PKA35V.V. BEHAVABS.8xv**TI83F* AppVariable file 12/08/02, 17:12. .BEHAVABS . .GCVcv'B &99>"#%?'(^*,Behaviors-AbsValueEd LaughbaumSeptember 12, 2006RedBank Publishing $ABSThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83or84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhocan'tfollowrules.PNowtypethe>cursorkeytocontinue--butfirst...Reminder:|x|-meansabsolutevalue#ofx*inf-meansinfinity(or1increasingwithoutbound)PYoumayfindthatwritingthemathematicssymbolsandnotationcontainedinthislessoninproperformwillhelpyouunderstandit#andrememberitlonger.P1Youmayhaveheardofabsolutevaluerelationships.Ormaybenot?Belowaresituationsinvolvingmathematical#relationships.Selectthe*oneyouthinkisabsolute1valueinnature.P1.timevs.temperatureduringa"typical"AlbertaClippercoldfront2.datevs.hoursofsunlight3.agevs.diameterofa#treetrunk*4.volumevs.amountof1pollutantinyourwaterPThisisatoughonesinceyoumaynothavehadlifeexperiencewithabsolutevaluerelationships.Butrather,youmaythinkofit#as"somethingyoudotoa*number."Asitturnsout,#11behaveslikeanabsolutePvaluefunction.Belowisagraphofthistime-temperaturerelationshipduringanAlbertaClippercoldfrontinColumbus,OH.#*1P 8` @@@@`@@ @@ @!`A@@@@@` P@HDbA@@@@ `@@@@@`@@@ @@@@`@@@@H`0@@@@`@@@@BB!!C@@P2Whichofthefollowingisthesymbolicformofanabsolutevaluefunction?1.y=dx+e2.y=d(x+e)^2+f#3.y=dsqr(x+e)+f*4.y=d|x+e|+f15.y=d/(x+e)+fP#4iscorrect,y=d|x+e|+fThefunctionparametersared,e,andf.The#functionvariableisxand*thefunctionissymbolized1asy.Px3Iny=d|x+e|+f,whatisthefunctionvariable?1.d2.e3.f#4.x*5.yPKThesymbolsd,e,andfarefunctionparameters.Theyareconstantsthatcontrolthebehaviorsofthefunction.Thefunction#variableisx,andthename*(andvalue)ofthefunction1issymbolizedasy.PBelowisthegraphofy=2|x+1|-2#*1P 8Rݻv  @@@@ @@@@@    @XP4Thegraphofwhichofthefollowingfunctionsopenup?1.y=-2|x+1|-32.y=-(1/3)|x-2|-1#3.y=-10|x|+2*4.y=3|x-1|+215.y=0|x+2|-1PDidyougrapheachfunction?Ithelps,butisnotnecessaryifyouknowtheconnectionbetweentheparametersand#resultingbehaviors.#4is*correct.1Whatparametersets#4Papartfromtherest?P5Thegraphofwhichofthefollowingfunctionsopensdown?1.y=-2|x+1|-32.y=(1/3)|x-2|-1#3.y=10|x|+2*4.y=3|x-1|+215.y=0|x+2|-1PDidyougrapheachfunction?Ithelps,butisnotnecessaryifyouknowtheconnectionbetweentheparametersand#resultingbehaviors.#1is*correct.1Whatparametersets#1Papartfromtherest?P6Giveny=d|x+e|+f,whatparametercontrolswhetheritopensupordown?1.d#2.e*3.f14.notenoughinformationPDidyounoticeasyougraphedfunctionsfrompreviouscards,thatwhendisnegativethegraphopensdownandwhenitis#positive,itopensup?*1Lookforconnections!!!!P7Theslopeoftheleftbranchofy=3|x-1|is:1.12.-13.-3#4.3*5.??HowdoIfindout??PUsingthetablefeatureofyourcalculator,findtheslope.#3iscorrect.Ifyouhaveforgottenhowtofindtheslope,usethestack#called"slope."*Doyouseeanyconnection1betweenthefunctionPparametersandresultingbehaviors?P8Theslopeoftherightbranchofy=3|x|is:1.12.-1#3.-3*4.315.OK,Igiveup.PUsingthetablefeatureofyourcalculator,findtheslope.Graphtoseethebranches.#4iscorrect.Doyouseeanyconnection#betweenafunction*parameterandtheslope1behavior?P9Theslopeoftheleftbranchofy=5|x+1|is:1.12.-13.-5#4.5*5.Iamclueless.PUsingthetablefeatureofyourcalculator,findtheslope.Graphtoseethebranches.#3iscorrect.Doyouseeanyconnection#betweenafunction*parameterandtheslope1behavior?P 10Theslopeoftherightbranchofy=-5|x+1|is:1.12.-13.-5#4.5*5.MomwouldbehappyifI1knew!P#3iscorrect.Ifyouknewwhatparametercontrolstheslopebehavior,youcould#answerthiswithoutusinga*graphingcalculator?P 11Giveny=d|x+e|+f,whatparametercontrolstheslopesofthebranches?1.d2.e#3.f*4.y15.xPSure,itisparameterd.IthasTWOconnectionstobehaviors!--upordownandslopes#ofthebranches--P 12Giveny=-2|x+1|-3,whendoesitchangefromincreasingtodecreasing?1.whenx=-22.whenx=1#3.whenx=-1*4.whenx=-315.whenx=3PIfyougraphthefunction(onadecimalwindow),youwillseethatitchangeswhenx=-1.#Butisthereaparameter*thatwilltellyouthesame1thing?P 13Giveny=-3|x-2|+1,whendoesitchangefromincreasingtodecreasing?1.whenx=-32.whenx=1#3.whenx=-1*4.whenx=-215.whenx=2PIfyougraphthefunction(onadecimalwindow),youwillseethatitchangeswhenx=2.#Butisthereaparameter*thatwilltellyouthesame1thing?YouneedtostartPthinking.P 14Giveny=2|x+4|-1,whendoesitchangefromdecreasingtoincreasing?1.whenx=22.whenx=4#3.whenx=-4*4.whenx=-115.whenx=1PIfyougraphthefunction,youwillseethatitchangeswhenx=-4.Butisthereafunction#parameterthatwilltell*youthesamething?Use1yourbraintofindthePpattern.P 15Thegraphofanabsolutevaluefunctionhasavertex.Thevertexfory=-|x+2|-3is:1.(2,-3)#2.(-2,3)*3.(-2,-3)14.(2,3)PAgraphusingtraceshouldhelpyouanswerthis.Itisat(-2,-3).Butaren'ttherefunctionparametersthatgivethe#sameinformation?Ifthis*wereatest,youmaynot1havetimetofinishwhenPusingthegraphingcalculatortoanswerthequestion.Yourbraincanfigureouttheconnectionbetweentheparameter#andtheconnected*behavior.P 16Thegraphofanabsolutevaluefunctionhasavertex.Thevertexfory=2|x-3|+1is:1.(3,1)#2.(-3,1)*3.(-3,-1)14.(3,-1)PAgraphusingtraceshouldhelpyouanswerthis.Itisat(3,1).Butaren'ttherefunction#parametersthatgivethe*sameinformation?Think1aboutthisforaminute.P 17Thegraphofanabsolutevaluefunctionhasavertex.Thevertexfory=d|x+e|+fis:1.(e,f)#2.(-e,f)*3.(-e,-f)14.(e,-f)PIfyoudon'tseethatthevertexisat(-e,f),gobackandgraphseveralabsolutevaluefunctionsandcheckitout.#*Lookfortheconnections1betweenfunctionPparametersandtheresultingbehaviors!Yourbrainisactuallyquitegoodatfindingpatterns-connections.P 18Whatistheminimumvalueofthefunctiony=3|x-1|+2?1.32.-1#3.1*4.215.-2PThelowestvaluethefunctionhasis2.Whatparameteristhekeytohelpingyoufindthis?Agraphwillgiveyouthe#information,butknowing*theparameter-behavior1connectionhelpstoo.P 19Whatistherangeofthefunctiony=3|x-1|+2?1.yison3,inf)2.yison-1,inf)#3.yison1,inf)*4.yison2,inf)15.yison-2,inf)PParameterf(alongwithd)providesinformationontheminimum,sotherangestartsattheminimumandincreases.#*Therangeis2,inf).P 20Whenisy=-2|x+1|zero?1.itisn'tzero2.whenx=-23.whenx=14.whenx=-1#5.itiszero,butIdon't*knowwhenPYoushouldknowthatthevertexisat(-1,0),andthatitisonthex-axis.Thatis,youknowthefunctionis0whenxis-1.#Thesearethekindsofmath*problemtohave-lookat1itandwritetheanswer!P  21Whatmathvocabularywasdiscussedinthisactivity?1.Absolutevaluefunctions2.Vertex3.Maximum#4.Minimum*5.Range16.SlopeP7.Increasing/Decreasing8.Functionparameters9.AlltheabovePYes,alltheideaslistedonthefrontofthecardwereincluded.Wedidnotdiscussdomainsinceabsolutevaluefunctionshave#domainsofallreal*numbers.(-inf,inf)1YoushouldlookinyourPtextbookformoreinformationoneachterm.P 22Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*Ed#Materialsfrom*"FoundationsforCollege1Mathematics2e"P`PKA35#+11 BEHAVEXP.8xv**TI83F* AppVariable file 11/27/02, 18:580 0BEHAVEXP00G;IVi| 0'rO "%($+b,-@/Behaviors-ExpEd LaughbaumSeptember 12, 2006RedBank Publishing .ExpThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhowanttodosomethingdifferent.PNowtypethe>cursorkeytocontinue,butfirst...Reminder:inf-meansinfinity(or#increasingwithoutbound)*1YoumayfindthatwritingPthemathematicssymbolsandnotationcontainedinthislessoninproperformwillhelpyouunderstanditandrememberitlonger.P2Youmayhaveheardorreadaboutexponentialgrowth.Belowaresituationsinvolvinggrowth.Select#theoneyouthinkis*exponentialgrowth.1P1.timevs.heightofatree2.hoursworkedvs.salary3.timevs.thenumberofwordsread(whilereadingagoodmathbook)#4.timevs.human*populationofEarthP zwTheonlyrelationshipthatchangesexponentiallyisthetimevs.thepopulationoftheEarth.Belowisthepopulationrelationship#expressedingraphical*form.1P 8`  !!!!!!! !!!AAAA AAA    @~PThegraphseemstotouchthetime-axisbutitdoesn't.Thepopulationin1000AD,wasabout200millionpeople.Todayisisabout#6,500millionpeople(6.5*billion).Soonthisscale,it1lookslikethegraphisPtouchingthetime-axis.P{3Fromthelistbelow,pickanotherrelationshipthatisexponentialinnature.1.heightabovesealevel#vs.airtemperature*2.amountofalcohol1consumedvs.thenumberPofbraincellsdestroyed3.timevs.thenationaldebt4.sizeofafilebeingdown-loadedvs.thetimeit#takestodownload.PTheonlyrelationshipthatchangesnearlyexponentiallyisthetimevs.thenationaldebt.Belowisthetime/debt#relationshipexpressedin*graphicalform.(timeisin1calendaryearsfrom1940P 8`   @@  @0@0 ~PF4Thesymbolicrepresentationofanexponentialfunction(relationship)isbestdescribedbywhichsetof#symbolslistedbelow?*(b,d,e,andfarefunction1parameters)P1.y=dx+e2.y=db^(x+e)+f3.y=d(x+e)^2+f4.y=d/(x+e)+fPThereareothervariations,butfromthelist,y=db^(x+e)+fbestdescribesanexponentialrelationship.Belowisthe#graphof*y=(1/2)2^(x-3)-51P 8R۷nݿvj@j@jjjjjjjjjjjjjjjjjjjjjjjjDTDjj j@j@jjjjjj0jjj۷nݤjjjjjjjjjjjjjjjjXP 4aTheexponentialfunctioncanhaveaformatof:y=db^(x+e)+fOfthesesymbols,whichisthefunctionvariable?#1.b*2.d13.eP4.x5.fPOfcourse,theonlyvariableisx.yisthenameofthefunction(oritrepresentsthevalueofthefunctionforallx's).The#restarefunction*parameters-constants1thatcontrolthebehaviorsPofthefunction.P5Whichofthefollowingexponentialfunctionsisdecreasing?1.y=2^x+32.y=5^(x+1)-4#3.y=(1/2)^(x-1)+2*4.y=12^(x+2)PSincethistopicmightbenewtoyou,didyougrapheachfunctionsothatyoucouldanswerthequestion?#Whenthebaseisbetween0*and1,thegraphoften1decreases.#3iscorrect.P6Whichofthefollowingexponentialfunctionsisincreasing?1.y=2^x+32.y=0.25^(x+1)-4#3.y=(1/2)^(x-1)+2*4.y=0.1^(x+2)PDidyougrapheachfunctionsothatyoucouldanswerthequestion?Ordidyoulearnsomethingabouttheparameter"b"#fromthelastcard?*Whenthebaseislarger1than1,thegraphoftenPincreases.#1iscorrect.Pj7Sofar,youhavelearnedthatifbisbetween0and1thefunctiondecreasesandifbislargerthan1,itincreases.Butconsiderthe#following:Whichofthe*followingisincreasing?1P1.y=-0.5^x+12.y=3^(x+1)-23.y=-(1/3)^(x+1)4.y=1.1^x-25.Alloftheabove.PAsyouseefromgraphingthefunctions,ifparameterdisnegativeandthebase"b"isbetween0and1,thegraph(function)also#increases.So,parameter*dinconjunctionwithb1controltheincreasingandPdecreasingbehaviors.P8Whichofthefollowingaredecreasing?1.y=-(1.3)^(x-1)+22.y=(0.4)^(x-1)-23.y=-(5/2)^x+1#4.y=(1/5)^(x+1)+1*5.Alltheabove.PAsyouseefromgraphingthefunctions,ifparameterdisnegativeandthebase"b"islargerthan1,thegraph(function)also#decreases.So,parameter*dinconjunctionwithb1controltheincreasingandPdecreasingbehaviors.Pj9Imagine50litersofRadongasinasealedcontainer.ItdecaysintootherelementsaccordingtothemodelR=50(1/2)^(t/3.8),#whereRistheamountof*Radonleftaftertdays.1HowmuchRadonisleftPafter3.8days?1.3.8liters2.25liters3.0.84liter4.NotenoughinformationPUsingthemodelR=50(1/2)^(t/3.8)iftisreplacedwith3.8,Rhasavalueof25liters.DidyouuseTABLEonyour#calculator?Itworksquite*welltohelpanswerthis1question.P Z10The50litersofRadongasdecaysintootherelementsaccordingtothemodelR=50(1/2)^(t/3.8),whereRistheamountof#Radonleftaftertdays.*HowmuchRadonisleft1after7.6days?P(Thisistwice3.8days)1.25liters2.12.5liters3.7.6liters4.3.8litersPUsingthemodelR=50(1/2)^(t/3.8)iftisreplacedwith7.6,Rhasavalueof12.5liters.DidyouuseTABLEonyour#calculator?Itworksquite*welltohelpanswerthis1question.P 11Wewillletanother3.8daysgobyandaskthesamequestion.At11.4days,howmuchRadonisleft?#1.50liters*2.25liters13.12.5litersP4.6.25litersPUsingthemodelR=50(1/2)^(t/3.8)iftisreplacedwith11.4,Rhasavalueof6.25liters.#DidyouuseTABLEonyour*calculator?Thepointis1thatthelastthreePquestionscouldhavebeenansweredusingTABLE.Youshouldconsidersettingxto0,thendeltaxto3.8.P 12Hopefully,youcanseeapatterndevelopingintheamountofRadonleftastimepasses.Whatifwelettimeincreasewithout#bound?Graphthefunction*R=50(1/2)^(t/3.8)and1observetheamountofPRadonleftastimeincreases.Yourobservationis:1.Itkeepsdropping.2.Itdropstozero.#3.Itapproaceszero.*4.Can'ttell,thenumbers1gettoosmall.PThegraphoftheamountofRadonleftastimeincreasesisapproachingzero.Thatis,astimepasses,thegraphisgetting#closerandclosertothe*liney=0.Whatisimportant1isthatoftheconceptofaPfunctionAPPROACHINGaline.P ?13TheconceptdevelopedonthepreviouscardisthatofthegraphofafunctionAPPROACHINGaline.Whatlinedoesthegraphof#y=2(1/3)^x+3*approach?11.Itapproachesy=0.P2.Itapproachesy=2.3.Itapproachesy=3.4.Itapproachesx=3.PYoushouldhavegraphedthefunctionandthentracedtolargexvalues.Itiskindoflikelettingxincreasewithoutbound.#Thefunctionapproaches3.*Theliney=3.1TECHNOLOGYCAUTION:DuePtorounding,yourcalculatormayactuallyshow3fory,asyoutracetotheright.Thesenumbersarewrong.#ynevergetsto3,it*approaches3.P 14Thelineagraph(function)approachesiscalledtheasymptote.Whatistheasymptoteofy=-(3)^(x-1)-4?#1.y=3*2.y=-113.y=-4P4.y=0PThisisalittledifferentthanthepreviousexamples.Thistime,asxdecreaseswithoutbound,thegraph(function)#approachesy=-4.Butdo*younoticethegraph1gettingcloserandcloserPtotheliney=-4forsmallerandsmallerxvalues.Tracemayshowthistoyou.#Y=-4istheasymptoteof*thisfunction.P 15Whatistheasymptoteofthefunctiony=-(1/2)^(x+1)-5?1.y=02.y=-5#3.y=1*4.y=-115.y=-(1/2)PTheasymptoteistheliney=-5.Haveyounoticedtheconectionbetweenthe#asymptoteandafunction*parameter???P 16Whatistheasymptoteofthefunctiony=db^(x+e)+f?1.y=b#2.y=d*3.y=e14.y=fPIfyoudidn'tknowthecorrectanswer,gobacktothepreviousfewcardsandtrytomaketheconnection.#*y=fistheasymptote.P 17Whatmathvocabularywasdiscussedinthisactivity?1.Exponentialfunctions2.Increasing/Decreasing3.Range#4.Functionparameters*5.Exponentialrelationship16.HorizontalasymptoteP7.AlltheabovePYes,alltheideaslistedonthefrontofthecardwereincluded.Youshouldlookinyourtextbookformoreinformationoneachterm.P 18Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"PPKA35~44 BEHAVLIN.8xv**TI83F* AppVariable file 11/24/02, 10:433 3BEHAVLIN33GAR_r `G!#$&N()*,./B2Behaviors-LinearEd LaughbaumSeptember 12, 2006RedBank Publishing1Thisactivityismeanttobeusedinagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhocan'tfollowrules.PNowtypethe>cursorkeytocontinue,butfirst...inf-meansinfinity(orincreasingwithoutbound)#*Youmayfindthatwriting1themathematicalsymbolsPandnotationcontainedinthislessoninproperformwillhelpyouunderstanditandrememberitlonger.P 1aYoumayhaveheardorreadaboutlinearrelationships.Belowaresituationsinvolvingmathematical#relationships.Selectthe*oneyouthinkislinearin1nature.P1.timevs.humanpopulation2.hoursworkedvs.salary(whenpaidaconstanthourlyrate)#3.timevs.heightofaball*thrownstraightup14.incomeearnedvs.PfederaltaxespaidPThelinearrelationshipis#2-hoursworkedvs.salary(whenpaidaconstanthourlyrate).#Belowisagraphofa*typicalgraphoftimevs.1salary.P 8`a``` 0 0 0 0``` @P h1bBelowaresituationsinvolvingmathematicalrelationships.Selecttheoneyouthinkislinearinnature.#*1.electricityusevs.1monthlyelectricchargesP(amountowed)2.timevs.heightofthespaceshuttleontake-off3.radiusvsareaofacircle#4.timevs.valueofacarPMostelectriccompanieschargeinalinearfashion.Thatis,theychargecustomersataconstantrate.#Belowisagraphofthe*electricchargesatvarious1usagelevels(inkWh).P 8`   a ` ` 0@0@00  `P82Thesymbolicrepresentationofalinearfunction(relationship)isbestdescribedbywhichsetofsymbolslistedbelow?#(d,e,andfarefunction*parameters)1P1.y=d(x+e)^2+f2.y=dsqr(x+e)+f3.y=dx+e4.y=d|x+e|+fPThefunctionwritteninsymbolicformasy=dx+eisthelinearfunction.#Theothersarequadratic,*squareroot,andabsolute1valuefunctions.P3Whichlinearfunction(iny=dx+eform)isincreasing?1.y=-2x+32.y=-(1/2)x-3#3.y=0x-5*4.y=4x-13PAsimplewayofidentifyingwhetherthefunctionisincreasingistolookatthegraphortable.#Thekeyindeciding*whetheralinearfunction1isincreasing,decreasing,Porneither,istheparameterdiny=dx+e.Ifitisontheinterval(0,inf),thenthefunctionisincreasing.#Number4iscorrectsince*thecoefficientofxison1theinterval(0,inf).P4{Whatkindofnumberistheparameterdinthelinearfunctiony=dx+e--whosegraphisshownbelow?#Thescreenbelowhasa*windowof-4.7,4.7]by1-3.1,3.1]P 8`````a``@@p``aP1.disnegative2.dispositive3.dis04.notenoughinformationPSincethegraphofthelinearfunctionisdecreasing,theparameterdisnegative.Itisontheinterval(-inf,0).#*Thecorrectansweris#1P5AGiventhefollowingnumericrepresentationofalinearfunction,whatkindofnumberistheparameterd?#*1P 8` W 0 8 000  88 08     P1.notenoughinformation2.dison(0,inf)3.dison(-inf,0)4.disanintegerPAsthevaraiblexincreases,thefunctionvaluesaredropping.Thismeansthelinearfunctionisdecreasingwhichmeans#theparameterdis*negative.Itisonthe1interval(-inf,0).P6Whichlinearfunction(iny=dx+eform)isdecreasing?1.y=-2x+32.y=(1/2)x-3#3.y=0x-5*4.y=4x-13PAgraphortablewilllikelytellyouifthefunctionisincreasingordecreasing,butthekeyindecidingwhetheralinear#functionisincreasing,*decreasing,orneither,is1theparameterdiny=dx+Pe.Ifitisontheinterval(-inf,0),thenthefunctionisdecreasing.#1iscorrect.P7Whichofthefollowinglinearfunctionsisalsoaconstantfunction?1.y=3000x-72.y=-0.001x-7#3.y=0.0004x-7*4.y=0x-7PDidyougraphthefunctions?Ormaketables?Response4isaconstant,thenumber-7.#Sincethefunctionis*alwaysthesamenumber1(aconstant),itiscalledaPconstantfunction,aswellaslinear.P8Thegraphofwhichlinearfunctionseemstorisethefastest?1.y=1x-12.y=2x-1#3.y=3x-1*4.y=6x-115.y=32x-1P6.y=(1/2)x-1POfcourse,itisnumber5.Agraphingcalculatorwillhelpyoudecide.Whatparameterdoyou#thinkcontrolstherate*(howfast)ofincrease?P9Afterlookingatthesymbolicformsofthelinearfunctionsinthepreviouscard,whatfunctionparametergivesa#clueastowhichgraph*risesfastest?(y=dx+e)11.eP2.d3.x4.Thereisnoclue.PHopefully,itisreallyobvious.Thebiggerdisthefasterthegraphrises.Didyougraphallofthefunctionsontheprevious#card?Ifnot,tryitnowto*convinceyourselfthat#21iscorrect.P 10Whichofthelinearfunctionsbelowfalls(decreases)thefastest?1.y=-0.5x+12.y=-1x+1#3.y=-2x+1*4.y=-3x+115.y=-12x+1P6.y=-75x+1PHopefully,itisreallyobvious.Thebiggerdis(inabsolutevalue)thefasterthegraphfalls(decreases).Youmayalso#recognizethatd<0.*1CanyouconfirmthiswithPyourcalculator?P 11Whichgraphdoesn'tincrease(rise)ORdecrease(fall)?1.y=-0.1x+22.y=-0.01x+2#3.y=0x+2*4.y=0.1x+215.y=0.01x+2Py=0x+2doesnotincreaseordecrease.Itisconstant.Thismakesthegraphahorizontalline.Whatparametercontrolsthis?P 12Sofarinthisactivity,wehaveconcentratedonalinearfunctionbeingincreasing,decreasing,orconstant.Anotherbehavior#istherateatwhichthe*linearfunctionis1increasing,decreasing,orPconstant.Inotherwords,therateatwhichitischanging.Foralessononthis,pleaseusethestackcalled"slope"when#finishedwiththislesson.*11.Typeonetocontinue.PNowtypethe>cursorkeytocontinue.P 13Wheredoeseachgraphcrossthey-axis?y=2x-1,y=x-1,y=4x-1,y=-4x-1,y=-2x-1,y=-x-1#1.Theydon'tcross.*2.At-113.Theyarealldifferent.PTheyallcrossthey-axisat-1.Isthereafunctionparameterthatmaycause#this?P 14Giveny=dx+e,wheredoesitcrossthey-axis?1.at12.atd#3.ate*4.Ican'ttellfromthe1informationgivenPDidyounoticeinallofthefunctionsyougraphedinthislesson,theyALWAYScrossedthey-axisatthenumbere?P 15Giventhelinearfunctiony=2x+3,whatisthevalueofthefunctionwhenxis-3/2?1.Howdoyoudothis?