Linear & Exponential Functions...20. Best Buy Shoes had a back to school special. The total bill for four pairs of shoes came to $69.24 (before tax.) What was the average price for

The Mathematics Vision Project

Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2016 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 2

Linear & Exponential Functions

SECONDARY

MATH ONE

An Integrated Approach

SECONDARY MATH 1 // MODULE 2

LINEAR & EXPONETIAL FUNCTIONS

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 2 - TABLE OF CONTENTS

LINEAR AND EXPONENTIAL FUNCTIONS

2.1 Piggies and Pools – A Develop Understanding Task

Introducing continuous linear and exponential functions (F.IF.3)

READY, SET, GO Homework: Linear and Exponential Functions 2.1

2.2 Shh! Please Be Discreet (Discrete!) – A Solidify Understanding Task

Connecting context with domain and distinctions between discrete and continuous functions

(F.IF.3, F.BF.1a, F.LE.1, F.LE.2)


2.3 Linear Exponential or Neither – A Practice Understanding Task

Distinguishing between linear and exponential functions using various representations (F.LE.3, F.LE.5)


2.4 Getting Down to Business – A Solidify Understanding Task

Comparing growth of linear and exponential models (F.LE.2, F.LE.3, F.LE.5, F.IF.7, F.BF.2)


2.5 Making My Point – A Solidify Understanding Task

Interpreting equations that model linear and exponential functions (A.SSE.1, A.CED.2, F.LE.5, A.SSE.6)


2.6 Form Follows Function – A Solidify Understanding Task

Building fluency and efficiency in working with linear and exponential functions in their various forms

(F.LE.2, F.LE.5, F.IF.7, A.SSE.6)



LINEAR & EXPONENTIAL FUNCTIONS – 2.1


2.1 Connecting the Dots: Piggies and Pools

A Develop Understanding Task

1. Mylittlesister,Savannah,isthreeyearsold.Shehasapiggybankthatshewantstofill.She

startedwithfivepenniesandeachdaywhenIcomehomefromschool,sheisexcitedwhenI

giveherthreepenniesthatareleftoverfrommylunchmoney.Useatable,agraph,andan

equationtocreateamathematicalmodelforthenumberofpenniesinthepiggybankondayn.

2. Ourfamilyhasasmallpoolforrelaxinginthesummerthatholds1500gallonsofwater.I

decidedtofillthepoolforthesummer.WhenIhad5gallonsofwaterinthepool,Idecidedthat

Ididn’twanttostandoutsideandwatchthepoolfill,soIhadtofigureouthowlongitwould

takesothatIcouldleave,butcomebacktoturnoffthewaterattherighttime.Icheckedthe

flowonthehoseandfoundthatitwasfillingthepoolatarateof2gallonseveryminute.Usea

table,agraph,andanequationtocreateamathematicalmodelforthenumberofgallonsof

waterinthepoolattminutes.

CCBYStockm

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1




3. I’mmoresophisticatedthanmylittlesistersoIsavemymoneyinabankaccountthatpaysme

3%interestonthemoneyintheaccountattheendofeachmonth.(IfItakemymoneyout

beforetheendofthemonth,Idon’tearnanyinterestforthemonth.)Istartedtheaccountwith

$50thatIgotformybirthday.Useatable,agraph,andanequationtocreateamathematical

modeloftheamountofmoneyIwillhaveintheaccountaftermmonths.

4. Attheendofthesummer,Idecidetodrainthe1500gallonswimmingpool.Inoticedthatit

drainsfasterwhenthereismorewaterinthepool.Thatwasinterestingtome,soIdecidedto

measuretherateatwhichitdrains.Ifoundthat3%wasdrainingoutofthepooleveryminute.

Useatable,agraph,andanequationtocreateamathematicalmodelofthegallonsofwaterin

thepoolattminutes.

2




5. Compareproblems1and3.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?



3

SECONDARY MATH I // MODULE 2


Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

2.1

READY Topic:RecognizingarithmeticandgeometricsequencesPredictthenext2termsinthesequence.Statewhetherthesequenceisarithmetic,geometric,orneither.Justifyyouranswer.1.4,-20,100,-500,... 2.3,5,8,12,...3.64,48,36,27,... 4.1.5,0.75,0,-0.75,...

5.40,10,!! , !! ,… 6.1,11,111,1111,...

