Algebra 2 Unit 4A: Exponentials Algebra 2 Unit 4A: Exponentials · 2018-11-29 · Algebra 2 Unit...

Algebra2Unit4A:Exponentials

Ms.Talhami 1


Name_________________


Ms.Talhami 2

INTEGER EXPONENTS COMMONCOREALGEBRAII

Wejustfinishedourreviewoflinearfunctions.Linearfunctionsarethosethatgrowbyequaldifferencesforequalintervals.Inthisunitwewillconcentrateonexponentialfunctionswhichgrowbyequalfactorsforequalintervals.Tounderstandexponentialfunctions,wefirstneedtounderstandexponents.Exercise#1:Thefollowingsequenceshowspowersof3byrepeatedlymultiplyingby3.Fillinthemissingblanks.Thispatterncanbeduplicatedforanybaseraisedtoanyintegerexponent.Becauseofthiswecannowdefinepositive,negative,andzeroexponentsintermsofmultiplyingthenumber1repeatedlyordividingthenumber1repeatedly. Exercise#2:Giventheexponentialfunction ( ) ( )20 2 xf x = evaluateeachofthefollowingwithoutusingyourcalculator.Showthecalculationsthatleadtoyourfinalanswer.(a) ( )2f (b) ( )0f (c) ( )2f − (d)Whenxincreasesby3,bywhatfactordoesyincrease?Explainyouranswer.

3 9

INTEGEREXPONENTDEFINITIONS

Ifnisanypositiveintegerthen:

1. 2. 3.

n-times n-times


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Therearemanybasicexponentpropertiesorlawsthatarecriticallyimportantandthatcanbeinvestigatedusingintegerexponentexamples.Twooftheveryimportantoneswewillseenext.Exercise#3:Foreachofthefollowing,writetheproductasasingleexponentialexpression.Write(a)and(b)asextendedproductsfirst(ifnecessary).(a) 3 42 2⋅ (b) 6 22 2⋅ (c)2 2m n⋅ It'sclearwhytheexponentlawthatyougeneralizedinpart(c)worksforpositiveintegerexponents.But,doesitalsomakesensewithinthecontextofournegativeexponents?Exercise#4:Considernowtheproduct 3 12 2−⋅ .

(c) Doyouranswersfrom(a)and(b)supporttheextensionoftheAdditionPropertyofExponentstonegative

powersaswell?Explain.Let'slookatanotherimportantexponentproperty.Exercise#5:Foreachofthefollowing,writetheexponentialexpression intheform 3x .Write(a)and(b)asextendedproductsfirst(ifnecessary).(a) ( )323 (b) ( )243 (c) ( )3 nm

Again,let'slookathowtheProductPropertyofExponentsstillholdsfornegativeexponents.Exercise#6:Considertheexpression ( )423− .Showthisexpressionisequivalentto 83− byfirstrewriting 23− in

fractionform.

(a) UsetheexponentlawfoundinExercise3(c)towritethisasasingleexponentialexpression.

(b) Evaluate by first rewriting and andthensimplifying.


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INTEGER EXPONENTS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. WriteeachofthefollowingexponentialexpressionswithouttheuseofexponentssuchaswedidinlessonExercise#1.

(a) (b)

2. Nowlet'sgotheotherwayaround.Foreachofthefollowing,determinetheintegervalueofnthatsatisfiestheequation.Thefirstisdoneforyou.

(a)

3

3

128122

2 23

n

n

n

n

−

=

=

== −

(b) 4 16n = (c) 1381

n = (d)7 1n =

(e) 1525

n = (f) 11010,000

n = (g)13 1n = (h) 1232

n =

2

4

5

25


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3. UsetheAdditionPropertyofExponentstosimplifyeachexpression.Then,findafinalnumericalanswerwithoutusingyourcalculator.

(a) 5 3 42 2 2− ⋅ ⋅ (b) 3 7 105 5 5−⋅ ⋅ (c) 3 7 210 10 10−⋅ ⋅ 4. UsetheProductPropertyofExponentstosimplifyeachexponentialexpression.Youdonotneedtofinda

finalnumericalanswer.

(a) ( )432 (b) ( )223− (c) ( )( ) 2425−−

5. Theexponentialexpression41

8⎛ ⎞⎜ ⎟⎝ ⎠

isequivalenttowhichofthefollowing?Explainyourchoice.

(1) 84− (3) 28− (2) 122− (4) 132− REASONING

6. Howcanyouusethefactthat 225 625= toshowthat 4 15625

− = ?Explainyourprocessofthinking.

7. We'veextendedthetwofundamentalexponentpropertiestonegativeaswellaspositiveintegers.What

wouldhappenifweextendedtheProductExponentPropertytoafractionalexponentlike 12?Let'splay

aroundwiththatidea.

(a) Use the Product Property of Exponents to

justifythat .

(b)What other number can you square thatresultsin9?Hmm...


Ms.Talhami 6

RATIONAL EXPONENTS COMMONCOREALGEBRAII

Whenyoufirstlearnedaboutexponents,theywerealwayspositiveintegers,andjustrepresentedrepeatedmultiplication.Andthenwehadtogoandintroducenegativeexponents,whichreallyjustrepresentrepeateddivision.Todaywewill introducerational(orfractional)exponentsandextendyourexponentialknowledgethatmuchfurther.Exercise#1:RecalltheProductPropertyofExponentsanduseittorewriteeachofthefollowingasasimplifiedexponentialexpression.Thereisnoneedtofindafinalnumericalvalue.

(a) ( )432 (b) ( )525− (c) ( )073 (d) ( )( )2224−

Wewill now use the Product Property to extend our understanding of exponents to includeunit fractionexponents(thoseoftheform 1n wherenisapositiveinteger).

Exercise#2:Considertheexpression1216 .

Thisisremarkable!Anexponentof 12 isequivalenttoasquarerootofanumber!!!

Exercise#3:Testtheequivalenceofthe 12 exponenttothesquarerootbyusingyourcalculatortoevaluate

eachofthefollowing.Becarefulinhowyouentereachexpression.

(a)1225 = (b)

1281 = (c)

12100 =

Wecanextendthisnowtoalllevelsofroots,thatissquareroots,cubicroots,fourthroots,etcetera.

(a) ApplytheProductPropertytosimplify

.Whatothernumbersquaredyields16?

(b) You can now say that is equivalent towhatmorefamiliarquantity?

UNITFRACTIONEXPONENTS

Forngivenasapositiveinteger:


Ms.Talhami 7

Exercise #4: Rewrite each of the following using roots instead of fractional exponents. Then, if necessary,evaluateusingyourcalculatortoguessandchecktofindtheroots(don'tusethegenericrootfunction).Checkwithyourcalculator.

(a)13125 (b)

1416 (c)

129− (d)

1532−

Wecannowcombinetraditionalintegerpowerswithunitfractionsinorderinterpretanyexponentthatisarationalnumber, i.e.theratiooftwointegers.Thenextexercisewill illustratethethinking.Remember,wewantourexponentpropertiestobeconsistentwiththestructureoftheexpression.

Exercise#5:Let'sthinkabouttheexpression324 .

Exercise#6:Evaluateeachofthefollowingexponentialexpressionsinvolvingrationalexponentswithouttheuseofyourcalculator.Showyourwork.Then,checkyourfinalanswerswiththecalculator.

(a)3416 (b)

3225 (c)

238−

(a) Fillinthemissingblankandthenevaluatethisexpression:

(b) Fillinthemissingblankandthenevaluatethisexpression:

(c) Verifyboth(a)and(b)usingyourcalculator. (d) Evaluate without your calculator. Showyourthinking.Verifywithyourcalculator.

RATIONALEXPONENTCONNECTIONTOROOTS

Fortherationalnumber wedefine tobe: or .


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RATIONAL EXPONENTS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Rewritethefollowingasequivalentrootsandthenevaluateasmanyaspossiblewithoutyourcalculator.

(a)1236 (b)

1327 (c)

1532 (d)

12100−

(e)14625 (f)

1249 (g)

1481− (h)

13343

2. Evaluateeachofthefollowingbyconsideringtherootandpowerindicatedbytheexponent.Doasmany

aspossiblewithoutyourcalculator.

(a)238 (b)

324 (c)

3416− (d)

5481

(e)524− (f)

37128 (g)

34625 (h)

35243

3. Giventhefunction ( ) ( )325 4f x x= + ,whichofthefollowingrepresentsitsy-intercept? (1)40 (3)4 (2)20 (4)30


Ms.Talhami 9

4. Whichofthefollowingisequivalentto12x− ?

(1) 12x− (3) 1

x

(2) x− (4) 12x

−

5. Writtenwithoutfractionalornegativeexponents,32x− isequalto

(1) 32x− (3)

3

1x

(2)3 2

1x

(4) 1x

−

6.Whichofthefollowingisnotequivalentto3216 ?

