algebra 2 unit 3: linear functions algebra 2 unit 3: linear … · 2018. 11. 29. · algebra 2 unit...
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Algebra2Unit3:LinearFunctions
Ms.Talhami 1
Algebra2Unit3:LinearFunctions
Name_________________
Algebra2Unit3:LinearFunctions
Ms.Talhami 2
DIRECT VARIATION COMMONCOREALGEBRAII
Webeginour linearunitby lookingatthesimplest linearrelationshipthatcanexistbetweentwovariables,namelythatofdirectvariation.Wesaythattwovariablesaredirectlyrelatedorproportionaltooneanotherifthefollowingrelationshipholds.
Exercise#1:Ineachofthefollowing,xandyaredirectlyrelated.Solveforthemissingvalue.
(a) 15 when 5
? when 9
y x
y x
= =
= =
(b) 6 when 4
? when 10
y x
y x
= − =
= = −
(c) 12 when 16
? when 24
y x
y x
= =
= =
Exercise#2:Thedistanceapersoncantravelvariesdirectlywiththetimetheyhavebeentravelingifgoingataconstantspeed.IfPhoenixtraveled78milesin1.5hourswhilegoingataconstantspeed,howfarwillhetravelin2hoursatthesamespeed?Exercise#3:Jennaworksajobwhereherpayvariesdirectlywiththenumberofhoursshehasworked.Inoneweek, sheworked35hoursandmade$274.75.Howmanyhourswouldsheneed towork inorder toearn$337.55?
PROPORTIONALORDIRECTRELATIONSHIPS
Twovariables,xandy,haveadirect(proportional)relationshipifforeveryorderedpair wehave:
Stated succinctly,ywill always be a constantmultiple of x. The value ofk is known as the constant ofvariation.
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Wewill now examine the graph of a direct relationship and seewhy it is indeed the simplest of all linearfunctions.
Exercise#4:Twovariables,xandy,varydirectly.When 6x = then 4y = .Thepointisshownplottedbelow.
(a) Findthey-valuesforeachofthefollowingx-values.Ploteachpointandconnect.
3x = 6x = − (b)Whatistheconstantofvariationinthisproblem?Whatdoes
itrepresentonthisline?(c)Writetheequationofthelineyouplottedin(a).Directrelationshipsoftenexistbetweentwovariableswhosevaluesarezerosimultaneously.Exercise#3:Themilesdrivenbyacar,d,variesdirectlywiththenumberofgallons,g,ofgasolineused.Abagailisabletodrive 336d = mileson 8g = gallonsofgasolineinherhybridvehicle.
(a) Calculate the constant of variation for the
relationship . Includeproperunits inyour
answer.
(b)Give a linear equation that represents therelationship betweend andg. Express youranswerasanequationsolvedford.
(c) HowfarcanAbagaildriveon gallonsofgas?
(d)HowmanygallonsofgaswillAbagailneed inordertodrive483miles?
y
x
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DIRECT VARIATION COMMONCOREALGEBRAIIHOMEWORK
FLUENCY
1. Ineachofthefollowing,thevariablepairgivenareproportionaltooneanother.Findthemissingvalue. (a) 8 when 16
? when 18
b a
b a
= =
= =
(b) 10 when 14
? when 21
y x
y x
= =
= =
(c) 2 when 6
? when 15
w u
w u
= − =
= = −
2. Inthefollowingexercises,thetwovariablesgivenvarydirectlywithoneanother. Solveforthemissing
value. (a) 12 when 8
? when 6
p q
p q
= =
= =
(b) 21 when 9
? when 6
y x
y x
= =
= = −
(c) 5 when 2
? when 8
z w
z w
= − =
= =
3. Ifxandyvarydirectlyand 16 when 12y x= = ,thenwhichofthefollowingequationscorrectlyrepresents
therelationshipbetweenxandy?
(1) 34
y x= (3) 192xy =
(2) 28y x+ = (4) 43
y x=
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APPLICATIONS
4. ThedistanceMax’sbikemoves isdirectlyproportional tohowmanyrotationshisbike’scrankshafthasmade.IfMax’sbikemoves25feetaftertworotations,howmanyfeetwillthebikemoveafter15rotations?
5. Forhisworkout,theincreaseinJacob’sheartrateisdirectlyproportionaltotheamountoftimehehasspent
workingout. Ifhisheartbeathasincreasedby8beatsperminuteafter20minutesofworkingout,howmuchwillhisheartbeathaveincreasedafter30minutesofworkingout?
6. Whenaphotographisenlargedorshrunken,itswidthandlengthstayproportionaltotheoriginalwidthand
length.Rojasisenlargingapicturewhoseoriginalwidthwas3inchesandwhoseoriginallengthwas5inches.Ifitsnewlengthistobe8inches,whatistheexactvalueofitsnewwidthininches?
7. Forasetamountoftime,thedistanceKirkcanrunisdirectlyrelatedtohisaveragespeed.IfKirkcanrun3
miles inwhile runningat6milesperhour,howfarcanherun in thesameamountof time ifhisspeedincreasesto10milesperhour?
REASONING8. Twovariablesareproportionaliftheycanbewrittenat y kx= ,wherekissomeconstant.Thisleadstothe
fact that when 0x = then 0y = as well. Is the temperature measured in Celsius proportional to thetemperaturemeasuredinFahrenheit?Explain.
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AVERAGE RATE OF CHANGE COMMONCOREALGEBRAII
Whenwemodelusingfunctions,weareveryofteninterestedintheratethattheoutputischangingcomparedtotherateoftheinput.Exercise#1:Thefunction ( )f x isshowngraphedtotheright.(a) Evaluateeachofthefollowingbasedonthegraph: (i) ( )0f (ii) ( )4f (iii) ( )7f (iv) ( )13f (b) Find the change in the function, fΔ , over each of the
following domain intervals. Find this both by subtractionandshowthisonthegraph.
(i)0 4x≤ ≤ (ii) 4 7x≤ ≤ (iii)7 13x≤ ≤ (c) Whycan'tyousimplycomparethechangesinffrompart(b)todetermineoverwhichintervalthefunction
changingthefastest?(d) Calculatetheaveragerateofchangeforthefunctionovereachoftheintervalsanddeterminewhichinterval
hasthegreatestrate. (i)0 4x≤ ≤ (ii) 4 7x≤ ≤ (iii)7 13x≤ ≤ (e) Usingastraightedge,drawinthelineswhoseslopesyoufoundinpart(d)byconnectingthepointsshown
onthegraph.Theaveragerateofchangegivesameasurementofwhatpropertyoftheline?
y
x
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Theaveragerateofchangeisanexceptionallyimportantconceptinmathematicsbecauseitgivesusawaytoquantifyhowfastafunctionchangesonaverageoveracertaindomaininterval.Althoughweuseditsformulainthelastexercise,westateitformallyhere:Exercise#2:Considerthetwofunctions ( ) 5 7f x x= + and ( ) 22 1g x x= + .(a) Calculatetheaveragerateofchangeforbothfunctionsoverthefollowingintervals.Doyourworkcarefully
andshowthecalculationsthatleadtoyouranswers.
(i) 2 3x− ≤ ≤ (ii)1 5x≤ ≤ (b) Theaveragerateofchangeforfwasthesameforboth(i)and(ii)butwasnotthesameforg.Whyisthat?Exercise#3:Thetablebelowrepresentsalinearfunction.Fillinthemissingentries.
AVERAGERATEOFCHANGE
Forafunctionoverthedomaininterval ,thefunction'saveragerateofchangeiscalculatedby:
x 1 5 11 45
y -5 1 22
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AVERAGE RATE OF CHANGE COMMONCOREALGEBRAIIHOMEWORK
FLUENCY
1. Forthefunction ( )g x giveninthetablebelow,calculatetheaveragerateofchangeforeachofthefollowing
intervals. (a) 3 1x− ≤ ≤ − (b) 1 6x− ≤ ≤ (c) 3 9x− ≤ ≤ (d)Explainhowyoucantellfromtheanswersin(a)through(c)thatthis isnotatablethatrepresentsa
linearfunction.2. Considerthesimplequadraticfunction ( ) 2f x x= .Calculatetheaveragerateofchangeofthisfunctionover
thefollowingintervals: (a)0 2x≤ ≤ (b) 2 4x≤ ≤ (c) 4 6x≤ ≤ (d)Clearlytheaveragerateofchangeisgettinglargeratxgetslarger.
Howisthisreflectedinthegraphoffshownsketchedtotheright?
x 4 6 9 8 13 12 5
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3. Whichhasagreateraveragerateofchangeovertheinterval 2 4x− ≤ ≤ ,thefunction ( ) 16 3g x x= − orthe
function ( ) 22f x x= ?Providejustificationforyouranswer.APPLICATIONS4. Anobjecttravelssuchthatitsdistance,d,awayfromitsstartingpointisshownasafunctionoftime,t,in
seconds,inthegraphbelow. (a) Whatistheaveragerateofchangeofdoverthe
interval 5 7t≤ ≤ ? Includeproperunits inyouranswer.
(b) Theaveragerateofchangeofdistanceovertime
(what you found in part (a)) is known as theaveragespeedofanobject.Istheaveragespeedofthisobjectgreaterontheinterval0 5t≤ ≤ or11 14t≤ ≤ ?Justify.
REASONING5. Whatmakestheaveragerateofchangeofalinearfunctiondifferentfromthatofanyotherfunction?What
isthespecialnamethatwegivetotheaveragerateofchangeofalinearfunction?
Time(seconds)
Distan
ce(fee
t)
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FORMS OF A LINE COMMONCOREALGEBRAII
Linear functions come ina varietyof forms. The two shownbelowhavebeen introduced inCommonCoreAlgebraIandCommonCoreGeometry.Exercise#1:Considerthelinearfunction ( ) 3 5f x x= + .
Exercise#2:Consideralinewhoseslopeis5andwhichpassesthroughthepoint ( )2, 8− .
Exercise#3:Whichofthefollowingrepresentsanequationforthelinethatisparallelto 3 72
y x= − andwhich
passesthroughthepoint ( )6, 8− ?
(1) ( )28 63
y x− = − + (3) ( )38 62
y x+ = −
(2) ( )38 62
y x− = + (4) ( )28 63
y x+ = − −
TWOCOMMONFORMSOFALINE
Slope-Intercept: Point-Slope:
wheremistheslope(oraveragerateofchange)ofthelineand representsonepointontheline.
(a) Determine they-interceptof this functionbyevaluating .
(b) Find its average rate of change over theinterval .
(a) Write theequationof this line inpoint-slopeform, .
(b)Write the equation of this line in slope-interceptform, .
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Exercise#4:Alinepassesthroughthepoints ( ) ( )5, 2 and 20, 4− .
Exercise#5:Thegraphofalinearfunctionisshownbelow.(a)Writetheequationofthislinein y mx b= + form.(b)Whatmustbetheslopeofalineperpendiculartothe
oneshown?(c)Drawalineperpendiculartotheoneshownthat
passesthroughthepoint ( )1, 3 .(d)Writetheequationofthelineyoujustdrewinpoint-
slopeform.
(a) Determinetheslopeofthislineinsimplestrationalform.
(b)Writeanequationofthislineinpoint-slopeform.
(c) Writeanequationforthislineinslope-interceptform.
(d) Forwhatx-valuewillthislinepassthroughay-valueof12?
y
x
(e)Doesthelinethatyoudrewcontainthepoint?Justify.
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FORMS OF A LINE COMMONCOREALGEBRAIIHOMEWORK
FLUENCY
1. Whichofthefollowinglinesisperpendicularto 5 73
y x= − andhasay-interceptof4?
(1) 5 43
y x= + (3) 345
y x= −
(2) 3 45
y x= − + (4) 3 45
y x= +
2. Whichofthefollowinglinespassesthroughthepoint ( )4, 8− − ?
(1) ( )8 3 4y x+ = + (3) ( )8 3 4y x+ = −
(2) ( )8 3 4y x− = − (4) ( )8 3 4y x− = +
3. Whichofthefollowingequationscoulddescribethegraphofthelinearfunctionshownbelow?
(1) 2 43
y x= − (3) 2 43
y x= − −
(2) 2 43
y x= + (4) 2 43
y x= − +
4. Foralinewhoseslopeis 3− andwhichpassesthroughthepoint ( )5, 2− :5.Foralinewhoseslopeis0.8 andwhichpassesthroughthepoint ( )3,1− :
y
x
(a) Writetheequationofthislineinpoint-slopeform, .
(b)Write the equation of this line in slope-interceptform, .
(a) Writetheequationofthislineinpoint-slopeform, .
(b)Write the equation of this line in slope-interceptform, .
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REASONING6. Thetwopoints ( ) ( )3, 6 and 6, 0− areplottedonthegridbelow.
(a)Find an equation, in y mx b= + form, for the linepassing through these two points. Use of the grid isoptional.
(b)Doesthepoint ( )30, 16− lieonthisline?Justify.7. Alinearfunctionisgraphedbelowalongwiththepoint ( )3,1 .
(a) Drawalineparalleltotheoneshownthatpassesthroughthepoint ( )3,1 .
(b)Writeanequationforthelineyoujustdrewinpoint-slope
form.
(c) Betweenwhattwoconsecutiveintegersdoesthey-interceptofthelineyoudrewfall?
(d)Determinetheexactvalueofthey-interceptofthelineyoudrew.
y
x
y
x
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LINEAR MODELING COMMONCOREALGEBRAII
InCommonCoreAlgebraI,youusedlinearfunctionstomodelanyprocessthathadaconstantrateatwhichonevariablechangeswithrespecttotheother,oraconstantslope.Inthislessonwewillreviewmanyofthefacetsofthistypeofmodeling.Exercise#1:DiawasdrivingawayfromNewYorkCityataconstantspeedof58milesperhour.Hestarted45milesaway.
InExercise#1, it isclear fromthecontextwhatboththeslopeandthey-interceptof this linearmodelare.Althoughthisisoftenthecasewhenconstructingalinearmodel,sometimestheslopeandapointareknown,inwhichcase,thepointslopeformofthealineismoreappropriate.Exercise#2:Edelynistryingtomodelhercell-phoneplan.Sheknowsthatithasafixedcost,permonth,alongwitha$0.15chargepercallshemakes.Inherlastmonth’sbill,shewascharged$12.80formaking52calls.
(a) Write a linear function that gives Dia’sdistance,D, fromNewYorkCityasafunctionofthenumberofhours,h,hehasbeendriving.
(b) IfDia’sdestinationis270milesawayfromNewYork City, algebraically determine to thenearesttenthofanhourhowlongitwilltakeDiatoreachhisdestination.
(a) Createalinearmodel,inpoint-slopeform,forthe amount Edelyn must pay, P, per monthgiventhenumberofphonecallsshemakes,c.
(b)How much is Edelyn’s fixed cost? In otherwords,howmuchwould shehave topay formakingzerophonecalls?
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Manytimeslinearmodelshavebeenconstructedandweareaskedonlytoworkwiththesemodels.Modelsintherealworldcanbemessyanditisoftenconvenienttouseourgraphingcalculatorstoplotandinvestigatetheirbehavior.Exercise#3:Afactoryproduceswidgets(genericobjectsofnoparticularuse).Thecost,C,indollarstoproducewwidgetsisgivenbytheequation 0.18 20.64C w= + .Eachwidgetsellsfor26cents.Thus,therevenuegained,R,fromsellingthesewidgetsisgivenby 0.26R w= .
(a) Use your graphing calculator to sketch andlabel each of these linear functions for theinterval . Besuretolabelyoury-axiswithitsscale.
(b)Use your calculator’s INTERSECT command todetermine the number of widgets, w, thatmustbeproducedfortherevenuetoequalthecost.
(c) If profit is defined as the revenueminus thecost,createanequationintermsofwfortheprofit,P.
w
Dollars
(e)Whatistheminimumnumberofwidgetsthatmustbesoldinorderfortheprofittoreachatleast$40?Illustratethisonyourgraph.
(d)Usingyourgraphingcalculator,sketchagraphof the profit over the interval .UseaTABLEonyourcalculatortodetermineanappropriateWINDOW forviewing. Label thexandyinterceptsofthislineonthegraph.
w
Dollars
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LINEAR MODELING COMMONCOREALGEBRAIIHOMEWORK
APPLICATIONS1. Whichofthefollowingwouldmodelthedistance,D,adriverisfromChicagoiftheyareheadingtowards
thecityat58milesperhourandstarted256milesaway?
(1) 256 58D t= + (3) 58 256D t= +
(2) 256 58D t= − (4) 58 256D t= − 2. Thecost,C,ofproducingx-bikesisgivenby 22 132C x= + .Therevenuegainedfromsellingx-bikesisgiven
by 350R x= .Iftheprofit,P,isdefinedas P R C= − ,thenwhichofthefollowingisanequationforPintermsofx?
(1) 328 132P x= − (3) 328 132P x= + (2) 372 132P x= + (4) 372 132P x= −
3. Theaveragetemperatureoftheplanetisexpectedtoriseatanaveragerateof0.04degreesCelsiusperyearduetoglobalwarming.Theaveragetemperatureintheyear2000was14.71degreesCelsius.TheaverageCelsiustemperature,C,isgivenby 14.71 0.04C x= + ,wherexrepresentsthenumberofyearssince2000.
(a) What will be the average temperature intheyear2100?
(b) Algebraically determine the number ofyears,x,itwilltakeforthetemperature,C,toreach20degreesCelsius.Roundtothenearestyear.
(c) Sketch a graph of the average yearlytemperature below for the interval
. Be sure to label your y-axisscaleaswellastwopointsontheline(they-interceptandoneadditionalpoint).
(d)What does this model project to be theaverageglobaltemperaturein2200?
x
C(Celsius)
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4. FabioisdrivingwestawayfromAlbanyandtowardsBuffaloalongInterstate90ataconstantrateofspeedof62milesperhour.Afterdrivingfor1.5hours,Fabiois221milesfromAlbany.
5. AparticularrockettakingofffromtheEarth’ssurfaceusesfuelataconstantrateof12.5gallonsperminute.
Therocketinitiallycontains225gallonsoffuel.
(a) Writea linearmodelforthedistance,D,thatFabioisawayfromAlbanyasafunctionofthenumberofhours,h,thathehasbeendriving.Write your model in point-slope form,
.
(b) Rewrite this model in slope-intercept form,.
(c) How far was Fabio from Albany when hestartedhistrip?
(d) IfthetotaldistancefromAlbanytoBuffalois290 miles, determine how long it takes forFabiotoreachBuffalo.Roundyouranswertothenearesttenthofanhour.
(a) Determinealinearmodel,in form,fortheamountoffuel,y,asafunctionofthenumber of minutes, x, that the rocket hasburned.
(b) Belowisageneralsketchofwhatthegraphofyour model should look like. Using yourcalculator,determinethexandyinterceptsofthis model and label them on the graph atpointsAandBrespectively.
(c)Therocketmuststillcontain50gallonsoffuelwhen it hits the stratosphere. What is themaximumnumber ofminutes the rocket cantaketohitthestratosphere?Showthispointonyourgraphbyalsographingthehorizontalline and showing the intersectionpoint. x
y
A
B
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INVERSES OF LINEAR FUNCTIONS COMMONCOREALGEBRAII
Recall that functions have inverses that are also functions if they are one-to-one. With the exception ofhorizontallines,alllinearfunctionsareone-to-oneandthushaveinversesthatarealsofunctions.Inthislessonwewillinvestigatetheseinversesandhowtofindtheirequations.
Exercise#1:Onthegridbelowthelinearfunction 2 4y x= − isgraphedalongwiththeline y x= .
(a) How can you quickly tell that 2 4y x= − is a one-to-onefunction?
(b)Graphtheinverseof 2 4y x= − onthesamegrid.Recallthat
thisiseasilydonebyswitchingthexandycoordinatesoftheoriginalline.
(c)Whatcanbesaidaboutthegraphsof 2 4y x= − anditsinverse
withrespecttotheline y x= ?
Aswecanseefrompart(e) inExercise#1, inversesof linearfunctionsincludetheinverseoperationsoftheoriginalfunctionbutinreverseorder.Thisgivesrisetoasimplemethodoffindingtheequationofanyinverse.Simplyswitchthexandyvariablesintheoriginalequationandsolvefory.
Exercise#2:Whichofthefollowingrepresentstheequationoftheinverseof 5 20y x= − ?
(1) 1 205
y x= − + (3) 1 45
y x= −
(2) 1 205
y x= − (4) 1 45
y x= +
y
x
(d)Findtheequationoftheinversein form.
(e) Find theequationof the inverse in
form.
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Althoughthisisasimpleenoughprocedure,certainproblemscanleadtocommonerrorswhensolvingfory.Careshouldbetakenwitheachalgebraicstep.
Exercise#3:Whichofthefollowingrepresentstheinverseofthelinearfunction 2 83
y x= + ?
(1) 3 82
y x= − (3) 3 82
y x= − +
(2) 3 122
y x= − (4) 3 122
y x= − +
Exercise#4:Whatisthey-interceptoftheinverseof 3 95
y x= − ?
(1) 15y = (3) 9y =
(2) 19
y = (4) 53
y = −
Sometimesweareaskedtoworkwithlinearfunctionsintheirpoint-slopeform.Themethodoffindingtheinverseandplottingit,though,donotchangejustbecausethelinearequationiswritteninadifferentform.
Exercise#5:Whichofthefollowingwouldbeanequationfortheinverseof ( )6 4 2y x+ = − ?
(1) ( )12 64
y x− = + (3) ( )6 4 2y x− = − +
(2) ( )12 64
y x− = − + (4) ( )2 4 6y x+ = − −
Exercise#6: Whichofthefollowingpoints liesonthegraphofthe inverseof ( )8 5 2y x− = + ?Explainyourchoice.
(1) ( )8, 2− (3) ( )10, 40−
(2) ( )8, 2− (4) ( )2, 8− Exercise#7:Whichofthefollowinglinearfunctionswouldnothaveaninversethatisalsoafunction?Explainhowyoumadeyourchoice. (1) y x= (3) 2y = (2) 2y x= (4) 5 1y x= −
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INVERSES OF LINEAR FUNCTIONS COMMONCOREALGEBRAIIHOMEWORK
FLUENCY
1. Thegraphofafunctionanditsinversearealwayssymmetricacrosswhichofthefollowinglines?
(1) 0y = (3) y x=
(2) 0x = (4) 1y =
2. Whichofthefollowingrepresentstheinverseofthelinearfunction 3 24y x= − ?
(1) 1 83
y x= + (3) 1 243
y x= − +
(2) 1 83
y x= − − (4) 1 13 24
y x= −
3. Ifthey-interceptofalinearfunctionis8,thenweknowwhichofthefollowingaboutitsinverse?
(1)Itsy-interceptis 8− . (3)Itsy-interceptis 18.
(2)Itsx-interceptis8. (4)Itsx-interceptis 8− .
4. Ifbothwereplotted,whichofthefollowinglinearfunctionswouldbeparalleltoitsinverse?Explainyourthinking.
(1) 2y x= (3) 5 1y x= −
(2) 2 43
y x= − (4) 6y x= +
5.Whichofthefollowingrepresentstheequationoftheinverseof 4 243
y x= + ?
(1) 4 243
y x= − − (3) 3 184
y x= −
(2) 3 184
y x= − + (4) 4 243
y x= −
6.Whichofthefollowingpointsliesontheinverseof ( )2 4 1y x+ = − ?
(1) ( )2, 1− (3)1 ,12
⎛ ⎞⎜ ⎟⎝ ⎠
(2) ( )1, 2− (4) ( )2,1−
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7. Alinearfunctionisgraphedbelow.Answerthefollowingquestionsbasedonthisgraph. (a)Writetheequationofthislinearfunctionin y mx b= + form. (b)Sketchagraphoftheinverseofthisfunctiononthesamegrid. (c)Writetheequationoftheinversein y mx b= + form. (d)Whatistheintersectionpointofthislinewithitsinverse?APPLICATIONS
8. Acartravelingataconstantspeedof58milesperhourhasadistanceofy-milesfromPoughkeepsie,NY,givenbytheequation 58 24y x= + ,wherexrepresentsthetimeinhoursthatthecarhasbeentraveling.
(c) Giveaphysicalinterpretationoftheansweryoufoundinpart(b).Considerwhattheinputandoutput
oftheinverserepresentinordertoanswerthisquestion.REASONING
9. Giventhegenerallinearfunction y mx b= + ,findanequationforitsinverseintermsofmandb.
(a) Findtheequationoftheinverseofthislinear
functionin form.
(b) Evaluatethefunctionyoufoundinpart(a)foraninputof .
y
x
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PIECEWISE LINEAR FUNCTIONS COMMONCOREALGEBRAII
Functions expressed algebraically can sometimes bemore complicated and involvedifferent equations fordifferentportionsoftheirdomains.Theseareknownaspiecewisefunctions(theycomeinpieces).Ifallofthepiecesarelinear,thentheyareknownaspiecewiselinearfunctions.
Exercise#1:Considerthepiecewiselinearfunctiongivenbytheformula ( )3 3 0
1 4 0 42
x xf x
x x
− − ≤ <⎧⎪= ⎨ + ≤ ≤⎪⎩
.
(a) Createatableofvaluesbelowandgraphthefunction.(b) Statetherangeoffusingintervalnotation.Notonlyshouldwebeabletographpiecewisefunctionswhenwearegiventheirequations,butweshouldalsobeabletotranslatethegraphsofthesefunctionsintoequations.Exercise#2:Thefunction ( )f x isshowngraphedbelow.Writeapiecewiselinearformulaforthefunction.Besuretospecifyboththeformulasandthedomainintervalsoverwhichtheyapply.
y
x
x 0 1 2 3 4
y
x
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Piecewiseequationscanbechallengingalgebraically.Sometimesinformationthatwefindfromthemcanbemisleadingorincorrect.Exercise #3: Consider the piecewise linear function
( ) 12
5 22 2x x
g xx x− <⎧
= ⎨ + ≥⎩.
(e) Howcanyouresolvethefactthatthealgebraseemstocontradictyourgraphicalevidenceofx-intercepts?
Exercise #4: For the piecewise linear function ( ) 2 10 05 1 0x x
f xx x
− + ≤⎧= ⎨ − >⎩
, find all solutions to the equation
( ) 1f x = algebraically.
(a) Determine the y-intercept of this functionalgebraically. Why can a function have onlyoney-intercept?
(b) Findthex-interceptsofeach individual linearequation.
(c) Graphthepiecewiselinearfunctionbelow.
(d)Whydoesyourgraphcontradict theanswersyoufoundinpart(b)?
y
x
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PIECEWISE LINEAR FUNCTIONS COMMONCOREALGEBRAIIHOMEWORK
FLUENCY
1. For ( )13
5 3 28 2 37 3
x xf x x x
x x
− < −⎧⎪= + − ≤ <⎨⎪ + ≥⎩
answerthefollowingquestions.
(a) Evaluateeachofthefollowingbycarefullyapplyingthecorrectformula: (i) ( )2f (ii) ( )4f − (iii) ( )3f (iv) ( )0f (b) Thethreelinearequationshavey-interceptsof 3, 8 and 7− respectively.Yet,afunctioncanhaveonly
oney-intercept.Whichoftheseisthey-interceptofthisfunction?Explainhowyoumadeyourchoice. (c) Calculatetheaveragerateofchangeoffovertheinterval 3 9x− ≤ ≤ .Showthecalculationsthatleadto
youranswer.
2. Determinetherangeofthefunction ( ) 32 64 2 29 2 x
x xg x
x < ≤
+ − ≤ ≤⎧= ⎨− +⎩
graphically.
y
x
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Ms.Talhami 25
3. Determineapiecewiselinearequationforthefunction ( )f x shownbelow.Besuretospecifynotonlytheequations,butalsothedomainintervalsoverwhichtheyapply.
REASONING
4. Stepfunctionsarepiecewisefunctionsthatareconstants(horizontallines)overeachpartoftheirdomains.Graphthefollowingstepfunction.
( )
2 0 33 3 57 5 105 10 12
xx
f xxx
− ≤ <⎧⎪ ≤ <⎪= ⎨ ≤ <⎪⎪ ≤ ≤⎩
5. Find all x-intercepts of the function ( ) 12
2 8 5 14 1 1
4 10 1 4
x xg x x x
x x
+ − ≤ < −⎧⎪= − − − ≤ <⎨⎪− + ≤ ≤⎩
algebraically. Justify your work by
showingyouralgebra.Besuretocheckyouranswersversusthedomainintervalstomakesureeachsolutionisvalid.
y
x
y
x
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SYSTEMS OF LINEAR EQUATIONS COMMONCOREALGEBRAIISystemsofequations,ormorethanoneequation,arisefrequentlyinmathematics.Tosolveasystemmeanstofindallsetsofvaluesthatsimultaneouslymakeallequationstrue.Ofspecialimportancearesystemsoflinearequations. You have solved them in your last two Common Core math courses, but we will add to theircomplexityinthislesson.Exercise#1:Solvethefollowingsystemofequationsby:(a)substitutionand(b)byelimination.(a)3 2 92 7x yx y+ = −+ = −
(b)3 2 92 7x yx y+ = −+ = −
You should be very familiar with solving two-by-two systems of linear equations (two equations and twounknowns).Inthislesson,wewillextendthemethodofeliminationtolinearsystemsofthreeequationsandthreeunknowns.TheselinearsystemsserveasthebasisforafieldofmathknownasLinearAlgebra.Exercise#2:Considerthethree-by-threesystemoflinearequationsshownbelow.Eachequationisnumberedinthisfirstexercisetohelpkeeptrackofourmanipulations.
2 156 3 354 4 14
x y zx y zx y z
+ + =− − =
− + − = −
(1)
(2)
(3)
(a) Theadditionpropertyofequalityallowsustoaddtwoequationstogethertoproduceathirdvalid equation. Create a system by addingequations (1)and (2)and (1)and (3).Why isthisaneffectivestrategyinthiscase?
(b)Usethisnewtwo-by-twosystemtosolvethethree-by-three.
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Just as with two by two systems, sometimes three-by-three systems need to be manipulated by themultiplicationpropertyofequalitybeforewecaneliminateanyvariables.Exercise#3: Consider the systemof equations shownbelow.Answer the followingquestions basedon thesystem.4 3 62 4 2 385 7 19
x y zx y zx y z
+ − = −− + + =
− − = −
Exercise#4:Solvethesystemofequationsshownbelow.Showeachstepinyoursolutionprocess.
4 2 3 235 3 37
2 4 27
x y zx y zx y z
− + =+ − = −
− + + =
(a) Which variable will be easiest to eliminate?Why? Use the multiplicative property ofequalityandeliminationtoreducethissystemtoatwo-by-twosystem.
(b) Solvethetwo-by-twosystemfrom(a)andfindthefinalsolutiontothethree-by-threesystem.
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SYSTEMS OF LINEAR EQUATIONS COMMONCOREALGEBRAIIHOMEWORKFLUENCY1. Thesumoftwonumbersis5andthelargerdifferenceofthetwonumbersis39.Findthetwonumbersby
settingupasystemoftwoequationswithtwounknownsandsolvingalgebraically.2. Algebraically,findtheintersectionpointsofthetwolineswhoseequationsareshownbelow.
4 3 136 8
x yy x
+ = −= −
3. Showthat 10, 4, and 7x y z= = = isasolutiontothesystembelowwithoutsolvingthesystemformally.
2 254 5 12 8 32
x y zx y zx y z
+ + =− − =
− − + =
4. Inthefollowingsystem,thevalueoftheconstantcisunknown,butitisknownthat 8x = − and 4y = are
thexandyvaluesthatsolvethissystem.Determinethevalueofc.Showhowyouarrivedatyouranswer.
5 2 3 811
2 35
x y zx y zx y cz
− + + =− + = −− + =
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Ms.Talhami 29
5. Solvethefollowingsystemofequations.Carefullyshowhowyouarrivedatyouranswers.
4 2 212 2 13
3 2 5 70
x y zx y zx y z
+ − =− − + =
− + =
6. Algebraicallysolvethefollowingsystemofequations.Therearetwovariablesthatcanbereadilyeliminated,
butyouranswerswillbethesamenomatterwhichyoueliminatefirst.
2 5 353 4 31
3 2 2 23
x y zx y zx y z
+ − = −− + =
− + + = −
7. Algebraically solve the following systemof equations. This systemwill takemoremanipulationbecause
therearenovariableswithcoefficientsequalto1.
2 3 2 334 5 3 546 2 8 50
x y zx y zx y z
+ − =+ + =
− − − = −