#2.Notenoughinformation*3.Itiszero14.Itis3PThefunction(y)iszerowhenx=-3/2.Ifyougraphthefunction,whendoesitcrossthe#x-axis?P 16Giventhelinearfunctiony=-2x+3,whatisthevalueofthefunctionwhenxis3/2?1.Howdoyoudothis?#2.Notenoughinformation*3.Itiszero14.Itis3PThefunction(y)iszerowhenx=3/2.Ifyougraphthefunction,whendoesitcrossthex-axis?#Doyouseeanywayof*answeringthisbyknowing1thefunctionparameters?P 17Thevalueofxthatcausesthefunctiontobezeroiscalledthezeroofthefunction.Whatisthezeroofthefunctiony=-2x+3?#1.3/2*2.-3/213.2/3P4.-2/3PThezeroofthefunctiony=-2x+3is3/2.Didyouusethezerofinderonyourcalculator?Did#youconvertthezeroin*decimalformtofraction1formwithMATHFrac?PCanyoudothisinyourheadusingthefunctionparameters?Tryit.P 18Thefunctionparameterstellyouthezero.Gobacktothepreviousfewcardsandtrytofigurethisout.Whatisthezeroof#y=dx+e?*1.e/d12.edP3.-e/d4.-d/ePAzeroisavalueforxthatcausesthefunction(y)tobezero.Inthiscase,andineverycase,itis-e/d.P " 19Whatmathvocabularywasdiscussedinthisactivity?1.Linearfunctions2.Increasing/Decreasing3.Functionparameters#4.Linearrelationships*5.Rate(howfast)16.y-interceptP7.Constantfunctions8.Zeros9.AlltheabovePYes,allthebehaviorslistedonthefrontofthiscardwereincluded.Domainandrangewerenotincludedbecausefor#linearfunctions,theyare*bothallrealnumbers.1YoushouldgotoyourPtextbookformoreinformationoneachterm.P 20Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"P7PKA35v66 BEHAVRTN.8xv**TI83F* AppVariable file 12/09/02, 08:16g6 V6BEHAVRTNV6T6GEXex2a R;-z #!%e'(*+-C0134Behaviors-RationalEd LaughbaumSeptember 12, 2006RedBank Publishing $RtnBehThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhocan'tfollowrules.PNowtypethe>cursorkeytocontinue,butfirst...Reminder:inf-meansinfinity(or#increasingwithoutbound)*sqr-meanssquareroot1MOREPYoumayfindthatwritingthemathematicssymbolsandnotationcontainedinthislessoninproperformwillhelpyouunderstandit#andrememberitlonger.P1Youmayhaveheardofrationalrelationships,ormaybenot.Belowaresituationsinvolvingmathematical#relationships.Selectthe*oneyouthinkisrationalin1nature.P1.volumevs.pressureofanenclosedgas(constanttemp)2.timeasasmokervs.%lungfunctionremaining#3.pizzasizevs.price*4.volumeofairputin1balloonvs.diameterofPballoonPThisisatoughonesinceyoumaynothavehadlifeexperiencewithrationalrelationships.However,simplerational#relationshipsarecalled*inversevariation.Maybe1youhaveexperiencedPthese.#1iscorrectandyouwillfindthegraphoftherelationshipbelow.20cc'sofairwereenclosedandthenthevolumewas#changedasthepressure*wasmeasured.1P 8`@@@@   @ 8P2Whichofthefollowingisthesymbolicformofarationalfunction?1.y=dx+e2.y=d(x+e)^2+f#3.y=dsqr(x+e)+f*4.y=d|x+e|+f15.y=d/(x+e)+fP#5iscorrect.Althoughtherecanbemorecomplicatedforms,wewillnotconsiderthem.#Thefunctionparameters*ared,e,andf.Px3Iny=d/(x+e)+f,whatisthefunctionvariable?1.d2.e3.f#4.x*5.yPThesymbolsd,e,andfarefunctionparameters.Theyareconstantsthatcontrolthebehaviorsofthefunction.Thefunction#variableisx,andthename*(andvalue)ofthefunction1issymbolizedwithy.PBelowisthegraphoftherationalfunctiony=2/(x+1)-3.Youwillnotethatitisalitlemorecomplicatedthanthe#volume-pressuregraphas*ithastwobranches.1P 8`             B%B! @`<`P4TECHNOLOGYCAUTION:Whengraphingrationalfunctionsusingtechnology,youmayfinderrorsinthelooksofthegraphdueto#theconsecutivepoints*beingconnectedbythe1technology.Type1toPcontinue.PYoumaywanttoaskyourteacherwhyithappens.Youcansolvetheproblembygoingtodotmode.Butthenthegraphdoesn't#"look"likeasolidcurve.*Usingadecimalwindow1maysolvetheproblem.P5Whichofthefunctionsbelowisdecreasing?1.y=-2/(x+1)-32.y=-1/(x-2)+13.y=1/(x+2)-1#4.y=-3/(x-3)+2*5.y=-4/(x+1)-3P #3iscorrect.Butdidyounoticesomethingstrangehappeningwhenxis-2inthefunctiony=1/(x+2)-1?Ifyoutraceto-2,you#willseethatthereisno*functionvalue.Ifyouseea1nearlyverticallineat-2,Pthisistheerrormentionedinthepreviouscard.THEREISNOVERTICALLINEINTHEGRAPH.Thisfunctionisdecreasing#everywherebutat-2.At*-2,thefunctionisnot1defined.MORE...PDoyouthinkasinglefunctionparametermightbecausingthefunctiontodecrease?P6Whichofthefunctionsbelowisincreasing?1.y=-2/(x+1)-32.y=1/(x-2)+13.y=3/(x+2)-1#4.y=4/(x-3)+2*5.y=2/(x+1)+3P #1iscorrect.Butdidyounoticesomethingstrangehappeningwhenxis-1inthefunctiony=-2/(x+1)-3?Ifyoutraceto-1,you#willseethatthereisno*functionvalue.Ifyouseea1nearlyverticallineat-1,Pthisistheerrormentionedtwocardsprevious.THEREISNOVERTICALLINEINTHEGRAPH.Thisfunctionisincreasing#everywherebutat-1.At*-1,thefunctionisnot1defined.MORE...PDoyouthinkasinglefunctionparametermightbecausingthefunctiontoincrease?P 6aWhatparameterinthefunctiony=d/(x+e)+fcontrolsincreasinganddecreasing?1.d#2.e*3.f14.xPItisd.Ifdison(-inf,0),thefunctionisincreasing.Ifdison(0,inf),thefunctionisdecreasing.xisnotaparameter,butit#isthefunctionvariable.P7Forthefunctionsbelowwhichonehasadomainof(-inf,3)U(3,inf)?1.y=-2/(x+1)-32.y=2/(x+1)+3#3.y=1/(x-3)+2*4.y=1/(x+3)-115.y=-3/(x-2)+1P#3iscorrect.Thereasonwemustexclude3fromthedomainisthatifitisleftin,thefunctionwouldnotbearealnumber#becauseofdivisionbyzero.*Haveyounoticedwhatthe1graphlookslikenearPx=3?P8Forthefunctionsbelowwhichonehasadomainof(-inf,-3)U(-3,inf)?1.y=-2/(x+1)-32.y=2/(x+1)+3#3.y=1/(x-3)+2*4.y=1/(x+3)-115.y=-3/(x-2)+1P#4iscorrect.Thereason-3mustbeexcludefromthedomainisthatifitisincluded,thefunctionwouldnotbeareal#numberbecauseofdivision*byzero.1HaveyounoticedwhatthePgraphlookslikenearx=-3?P9Whatisthedomainofthefunctiony=-2/(x+1)-3?1.(-inf,-3)U(-3,inf)2.(-inf,3)U(3,inf)#3.(-inf,-2)U(-2,inf)*4.(-inf,-1)U(-1,inf)15.(-inf,1)U(1,inf)PSincedivisionbyzeroisnotarealnumber,xmaynotbe-1.So#4iscorrect.Haveyounoticedwhatthe#graphlookslikenear*x=-1?P 10Whatisthedomainofy=d/(x+e)+f?1.(-inf,-d)U(-d,inf)2.(-inf,d)U(d,inf)3.(-inf,-e)U(-e,inf)#4.(-inf,e)U(e,inf)*5.(-inf,-f)U(-f,inf)16.(-inf,f)U(f,inf)Pxcannotbe-ebecauseitwouldcausedivisionbyzeroandmakeyanon-realnumber.Infunctions,xandymustbereal#numbers.Thedomainis*(-inf,-e)U(-e,inf).P 11Describethebehaviorofthegraphnearx=1fory=3/(x-1)+2.(Note:Tryadecimalwindowwithyon-10,10].)#*1.thegraphgoesoffthe1screenP2.thegraphdropsto-infleftof1andtoinfrightof13.asxgetscloserto1,thegraphseemstogetcloser#toaverticallinethrough*1.14.alltheabovePThelinethegraphgetscloserandclosertoiscalledanasymptote.Sinceitisvertical,itiscalledaverticalasymptote.#Didyounoticethevertical*asymptoteisatthevalue1thatmakesthefunctionPundefined?TheverticalasymptoteisNOTapartofthegraphofthefunction,butsimplya#linethegraphapproaches.P 12Whereistheverticalasymptoteofy=-3/(x-2)+1?1.atx=-32.atx=3#3.atx=-2*4.atx=215.atx=1PTheverticalasymptoteisatx=2.(Wherethefunctionisundefined.)Theverticalasymptoteis#NOTapartofthegraphof*thefunction,butsimplya1linethegraphapproaches.P U13Graphy=1/(x-1)+2andtheliney=2.Describetheconnectionbetweenthetwographs.1.theyaren'tconnected#2.thegraphoftherational*functionseemstoapproach1theliney=2P3.thegraphoftherationalfunctioncrossestheliney=2.4.noneofthese.PTheideaisthatthegraphoftherationalfunction"approaches"theline.Whenthishappens,thelinethegraphisapproachingis#theasymptote.Inthis*case,y=2isahorizontal1asymptote.P 14Whereisthehorizontalasymptoteofthefunctiony=-1/(x+1)-3?1.aty=-12.aty=1#3.aty=-3*4.aty=3PTrygraphingthefunctionandtheasymptotetoconfirmthaty=-3isthehorizontalasymptote.#Doyouseeaparameter*connectedtothe1horizontalasymptote?P 15Whereisthehorizontalasymptoteofy=d/(x+e)+f?1.aty=d2.aty=-e#3.aty=e*4.aty=f15.aty=-fPHopefullyyouhavemadetheconnectionbetweentheparameterandthehorizontalasymptote.#Itisy=f.P 16Findtherangeofthefunctiony=-2/(x+3)-5.1.yis(-inf,-3)U(-3,inf)2.yis(-inf,3)U(3,inf)#3.yis(-inf,5)U(5,inf)*4.yis(-inf,-5)U(-5,inf)15.yis(-inf,-2)U(-2,inf)PSincethefunction(graph)onlyapproachesy=-5,andgoestoinfinityabove-5and-infinitybelow-5,therangemustbeallreals#but-5.Thatis:ybelongsto*theinterval1(-inf,-5)U(-5,inf).P 17Findthedomainofthefunctiony=-2/(x+3)-5.1.xis(-inf,-3)U(-3,inf)2.xis(-inf,3)U(3,inf)#3.xis(-inf,5)U(5,inf)*4.xis(-inf,-5)U(-5,inf)15.xis(-inf,-2)U(-2,inf)PSincethefunctionisrealforallx,but-3,thisisthedomain.Thatis:xbelongstotheinterval(-inf,-3)U(-3,inf).#*Theparameterecontrols1thedomain.P 18Intherationalfunctiony=d/(x+e)+f,whatistherange?1.yison(-inf,d)U(d,inf)2.yison(-inf,e)U(e,inf)#3.yison(-inf,-e)U(-e,inf)*4.yison(-inf,f)U(f,inf)15.yison(-inf,-f)U(-f,inf)PKeepinmindthatthefunctionapproachestheliney=f(seecard15).Youcanalsoseethatthegraphgoesto-infandinfnear#theverticalasymptote.So*they-valuesareallreals1exceptf.Thatis,yisonP(-inf,f)U(f,inf).Theparameterfcontrolstherangeoftherationalfunctiony=d/(x+e)+f.P 19Findthezerosofthefunctiony=-1/(x+3)1.02.-1/3#3.-3*4.therearenozerosPPleasenotethatparameterfiszero,sothereisahorizontalasymptoteaty=0(thex-axis).Sothefunctiononly#APPROACHESthex-axis.*Conclusion:nozeros.#4is1correct.P 20Whatisthemaximumvalueofthefunctiony=1/(x+2)-1?1.-12.-2#3.2*4.inf15.Thereisnomaximum.PIfinfmeansincreasingwithoutbound,andthefunctionincreaseswithoutboundasxapproaches-2fromtheright,thereisno#maximumvalueofthe*function.P ) 21Whatmathvocabularywasdiscussedinthisactivity?1.Rationalfunctions2.Increasing/Decreasing3.Range#4.Domain*5.Functionparameters16.RationalrelationshipP7.Horizontal&verticalasymptotes8.Zeros9.AlltheabovePYes,allthebehaviorslistedonthefrontofthecardwereincluded.Youshouldgotoyourtextbookformoreinformationon#eachterm.P 22Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"PPKA35M )) BEHAVSQR.8xv**TI83F* AppVariable file 12/08/02, 12:19m) \)BEHAVSQR\)Z)G?R_r/ *Q3u AA@!";$$&'Behaviors-SquareRtEd LaughbaumSeptember 12, 2006RedBank Publishing(SqrRootBehThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhocan'tfollowrules.PNowtypethe>cursorkeytocontinue--butfirst...Reminder:sqr-meanssquareroot#inf-meansinfinity(or*increasingwithoutbound)1MOREPYoumayfindthatwritingthemathematicssymbolsandnotationcontainedinthislessoninproperformwillhelpyouunderstandit#andrememberitlonger.P1Youmayhaveheardofsquarerootrelationships.Ormaybenot.Belowaresituationsinvolvingmathematical#relationships.Selectthe*oneyouthinkissquare1rootinnature.P1.timevs.distancetraveledataconstantrate2.distancefromalightsourcevs.intensityoflight3.timevs.velocityduring#freefall*4.volumevs.pressureof1anenclosedgasP|Thisisatoughonesinceyoumaynothavehadlifeexperiencewithsquarerootrelationships.Butrather,youmaythinkofit#as"somethingyoudotoa*number."Asitturnsout,#13behaveslikeasquareProotfunction.Belowisagraphofthistime-velocityduringfree-fallrelationship.#*1P 8`p p0`` `0@  @@P2Whichofthefollowingisthesymbolicformofasquarerootfunction?1.y=dx+e2.y=d(x+e)^2+f#3.y=dsqr(x+e)+f*4.y=d|x+e|+f15.y=d/(x+e)+fP#3iscorrect.Recallthatwecannotusethesquarerootsymbollikeisonyourcalculatororinyourtextbook.#Thefunctionparameters*ared,e,andf.P3Iny=dsqr(x+e)+f,whatisthefunctionvariable?1.d2.e#3.f*4.x15.yPOThesymbolsd,e,andfarefunctionparameters.Theyareconstantsthatcontrolthebehaviorsofthefunction.Thefunction#variableisx,andthename*(andvalue)ofthefunction1issymbolizedwithy.PBelowisthegraphofy=2sqr(x+3)-1#*1P 8`}><x<` 0@D@  @@P4Whichofthefollowingfunctionsaredecreasing?1.y=2sqr(x-1)+32.y=-2sqr(x-1)+3#3.y=5sqr(x)-3*4.y=sqr(x)15.y=(1/2)sqr(x+1)PDidyougraphormakeatableforeachfunction?Ithelps(assumingyouknowwhatdecreasingmeans.)#2iscorrect.#Whatparametersets#2*apartfromtherest?P5Whichofthefollowingfunctionsareincreasing?1.y=-2sqr(x-1)+32.y=2sqr(x-1)+3#3.y=-5sqr(x)-3*4.y=-sqr(x)15.y=-(1/2)sqr(x+1)PDidyougraphormakeatableforeachfunction?Ithelps(assumingyouknowwhatincreasingmeans.)#2iscorrect.#Whatparametersets#2*apartfromtherest?P6Giveny=dsqr(x+e)+f,whatparametercontrolsincreasinganddecreasing?1.d2.e#3.f*4.dande15.dandfP Ifyouhavegraphedallofthefunctionssofarinthisactivity,youmayknowthatparameterdcontrolswhethery=dsqr(x+e)+f#isincreasingordecreasing.*Tryit.Ifdison(-inf,0),1thefunctiondecreases.IfPdison(0,inf)thefunctionincreases.P7Whatisthesmallestnumberinthedomainofy=2sqr(x-3)-4?1.22.-3#3.3*4.-415.4PMakingatableorgraph(onadecimalwindow)mayhelpanswerthis.#3iscorrect.#Whatparameterseemsto*helpyouanswerthe1question?P8Whatisthesmallestnumberinthedomainofy=2sqr(x+3)-4?1.22.-3#3.3*4.-415.4PMakingatableorgraph(onadecimalwindow)mayhelpanswerthis.#2iscorrect.#Whatparameterseemsto*helpyouanswerthe1question?P9Whatisthesmallestnumberinthedomainofy=sqr(x+1)?1.12.-1#3.Thisisatrickquestion*becausetwoofthe1parametersaremissing!PNo,itisnotatrick.Parameterdis1andfis0.Didyougraphandtraceonadecimalwindow?#2is#correct.P 10Whatisthesmallestnumberinthedomainofy=dsqr(x+e)+f?1.d2.e#3.-e*4.f15.-fPSure,-eisthesmallestnumberinthedomain.dcontrolstheincreasinganddecreasing,andwe#haven'tlookedat*parameterfyet.P 11Usingthesmallestnumberinthedomain,howfarfromthex-axisisy=2sqr(x-1)+3?1.2units#2.-1units*3.3units14.TherearenounitsP5.What'saunit?PThegraphonadecimalwindowmaygiveyoutheanswer.Itis#3.Butwhatisimportanthere#isthatitiscontrolledbya*parameter.Whichone?P 12Usingthesmallestnumberinthedomain,howfarfromthex-axisisy=-2sqr(x-1)+3?1.2units#2.-1units*3.3units14.Therearenounits,soIPcan'tanswer.PWhile#3iscorrect,whatisimportantisthatthevalueiscontroledbyafunctionparameter.Whichone?P 13Usingthesmallestnumberinthedomain,howfarfromthex-axisisy=dsqr(x+e)+f?1.itstartsatd#2.itstartsate*3.itstartsat-e14.itstartsatfP5.itstartsat-fPThefunction,whetherincreasingordecreasing,isfunitsfromthex-axisatthebeginningofthedomain.#*Didyouknowallthisstuff?P 14Whatistherangeofthefunctiony=-3sqr(x+1)-5?1.ybelongsto(-inf,-5]2.ybelongsto-5,inf)#3.ybelongsto-1,inf)*4.ybelongsto1,inf)15.ybelongsto5,inf)PSinceparameterfindicatesthefunctionisat-5forthesmallestnumberofthedomain,andparameterdshowsthe#functiondecreasesfrom*there,#1iscorrect.1CheckthisonyourPcalculator!P 15Whatisthemaximumvalueofthefunctiony=-3sqr(x+1)-5?1.itis52.itis-5#3.itis-1*4.itis115.itdoesn'thaveamaxPThebiggestvaluethefunctionhasis-5asindicatedbyparametersdandf.They-valuesstartat-5anddecreasefrom#there.Thusithasa*maximumof-5.P 16Whatistheminimumvalueofthefunctiony=3sqr(x+1)-5?1.itis52.itis-5#3.itis-1*4.itis115.itdoesn'thaveaminPSincedispositive,thefunctionincreases,andfisthesmallestvalue.Sotheminimumis-5.#Graphittocheck.P 17Whatisthedomainofy=2sqr(3-x)+1?1.xison3,inf)2.xison-3,inf)3.xison(-inf,3)#4.xison(-inf,3]*5.youcan'ttellsincethe1functionisnotinstandardPformPTrue,thefunctionisnotintheformy=dsqr(x+e)+f,butthegraphingcalculatorwillhelpyouanswerthe#question.Thedomainis*(-inf,3].P 18Whatisthemaximumvalueofthefunctiony=-(1/2)sqr(3-x)+1?1.itdoesn'thaveamax2.itis-1#3.itis1*4.youcan'ttellsincethe1functionisnotinstandardPformPTrue,thefunctionisnotintheformy=dsqr(x+e)+f,butthegraphingcalculatorwillhelpyouanswerthe#question.Themaxis1.Did*youthinkabouthowthe1functionparametersmightPhelpyou?P  19Whatmathvocabularywasdiscussedinthisactivity?1.Squarerootfunctions2.Increasing/Decreasing3.Maximum#4.Minimum*5.Range16.DomainP7.Functionparameters8.Squarerootrelationship9.AlltheabovePYes,allthebehaviorslistedonthefrontofthecardwereincluded.Youshouldgotoyourtextbookformoreinformationon#eachterm.P 20Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"PbSPKA35y// BEHVQUAD.8xv**TI83F* AppVariable file 11/24/02, 18:51/ /BEHVQUAD//GCVcv L >h:p !b#$v& (*N,'.Behaviors-QuadratiEd LaughbaumSeptember 12, 2006RedBank Publishing%QuadBehThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhocan'tfollowrules.PNowtypethe>cursorkeytocontinue.Reminder:inf-meansinfinity(or#increasingwithoutbound)*1YoumayfindthatwritingPthemathematicssymbolsandnotationcontainedinthislessoninproperformwillhelpyouunderstanditandrememberitlonger.P1Youmayhaveheardofquadraticrelationships.Ormaybenot.Belowaresituationsinvolvingmathematical#relationships.Selectthe*oneyouthinkisquadratic1innature.P1.timevs.theworld'spopulation2.hoursworkedvs.salary(whenpaidaconstanthourlyrate)#3.timevs.heightofaball*thrownstraightup14.incomeearnedvs.PfederaltaxespaidP5Visualizeaballbeingtossedstraightup.Nowthinkabouttherelationshipbetweentimeandtheheightoftheball.#Thisisaquadratic*relationship.Belowisthe1graphoftimevsheightofaPballtossedSTRAIGHTup.#*1P 8`  @ @@@@  @@@ !P |1aBelowaresituationsinvolvingmathematicalrelationships.Selecttheoneyouthinkisquadraticinnature.#*1.monthlyelectricityuse1vs.electricchargesP(amountowed)2.timevs.heightoftheplaneonacommercialairlineflight3.radiusvsareaofa#circle*4.timevs.valueofacarPThisisatoughoneifyoudon'tthinkaboutmathematicalrelationshipsintheworldaroundyou.#Theradius-area*(A=Pi*r^2)relationshipis1quadratic.BelowisagraphPofthisrelationship.#*1P 8`  @@  @ @ @@P@2Thesymbolicrepresentationofaquadraticfunction(relationship)isbestdescribedbywhichsetof#symbolslistedbelow?*(d,e,andfarefunction1parameters)P1.y=d(x+e)^2+f2.y=dsqr(x+e)+f3.y=dx+e4.y=d|x+e|+fPThefunctionwritteninsymbolicformasy=d(x+e)^2+fisthequadraticfunction.#Theothersare2.square*root,3.linear,and4.1absolutevaluefunctions.P3Graphthequadraticfunctiony=(x-2)^2-1andfindthevertexoftheparabola.Itis:1.(-2,-1)#2.(2,-1)*3.(-2,1)14.(2,1)PInthiscase,ifyouuseadecimalwindow,youcantracetothevertex.Itisat(2,-1).Otherwise,youcanusetheminimumfinderon#yourcalculator.*Doyouseeanyconnection1betweenthevertexandPfunctionparameters?P4Graphthequadraticfunctiony=(x+3)^2-2andfindthevertexoftheparabola.Itis:1.(-3,-2)#2.(3,-2)*3.(3,2)14.(-3,2)PInthiscase,ifyouuseadecimalwindow,youcantracetothevertex.Itisat(-3,-2).Otherwise,youcanusetheminimum#finderonyourcalculator.*Doyouseeanyconnection1betweenthevertexandPfunctionparameters?P5Graphthequadraticfunctiony=(x-2)^2-3andfindthevertexoftheparabola.Itis:1.(-2,-3)#2.(2,-3)*3.(2,3)14.(-2,3)PInthiscase,ifyouuseadecimalwindow,youcantracetothevertex.Itisat(2,-3).Otherwise,youcanusetheminimumfinderon#yourcalculator.*Doyouseeanyconnection1betweenthevertexandPfunctionparameters?P6Giventhequadraticfunctiony=(x+e)^2+fandfindthevertexofthegraph.Thegraphisaparabola.#1.(-e,-f)*2.(e,-f)13.(e,f)P4.(-e,f)PHopefully,youhavebeenpayingattentiontotheconnectionbetweenthefunctionparametersandthevertex.Ifnot,#reconsideryourthinking.*Itis(-e,f).Lookatthe1patternonthelast3cards.P7Thevertexoftheparabola(thegraphofthequadraticfunction)holdsaconnectiontootherfunctionbehaviors.For#example,youjustlearned*thaty=(x+e)^2+fhasa1vertexat(-e,f).PForwhatvalueofx,doesthegraphchangefromdecreasingtoincreasing(d>0)?1.f#2.-e*3.-f14.ePLookatthegraphofanyquadraticfunction(whered>0),itfirstdecreasesandthenitincreases.Itchangeswhenxhasavalue#of-e.P8Whendoesy=(x-1)^2-3changefromdecreasingtoincreasing?1.whenxis-1#2.whenxis1*3.whenxis-314.whenxis3PUsingtraceshouldhelpyouseethatthey-valueschangefromdecreasingtoincreasingatx=1.P9Whenisy=(x+1)^2-2decreasing?1.notenoughinformation2.whenxison(-inf,1)3.whenxison(-inf,-1)#4.whenxison(-inf,-2)PThegraphisdropping(decreasing)whenxison(-inf,-1),thusthefunctionisdecreasingwhenxison(-inf,-1).#Note:ifyouarenot*familiarwithinterval1notation,pleasereviewPthecardstackcalledIntervalNotation.P 10Whenisy=(x+1)^2-2increasing?1.notenoughinformation2.whenxison(1,inf)3.whenxison(-1,inf)#4.whenxison(-2,inf)PDidyouusetable?Didyouusetraceonyourcalculator?Ifyouhad,youwouldhavenotedthatthegraphisrisingwhenx#ison(-1,inf),thusthe*functionisincreasingwhen1xison(-1,inf).P 11Whatistheminimumvalueofthefunctiony=(x+1)^2-2?1.Ithasnomimimum.#2.Itis-2*3.Itis-1.14.Werewetaughtthis?PThelowestvaluethefunctionhasis-2,soitiscalledtheminimum.Haveyoubeenwatchingtheyvaluesasyoutrace?#Youweren'tasked,butthe*minimumhappenswhenx1is-1.P 12Whatistheminimumvalueofthefunctiony=(x-1)^2+3?1.Itis-3#2.Itis3.*3.Itis1.14.Werewetaughtthis?PThelowestvaluethefunctionhasis3,soitiscalledtheminimum.Youweren'tasked,butthe#minimumhappenswhenx*is3.P 13Whatistheminimumvalueofthefunctiony=(x+e)^2+f?1.Itis-e#2.Itise.*3.Itisf.14.Itis-f.PUsingthepreviouscards,youshouldhavemadetheconnectionthattheminimumfunctionvalue(y)isf.P 14WhatistheMAXIMUMvalueofthefunctiony=-(x-1)^2+3?1.Itis-3#2.Itis3.*3.Itis1.14.Werewetaughtthis?PThisisnew.Theparameterdnowhasavalueof-1andthismakesthegraphopendown,thusthereisamaximuminsteadofa#minimum.Itis3,andit*occurswhenxis1.P 15WhatistheMAXIMUMvalueofthefunctiony=-4(x-1)^2+3?1.Itis-3#2.Itis3.*3.Itis1.14.Werewetaughtthis?PThisisnew.Theparameterdnowhasavalueof-4andthismakesthegraphopendown,thusthereisamaximuminsteadofa#minimum.Itis3,andit*occurswhenxis1.P 16Aquadraticfunctiony=d(x+e)^2+fhasamaximumif:1.dison(0,inf)2.dison0,inf)#3.dison(-inf,0)*4.dison(-inf,0]PAnimportantideainmathematicsistheconnectionbetweenfunctionparametersandfunctionbehaviors.#Thisoneisthatifdis*negative(d<0),thenthe1functionhasamaximum.P 17Aquadraticfunctiony=d(x+e)^2+fhasaminimumif:1.dison(0,inf)2.dison0,inf)#3.dison(-inf,0)*4.dison(-inf,0]PAnimportantideainmathematicsistheconnectionbetweenfunctionparametersandfunctionbehaviors.#Thisoneisthatifdis*positive(d>0),thenthe1functionhasaminimum.P a18Therangeofthequadraticfunctionisthesetofally-values.So,ifyouknowthemaximumorminimum,youknowtherange.#*Whatistherangeofthe1functiony=2(x-1)^2+7?P1.(7,inf)2.7,inf)3.(-inf,7)4.(-inf,7]5.(-inf,inf)#6.(1,inf)*7.1,inf)PSinceyouknowtheminimumvalueofthefunction(y)is7,thismeansally-valuesmustbe7andlarger.#*Therangeis7,inf).P u19Therangeofthequadraticfunctionisthesetofally-values.So,ifyouknowthemaximumorminimum,youknowtherange.#Whatistherangeofthe*function1y=-2(x+3)^2+6?P1.(-inf,6]2.(-inf,6)3.(-inf,-6)4.(-inf,-6]5.(-inf,3]#6.(-inf,-3]*7.6,inf)18.-3,inf)PSinceyouknowthemaximumvalueofthefunction(y)is6,thismeansally-valuesmustbe6andsmaller.#Therangeof*y=-2(x+3)^2+61is(-inf,6].P  20Whatmathvocabularywasdiscussedinthisactivity?1.Quadraticfunctions2.Vertex3.Maximum#4.Minimum*5.Range16.QuadraticrelationshipsP7.Functionparameters8.Connections9.AlltheabovePYes,allthebehaviorslistedonthefrontofthecardwereincluded.Wedidnotdiscussdomainsinceallquadraticfunctionshave#domainsofallreal*numbers.(-inf,inf)P 21Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"PPKA35Kf$$ CHANGE.8XV**TI83F* AppVariable file 06/03/02, 12:09$ $CHANGE$$G9@M`s. M  DUg`%!#ChangeEd LaughbaumSeptember 12, 2006RedBank Publishing pIntroIntheworldaroundyou,andoutsidethemathclassroom,therearenumerousreferencestothemathematicalideaof#rateofchange.*1SeeifyoucanexplainthePmathematicalmeaningofeachexampleinthislesson.1.Enter1tocontinue.#2.Nochoicehere,enter1*tocontinuePOK,typethe>keynext.Butfirst:Youmayfindthatwritingthemathematicssymbolsandnotationcontainedin#thislessoninproperform*willhelpyouunderstandit1andrememberitlonger.Pc1Ataxicompanycharges$0.45per1/8 mile.Whatdoes"$0.45per1/8 mile"#mean?*1P1.Gee,thatischeap!2.ItmeansIwillpay45centsifmytripis1/8mile.3.Itmeansthetaxicompanychargesmeata#rateof$0.45per1/8mile*driven.14.Idon'tknow.PItmeansthetaxicompanychargesapassengerattherateof$0.45per1/8miledriven.#Or,itmeasurestherateat*whichyourfareis1increasing.PG2Whatisthemeaningof"60milesperhour?"1.ItmeansIamchangingpositionatarateof60#milesforeveryhour*driven.12.ItmeansmywifeiswithPmeandIcan'tgoanyfaster.3.Isn'titobvious?4.ItmeansIamonalimitedaccesshighway.PItmeansIamchangingpositionatarateof60milesforeveryhourdriven.#Itisameasureof"how*fast"youarechanging1yourposition.Pe3AhospitalpatientIVdripissettorelease20dripsperminute.Whatisthemeaningof"20#dripsperminute?"*1P1.Thepatientisreallysick.2.Anunknownfluidisenteringhis/herbodyatarateof20dripsforevery#minuteitisattached.*3.What'sanI.V.drip?P#2iscorrect.Anunknownfluidisenteringhis/herbodyatarateof20dripsforeveryminuteitisattached.#Itisameasureofhowfast*thefluidisentering1his/herbody.Pp4Aspeed-readercanread500wordsperminute.Whatisthemeaningof"500wordsperminute?"#*1P1.Itmeanshe/shemissesalotofcontent.2.Itmeanshe/sheis10timesfasterthanme.3.Itmeansthatthisperson#canreadatarateof500*wordsforeachminuteof1reading.PItmeansthatthispersoncanreadatarateof500wordsforeachminuteofreading.#Ittellsyoutherateat*whichhe/sheisreading.PQ5Myreal-estatepropertyistaxed$54per$1000inassessedvalue.Whatisthemeaningof#"$54per$1000?"*1P1.TherateatwhichIamtaxedis$54oneach$1000ofpropertyvalue.2.Mayberentingisn'tsobad.#3.Realestatetaxesgoto*education.PTherateatwhichIamtaxedis$54oneach$1000ofpropertyvalue.Itiscalledataxrate.P?6tItake3vitaminpillsperday.Whatisthemeaningof"3#pillsperday?"*1P1.ImustbeanoldpersonifItakevitamins.2.3pillsperday,hummm?3.ItmeansItakevitaminsatarateof3pillsfor#everyday.*4.AretheyvitaminC?PItmeansItakevitaminsatarateof3pillsforeveryday.ItistherateatwhichIam#consumingvitaminpills.P7Onatypicalcommercialairlineflight,theplaneascendsabout1200feetperminuteonitswaytoacruisingaltitude.#*Explainwhat"1200feet1perminute"means.P1.Itistheratethattheplaneisascending,1200feetperoneminuteinflight.2.Itmustmeanhowfast#theplaneisgoing?*3.Thisisatrickquestion.1Planescan'tascendthatPfast.PItistheratethattheplaneisascending,1200feetforeachminuteinflight(untilatthecruisingaltitude).#*Itishowfasttheplaneis1increasinginaltitude.P8Theauthorofatextbookmayearn$1.50forevery$10inoriginalsalesofthebook.#Whatdoes"$1.50forevery*$10"mean?1P1.Gee,authorsdon'tmakethatmuch.2.Mybookcosts$90,sotheauthormustmake$13.50.#3.Itmeansthattheauthor*ispaidattherateof$1.501forevery$10insellingPprice.PItmeansthattheauthorispaidattherateof$1.50forevery$10insellingprice.#Itisanindicationofhow*fasthis/herroyalitiesare1changing,basedonsales.P_9Theroofonmyhouserises12feetverticallyforevery12feetofhorizontalwidthofthehouse.#Explain"rises12feetfor*every12feetofhorizontal1width."P1.Itisasteeproof.2.Itistherateatwhichtheroofrises.3.Thiswouldbeconfusingtoacarpenter.PItistherateatwhichtheroofrises.Comparethistoalineatanangletothehorixontal.It#istherateatwhichthe*lineisrising(increasing).P 310Thebankchargesme$0permonthforcheckingaccountservices.#Whatdoes"$0permonth"*mean?1P1.Itisarate($0permonth)forservicesrendered.2.ItmeansIamchargedamonthlyfee.#3.Banksdon'tdothat*anymore.PItisarate($0permonth)atwhichthebankchargesyouforservicesrendered.P 11}Myprintercanprint4pagesperminute.Whatdoes"4pagesper#minute"mean?*1P1.Itprints4pagesperminute.2.Itprintsatarateof4pagesperminute.3.Itprintsatarateof4#pagesperhourPItprintsatarateof4pagesperminute.Ittellsyouhowfastyourprinterprints.P k12MyInternetbrowsercandownloadinformationatabout25Kpersecondwhenusingamodem.#Explain"25Kpersecond."*1P1.Thefileisbeingtransferredatarateof25Kpersecond.Orameasureofhowfast.2.Browswerscan'tdo#that.*3.What'sabrowser?14.Itisarate.PAfileisbeingtransferredatarateof25Kpersecond.#Orameasureofhowfast*dataisbeingtransferred1toyourcomputer.P t13Theearthisspinningonitsaxisandmakes1revolutionperday.#Whatdoes"1revolution*perday"mean?1P1.ItistherateatwhichtheEarthisturning.2.Itmeans1revolutionperday.3.Theearthisn'trotating#becauseIcan'tfeelit.*4.Nooneknowsthiskindof1stuff.PItistherateatwhichtheEarthisturning.(Alittleover1000milesperhourattheequator.)#*Yeah,youaremoving.P P14TheUSbirthrateiscurrently6850peopleperday.Whatdoes"6850peopleperday"mean?#*1P1.Therearetoomanypeopleborneachday.2.Isn'tthiscausingapopulationproblem?3.Itistherateatwhich#peoplearebeingaddedto*theUSpopulation.PItistherateatwhichpeoplearebeingaddedtotheUSpopulation.Itisameasureofhowfast#theUSpopulationis*increasing.P G15~TheJapanesebirthrateis822peopleperday.Whatdoes"822peopleperday"#mean?*1P1.Therepopulationismuchsmaller.2.ItistheJapanesecontributiontotherateinwhichpeoplearebeing#addedtotheplanet.*3.Whatwasthequestion?PItistheJapanesecontributiontotherateinwhichpeoplearebeing#addedtotheplanet.P 216rDescribeonecommonfeaturetoallofthequestionsinthisgroup.#*1P1.Theyallhavesomethingtodowithspeed.2.Theyhadverylittleincommon.3.Theyallshowedthe#manywaysaconstantrate*ofchangeisused.PTheyallshowedthemanywaysaconstantrateofchangeisused.Wheredoyouuserateof#changeinalgebra?Surely*youuserateofchange!1Don'tyou?P 17Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"P3PKA35B]r r CREATABS.8xv**TI83F* AppVariable file 12/02/02, 15:349 ( CREATABS( & G/BOb uMR N w68qCreateAbsoluteValuEd LaughbaumSeptember 12, 2006RedBank Publishing /AbsThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhowanttodosomethingdifferent.PNowtypethe>cursorkeytocontinue,butfirst...Reminder:|x|meanstheabsolute#valueofx*infmeansinfinity(or1increasingwithoutbound)PTheformofanabsolutevaluefunctionis:y=d|x+e|+fwithfunctionparametersofd,e,andf.#*Youmayfindthatwriting1themathematicssymbolsPandnotationcontainedinthislessoninproperformwillhelpyouunderstanditandrememberitlonger.P|1Whatisthefunctionvariablein:y=d|x+e|+f?1.d2.x#3.e*4.f15.yPThefunctionvariableisx.d,e,andfareconstantfunctionparametersthatcontrolthegeometricbehaviorsofeachfunction.#yrepresentsthefunction*values.1P 1aFromthelistbelow,selectanabsolutevaluefunctionthatopensdown.1.y=-2|x+1|-32.y=2|x-3|-4#3.y=|x|*4.y=50|x+7|-50015.y=77|x+300|+5000PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidewhichfunction*opensdown.Studythe1parametersineachPfunction.#1iscorrect.Createanabsolutevaluefunctionthatmeetsthe#conditions,andputiton*papermarkedas#1.P2Fromthelistbelow,selectanabsolutevaluefunctionthatisdecreasingonlytotheleftof-3.1.y=-|x+3|-1#2.y=2|x+4|-3*3.y=3|x+3|+214.y=-|x-3|-3PYoucangrapheachfunction,oryoucanlearntheparameter-behaviorconnectiontoanswerthisquestion.#Tobedecreasingleftof*thevertex,thegraphmust1openup;thus,thisisPcontrolledbyparameterd.Also,parameterecontrolswherethegraphchangesfromdecreasingtoincreasing.##3iscorrect.*Createanotherabsolute1valuefunctionthatmeetsPtheconditionsontheflipsideofthiscard,andwriteitonpapermarkedas#2.P3Fromthelistbelow,selectanabsolutevaluefunctionthatisonlyincreasingtotheleftof-3.1.y=-|x+3|-1#2.y=2|x+4|-3*3.y=3|x+3|+214.y=-|x-3|-3PYoucangrapheachfunction,oryoucanlearntheparameter-behaviorconnectiontoanswerthisquestion.#Tobeincreasingleftofthe*vertex,thegraphmust1opendown;thus,thisisPcontrolledbyparameterd.Also,parameterecontrolswhenthegraphchangesfromincreasingtodecreasing.#1iscorrect.#Createanabsolutevalue*functionthatmeetsthe1conditionsontheflipsidePofthiscard,andwriteitonpapermarkedas#3.P4Fromthelistbelow,selectanabsolutevaluefunctionthathasavertexat(-3,2).1.y=-|x+3|-2#2.y=2|x+4|-3*3.y=3|x+3|+214.y=-|x-3|-3PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#findthevertexofthe*function.Studythe1parametersineachPfunction.#3iscorrect.Createanabsolutevaluefunctionthatmeetstheconditionsontheflipside#ofthiscard,andputiton*papermarkedas#4.P5Fromthelistbelow,selectanabsolutevaluefunctionthathasarangeof2,inf).1.y=-|x+3|+22.y=2|x+4|+3#3.y=3|x+3|+2*4.y=-|x-3|+2PItmighthelptographeachfunction,butknowingtheparameter-behaviorconnectionwouldbesimpler.Parametersdand#fholdtheanswer.dmust*bepositivetomakethe1functionhaveaminimum,Pandfisthentheminimumvalue.Thisgivestherange.#3iscorrect.Createanabsolutevaluefunctionthatmeetsthe#conditionsontheflipside*ofthiscard,andwriteit1onpapermarkedas#5.P6Fromthelistbelow,selectanabsolutevaluefunctionthathasarangeof(-inf,2].1.y=-|x+3|+2#2.y=2|x+4|+3*3.y=3|x+3|+214.y=-|x-3|-2PItmighthelptographeachfunction,butknowingtheparameter-behaviorconnectionwouldbesimpler.Parametersdand#fgiveyoutheanswer.d*mustbenegativetomake1thefunctionhaveaPmaximum,andfisthenthemaximumvalue.Thisgivestherange.#1iscorrect.Createanabsolutevaluefunctionthatmeetsthe#conditionsontheflipside*ofthiscard,andwriteit1onpapermarkedas#6.P7Fromthelistbelow,selectanabsolutevaluefunctionthathasamaximumvalueof3.1.y=-|x+3|+2#2.y=-|x+4|+3*3.y=3|x+3|+214.y=-|x-3|-3PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidewhichfunctionhasa*maximumat3.Studythe1parametersineachPfunction.#2iscorrect.Createanabsolutevaluefunctionthatmeetstheconditionsontheflipside#ofthiscard,andwriteit*onpapermarkedas#7.P8Fromthelistbelow,selectanabsolutevaluefunctionthathasaminimumvalueof3.1.y=-|x+3|+3#2.y=-|x+4|+3*3.y=6|x+2|+314.y=-|x-3|-3PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidewhichfunctionhasa*minimumat3.Studythe1parametersineachPfunction.#3iscorrect.Createanabsolutevaluefunctionthatmeetsthe#conditionsontheflipside*ofthiscard,andwriteit1onpapermarkedas#8.P9Fromthelistbelow,selectanabsolutevaluefunctionthathasaslopeof-2ontheleftbranch.1.y=-2|x+3|+3#2.y=-|x+4|+3*3.y=2|x+2|+314.y=-|x-3|-3PParameterdcontrolsthebehavior.Ifdispositive,thentheleftbranchdecreases(negativeslope).Thedvalueof2makesit#-2.#3iscorrect.*Createanabsolutevalue1functionthatmeetsthePconditionsontheflipsideofthiscard,andwriteitonpapermarkedas#9.P 10Fromthelistbelow,selectanabsolutevaluefunctionthathasaslopeof-2ontherightbranch.1.y=-2|x+3|+3#2.y=-|x+4|+3*3.y=2|x+2|+314.y=-|x-3|-3PParameterdcontrolsthebehavior.Ifdisnegative,thentherightbranchdecreases(negativeslope).Thedvalueof2makesit#-2.#1iscorrect.*Createanabsolutevalue1functionthatmeetsthePconditionsontheflipsideofthiscard,andwriteitonpapermarkedas#10.P 11Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"PPKA35T}&& CREATEXP.8xv**TI83F* AppVariable file 12/01/02, 14:57 CREATEXPG-@M` sD*\  nmtaCreate ExponentialEd LaughbaumSeptember 12, 2006RedBank Publishing 1CreExpThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhowanttodosomethingdifferent.PNowtypethe>cursorkeytocontinue-butfirst...Reminder:sqrmeanssquareroot#infsymbolizesinfinity(or*increasingwithoutbound)1TheformofanexponentialPfunctionis:y=db^(x-e)+fwhered,b,e,andfarethefunctionparameters.#*Youmayfindthatwriting1themathematicssymbolsPandnotationcontainedinthislessoninproperformwillhelpyouunderstanditandrememberitlonger.P1Exponentialfunctionshaveasymbolicformsomethinglikey=db^(x+e)+f.Thefuctionvariableis:1.d#2.b*3.x14.eP5.f6.yP#3iscorrect,xisthefunctionvariable.Thesymbolsb,d,e,andfarefunctionparametersthatcontrolthebehaviorsof#thefunction.yisthename*(orvalue)ofthefunction.P2Fromthelistbelow,selectanexponentialfunctionthatisdecreasing.1.y=(1/2)^(x+2)-12.y=-3^(x+1)+3#3.y=-10^x-2*4.y=(2/3)^(x-1)-2.515.alloftheabovePIffunctionparameterbisontheinterval(0,1)andparameterdispositive,thefunctionisdecreasing.Likewise,ifparameterbis#ontheinterval(1,inf),*anddisnegative,the1functionalsodecreases.P#5iscorrect.Youwouldalsoknowtheanswerifyougraphedallthefunctions.#Createanexponential*functionthatmatchesthe1behavioronthefrontofPthiscardandwriteitonpapermarkedas#1.P3Fromthelistbelow,selectanexponentialfunctionthathasahorizontalasymptoteaty=3.#1.y=2^x-3*2.y=313.y=3^(x+1)+2.9P4.y=1.4^(x-3)+3PTheparamenter-behaviorconnectionisthatfcontrolsthehorizontalasymptote.#4iscorrect.Youwouldalsohaveknown#thisifyougraphedallthe*functions.1CreateanexponentialPfunctionthathasahorizontalasymptoteaty=3,andwriteitonpapermarkedas#2.P4Fromthelistbelow,selectanexponentialfunctionthatpassesthroughthepoint(0,1).1.y=2^x#2.y=(1/2)^x*3.y=4^x14.alltheabovePYouwouldhaveknownthecorrectanswerifyougraphedallthefunctions.Theyallgothrough(0,1).Butseveralparameters#areinvolved.*1CreateoneyourselfandPwriteitonpapermarkedas#3.P5Fromthelistbelow,selectanexponentialfunctionthatisalwaysabovethegraphofy=2^(x-1)-3.#1.y=2^(x-1)+3*2.y=2^(x-1)-413.y=5^(x-1)-3P4.y=1000^(x-1)-3PYouwouldhaveknowntheanswerifyougraphedallthefunctions.Sincethehorizontalasymptoteisatf,didyoulookfor#somethingwiththesame*baseandwhoseasymptote1isabove-3?ParameterePmustalsobeconsidered.Createoneyourselfandwriteitonpapermarkedas#4.P6Fromthelistbelow,selectanexponentialfunctionthatisdecreasingandhasanasymptoteat2.1.y=2^(x-2)+2#2.y=(1/2)^x+2*3.y=-4^(x+1)-214.y=-(1/3)^(x-1)+2PTheonlyonethatmeetsthecriteriais#2.Allparameterscanplayapartofthis.#Createanexponential*functionthatmeetsthe1conditionsdescribedonthePfrontofthiscardandwriteitonpapermarkedas#5.P7Fromthelistbelow,selectanexponentialfunctionthatnevercrossesthey-axis.1.y=2^(x+2000)#2.y=2^(x-2000)*3.y=2^(x-2000)+50000014.Noneofthese.PAllexponentialfunctionsoftheformy=db^(x+e)+fcrossthey-axis.Whataboutthisone?#y=2^(x+1)-3+*0sqr(x+1)1CreateanexponentialPfunctionyourselfthatnevercrossesthey-axisandwriteitonpapermarkedas#6.P8Fromthelistbelow,selectanexponentialfunctionthathasarangeof(-2,inf).1.y=-(1/2)^(x+1)-3#2.y=3^(x-2)-1*3.y=4^(x+7)-214.y=-(2/3)^(x+4)-1P5.y=6^x-3PWhilegraphingallthefunctionsmighthelpanswerthequestion,knowingthatfunctionparameterfisthe#horizontalasymptote*shouldhavegivenyouthe1answer.Itis#3.PCreateanexponentialfunctionyourselfwitharangeof(-2,inf)andwriteitonpapermarkedas#7.PV9Amathematicalrelationshipthatbestdisplays"exponential"behavioris:1.timevs.heightofaball#tossedstraightup*2.temperaturein1Fahrenheitvs.PtemperatureinCelsius3.timevs.humanpopulationofearth4.dayofyearvs.numberofminutesofsunlightPThehumanpopulationofearthhasbeenbehavinglikeanexponentialfunction.(Thinkabouttheimplications.)#Listareal-world*exponentialrelationship1andwriteitonpaperPmarkedas#8.P 10Fromthelistofexponentialrelationshipslistedbelow,whichoneisdecreasinginbehavior?1.timevs.accountbalance#undercompoundedinterest*2.timevs.numberof1bacteriainapetridishPunderunrestrictedgrowth3.timevs.populationoftheearth4.ageofacarvs.valueusingadepreciationrate#of20%P#4isdecreasingandhasanexponentialbehavior.Listareal-worldexponentialrelationship#thatisdecreasingand*writeitonpapermarked1as#9.P 11Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"PFRPKA35*DHH CREATLIN.8xv**TI83F* AppVariable file 11/29/02, 18:52 CREATLING-;H[ n g 1Create LinearEd LaughbaumSeptember 12, 2006RedBank Publishing,1Thisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhowanttodosomethingdifferent.PNowtypethe>cursorkeytocontinue.Butfirst...Youmayfindthatwritingthemathematicssymbols#andnotationcontainedin*thislessoninproperform1willhelpyouunderstanditPandrememberitlonger.P2Fromthelistbelow,selectalinearfunctionthatisincreasingandisneverinQuadrantII.1.y=2x+1#2.y=32x-2*3.y=-3x-4814.y=0x-1Py=32x-2iscorrect,butitisrelativelyeasytoanswerthequestionbygraphingallthefunctionstoseewhichonemeetsthe#conditions.*1CreateoneyourselfthatPmeetsthesameconditionsandwriteitonpapermarkedas#1.P3Fromthelistbelow,selectalinearfunctionthatisdecreasingandcrossesthey-axisat2.1.y=0x+2#2.y=3x+2*3.y=-4x-214.y=-2x+2Py=-2x+2iscorrect,butitisrelativelyeasytoanswerthequestionbygraphingallthefunctionstoseewhichonemeetsthe#conditions.Butcouldn'tyou*answerthisquestionby1justknowingwhatPbehaviorstheparameterscontrol?Createalinearfunctionthatmatchestheconditions#onthefrontofthecard*andwriteitonpaper1markedas#2.P4Fromthelistbelow,selectalinearfunctionthatisneitherdecreasingorincreasingandcrossesthey-axisat2.#1.y=2x+2*2.y=-2x+213.y=0x-2P4.y=0x+2Py=0x+2iscorrect,butitisrelativelyeasytoanswerthequestionbygraphingallthefunctionstoseewhichonemeetsthe#conditions.Ormaybethe*parameterswouldbeable1tohelpyoudoitmentally?P5Fromthelistbelow,selectalinearfunctionwheretheyvaluechanges3unitsforeverychangeinxof1.1.y=1x+2#2.y=3x+7*3.y=-3x+114.y=(1/3)x+1Py=3x+7iscorrect,butitisrelativelyeasytoanswerthequestionbymakingtablesforallthefunctions,toseewhichone#meetstheconditions.But*don'tthefunction1parametershelpyouPanswerthismentally?Createalinearfunctionthatmatchestheconditionsonthefrontofthecard#andwriteitonpaper*markedas#3.P6Fromthelistbelow,selectalinearfunctionthatcrossesthex-axisat-2andisincreasing.1.y=x+2#2.y=3x+6*3.y=7x+1414.Alltheabove.PNumber4iscorrect,butitisrelativelyeasytoanswerthequestionbygraphingallthefunctions,toseewhichonemeetsthe#conditions.ALSO,ifyou*knowtheparameter-1behaviorconnection,youPcandothismentally.Createalinearfunctionthatmatchestheconditionsonthefrontofthecard#andwriteitonpaper*markedas#4.P7Fromthelistbelow,selectalinearfunctionthatcrossesthex-axisat2andisdecreasing.1.y=-x+2#2.y=-3x+6*3.y=-7x+1414.Alltheabove.P#4iscorrect,butitisrelativelyeasytoanswerthequestionbygraphingallthefunctions,toseewhichonemeetsthe#conditions.Doparameters*dandfhelpyou?1PCreatealinearfunctionthatmatchesthegivenconditionsonthefrontofthecartdandwriteitonpapermarkedas#5.P8Fromthelistbelow,selectalinearfunctionwheretheyvaluedecreases2unitsforeverychangeinxof1.#1.y=2x+1*2.y=-x-213.y=-(4/2)x+54P4.y=-(1/2)x+1Py=-(4/2)x+54iscorrect.Ifyouknowtheparameter-behaviorconnection,oneparameterholdsthisinformation.#Createalinearfunction*thatmatchesthegiven1conditionsonthefrontofPthecardandwriteitonpapermarkedas#6.P9Fromthelistbelow,selectalinearfunctionthatrisesatarateof50%.1.y=.5x+02.y=1.5x-3#3.y=50x+0*4.y=x+50PWhatdoesariseof50%mean?Well,50%means50/100,or1/2,or0.5.Thisisameasureofhowfastitisrising,butthisisrateof#change(orslope).So#1is*correct.1CreatealinearfunctionPthatmatchestheconditionsonthefrontofthecardandwriteitonpapermarkedas#7.P 10Fromthelistbelow,selectalinearfunctionthatfallsatthesamerateasdoesy=-2x+1.1.y=2x+1#2.y=-(1/2)x+1*3.y=-2x+514.Notenoughinfo.PFallingatthesameratemeanstheirratesofchange(orslopes)arethesamenegativenumber.#3iscorrect.Parameterd#helpsyouanswerthis*questionwithouta1graphingcalculator.PCreatealinearfunctionthatmatchestheconditionsonthefrontofthecardandwriteitonpaper#markedas#8.P 11Fromthelistbelow,selectalinearfunctionthatpassesthroughthepoint(-1,2).1.y=3x+5#2.y=-2x+0*3.y=x+314.y=-4x-2P5.Alloftheabove.PApointonthegraphmeansthatyouknowapairof(x,y)valuesthatmustsatisfythe"equation"ofthefunction.#*Createalinearfunction1thatmatchestheconditionsPonthefrontofthecardandwriteitonpapermarkedas#9.P 12Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"PBPKA35 Nh!! CREATQDR.8xv**TI83F* AppVariable file 11/30/02, 14:25 CREATQDR G1BObu*v  "&UCreate QuadraticEd LaughbaumSeptember 12, 2006RedBank Publishing2CreQuadThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhowanttodosomethingdifferent.PNowtypethe>cursorkeytocontinue,butfirst...Reminder:Thequadraticfunctionmaylooklike#y=d(x+e)^2+fOR*y=ax^2+bx+c1MORE...PYoumayfindthatwritingthemathematicssymbolsandnotationcontainedinthislessoninproperformwillhelpyouunderstandit#andrememberitlonger.P}1Whatisthefunctionvariablein:y=d(x+e)^2+f?1.d2.x#3.e*4.f15.yPThefunctionvariableisx.d,e,andfareconstantfunctionparametersthatcontrolthebehaviorofthefunction.yrepresentsthe#function,orfunction*values.1P2Fromthelistbelow,selectaquadraticfunctionthatisincreasingleftofx=2.#1.y=(x+2)^2-1*2.y=-(x-2)^2+713.y=-(x+4)^2-3P4.y=(x-2)^2-45.Noneoftheabove.PIfthequadraticincreasestotheleftof2,theleadingcoefficient(d)mustbenegative.Theeparametermustbetheoppositeof2.#So,y=-(x-2)^2+7isa*correctresponse(#2).1CreateyourownquadraticPfunction(insymbolicform)thatmeetstheconditionsonthereversesideofthiscardandputitonpapermarkedas#1.P3Fromthelistbelow,selectaquadraticfunctionwhosegraphhasamaximumpointat(-1,3).1.y=-(x-1)^2+3#2.y=-(x-1)^2-3*3.y=(x+1)^2+314.y=-(x+1)^2+3PDidyougraphthefunctionstoseewhichoneiscorrect?Didyouknowthatparameterseandfcontrolthemaximumpoint,while#dcontrolswhetherthere*isamaxormin?#4is1correcty=-(x+1)^2+3.PCreateyourownquadraticfunction(insymbolicform)thatmeetstheconditionsonthereversesideofthiscardandwriteitonpaper#markedas#2.P 3aGiventhesymbolicformofaquadraticfunction,y=d(x+e)^2+f,whatparametercontrolswhetherthegraphopens#upordown?*1.d12.eP3.f4.NoneofthesePParameterdcontrolsthegraphopeningupordown.Didyoutrygraphingseveralquadraticfunctionswithvarious#parameterchanges?It*mayhavebeenhelpful.1CreateyourownquadraticPfunction(insymbolicform)thatmeetstheconditionsonthereversesideofthiscardandwriteitonpapermarkedas#3.P4 $5䔒ZII/0TմZ@  @ꪪ;Supposea;quadratic;function;lookslike;that#;displayed.*Whatisparameterd?11.dison(-inf,0)P2.dison(0,inf)3.d=-14.d=15.Notenoughinformation.PBecausethegraphopensdown,youknowthatdisnegative.Answer1iscorrect,dison(-inf,0).#Createyourownquadratic*function(insymbolicform)1thatmeetstheconditionsPonthereversesideofthiscardandwriteitonpapermarkedas#4.P5Usingthesamegraphasinthepreviouscard,whatisparametera?1.aison(-inf,0)2.aison(0,inf)#3.a=-1*4.a=115.Notenoughinformation.PBecausethegraphopensdown,youshouldknowthataisnegative.Answer1iscorrect,aison(-inf,0).#*Createaquadratic1functionthatopensdownPandwriteitonpapermarkedas#5.P6Fromthelistbelow,selectaquadraticfunctionthatisflatterthany=2(x+1)^2-1.1.y=(1/2)(x+1)^2-3#2.y=(1/4)(x+1)^2+4*3.y=x^214.Alltheabove.PAslongasdisbetween0and2,thegraphisflatter.Likewiseiftheabsolutevalueofdisbetween0and2,thegraphisflatter.#*Createyourownquadratic1function(insymbolicform)Pthatmeetstheconditionsonthereversesideofthiscardandwriteitonpapermarkedas#6.P7Fromthelistbelow,selectaquadraticfunctionthathaszerosof-2and3,andhasamaximum.1.y=(x-2)(x+3)#2.y=-(x-2)(x+3)*3.y=-(x+2)(x-3)14.y=(x+2)(x-3)PIfyouhavenoideahowtoanswer,whynotgraphthefunctionsandfindthezeros?Thinkabouthowtheyareconnectedtothe#functionparameters.#3is*correct.1CreateyourownquadraticPfunction(insymbolicform)thatmeetstheconditionsonthereversesideofthiscardandwriteitonpapermarkedas#7.P 7aFromthelistbelow,selectaquadraticfunctionthathaszerosof-2and3,andhasamimimum.1.y=(x-2)(x+3)#2.y=-(x-2)(x+3)*3.y=-(x+2)(x-3)14.y=(x+2)(x-3)PIfyouhavenoideahowtoanswer,whynotgraphthefunctionsandfindthezeros?Thinkabouthowtheyareconnectedtothe#functionparameters.#4is*correct.1CreateyourownquadraticPfunction(insymbolicform)thatmeetstheconditionsonthereversesideofthiscardandwriteitonpapermarkedas#8.P8Fromthelistbelow,selectaquadraticfunctionthatisnegativebetweenx=-2andx=3.1.y=(x-2)(x+3)#2.y=-(x-2)(x+3)*3.y=(x+2)(x-3)14.y=-(x+2)(x-3)PWhileagraphingcalculatorwouldhelpyouanswerthisquestion,doyouseethatparameterdmustbepositivesothatthe#graphhasaminimum*betweenthezeros.The1zerosarecontrolledbyPtheparameters2and-3.#3iscorrect.Createyourownquadraticfunction(insymbolicform)thatmeetstheconditions#ontheflipsideofthiscard*andwriteitonpaper1markedas#9.P9Fromthelistbelow,selectaquadraticfunctionthathasarangeof-1,inf).1.y=x^2+1#2.y=16(x-3)^2-1*3.y=-x^2-114.y=-4x^2+7x-3PAllyouneedisaquadraticthathasaminimumat-1.y=16(x-3)^2-1hasaminimumat-1andthus,arangeof-1,inf).Thatis,#ifparameterfis-1anddis*positive,youhavemetthe1conditions.PCreateyourownquadraticfunction(insymbolicform)thatmeetstheconditionsontheflipsideofthiscardandwriteitonpaper#markedas#10.P 9aFromthelistbelow,selectaquadraticfunctionthathasavertexat(-1,2).1.y=-16(x+1)^2+2#2.y=7(x+1)^2+2*3.y=0.3(x+1)^2+214.alloftheabove.PThevertexisat(-1,2)inthefirst3responsesbecausetheparameterseandfcontrolit,andtheyarethesamein#'s1-3.#Createyourownquadratic*function(insymbolicform)1thatmeetstheconditionsPontheflipsideofthiscardandwriteitonpapermarkedas#11.P 10Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"P&PKA35LXC@ CREATRTN.8xv**TI83F* AppVariable file 12/03/02, 13:36q `CREATRTN`^G->K^ qG & }JCreate RationalEd LaughbaumSeptember 12, 2006RedBank Publishing 5CreRtnThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhowanttodosomethingdifferent.PhNowtypethe>cursorkeytocontinue,butfirst...Reminder:infmeansinfinity(or#increasingwithoutbound)*1TheformofarationalPfunctionwewillstudyis:y=d/(x+e)+fwithfunctionparametersofd,e,andf.#*Youmayfindthatwriting1themathematicssymbolsPandnotationcontainedinthislessoninproperformwillhelpyouunderstanditandrememberitlonger.P|0Whatisthefunctionvariablein:y=d/(x+e)+f?1.d2.x#3.e*4.f15.yPThefunctionvariableisx.d,e,andfareconstantfunctionparametersthatcontrolthebehaviorofeachfunction.yrepresents#thefunction,orfunction*values.P1Fromthelistbelow,selectarationalfunctionthatisincreasingonitsdomain.1.y=-2/(x+1)-32.y=3/(x-2)+1#3.y=1/x*4.y=4/(x+1)-215.y=5/(x-3)-1PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidewhichfunctionis*increasing.Studythe1parametersineachPfunction.#1iscorrect.Createarationalfunctionthatmeetstheconditionsonthefrontofthiscard,andwriteitonpaper#markedas#1.P2Fromthelistbelow,selectarationalfunctionthatisdecreasingonitsdomain.1.y=-2/(x+1)-32.y=-3/(x-2)+1#3.y=1/x*4.y=-4/(x+1)-215.y=-5/(x-3)-1PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidewhichfunctionis*decreasing.Studythe1parametersineachPfunction.#3iscorrect.Createarationalfunctionthatmeetstheconditionsonthefrontofthecard,#andwriteitonpaper*markedas#2.P3Fromthelistbelow,selectarationalfunctionthathasaverticalasymptoteatx=3.1.y=-2/(x+3)-3#2.y=-3/(x-2)+1*3.y=1/x+314.y=-4/(x+1)-2P5.y=-5/(x-3)-1PYoucangrapheachfunctiontohelpyouanswerthis,buttheparameter-behaviorconnectiontellsyouthat#thereisavertical*asymptoteat-e.The1correctansweris#5.PCreatearationalfunctionthatmeetstheconditionsonthereversesideofthiscard,andwriteitonpapermarkedas#3.P4Fromthelistbelow,selectarationalfunctionthathasahorizontalasymptoteaty=3.1.y=-2/(x+3)-3#2.y=-3/(x-2)+1*3.y=1/x+314.y=-4/(x+1)-2P5.y=-5/(x-3)-1PYoucangrapheachfunctiontohelpanswerthequestion,buttheparameter-behaviorconnectiontellsyouthere#isahorizontalasymptote*atf.#3iscorrect.1CreatearationalfunctionPthatmeetstheconditionsonthereversesideofthiscard,andwriteitonpapermarkedas#4.P5Fromthelistbelow,selectarationalfunctionthathasadomainofallrealnumbersexcept-1.1.y=-2/(x+3)-3#2.y=-3/(x-2)+1*3.y=1/x+314.y=-4/(x+1)-2P5.y=-5/(x-3)-1PYoucangrapheachfunctiontohelpanswerthequestion,buttheparameter-behaviorconnectiontellsyouthat#whenthedenominatoris*zero(x=-e)thefunction1isnotreal.#4iscorrect.PCreateanotherrationalfunctionthatmeetstheconditionsonthefrontofthiscard,andwriteitonpapermarkedas#5.P6Fromthelistbelow,selectarationalfunctionthathasadomainofallrealnumbersexcept3.1.y=-2/(x+3)-3#2.y=-3/(x-2)+1*3.y=1/x+314.y=-4/(x+1)-2P5.y=-5/(x-3)-1PYoucangrapheachfunctiontohelpanswerthequestion,buttheparameter-behaviorconnectiontellsyouthat#whenthedenominatoris*zero(x=-e)thefunction1isnotreal.#5iscorrect.PCreatearationalfunctionthatmeetstheconditionsonthereversesideofthiscard,andwriteitonpapermarkedas#6.P7Fromthelistbelow,selectarationalfunctionthathasarangeofallrealnumbersexcept3.1.y=-2/(x+3)-3#2.y=-3/(x-2)+1*3.y=1/x+314.y=-4/(x+1)-2P5.y=-5/(x-3)-1PYoucangrapheachfunctiontohelpanswerthequestion,buttheparameter-behaviorconnectiontellsyouthere#isahorizontalasymptote*atf,thusyvaluesare1neverf.#3iscorrect.PCreateanotherrationalfunctionthatmeetstheconditionsontheothersideofthuscard,andwriteitonpapermarkedas#7.P8Fromthelistbelow,selectarationalfunctionthatdoesnotcrossthex-axis.1.y=-2/(x+3)-32.y=-3/(x-2)+1#3.y=1/x+3*4.y=-4/(x+1)15.y=-5/(x-3)-1PAgraphmayhelpanswerthis,butmaybeyoushouldhavethoughtitthrough.#4iscorrect.Ifyouwantytonever=0,thenyou#musthaveahorizontal*asymptotethere,so1parameterfis0.PCreateanotherrationalfunctionthatmeetstheseconditions,andwriteitonpapermarkedas#8.P9Fromthelistofmathematicalrelationshipsbelow,whichoneismostlikearationalfunction?1.numberofmovietickets#soldvs.profitearned*2.amountofalcoholina1drinkvs.thestrengthofPthedrink3.numberofkeysonacalculatorvs.itscost4.thenumberofpagesofabookvs.theweightofthe#bookPTofindthestrengthofasolution,itisaquotientoftheamountofdilutant(alcohol)andthetotalamountofsolution.#Findanotherrational*relationshipfromthereal1worldandwriteitonyourPpapermarkedas#9.P 10Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"PPKA35s* CREATSQR.8xv**TI83F* AppVariable file 12/03/02, 12:52_ NCREATSQRNLG+<I\ o= L A7CreateSquareRootEd LaughbaumSeptember 12, 2006RedBank Publishing 2CreSqrThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhowanttodosomethingdifferent.P_Nowtypethe>cursorkeytocontinue,butfirst...Reminder:sqrmeanssquareroot#infmeansinfinity(or*increasingwithoutbound)1TheformofasquarerootPfunctionis:y=dsqr(x+e)+fwithfunctionparameters#ofd,e,andf.*1YoumayfindthatwritingPthemathematicssymbolsandnotationcontainedinthislessoninproperformwillhelpyouunderstanditandrememberitlonger.#P0Whatisthefunctionvariablein:y=dsqr(x+e)+f?1.d2.x#3.e*4.f15.yPThefunctionvariableisx.d,e,andfareconstantfunctionparametersthatcontrolthebehaviorofeachfunction.yrepresents#thefunction,orfunction*values.P1Fromthelistbelow,selectasquarerootfunctionthatisdecreasing.1.y=-2sqr(x+1)-32.y=2sqr(x-3)-4#3.y=sqr(x)*4.y=50sqr(x+7)-50015.y=77sqr(x+300)+5000PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidewhichfunctionis*decreasing.#1iscorrect.1CreateasquarerootPfunctionthatmeetstheconditionsonthefrontofthiscard,andwriteitonpapermarkedas#1.P2Fromthelistbelow,selectasquarerootfunctionwithadomainof-2,inf).1.y=-2sqr(x+1)-32.y=2sqr(x-2)-4#3.y=sqr(x)-2*4.y=50sqr(x+2)-515.y=-2sqr(x+3)+5PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidethedomainofthe*function.Studythe1parametersineachPfunction.#4iscorrect.Createasquarerootfunctionthatmeetstheconditionsonthefrontof#thiscard,andwriteiton*papermarkedas#2.P3Fromthelistbelow,selectasquarerootfunctionwitharangeof-2,inf).1.y=-sqr(x+1)-2#2.y=2sqr(x-2)-4*3.y=6sqr(x)-214.y=5sqr(x+2)-5P5.y=-2sqr(x+3)-2PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidetherangeofthe*function.Studythe1parametersineachPfunction.#3iscorrect.Createasquarerootfunctionthatmeetstheconditionsonthefrontofthiscard,andwriteiton#papermarkedas#3.P4Fromthelistbelow,selectasquarerootfunctionwithamaximumof-5.1.y=-sqr(x+5)-22.y=2sqr(x-2)-5#3.y=6sqr(x)-5*4.y=-sqr(x+2)-515.y=-2sqr(x-5)-2PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidethemaximumofthe*function.Studythe1parametersineachPfunction.#4iscorrect.Createasquarerootfunctionthatmeetstheconditionsonthefrontofthiscard,andwriteiton#papermarkedas#4.P5Fromthelistbelow,selectasquarerootfunctionwithaminimumof-5.1.y=-sqr(x+5)-22.y=2sqr(x-2)-5#3.y=6sqr(x)+5*4.y=-sqr(x+2)-515.y=-2sqr(x-5)-2PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectionshouldhelpyou#decidetheminimumofthe*function.Studythe1parametersineachPfunction.#2iscorrect.Createasquarerootfunctionthatmeetstheconditionsonthefrontofthiscard,andwriteiton#papermarkedas#5.P6Fromthelistbelow,selectasquarerootfunctionwithazeroof-5.1.y=-3sqr(x+5)2.y=2sqr(x-2)-5#3.y=6sqr(x)+5*4.y=-sqr(x+2)-515.y=-2sqr(x-5)-2PYoucangrapheachfunctiontohelpanswerthequestion,buttheparameter-behaviorconnectiontellsyouthatif#fis0,thenthegraph*(function)startsaty=01(onthex-axis).#1isPcorrect.Createasquarerootfunctionthatmeetstheconditionsonthefrontofthiscard,andwriteiton#papermarkedas#6.P7Fromthelistbelow,selectasquarerootfunctionwithnozero.1.y=-3sqr(x-1)2.y=-2sqr(x-2)-1#3.y=6sqr(x+5)*4.y=-sqr(x+2)+215.y=2sqr(x-5)-1PYoucangrapheachfunctiontohelpyouanswerthequestion,buttheparameter-behaviorconnectiontellsyouthatif#d>0andf>0,thenthere*isnozero.Likewise,ifd<10andf<0thenthereisnoPzero.Why?#2iscorrect.Createasquarerootfunctionthatmeetstheconditionsonthefrontof#thiscard,andwriteiton*papermarkedas#7.P8Fromthelistbelow,selectasquarerootfunctionwithadomainof(-inf,2].1.y=-3sqr(x-2)2.y=-2sqr(2-x)-1#3.y=6sqr(x+2)*4.y=-sqr(3-x)+215.y=2sqr(-1-x)-1PYoucangrapheachfunctiontohelpyouanswerthequestion,butinthiscasetheparameter-behaviorconnection#doesn'tentirelyhelpsince*thefunctionisnotinthe1standardform.UseyourPbrainalittlemore.#2iscorrect.Createasquarerootfunctionthatmeetstheconditionsonthefrontof#thiscard,andwriteiton*papermarkedas#8.P9Forcommentsonthisstack,pleasecontactEdbywhichofthefollowingmethods?#1.(614)292-7223*2.elaughba@math.ohio-1state.eduP3.(614)292-0694F4.www.math.ohio-state.edu/~elaughba/PEmailisbest.Regards,*EdMaterialsfrom#"FoundationsforCollege*Mathematics2e"P^SPKA35U ;; DOMRANGE.8xv**TI83F* AppVariable file 07/31/03, 15:28; ;DOMRANGE;;G9HUh{ gt #&|*-`0[24-7':Domain & RangeEd LaughbaumSeptember 12, 2006RedBank Publishing&DOMRANGEThisactivityismeanttobeusedbyagroupoftwopeople.Onepersonwillusethisstack,andtheotherwillneedtheTI-83/84#Plusfunctionality.*1Type1tocontinue.P1.Continue2.Forthosewhocan'tfollowrules.PNowtypethe>cursorkeytocontinue.7Butfirst.....Reviewyourintervalnotation:#1,5]meansallreal*numbersfrom1to51including1and5.MOREP(1,5)meansallrealnumbersfrom1to5notincluding1or5.(1,5]meansallrealnumbersfrom1to5,#including5.*1,inf)meansallreal1numbersfrom1toinfinity,Pincluding1.Youmayfindthatwritingthemathematicssymbolsandnotationcontainedin#thislessoninproperform*willhelpyouunderstandit1andrememberitlonger.PR1 ConsiderthesituationinwhichyouuseaCBR2tocollecttime-heightdataasyoutossaballstraightup.Attimezero,theball#leavesyourhandatabout*5.5feetabovethefloor.It1travelsstraightupandatP0.6seconditreachesamaximumheightof11.4feet.Itthendecreasesinheight,untilatabout1.4secondsithitsth