7.-3.6,-5.4,-8.1,-12.15,... 8.-64,-47,-30,-13,...9.Createapredictablesequenceofatleast4numbersthatisNOTarithmeticorgeometric. SET Topic:DiscreteandcontinuousrelationshipsIdentifywhetherthefollowingstatementsrepresentadiscreteoracontinuousrelationship.10.Thehaironyourheadgrows½inchpermonth.11.Foreverytonofpaperthatisrecycled,17treesaresaved.12.Approximately3.24billiongallonsofwaterflowoverNiagaraFallsdaily.13.Theaveragepersonlaughs15timesperday.14.ThecityofBuenosAiresadds6,000tonsoftrashtoitslandfillseveryday.15.DuringtheGreatDepression,stockmarketpricesfell75%.

READY, SET, GO! Name PeriodDate

4




Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

2.1

GO Topic:Solvingone-stepequationsEitherfindorusetheunitrateforeachofthequestionsbelow.16.Applesareonsaleatthemarket4poundsfor$2.00.Whatistheprice(incents)foronepound?17.Threeapplesweighaboutapound.Abouthowmuchwouldoneapplecost?(Roundtothenearestcent.)18.Onedozeneggscost$1.98.Howmuchdoes1eggcost?(Roundtothenearestcent.)19.Onedozeneggscost$1.98.Ifthechargeattheregisterforonlyeggs,withouttax,was$11.88,howmanydozenwerepurchased?

20.BestBuyShoeshadabacktoschoolspecial.Thetotalbillforfourpairsofshoescameto$69.24(beforetax.)Whatwastheaveragepriceforeachpairofshoes?21.Ifyouonlypurchased1pairofshoesatBestBuyShoesinsteadofthefourdescribedinproblem20,howmuchwouldyouhavepaid,basedontheaverageprice?Solveforx.Showyourwork.22.6! = 72

23.4! = 200 24.3! = 50

25.12! = 25.80

26.!! ! = 17.31 27.4! = 69.24

28.12! = 198

29.1.98! = 11.88 30.!! ! = 2

31.Someoftheproblems22–30couldrepresenttheworkyoudidtoanswerquestions16–21.Writethe

numberoftheequationnexttothestoryitrepresents.

5




2.2 Shh! Please Be Discreet (Discrete)!

A Solidify Understanding Task

1. TheLibraryofCongressinWashingtonD.C.isconsideredthelargestlibraryintheworld.Theyoftenreceiveboxesofbookstobeaddedtotheircollection.Sincebookscanbequiteheavy,theyaren’tshippedinbigboxes.If,onaverage,eachboxcontainsabout8books,howmanybooksarereceivedbythelibraryin6boxes,10boxes,ornboxes?a. Useatable,agraph,andanequationtomodelthissituation.

b. Identifythedomainofthefunction.

2. ManyofthebooksattheLibraryofCongressareelectronic.Ifabout13e-bookscanbedownloadedontothecomputereachhour,howmanye-bookscanbeaddedtothelibraryin3hours,5hours,ornhours(assumingthatthecomputermemoryisnotlimited)?a. Useatable,agraph,andanequationtomodelthissituation.


CCBYRoche

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3. Thelibrariansworktokeepthelibraryorderlyandputbooksbackintotheirproperplacesaftertheyhavebeenused.Ifalibrariancansortandshelve3booksinaminute,howmanybooksdoesthatlibrariantakecareofin3hours,5hours,ornhours?Useatable,agraph,andanequationtomodelthissituation.

4. Woulditmakesenseinanyofthesesituationsfortheretobeatimewhen32.5bookshadbeenshipped,downloadedintothecomputerorplacedontheshelf?

5. Whichofthesesituations(inproblems1-3)representadiscretefunctionandwhichrepresentacontinuousfunction?Justifyyouranswer.

7




6. Agiantpieceofpaperiscutintothreeequalpiecesandtheneachofthoseiscutintothreeequalpiecesandsoforth.Howmanypaperswilltherebeafteraroundof10cuts?20cuts?ncuts?

ZeroCuts OneCut TwoCuts

a. Useatable,agraph,andanequationtomodelthissituation.


c. Woulditmakesensetolookforthenumberofpiecesofpaperat5.2cuts?Why?d.Woulditmakesensetolookforthenumberofcutsittakestomake53.6papers?Why?

8




7. Medicinetakenbyapatientbreaksdowninthepatient’sbloodstreamanddissipatesoutofthepatient’ssystem.Supposeadoseof60milligramsofanti-parasitemedicineisgiventoadogandthemedicinebreaksdownsuchthat20%ofthemedicinebecomesineffectiveeveryhour.Howmuchofthe60milligramdoseisstillactiveinthedog’sbloodstreamafter3hours,after4.25hours,afternhours?a. Useatable,agraph,andanequationtomodelthissituation.


c. Woulditmakesensetolookforanamountofactivemedicineat3.8hours?Why?

d. Woulditmakesensetolookforwhenthereis35milligramsofmedicine?Why?

9




8. Whichofthefunctionsmodeledin#6and#7arediscreteandwhicharecontinuous?Why?

9. Whatneedstobeconsideredwhenlookingatasituationorcontextanddecidingifitfitsbestwithadiscreteorcontinuousmodel?

10. Describethedifferencesineachrepresentation(table,graph,andequation)fordiscreteandcontinuousfunctions.

11. Whichofthefunctionsmodeledabovearelinear?Whichareexponential?Why?

10




Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

2.2

READY

Topic:Comparingratesofchangeinlinearsituations.

Statewhichsituationhasthegreatestrateofchange

1.Theamountofstretchinashortbungeecordstretches6incheswhenstretchedbya3poundweight.Aslinkystretches3feetwhenstretchedbya1poundweight.2.Asunflowerthatgrows2incheseverydayoranamaryllisthatgrows18inchesinoneweek.3.Pumping25gallonsofgasintoatruckin3minutesorfillingabathtubwith40gallonsofwaterin5minutes.4.Ridingabike10milesin1hourorjogging3milesin24minutes.

SET

Topic:Discreteandcontinuousrelationships

Identifywhetherthefollowingitemsbestfitwithadiscreteoracontinuousmodel.Thendeterminewhetheritisalinear(arithmetic)orexponential(geometric)relationshipthatisbeingdescribed.5. Thefreewayconstructioncrewpours300ftofconcreteinaday.

6. Foreveryhourthatpasses,theamountofareainfectedbythebacteriadoubles.

7. Tomeetthedemandsplacedonthemthebricklayershavestartedlaying5%morebrickseachday.

8. Theaveragepersontakes10,000stepsinaday.

9. ThecityofBuenosAireshasbeenadding8%toitspopulationeveryyear.

10.AttheheadwatersoftheMississippiRiverthewaterflowsatasurfacerateof1.2milesperhour.

11.a.! ! = ! ! − 1 + 3; ! 1 = 5b. c.! ! = 2!(7)


11






2.2

GO

Topic: Solving one-step equations Solve the following equations. Remember that what you do to one side of the equation must also be done to the other side. (Show your work, even if you can do these in your head.) Example: Solve for x . 1! + 7 = 23 !"" − 7 !! !"#ℎ !"#$! !" !ℎ! !"#$%&'(.

Example: Solve for x. 9! = 63 !"#$%&#' !"#ℎ !"#$! !" !ℎ! !"#$%&'( !" !! .

Note that multiplying by !! gives the same result as dividing everything by 9.

11. 1! + 16 = 36 12. 1! − 13 = 10 13. 1! − 8 = −3 14. 8! = 56 15. −11! = 88 16. 425! = 850

17. !! ! = 10 18. − !! ! = −1 19.

!! ! = −9

12




2.3 Linear, Exponential or Neither?

A Practice Understanding Task

Foreachrepresentationofafunction,decideifthefunctionislinear,exponential,orneither.Giveatleast2reasonsforyouranswer.

1.

Linear Exponential Neither

Why?

2. TennisTournament

Rounds 1 2 3 4 5Number

ofPlayersleft

64 32 16 8 4

Thereare4playersremainingafter5rounds


Why?

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13




3.! = 4!


Why?

4.Thisfunctionisdecreasingataconstantrate.


Why?

5. Linear Exponential Neither

Why?

14




6.Aperson'sheightasafunctionofaperson'sage(fromage0to100)


Why?

7.−3! = 4! + 7


Why?

8.x y-2 230 52 -134 -316 -49


Why?

15




9.HeightinInches ShoeSize

62 674 1370 967 1153 458 7


Why?

10.ThenumberofcellphoneusersinCentervilleasa

functionofyears,ifthenumberofusersis

increasingby75%eachyear.


Why?

11. Linear Exponential Neither

Why?

16




12.Thetimeittakesyoutogettoworkasafunctionthespeedatwhichyoudrive


Why?

13.! = 7!!


Why?

14.Eachpointonthegraphisexactly1/3ofthepreviouspoint.


Why?

17




15.! 1 = 7, ! 2 = 7, ! ! = ! ! − 1 + !(! − 2)


Why?

16.

! 1 = 1, ! ! = 23 !(! − 1)


Why?

18






2.3

READY Topic:Comparingratesofchangeinbothlinearandexponentialsituations.Identifywhethersituation“a”orsituation“b”hasagreaterrateofchange.

1. x y

-10 -48

-9 -43

-8 -38

-7 -33

a.

b.

2. a.

b.

3. a.Leehas$25withheldeachweekfromhissalarytopayforhissubwaypass.

b.Joseoweshisbrother$50.Hehaspromisedtopayhalfofwhatheoweseachweekuntilthedebtispaid.

4. a.

x 6 10 14 18

y 13 15 17 19

b.Thenumberofrhombiineachshape.

Figure1Figure2Figure3

5. a.! = 2(5)! b.Inthechildren'sbook,TheMagicPot,everytimeyouputoneobjectintothepot,twoofthesameobjectcomeout.Imaginethatyouhave5magicpots.


-5

-5

5

5

19






2.3

SET Topic:Recognizinglinearandexponentialfunctions.Basedoneachofthegivenrepresentationsofafunctiondetermineifitislinear,exponentialorneither.6.Thepopulationofatownisdecreasingatarateof1.5%peryear.

7.Joanearnsasalaryof$30,000peryearplusa4.25%commissiononsales.

8.3x+4y=-3

9.Thenumberofgiftsreceivedeachdayof”The12DaysofChristmas”asafunctionoftheday.(“Onthe4thdayofChristmasmytruelovegavetome,4callingbirds,3Frenchhens,2turtledoves,andapartridgeinapeartree.”)

10.

11.

Sideofasquare Areaofasquare1inch 1in22inches 4in2

3inches 9in2

4inches 16in2

GO Topic:GeometricmeansForeachgeometricsequencebelow,findthemissingtermsinthesequence.

12. 13. 14.

x 1 2 3 4 5

y 2 162

x 1 2 3 4 5

y 1/9 -3

x 1 2 3 4 5

y 10 0.625

20






2.3

15. 16. Findtherateofchange(slope)ineachoftheexercisesbelow.

17. x g(x)

-5 11

-3 4

-2 0.5

0 -6

18. t h(t)

3 13

8 23

18 43

23 53

19.

n f(n)

-7 20

-5 24

-1 32

2 38

20. (2,5)(8,29) 21. 22. (-3,7)(8,29)

x 1 2 3 4 5

y g gz4

x 1 2 3 4 5

y -3 -243

21




2.4 Getting Down to Business


Calcu-ramahadanetincomeof5milliondollarsin2010,whileasmallcompetingcompany,

Computafest,hadanetincomeof2milliondollars.ThemanagementofCalcu-ramadevelopsa

businessplanforfuturegrowththatprojectsanincreaseinnetincomeof0.5millionperyear,

whilethemanagementofComputafestdevelopsaplanaimedatincreasingitsnetincomeby15%

eachyear.

a. Createstandardmathematicalmodels(table,graphandequations)fortheprojectednet

incomeovertimeforbothcompanies.(Attendtoprecisionandbesurethateachmodelis

accurateandlabeledproperlysothatitrepresentsthesituation.)

b. Comparethetwocompanies.Howaretherepresentationsforthenetincomeofthetwo

companiessimilar?Howdotheydiffer?Whatrelationshipsarehighlightedineach

representation?

CCBYreynermed

ia

https://flic.kr/p/ifz6w

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c. Ifbothcompanieswereabletomeettheirnetincomegrowthgoals,whichcompanywould

youchoosetoinvestin?Why?

d. When,ifever,wouldyourprojectionssuggestthatthetwocompanieshavethesamenet

income?Howdidyoufindthis?Willtheyeverhavethesamenetincomeagain?

e. Sincewearecreatingthemodelsforthesecompanieswecanchoosetohaveadiscrete

modeloracontinuousmodel.Whataretheadvantagesordisadvantagesforeachtypeof

model?

23






2.4

READY Topic:Comparingarithmeticandgeometricsequences.Thefirstandfifthtermsofasequencearegiven.Fillinthemissingnumbersifitisanarithmeticsequence.Thenfillinthenumbersifitisageometricsequence.

Example:

Arithmetic 4 84 164 244 324

Geometric 4 12 36 108 324

1.

Arithmetic 3 48

Geometric 3 48

2.

Arithmetic -6250 -10

Geometric -6250 -10

3.

Arithmetic -12 -0.75

Geometric -12 -0.75

SET Topic:Distinguishingspecificsbetweensequencesandlinearorexponentialfunctions.Answerthequestionsbelowwithrespecttotherelationshipbetweensequencesandthelargerfamiliesoffunctions.4.Ifarelationshipismodeledwithacontinuousfunctionwhichofthedomainchoicesisapossibility? A. ! | ! ∈ !, ! ≥ 0 B. ! | ! ∈ ! C. ! | ! ∈ !, ! ≥ 0 D. ! | ! ∈ ! 5.WhichoneoftheoptionsbelowisthemathematicalwaytorepresenttheNaturalNumbers? A. ! | ! ∈ !, ! ≥ 0 B. ! | ! ∈ !, ! ≥ 0 C. ! | ! ∈ !, ! ≥ 0 D. ! | ! ∈ !


x 3 x 3 x 3 x 3

+80 +80 +80 +80

24






2.4

6.Onlyoneofthechoicesbelowwouldbeusedforacontinuousexponentialmodel,whichoneisit? A.! ! = ! ! − 1 ∙ 4, ! 1 = 3 B.! ! = 4!(5) C.ℎ ! = 3! − 5 D.! ! = ! ! − 1 − 5, ! 1 = 327.Onlyoneofthechoicesbelowwouldbeusedforacontinuouslinearmodel,whichoneisit? A.! ! = ! ! − 1 ∙ 4, ! 1 = 3 B.! ! = 4!(5) C.ℎ ! = 3! − 5 D.! ! = ! ! − 1 − 5, ! 1 = 328.Whatdomainchoicewouldbemostappropriateforanarithmeticorgeometricsequence? A. ! | ! ∈ !, ! ≥ 0 B. ! | ! ∈ !, ! ≥ 0 C. ! | ! ∈ !, ! ≥ 0 D. ! | ! ∈ ! 9.Whatattributeswillarithmeticorgeometricsequencesalwayshave? (Therecouldbemorethanonecorrectchoice.Circleallthatapply.) A.Continuous B.Discrete C.Domain: ! | ! ∈ ! D.Domain: ! | ! ∈ ! E.Negativex-values F.Somethingconstant G.RecursiveRule10.Whattypeofsequencefitswithlinearmathematicalmodels? Whatisthedifferencebetweenthissequencetypeandtheoverarchingumbrellaoflinear relationships?(Usewordslikediscrete,continuous,domainandsoforthinyourresponse.)11.Whattypeofsequencefitswithexponentialmathematicalmodels? Whatisthedifferencebetweenthissequencetypeandtheoverarchingumbrellaofexponential relationships?(Usewordslikediscrete,continuous,domainandsoforthinyourresponse.)

25






2.4

GO Topic:Writingexplicitequationsforlinearandexponentialmodels.Writetheexplicitequationsforthetablesandgraphsbelow.Thisissomethingyoureallyneedtoknow.Persevereanddoallyoucantofigurethemout.Rememberthetoolswehaveused.(#21isbonusgiveitatry.)12. x f (x)

2

3

4

5

-4

-11

-18

-25

13. x f (x)

-1

0

1

2

2/5

2

10

50

14. x f (x)

2

3

4

5

-24

-48

-96

-192

15. x f (x)

-4

-3

-2

-1

81

27

9

3

16

17.

18

19.

20.

21.

26




2.5 Making My Point


ZacandSionewereworkingonpredictingthenumberofquiltblocksinthispattern:

Whentheycomparedtheirresults,theyhadaninterestingdiscussion:

Zac:Igot! = 6! + 1 becauseInoticedthat6blockswereaddedeachtimesothepatternmusthavestartedwith1blockatn=0.

Sione:Igot! = 6 ! − 1 + 7becauseInoticedthatatn=1therewere7blocksandatn=2therewere13,soIusedmytabletoseethatIcouldgetthenumberofblocksbytakingonelessthanthen,multiplyingby6(becausethereare6newblocksineachfigure)andthenadding7becausethat’showmanyblocksinthefirstfigure.Here’smytable:

1 2 3 n7 13 19 6 ! − 1 + 7

©201

2www.flickr.com

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1. WhatdoyouthinkaboutthestrategiesthatZacandSioneused?Areeitherofthemcorrect?Whyorwhynot?Useasmanyrepresentationsasyoucantosupportyouranswer.

ThenextproblemZacandSioneworkedonwastowritetheequationofthelineshownonthegraphbelow.

Whentheywerefinished,hereistheconversationtheyhadabouthowtheygottheirequations:

Sione:Itwashardformetotellwherethegraphcrossedtheyaxis,soIfoundtwopointsthatIcouldreadeasily,(-9,2)and(-15,5).Ifiguredoutthattheslopewas-!!andmadeatableandcheckeditagainstthegraph.Here’smytable:

x -15 -13 -11 -9 n

f(x) 5 4 3 2

− 12 ! + 9 + 2

28




Iwassurprisedtonoticethatthepatternwastostartwiththen,add9,multiplybytheslopeandthenadd2.

Igottheequation:! ! = − !! ! + 9 + 2.

Zac:Hey—IthinkIdidsomethingsimilar,butIusedthepoints,(7,-6)and(9,-7).

Iendedupwiththeequation:! ! = − !! ! − 9 − 7.Oneofusmustbewrongbecauseyours

saysthatyouadd9tothenandminesaysthatyousubtract9.Howcanwebothberight?

2. Whatdoyousay?Cantheybothberight?Showsomemathematicalworktosupportyourthinking.

Zac:Myequationmademewonderiftherewassomethingspecialaboutthepoint(9,-7)sinceitseemedtoappearinmyequation! ! = − !

! ! − 9 − 7whenIlookedatthenumberpattern.NowI’mnoticingsomethinginteresting—thesamethingseemstohappenwithyourequation,! ! = − !

! ! + 9 + 2andthepoint(-9,2)

3. DescribethepatternthatZacisnoticing.

4. FindanotherpointonthelinegivenaboveandwritetheequationthatwouldcomefromZac’spattern.

5. Whatwouldthepatternlooklikewiththepoint(a,b)ifyouknewthattheslopeofthelinewasm?

29




6. Zacchallengesyoutousethepatternhenoticedtowritetheequationoflinethathasaslopeof3andcontainsthepoint(2,-1).What’syouranswer?

Showawaytochecktoseeifyourequationiscorrect.

7. Sionechallengesyoutousethepatterntowritetheequationofthelinegraphedbelow,usingthepoint(5,4).


8. Zac:“I’llbetyoucan’tusethepatterntowritetheequationofthelinethroughthepoints(1,-3)and(3,-5).Tryit!”


30




9. Sione:Iwonderifwecouldusethispatterntographlines,thinkingofthestartingpointandusingtheslope.Tryitwiththeequation:! ! = −2 ! + 1 − 3.Startingpoint:Slope:

Graph:

10. Zacwonders,“Whatisitaboutlinesthatmakesthiswork?”HowwouldyouanswerZac?

11. Couldyouusethispatterntowritetheequationofanylinearfunction?Whyorwhynot?

31






2.5

READY Topic:Writingequationsoflines.

Writetheequationofalineinslope-interceptform:y=mx+b,usingthegiveninformation.

1.m=-7,b=4 2.m=3/8,b=-3 3.m=16,b=-1/5

Writetheequationofthelineinpoint-slopeform:y=m(x–x1)+y1,usingthegiveninformation.

4.m=9,(0.-7) 5.m=2/3,(-6,1) 6.m=-5,(4,11)

7.(2,-5)(-3,10) 8.(0,-9)(3,0) 9.(-4,8)(3,1)


32






2.5

SET

Topic:Graphinglinearandexponentialfunctions

Makeagraphofthefunctionbasedonthefollowinginformation.Addyouraxes.Choosean

appropriatescaleandlabelyourgraph.Thenwritetheequationofthefunction.

10.Thebeginningvalueis5anditsvalueis3

unitssmallerateachstage.

Equation:

11.Thebeginningvalueis16anditsvalueis¼

smallerateachstage.

Equation:

12.Thebeginningvalueis1anditsvalueis10

timesasbigateachstage.

Equation:

13.Thebeginningvalueis-8anditsvalueis2

unitslargerateachstage.

Equation:

33






2.5

GO Topic:Equivalentequations

Provethatthetwoequationsareequivalentbysimplifyingtheequationontherightsideofthe

equalsign.Thejustificationintheexampleistohelpyouunderstandthestepsforsimplifying.

YoudoNOTneedtojustifyyoursteps.

Example: Justification 2! − 4 = 8 + ! − 5! + 6 ! − 2 Add! − 5!anddistributethe6over ! − 2 = 8 − 4! + 6! − 12 Combineliketerms. = −4 + 2! 2! − 4 = 2! − 4 Commutativepropertyofaddition

14.! − 5 = 5! − 7 + 2 3! + 1 − 10!

15.6 − 13! = 24 − 10 2! + 8 + 62 + 7!

16.14! + 2 = 2! − 3 −4! − 5 − 13

17. ! + 3 = 6 ! + 3 − 5 ! + 3

18.4 = 7 2! + 1 − 5! − 3 3! + 1

19.! = 12 + 8! − 3 ! + 4 − 4!

20.Writeanexpressionthatequals ! − 13 . Itmusthaveatleasttwosetsofparenthesesandoneminussign.Verifythatitisequalto ! − 13 .

34




2.6 Form Follows Function

A Practice Understanding Task

Inourworksofar,wehaveworkedwithlinearand

exponentialequationsinmanyforms.Someofthe

formsofequationsandtheirnamesare:

LinearFunctionsEquation Name

! = 12 ! + 1

SlopeInterceptForm! = !" + !,wheremistheslopeandbisthe

y-intercept

! = 12 ! − 4 + 3

PointSlopeForm! = ! ! − !! + !!,wheremistheslopeand(x1,y1)thecoordinatesofapointontheline

! 0 = 1, ! ! = ! ! − 1 + 12

RecursionFormula! ! = ! ! − 1 +D,

Givenaninitialvaluef(a)D=constantdifferenceinconsecutiveterms

(usedonlyfordiscretefunctions)

ExponentialFunctionsEquation Name

! = 10(3)! ExplicitForm! = !(!)!

! 0 = 10, ! ! + 1 = 3! !

RecursionFormula! ! + 1 = !" !

Givenaninitialvaluef(a)R=constantratiobetweenconsecutiveterms

(usedonlyfordiscretefunctions)

CCBYPaulDow

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Knowinganumberofdifferentformsforwritingandgraphingequationsislikehavinga

mathematicaltoolbox.Youcanselectthetoolyouneedforthejob,orinthiscase,theformofthe

equationthatmakesthejobeasier.Anymasterbuilderwilltellyouthatthemoretoolsyouhave

thebetter.Inthistask,we’llworkwithourmathematicaltoolstobesurethatweknowhowtouse

themallefficiently.Asyoumodelthesituationsinthefollowingproblems,thinkaboutthe

importantinformationintheproblemandtheconclusionsthatcanbedrawnfromit.Isthe

functionlinearorexponential?Doestheproblemgiveyoutheslope,apoint,aratio,ay-intercept?

Isthefunctiondiscreteorcontinuous?Thisinformationhelpsyoutoidentifythebesttoolsandget

towork!

1. Inhisjobsellingvacuums,Joemakes$500eachmonthplus$20foreachvacuumhesells.WritetheequationthatdescribesJoe’smonthlyincomeIasafunctionofthen,thenumberofvacuumssold.

Nametheformoftheequationyouwroteandwhyyouchosetousethatform.

Thisfunctionis: linear exponential neither (chooseone)

Thisfunctionis: continuous discrete neither (chooseone)

2. Writetheequationofthelinewithaslopeof-1throughthepoint(-2,5)Nametheformoftheequationyouwroteandwhyyouchosetousethatform.



36




3.Writetheequationofthegeometricsequencewithaconstantratioof5andafirsttermof-3.




3. Writetheequationofthefunctiongraphedbelow:




4. ThepopulationoftheresorttownofJavaHotSpringsin2003wasestimatedtobe35,000

peoplewithanannualrateofincreaseofabout2.4%.Writetheequationthatmodelsthe

numberofpeopleinJavaHotSprings,witht=thenumberofyearsfrom2003?




37




5. Yessica’ssciencefairprojectinvolvedgrowingsomeseedstoseewhatfertilizermadethe

seedsgrowfastest.Oneideashehadwastouseanenergydrinktofertilizetheplant.(She

thoughtthatiftheymakepeopleperky,theymighthavethesameeffectonplants.)Thisis

thedatathatshowsthegrowthoftheseedeachweekoftheproject.

Week 1 2 3 4 5

Height(cm) 1.7 2.9 4.1 5.3 6.5

Writetheequationthatmodelsthegrowthoftheplantovertime.




Anequationgivesusinformationthatwecanusetographthefunction.Pickouttheimportant

informationgivenineachofthefollowingequationsandusetheinformationtographthefunction.

6. ! = !! ! − 5

Whatdoyouknowfromtheequationthathelpsyoutographthefunction?

38




7. ! = 2!

8. ! = −2 ! + 6 + 8

9. ! 1 = −5, ! ! = ! ! − 1 + 1




39






2.6

READY Topic:Comparinglinearandexponentialmodels.Comparingdifferentcharacteristicsofeachtypeoffunctionbyfillinginthecellsofeachtableascompletelyaspossible.

y=4+3x y=4(3x)

1.Typeofgrowth

2.Whatkindofsequencecorrespondstoeachmodel?

3.Makeatableofvalues

x y

x y

4.Findtherateofchange

5.Grapheachequation.

Comparethegraphs.

Whatisthesame?

Whatisdifferent?

6.Findthey-interceptfor

eachfunction.

20

18

16

14

12

10

8

6

4

2

5 10 15 20

20

18

16

14

12

10

8

6

4

2

5 10 15 20


40






2.6

7.Findthey-interceptsforthefollowingequations

a)y=3x

b)y=3x

8.Explainhowyoucanfindthey-interceptofalinearequationandhowthatisdifferentfromfinding

they-interceptofageometricequation.

SET Topic:Efficiencywithdifferentformsoflinearandexponentialfunctions.Foreachexerciseorproblembelowusethegiveninformationtodeterminewhichoftheformswouldbethemostefficienttouseforwhatisneeded.(Seetask2.6,Linear:slope-intercept,point-slopeform,recursive,Exponential:explicitandrecursiveforms)

9. Jasmine has been working to save money and wants to have an equation to model the amount of

money in her bank account. She has been depositing $175 a month consistently, she doesn’t remember

how much money she deposited initially, however on her last statement she saw that her account has

been open for 10 months and currently has $2475 in it. Create an equation for Jasmine.

Whichequationformdoyouchose? Writetheequation.

10.

The table below shows the number of rectangles created every time there is a fold made through the

center of a paper. Use this table for each question.

Folds Rectangles

1 2

2 4

3 8

4 16

A. Find the number or rectangles created with 5 folds.

Whichequationformdoyouchose?Writetheequation.

B.Findthenumberofrectanglescreatedwith14folds.

Whichequationformdoyouchose?Writetheequation.

41






2.6

11.UsinganewappthatIjustdownloadedIwanttocutbackonmycalorieintakesothatIcanloseweight.Icurrentlyweigh90kilograms,myplanistolose1.2kilogramsaweekuntilIreachmygoal.HowcanImakeanequationtomodelmyweightlossforthenextseveralweeks.Whichequationformdoyouchose?Writetheequation.12.SinceScottstarteddoinghisworkoutplanJanethasbeeninspiredtosetherselfagoaltodomoreexerciseandwalkalittlemoreeachday.Shehasdecidedtowalk10metersmoreeveryday.Ontheday20shewalked800meters.Howmanymeterswillshewalkonday21?Onday60?Whichequationformdoyouchose?Writetheequation.Foreachequationprovidedstatewhatinformationyouseeintheequationthatwillhelpyougraphit,thengraphit.Also,usetheequationtofillinanyfourcoordinatesonthetable.13. ! = !

!!8

14. ! = 5 ! − 2 − 6

Whatdoyouknowfromtheequation

thathelpsyoutographthefunction?

Whatdoyouknowfromtheequation

thathelpsyoutographthefunction?

42






2.6

GO Topic:Solvingone-stepequationswithjustification.Recallthetwopropertiesthathelpussolveequations.TheAdditivepropertyofequalitystates:Youcanaddanynumbertobothsidesofanequationandtheequationwillstillbetrue.TheMultiplicativepropertyofequalitystates:Youcanmultiplyanynumbertobothsidesofanequationandtheequationwillstillbetrue.Solveeachequation.Justifyyouranswerbyidentifyingtheproperty(s)youusedtogetit.Example1: ! − 13 = 7Justification+13+13additivepropertyofequality ! + 0 = 20addition ! = 20additiveidentity(Youadded0andgotx.)Example2:5! = 35Justification !! ! = !"! multiplicativepropertyofequality(multipliedby

!!)

1! = 7multiplicativeidentity(Anumbermultipliedbyitsreciprocal=1)15. 3! = 15Justification

16. ! − 10 = 2Justification

17. −16 = ! + 11Justification

18. 6 + ! = 10Justification

19. 6! = 18Justification

20. −3! = 2Justification

43

Linear & Exponential Functions...20. Best Buy Shoes had a back to school special. The total bill for four pairs of shoes came to $69.24 (before tax.) What was the average price for

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