(1) 4096 (3)64 (2) 38 (4) 316 REASONING7. Marleneclaimsthatthesquarerootofacuberootisasixthroot?Isshecorrect?Tostart,tryrewritingthe

expressionbelowintermsoffractionalexponents.ThenapplytheProductPropertyofExponents.

3 a

8. Weshouldknowthat 3 8 2= .Toseehowthisisequivalentto138 2= wecansolvetheequation8 2n = .To

dothis,wecanrewritetheequationas: ( )3 12 2

n=

Howcanwenowusethisequationtoseethat138 2= ?


Ms.Talhami 10

EXPONENTIAL FUNCTION BASICS COMMONCOREALGEBRAII

YoustudiedexponentialfunctionsextensivelyinCommonCoreAlgebraI.Today'slessonwillreviewmanyofthebasiccomponentsoftheirgraphsandbehavior.Exponentialfunctions,thosewhoseexponentsarevariable,areextremelyimportantinmathematics,science,andengineering.Exercise#1:Considerthefunction 2xy = .Fillinthetablebelowwithoutusingyourcalculatorandthensketchthegraphonthegridprovided.

Exercise#2:Nowconsiderthefunction ( )12

xy = .Usingyourcalculatortohelpyou,filloutthetablebelow

andsketchthegraphontheaxesprovided.

BASICEXPONENTIALFUNCTIONS

where

x

0

1

2

3

y

x

x

0

1

2

3

y

x


Ms.Talhami 11

Exercise#3:BasedonthegraphsandbehavioryousawinExercises#1and#2,statethedomainandrangeforanexponentialfunctionoftheform xy b= . Domain(inputset): Range(outputset):Exercise #4: Are exponential functionsone-to-one?How can you tell?Whatdoes this tell you about theirinverses?Exercise#5:Nowconsiderthefunction ( )7 3 xy = .

(a) Determinethey-interceptofthisfunctionalgebraically.Justifyyouranswer.

(b) Does the exponential function increase or decrease?

Explainyourchoice.(c) Create a rough sketch of this function, labeling its y-

intercept.

Exercise#6:Considerthefunction ( )1 43x

y = + .

(a) How does this function’s graph compare to that of

( )13

xy = ?Whatdoesadding4dotoafunction'sgraph?

(b) Determine this graph’sy-intercept algebraically. Justify

youranswer.(c) Create a rough sketch of this function, labeling its y-

intercept.

y

x

y

x


Ms.Talhami 12

EXPONENTIAL FUNCTION BASICS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Whichofthefollowingrepresentsanexponentialfunction?

(1) 3 7y x= − (3) ( )3 7 xy =

(2) 37y x= (4) 23 7y x= +

2. If ( ) ( )6 9 xf x = then ( )12f = ?(Rememberwhatwejustlearnedaboutfractionalexponentsanddowithou

acalculator.)

(1)72 (3)27

(2)18 (4)152

3. If ( ) 3xh x = and ( ) 5 7g x x= − then ( )( )2h g = (1)18 (3)38 (2)12 (4)274. Whichofthefollowingequationscoulddescribethegraphshownbelow? (1) 2 1y x= + (3) 2 1y x= − +

(2) ( )23

xy = (4) 4xy =

5. Whichofthefollowingequationsrepresentsthegraphshown?

(1) 5xy = (3) ( )1 22x

y = +

(2) 4 1xy = + (4) 3 2xy = +

y

x

1 2 3-1-2-3

5

10


Ms.Talhami 13

6. Sketchgraphsoftheequationsshownbelowontheaxesgiven.Labelthey-interceptsofeachgraph.

(a) ( )118 3x

y = (b) ( )25 4 xy =

APPLICATION

7. TheFahrenheittemperatureofacupofcoffee,F,startsatatemperatureof185 F .Itcoolsdownaccording

totheexponentialfunction ( )201113 72

2

m

F m ⎛ ⎞= +⎜ ⎟⎝ ⎠,

wheremisthenumberminutesithasbeencooling.

REASONING

8. Thegraphbelowshowstwoexponential functions,withrealnumberconstantsa,b,c,andd.Giventhegraphs,onlyonepairoftheconstantsshownbelowcouldbeequalinvalue.Determinewhichpaircouldbeequalandexplainyourreasoning.

bandd aandb aandc

9. Explainwhytheequationbelowcanhavenorealsolutions.Ifyouneedto,graphbothsidesoftheequationusingyourcalculatortovisualizethereason.

3 5 2x + =

(a) How do you interpret the statement that?

