4.5$Watch$Your$ Behavior $ A"Develop!Understanding+Task!

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1 23 4.5$Watch$Your$ Behavior $ A"DevelopUnderstanding+Task Inthistask,youwilldevelopyourunderstandingoftheendbehaviorofrationalfunctionsaswellas discoverthebehaviorofevenandoddfunctions. PartI:Endbehaviorofrationalfunctions AftercompletingthetaskThe$Gift,MarcusandHannahweretalkingaboutthediscussionregarding theendbehavioroftheparentfunction =.Marcussaid Ithoughttheendbehaviorofall functionswasthatyoueitherendedupgoingtopositiveornegativeinfinity. Hannahagreed, adding Nowwehaveafunctionthatapproacheszero.Iwonderifallrationalfunctionswillalways approachzeroasxapproaches±. Marcusreplied Iamsuretheydo.Justlikeallpolynomial functionsendbehaviorapproacheseither±,ithinktheendbehaviorforallrationalfunctions mustapproachzero. 1. CouldMarcusberight?Makeaconjectureabouttheendbehaviorofrationalfunctionsand testit.hint:thisshouldtakeawhileobesuretothinkaboutthevariousrational expressionswehavestudied).asyouanalyzetheendbehaviorofdifferentrational functions,trytogeneralizethepatternsyounoticeregardingendbehavior. MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

2 24 PartII:EvenandOddFunctions Belowarethreegraphs:onerepresentsanevenfunction,onerepresentsanoddfunction,andone isneitherevennorodd. evenfunction: = oddfunction: = neither: = 4) Usethegraphsandtheircorrespondingfunctionstowriteadefinitionforanevenfunction andanoddfunction. Afunctionisanevenfunctionif Afunctionisanoddfunctionif MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

3 $ Belowaremorefunctions.Basedonyourdefinition,classifyaseithereven,odd,orneither. a. x y b. = c. 42$ 10$ 41$ 5$ 0$ 44$ d. g.$$ 1$ 5$ 2$ 10$ x y 42$ 410$ 41$ 45$ 0$ 0$ e. h. = 3 = + 2 f. i. = = 2) + 2) 1$ 5$ 2$ 10$ 4. Theanswerstoquestionthreeareatthebottomofthispage.Checkyoursolutionsand adjustyourdefinitionsofevenandoddfunctions,asneeded. MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

4 4.5$WatchYourBehavior Teacher#Notes$ A"DevelopUnderstanding+Task Purpose:$$The$purpose$of$this$task$is$for$students$to$surface$their$thinking$about$the$end$behavior$ asymptote$of$rational$functions$both$proper$and$improper)$and$also$to$examine$whether$a$ function$is$even,$odd,$or$neither.$$ Teachernote:Prior$to$this$task,$students$have$become$familiar$with$the$end$behavior$of$various$ functions$and$have$also$become$familiar$with$proper$and$improper$rational$expressions.$in$this$ task,$we$are$asking$students$to$think$about$what$conditions$make$it$so$that$the$end$behavior$ asymptote$is$y=0,and$are$there$times$when$it$would$be$something$else?$can$they$explain$under$ what$conditions$the$end$behavior$asymptote$would$be$something$different?$it$is$important$to$note$ here$that$while$you$want$to$press$your$students$to$come$up$with$different$examples$and$to$try$to$ find$examples$of$rational$functions$with$different$end$behaviors,$if$your$class$does$not$generate$all$ three$scenarios$for$end$behavior$in$this$task,$it$is$ok$as$the$next$few$tasks$solidify$this$idea.$$ In$Part$II,$students$are$introduced$to$even$and$odd$functions$and$will$generate$definitions$for$even$ and$odd$functions$based$on$observations$provided$using$multiple$representations).$$ StandardsFocus: F.IF.7dGraph$functions$expressed$symbolically$and$show$key$features$of$the$graph,$by$hand$in$ simple$cases$and$using$technology$for$more$complicated$cases.*$ d.$graph$rational$functions,$identifying$zeros$when$suitable$factorizations$are$available,$and$ showing$end$behavior.$ $ F.BF.3Identify$the$effect$on$the$graph$of$replacing$fx)byfx)+k,kfx),fkx),andfx+k)for$specific$ values$of$k$both$positive$and$negative);$find$the$value$of$k$given$the$graphs.$experiment$with$cases$ and$illustrate$an$explanation$of$the$effects$on$the$graph$using$technology.$include$recognizing$even$ and$odd$functions$from$the$graphs$and$algebraic$expressions$for$them.$ $ LaunchPartIWholeClass):Startthistaskbyreadingthescenarioandthenlettingstudents knowthattheyshouldworkinpartnersandtrytocomeupwithvariousrationalfunctionsthat createdifferentendbehaviors.foreachrationalfunctiontheycreate,theyshouldstatetheend behaviorandcomeupwithanequationthatmodelstheendbehaviorasymptote.theyshouldalso beabletoprovideevidencethattheirendbehavioriscorrect. ExplorePartIsmallgroups):Monitorstudentthinkingastheyworkinsmallgroups.Press studentstofirstcreatedifferentrationalfunctionsandexplaintheendbehavior,thenseeifthey canidentifywhytheendbehaviorforaparticularfunctionexistsmovingthemtoageneralization MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

5 tofindtheendbehaviorasymptotesforvariousrationalfunctions).forstudentswhostruggle,ask themtostartsimpleandcreatearationalfunction,explainwhyitisrationalandthenfind/explain theendbehavior.tellthemtorepeattheprocessbychangingthefunctionabitandseeifthis changestheendbehavior. Asyoumonitor,lookforstudentswhonoticetheendbehaviorgoestoeitherpositiveornegative infinitywhenthedegreeofthenumeratorishigherthanthedenominator,forstudentswhonotice theendbehaviorisaconstantwhenthedegreeofboththenumeratoranddenominatorarethe sameandthattheconstantisthecooefficients),forstudentswhocanexplainthattheendbehavior isalwaysgoingtozerowhenthedegreeofthenumeratorislessthanthedenominator,andalso lookforstudentswhomayextendthisthinkingtoconnecttotheworktheydidinthetask4.4$ Rational$Expressions$andcomeupwiththeendbehaviorasymptoteequation.Ineachsituation, makesurethestudentscanexplain/provideevidenceoftheirclaim.itismoreimportantinthis taskthattheyknowwhatdifferentrationalfunctionslooklikeandthattheycandeterminetheend behaviorthanitisthattheysolidifytheirunderstandingofendbehaviorasymptotesofallrational functions. DiscussPartIWholeClass):Oncestudentshavedonetheirbest,bringthewholegroup togethertosharetheirfindings.onewaytosequencestudentworkmaybetoselectacoupleof studentstosharethathavecomeupwithdifferentequationsandyethavethesameendbehavior thosewithanendbehaviorofpositiveinfinityornegativeinfinity).doeseveryoneintheclass agreethattheendbehavioristhesamefortheseequations?howaretheseequationssimilar?at thistime,focusontheendbehaviorandnottheasymptoteequation.next,selectastudentwhose functionendbehaviorisaconstantandhavethemexplainwhy.theextentofthiswholegroup discussiondependsonwhatideasandfindingsyourstudentscameupwith. LaunchPartIIWholeClass):AfterthewholegroupdiscussionfromPartI,havestudents lookatthethreefunctionsinpartiiandhavethemwritedowntheirideasforwhattheythinkthe definitionisforanevenfunction,anoddfunction,andafunctionthatisneitherevennorodd. Gatherideasfromthegroup,thenhavethemworkintheirsmallgroupstoanswerquestion3. ExplorePartIISmallGroup):Monitorstudentsastheywork.Whenappropriate,letthem knowtheanswerstoquestion3areatthebottomofthepagesotheycanchecktoseeiftheir solutionsmatch.redirectstudentstoadjusttheirdefinitionforevenfunction,oddfunction,and thosethatwouldbeneitherbasedonwhattheyhavelearned. DiscussWholeClass):Forthewholegroupdiscussion,havetheclassclarifythedefinitions. Makesurethedefinitionincludesthefollowing: EVEN:$$ A$function$fx)is$classified$as$an$evenfunctionif$it$is$symmetric$about$the$y4axis.$ A$function$fx)is$classified$as$an$evenfunctionif$the$output$is$the$same$for$both$xand x. in$other$words,$a$function$is$an$even$function$if$f2θ)=fθ)). MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

6 ODD:$$ A$function$fx)is$classified$as$an$oddfunctionif$it$is$symmetric$about$the$origin$or$rotates$ 180$degrees$about$the$origin$onto$itself$or$reflects$about$the$xandyaxisonto$itself).$ If$afunctionis$an$odd$function,$then$if$the$point$a,b)satisfies$the$function,$then$so$does$ the$point$2a,2b). A$function$fx)is$classified$as$an$oddfunctionif$f2θ)=2fθ). IFafunctionisnotevenorodd,thenitisneither. AlignedReady,Set,GoHomework:RationalFunctions4.5 MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

7 NameRationalExpressionsandFunctions 4.5$ Ready,$Set,$Go$ Ready$ Topic:FeaturesofFunctions Basedonthegraphgivenineachproblembelow,identifyattributesofthefunctionsuchasthe domain,rangeandwhetherornotthefunctionisincreasingordecreasing,etc MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense

8 NameRationalExpressionsandFunctions 4.5$ Set$ Topic:Determineendbehaviorforrationalfunctions Foreachofthegivenfunctionsdeterminetheoutputorrangevaluethatisapproachedbythe functionasthex>valueapproaches+ andalso ) + 2) = = ) + 2) h = = ) = = Topic:EvenandOddfunctions 13.Determinewhichofthefollowingfunctionsareeven,oddorneither.Labelthemaccordingly. a. b. c. = 3 = 1 = d. e. f. = = + 7 = Usetechnologytographeachofthefunctionsfromnumber13.Whatgraphicalcharacteristics gowithanevenfunctionandwhatcharacteristicsgowithanoddfunction. 27 MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense

9 NameRationalExpressionsandFunctions 4.5$ Giventhatthepartialgraphsbelowareevenandoddfunctions,drawintherestofthegraph. 15.Evenfunction 16.OddFunction 28 Topic:Findingasymptotesofrationalfunctions. Findtheasymptotesforthefunctionsbelow fx)= x 2 >9 18.fx)= 3 x 3 +2x 2 >3x $ 19. fx)= 1 x 2 20.fx)= 3 x 2 x+3)x>1) Go$ Topic:Solveeachrationalequationandinequality ) + 2) 0 = 0 = = = ) 1) + 2) > < 0 MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense

10 NameRationalExpressionsandFunctions 4.5$ 21. TrueorFalse:Allpolynomialfunctionsarealsorationalfunctions TrueorFalse:Allrationalfunctionsareoddfunctions. 23. TrueorFalse:Allrationalfunctionsapproachzeroas ±. MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense