3.6 Interpreting Functions - uen.org · PDF fileSECONDARY MATH I // MODULE 3 FEATURES OF FUNCTIONS Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY

SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

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3.6 Interpreting Functions A Practice Understanding Task

Giventhegraphoff(x),answerthefollowingquestions.Unlessotherwisespecified,

restrictthedomainofthefunctiontowhatyouseeinthegraphbelow.Approximations

areappropriateanswers.

1. Whatisf(2)?

2. Forwhatvalues,ifany,doesf(x)=3?

3. Whatisthex-intercept?

4. Whatisthedomainoff(x)?

5. Onwhatintervalsisf(x)>0?

6. Onwhatintervalsisf(x)increasing?

7. Onwhatintervalsisf(x)decreasing?

8. Forwhatvalues,ifany,isf(x)>3?

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Considerthelineargraphoff(t)andthenonlineargraphofg(t)toanswerquestions9-14.Approximationsareappropriateanswers.

9. Whereisf(t)=g(t)?

10. Whereisf(t)>g(t)?

11. Whatisf(0)+g(0)?

12. Whatisf(-1)+g(-1)?

13. Whichisgreater:f(0)org(-3)?

14. Graph:f(t)+g(t)from[-1,3]

Thefollowingtableofvaluesrepresentstwocontinuousfunctions,f(x)andg(x).Usethetabletoanswerthefollowingquestions:

15. Whatisg(-3)?

16. Forwhatvalue(s)isf(x)=0?

17. Forwhatvaluesdoesf(x)seemtobeincreasing?

18. Onwhatintervalisg(x)>f(x)

19. Whichfunctionischangingfasterintheinterval[-5,-1]? Why?

x f(x) g(x)-5 44 -13-4 30 -9-3 20 -5-2 12 -1-1 6 30 2 71 0 112 0 153 2 194 6 235 12 276 20 31

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Usethefollowingrelationshipstoanswerthequestionsbelow.

ℎ ! = 2! ! ! = 3! − 2 ! ! = 8 ! = 4 ! = 5! + 1

20. Whichoftheaboverelationsarefunctions?Explain.

21. Findf(2),g(2),andh(2).

22. Writetheequationforg(x)+h(x).

23. Whereisg(x)<h(x)?

24. Whereisf(x)increasing?

25. Whichoftheabovefunctionshasthefastestgrowthrate?

Createagraphforeachofthefollowingfunctions,usingthegivenconditions

26. Thisfunctionhasthefollowingfeatures:f(2)ispositive;f(-2)=0,f(x)isalwaysIncreasingandhasadomainofAllRealNumbers.

27. Thisfunctionhasthefollowingfeatures:f(3)>f(6);f(1)=0;f(2)=4;f(x)isincreasingfrom[-5,3);hasadomainfrom[-5,10]

28. Thisfunctionhasthefollowingfeatures:f(x)hasaconstantrateofchange;f(5)=0

29. Createyourownconditions-haveatleastthreeandthencreateexampleswherethesolutioncouldbedifferentgraphs.

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3 . 6 Interpreting Functions – Teacher Notes A Practice Understanding Task

Purpose:Studentshavebeenusingfunctionnotationinvariousformsandhavebecome

morecomfortablewithfeaturesoffunctions.Inthistask,thepurposeisforstudentsto

practicetheirunderstandingofthefollowing:

• Distinguishbetweeninputandoutputvalueswhenusingnotation• Evaluatefunctionsforinputsintheirdomains• Determinethesolutionwherethegraphsoff(x)andg(x)intersectbasedontablesof

valuesandbyinterpretinggraphs• Combinestandardfunctiontypesusingarithmeticoperations(findingvaluesoff(x)+

g(x))• Creategraphsoffunctionsgivenconditions.

CoreStandardsFocus:

F.IF.2Usefunctionnotation,evaluatefunctionsforinputsintheirdomains,andinterpret

statementsthatusefunctionnotationintermsofacontext.

F.IF.4Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionisincreasing,decreasing,positive,ornegative;relativemaximumsandminimums;symmetries;endbehavior;andperiodicity.�

F.IF.5Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitative

relationshipitdescribes.Forexample,ifthefunctionh(n)givesthenumberofperson-hoursit

takestoassemblenenginesinafactory,thenthepositiveintegerswouldbeanappropriate

domainforthefunction.�

F.IF.7Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesandusingtechnologyformorecomplicatedcases.�a.Graphlinearandquadraticfunctionsandshowintercepts,maxima,andminima.






e.Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions,showingperiod,midline,andamplitude.F.BF.1bWriteafunctionthatdescribesarelationshipbetweentwoquantities.Combinestandardfunctiontypesusingarithmeticoperations.A.REI.11Explainwhythex-coordinatesofthepointswherethegraphsoftheequationsy=f(x)andy=g(x)intersectarethesolutionsoftheequationf(x)=g(x);findthesolutionsapproximately,e.g.,usingtechnologytographthefunctions,maketablesofvalues,orfindsuccessiveapproximations.Includecaseswheref(x)and/org(x)arelinear,polynomial,rational,absolutevalue,exponential,andlogarithmicfunctions.�A.CED.3Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/orinequalities,andinterpretsolutionsasviableornon-viableoptionsinamodelingcontext.Forexample,representinequalitiesdescribingnutritionalandcostconstraintsoncombinationsofdifferentfoods.RelatedStandards:F.IF.1,A.REI.10,A.REI.11,N.Q.1,A.CED.2

StandardsforMathematicalPracticeofFocusintheTask:

SMP7–Lookandmakeuseofstructure

SMP8–Lookforandexpressregularityinrepeatedreasoning

TheTeachingCycle:

Launch(WholeClass):

Studentsshouldbeabletogetstartedonthistaskwithoutadditionalsupport,sinceitissimilarin

naturetotheworktheydidon“TheWaterPark”and“PoolingItTogether”.Ifpreferred,youmay

wishtosketchagraphontheboardandaskstudentsacoupleofquestionsusingfunctionnotation

beforehavingthembeginthetask.Thiswouldbeagoodtasktohavestudentsstartontheirown,

thenhavethempairupaftermosthavecompletedthefirstsetofquestions.

Explore(SmallGroup):

Watchforstudentswhoconfuseinput/outputvalues.Withoutcontext,keepingtrackofthisisa

commonmistake.Encouragestudentstoexplaintheirreasoningtoeachotherwhileworking

throughsolutionstoproblems.Ifstudentsareincorrectintheirthinking,besuretoredirecttheir






thinking.Asyoumonitor,makenoteoftheareaswherestudentsarestrugglingandselectstudents

whoexplain/clarifydetails.

Discuss(WholeClass):

Gooverproblemsthatseemtobecommonissuesthatstudentsarestillgrapplingwithfirst.After

this,choosestudentstosharetheirmethodforgraphingnumber14.Comparestudentswhoused

pointbypointtothosewhoaddedonfromonegraphtothenext.Last,choosestudentswhohave

correctbutdifferentgraphsforoneofthefollowingproblems(either26or27).Havestudents

compare/contrastgraphsandexplainwhybothareappropriategiventheconditions.

ThegoalofthiswholegroupdiscussionisthatALLstudentscanevaluatefunctionsusingnotation,

caninterpretfeaturesoffunctionsusingagraphortableofvalues,cancreateagraphgiven

conditions,andcancombinetwofunctionstomakeanotherfunction.

AlignedReady,Set,Go:Features3.6


FEATURES OF FUNCTIONS – 3.6




3.6

READY Topic:SolvingSystemsbySubstitutionInpriorworkthemeaningof! ! = ! ! wasdiscussed.Thismeanstofindthepointwherethetwoequationsareequalandwherethetwographsintersect.Itispossibletofindthepointofintersectionalgebraicallyinsteadofgraphingthetwolines.Since! ! = ! ! ,it’spossibletoseteachequationequaltotheotherandsolveforx.

Example:Findthepointofintersectionoffunction! ! = 3! + 4andfunction! ! = 4! + 1.Since,! x = g x , !"# 3! + 4 = 4! + 1.Thensolveforx.

3! + 4 = 4! + 1 Subtract3xand1frombothsidesoftheequation.−3! − 1 = −3! − 10! + 3 = 1! + 0 3 = 1!Nowletx=3ineachequationtofind! ! !"# ! ! !ℎ!" ! = 3. ! 3 = 3 3 + 4 → 9 + 4 = 13 !"#! 3 = 4 3 + 1 → 12 + 1 = 13When! = 3, ! 3 !"# ! 3 !"#ℎ !"#$% 13. !ℎ! !"#$% !" !"#$%&$'#!(" !" 3, 13 .

Findthepointofintersectionfor! ! !"# ! ! usingthealgebraicmethodintheexampleabove.1.! ! = −5! + 12and! ! = −2! − 3

2.! ! = !! ! + 2and! ! = 2! − 7

3.! ! = − !! ! + 5and! ! = −! + 7

4.! ! = ! − 6and! ! = −! − 6

READY, SET, GO! Name PeriodDate

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3.6

SET Topic:DescribingattributesofafunctionsbasedongraphicalrepresentationUsethegraphofeachfunctionprovidedtofindtheindicatedvalues.

5.f(x)

a.f(4)=________b.f(-4)=__________

c.f(x)=4,x=_________d.f(x)=7,x=________

6.g(x)

a.g(-1)=__________b.g(-3)=___________

c.g(x)=-4x=_______d.g(x)=-1,x=_________

7.h(x)

a.h(0)=________b.h(3)=__________

c.h(x)=1,x=_________d.h(x)=-2,x=______

8.d(x)

a.d(-5)=________b.d(4)=__________

c.d(x)=4,x=_________d.d(x)=0,x=______

6

4

2

–2

–5 5

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3.6

Foreachsituationeithercreateafunctionorusethegivenfunctiontofindandinterpretsolutions.9.Francollecteddataonthenumberoffeetshecouldwalkeachsecondandwrotethefollowingruletomodelherwalkingrate! ! = 4!.

a.WhatisFranlookingforifshewrites! 12 = _______?

b.Inthissituationwhatdoes! ! = 100tellyou?

c.Howcanthefunctionrulebeusedtoindicateatimeof16secondswaswalked?

d.Howcanthefunctionrulebeusedtoindicatethatadistanceof200feetwaswalked? 10.Mr.Multbankhasdevelopedapopulationgrowthmodelfortherodentsinthefieldbyhishouse.Hebelievesthatstartingeachspringthepopulationcanbemodeledbasedonthenumberofweekswiththefunction! ! = 8 2! .

a.Find! ! = 128.

b.Find! 4 .

c.Find! 10 .

d.Findthenumberofweeksitwilltakeforthepopulationtobeover20,000.e.Inayearwith16weeksofsummer,howmanyrodentswouldheexpectbytheendofthesummerusingMr.Multbank’smodel?Whataresomefactorsthatcouldchangetheactualresultfromyourestimate?

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3.6

GO Topic:Describefeaturesoffunctionsfromthegraphicalrepresentation.Foreachgraphgivenprovideadescriptionofthefunction.Besuretoconsiderthefollowing:decreasing/increasing,min/max,domain/range,etc.11. Descriptionoffunction:

12. Descriptionoffunction:

13. Descriptionoffunction:

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3.6 Interpreting Functions - uen.org · PDF fileSECONDARY MATH I // MODULE 3 FEATURES OF FUNCTIONS Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY