exam Review G (Study Guide Questions 64-72) Converting Equations and Graphing Quadratics To rewrite a quadratic from modified vertex form into standard form (problems 64–66): Start by rewriting the ℎ ! as (ℎ)(). Leave the number at the end (k) where it is. You will add or subtract it last. Next, distribute every part of the first parentheses with every part of the second parentheses. If you did this step correctly, you will have 4 terms (and one extra (the “k”) hanging out in the back). Lastly, combine like terms to write the equation in the form = ! + + . Add or subtract terms with the same variables ( ! & ! , & , & ). 1. Rewrite the quadratic equation in standard form. = 7 + 4 ! + 1 2. Rewrite the quadratic equation in standard form. = 2 − 8 ! + 3 3. Rewrite the quadratic equation in standard form. = 9 − 8 ! − 5 4. Rewrite the quadratic equation in standard form. = 6 − 5 ! − 2 5. Rewrite the quadratic equation in standard form. = 5 + 1 ! − 4 6. Rewrite the quadratic equation in standard form. = 8 + 2 ! + 3 7. Rewrite the quadratic equation in standard form. = 2 − 9 ! − 3 8. Rewrite the quadratic equation in standard form. = 6 − 1 ! − 2 9. Rewrite the quadratic equation in standard form. = 3 + 8 ! + 9 To complete the square to find p and q (problems 67–69): There are two ways to solve this problem. OPTION 1: Determine the vertex to write the problem in vertex form. ℎ = − ! !! , then plug ℎ in for all the -values in the original equation to find . Rewrite the problem in the form − ℎ ! + = 0 Then, move k to the other side by adding or subtracting it. Identify & based off of the equation − ! = Basically, = ℎ, and = − (switch k’s sign). OPTION 2 (see #19-27 on Review F): Start by moving to the other side of the equal sign. Then, rewrite ! + = − as + ! ! ! = − + ! ! ! Divide the number in front of the x-term by 2. Write that number in squared parentheses on the left: 2 ! = Whatever number you put on the left, square it and add that number to the number on the right. Then simplify the right side & identify & based off of the equation − ! = .
Transcript of 63. IM2 Semester 1 Final Exam Review Gmathbygrosvenor.weebly.com/.../4/8/...1_final_exam.pdf · IM2...
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IM2Sem1FinalReviewG 1
IM2Semester1FinalExamReviewG(StudyGuideQuestions64-72)
ConvertingEquationsandGraphingQuadratics
Torewriteaquadraticfrommodifiedvertexformintostandardform(problems64–66):
Startbyrewritingthe 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 !as(𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔)(𝑖𝑡𝑠𝑒𝑙𝑓). Leavethenumberattheend(k)whereitis.Youwilladdorsubtractitlast.Next,distributeeverypartofthefirstparentheseswitheverypartofthesecondparentheses. Ifyoudidthisstepcorrectly,youwillhave4terms(andoneextra(the“k”)hangingoutintheback).Lastly,combineliketermstowritetheequationintheform𝑓 𝑥 = 𝑎𝑥! + 𝑏𝑥 + 𝑐. Addorsubtracttermswiththesamevariables(𝑥! & 𝑥!, 𝑥 & 𝑥, 𝑛𝑢𝑚𝑏𝑒𝑟 & 𝑛𝑢𝑚𝑏𝑒𝑟).
1.Rewritethequadraticequationinstandardform.
𝑓 𝑥 = 7𝑥 + 4 ! + 1
2.Rewritethequadraticequationinstandardform.
𝑓 𝑥 = 2𝑥 − 8 ! + 3
3.Rewritethequadraticequationinstandardform.
𝑓 𝑥 = 9𝑥 − 8 ! − 5
4.Rewritethequadraticequationinstandardform.
𝑓 𝑥 = 6𝑥 − 5 ! − 2
5.Rewritethequadraticequationinstandardform.
𝑓 𝑥 = 5𝑥 + 1 ! − 4
6.Rewritethequadraticequationinstandardform.
𝑓 𝑥 = 8𝑥 + 2 ! + 3
7.Rewritethequadraticequationinstandardform.
𝑓 𝑥 = 2𝑥 − 9 ! − 3
8.Rewritethequadraticequationinstandardform.
𝑓 𝑥 = 6𝑥 − 1 ! − 2
9.Rewritethequadraticequationinstandardform.
𝑓 𝑥 = 3𝑥 + 8 ! + 9
Tocompletethesquaretofindpandq(problems67–69):
Therearetwowaystosolvethisproblem.OPTION1:Determinethevertextowritetheprobleminvertexform. ℎ = − !
!!,thenplugℎinforallthe𝑥-valuesintheoriginalequationtofind𝑘.
Rewritetheproblemintheform𝑎 𝑥 − ℎ ! + 𝑘 = 0Then,movektotheothersidebyaddingorsubtractingit.Identify𝑝&𝑞basedoffoftheequation 𝑥 − 𝑝 ! = 𝑞 Basically,𝑝 = ℎ,and𝑞 = −𝑘(switchk’ssign).OPTION2(see#19-27onReviewF):Startbymoving𝑐totheothersideoftheequalsign.
Then,rewrite𝑥! + 𝑏𝑥 = −𝑐as 𝑥 + !!
!= −𝑐 + !
!
!
Dividethenumberinfrontofthex-termby2.Writethatnumberinsquaredparenthesesontheleft:
𝑥 𝑠𝑎𝑚𝑒 𝑠𝑖𝑔𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑏𝑦 2 !=
Whatevernumberyouputontheleft,squareitandaddthatnumbertothenumberontheright.Thensimplifytherightside&identify𝑝&𝑞basedoffoftheequation 𝑥 − 𝑝 ! = 𝑞.
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IM2Sem1FinalReviewG 2
Forexample:𝑥! − 100𝑥 − 8 = 0 𝑥! − 100𝑥 = 8100 ÷ 2 = 50 & 50 ! = 2500
𝑥 − 50!= 8 + 2500
𝑥 − 50 ! = 2508 𝑝 = 50, 𝑞 = 2508 10.Theequation𝑥! + 10𝑥 + 1 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.
Constant Value𝑝 𝑞
11.Theequation𝑥! − 6𝑥 − 24 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.
Constant Value𝑝 𝑞
12.Theequation𝑥! + 18𝑥 − 5 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.
Constant Value𝑝 𝑞
13.Theequation𝑥! + 6𝑥 + 15 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.
Constant Value𝑝 𝑞
14.Theequation𝑥! − 8𝑥 + 23 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.
Constant Value𝑝 𝑞
15.Theequation𝑥! + 34𝑥 + 2 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.
Constant Value𝑝 𝑞
16.Theequation𝑥! + 20𝑥 − 13 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.
Constant Value𝑝 𝑞
17.Theequation𝑥! − 22𝑥 − 17 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.
Constant Value𝑝 𝑞
18.Theequation𝑥! − 12𝑥 + 4 = 0canbetransformedintoanequationoftheform 𝑥 − 𝑝 ! = 𝑞,where𝑝and𝑞arerationalnumbers.Completethetablebelowwiththevaluesof𝑝and𝑞.
Constant Value𝑝 𝑞
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IM2Sem1FinalReviewG 3
Tographaquadraticinvertexform(problems70-72):
Startbydeterminingyourvertex(ℎ, 𝑘),whichisgiventoyouinvertexform:𝑓 𝑥 = 𝑎 𝑥 − ℎ ! + 𝑘.Graphthatpoint.Youneedaminimumof5points.Youmusthavethevertexandcrossortouchboththex-andthey-axes. Theonlyreasonsforyourgraphnottocrossbothaxeswouldbe
1. Iftheprovidedgraphingspacewasnotbigenoughtoallowyoutocrossthem.2. Ifthegraphcannevercrossaxis(therootsareimaginary).
Thereareseveralwaystofindtheotherpointsthatyouneed.Theprocessbelowusesanxytable.x y
Putyourvertexinthemiddleofanx-ytable
Onthexside,includeasmanyx’sinbothdirectionsasyouthinkyouneed(movebyoneeachtime).
Now,startwithoneofthex’sthatis1awayfromthevertex.
Plugitintotheoriginalequation.Simplifytogety.Putthaty-valueinthetable
©ittothematchingxontheothersideofthevertex.Graphthosetwopoints.
Repeatwiththex’sthatare2away,andsoonuntilyouhavetouched
orcrossedeachaxis.
Ifyourgraphlooksstrange,youhavelikelymadeamistake–gobackandcheckyourwork.
3less
Same
Same
2less
Same
1lessthanthevertexx
Vertexx Vertexy
1morethanthevertexx
2more
3more
19.Graph.Labelthevertexandaxisofsymmetry.
𝑓 𝑥 = − 𝑥 − 3 ! + 8
20.Graph.Labelthevertexandaxisofsymmetry.
𝑓 𝑥 = − 𝑥 + 4 ! + 10
21.Graph.Labelthevertexandaxisofsymmetry.
𝑓 𝑥 = 𝑥 + 1 ! − 2
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IM2Sem1FinalReviewG 4
22.Graph.Labelthevertexandaxisofsymmetry.
𝑓 𝑥 = 2 𝑥 − 1 ! − 8
23.Graph.Labelthevertexandaxisofsymmetry.
𝑓 𝑥 = − 𝑥 − 4 ! + 3
24.Graph.Labelthevertexandaxisofsymmetry.
𝑓 𝑥 = 𝑥 + 2 ! − 4
25.Graph.Labelthevertexandaxisofsymmetry.
𝑓 𝑥 = − 𝑥 − 2 ! − 3
26.Graph.Labelthevertexandaxisofsymmetry.
𝑓 𝑥 = 𝑥! − 1
27.Graph.Labelthevertexandaxisofsymmetry.
𝑓 𝑥 = − 𝑥 + 3 ! + 5
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IM2Sem1FinalReviewG 5
IM2Semester1FinalExamReviewGAnswers
1.𝑓 𝑥 = 49𝑥! + 56𝑥 + 17 2.𝑓 𝑥 = 4𝑥! − 32𝑥 + 67 3.𝑓 𝑥 = 81𝑥! − 144𝑥 + 594.𝑓 𝑥 = 36𝑥! − 60𝑥 + 23 5.𝑓 𝑥 = 25𝑥! + 10𝑥 − 3 6.𝑓 𝑥 = 64𝑥! + 32𝑥 + 77.𝑓 𝑥 = 4𝑥! − 36𝑥 + 78 8.𝑓 𝑥 = 36𝑥! − 12𝑥 − 1 9.𝑓 𝑥 = 9𝑥! + 48𝑥 + 7310.𝑝 = −5; 𝑞 = 24 11.𝑝 = 3; 𝑞 = 33 12.𝑝 = −9; 𝑞 = 8613.𝑝 = −3; 𝑞 = −6 14.𝑝 = 4; 𝑞 = −7 15.𝑝 = −17; 𝑞 = 28716.𝑝 = −10; 𝑞 = 113 17.𝑝 = 11; 𝑞 = 138 18.𝑝 = 6; 𝑞 = −3219.
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