(b)Determinethetemperatureofthecoffeeafterone day using your calculator.What do youthink this temperature represents about thephysicalsituation?

y

x

y

x


Ms.Talhami 14

FINDING EQUATIONS OF EXPONENTIAL FUNCTIONS COMMONCOREALGEBRAII

OneoftheskillsthatyouacquiredinCommonCoreAlgebraIwastheabilitytowriteequationsofexponentialfunctionsifyouhadinformationaboutthestartingvalueandbase(multiplierorgrowthconstant).Let'sreviewaverybasicproblem.Exercise#1:Anexponentialfunctionoftheform ( ) ( )xf x a b= ispresentedinthetablebelow.Determinethevaluesofaandbandexplainyourreasoning. a = _________ b = _________ FinalEquation:_____________________ Explanation:Findinganexponentialequationbecomesmuchmorechallengingifwedonothaveoutputvaluesforinputsthatareincreasingbyunitvalues(increasingby1unitatatime).Let'sstartwithabasicproblem.Exercise#2:Foranexponentialfunctionoftheform ( ) ( )xf x a b= ,itisknownthat ( )0 8f = and ( )3 1000f =.

Exercise#3:Anexponentialfunctionexistssuchthat ( ) ( )4 3 and 6 48f f= = ,whichofthefollowingmustbethevalueofitsbase?Explainorillustrateyourthinking.(1) 16b = (3) 6b = (2) 2b = (4) 4b =

x 0 1 2 3

5 15 45 135

(a) Use the fact that to determine thevalueofa.Showyourthinking.

(b)Useyouanswerfrompart(a)andthefactthattosetupanequationtosolvefor

b.Youwillsolveforbinpart(c).

(c) Solve for the value of b using properties ofexponents.

(d)Determinethevalueof


Ms.Talhami 15

Now,let'sworkwiththemostgenerictypeofproblem.Justlikewithlines,anytwo(non-verticallyaligned)pointswilluniquelydeterminetheequationofanexponentialfunction.Exercise#4:Anexponentialfunctionoftheform ( )xy a b= passesthroughthepoints ( )2, 36 and ( )5,121.5 .

Let'snowgetsomepracticeonthiswithadecreasingexponentialfunction.Exercise#5:Findtheequationoftheexponentialfunctionshowngraphedbelow.Becarefulintermsofyourexponentmanipulation.Stateyourfinalanswerintheform ( )xy a b= .Exercise#6:Abacterialcolonyisgrowingatanexponentialrate.Itisknownthatafter4hours,itspopulationisat98bacteriaandafter9hoursitis189bacteria.Determineanequationin ( )xy a b= formthatmodelsthepopulation,y,asafunctionofthenumberofhours,x.Atwhatpercentrateisthepopulationgrowingperhour?

(a) By substituting these two points into thegeneral form of the exponential, create asystemofequationsintheconstantsaandb.

(b)Divide these two equations to eliminate theconstant a. Recall that when dividing to likebases,yousubtracttheirexponents.

(c) Solve the resulting equation from (b) for thebase,b.

(d)Useyourvaluefrom(c)todeterminethevalueofa.Statethefinalequation.

Y

x( )2, 0.5


Ms.Talhami 16

FINDING EQUATIONS OF EXPONENTIAL FUNCTIONS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. For each of the following coordinate pairs, find the equation of the exponential function, in the form

( )xy a b= thatpassesthroughthepair.Showtheworkthatyouusetoarriveatyouranswer. (a) ( ) ( )0,10 and 3, 80 (b) ( ) ( )0,180 and 2, 80 2. For each of the following coordinate pairs, find the equation of the exponential function, in the form

( )xy a b= thatpassesthroughthepair.Showtheworkthatyouusetoarriveatyouranswer.

(a) ( ) ( )2,192 and 5,12288 (b) ( ) ( )1,192 and 5, 60.75 3. Eachofthepreviousproblemshadvaluesofaandbthatwererationalnumbers.Theydonotneednotbe.

Find the equation for an exponential function that passes through the points ( )2,14 and ( )7, 205 in

( )xy a b= form.Whenyoufindthevalueofbdonotroundyouranswerbeforeyoufinda.Then,findbothtothenearesthundredthandgivethefinalequation.Checktoseeifthepointsfallonthecurve.


Ms.Talhami 17

APPLICATIONS4. A population of koi goldfish in a pondwasmeasured over time. In the year 2002, the populationwas

recordedas380andin2006itwas517.Giventhatyisthepopulationoffishandxisthenumberofyearssince2000,dothefollowing:

5. Engineersaredrainingawaterreservoiruntilitsdepthisonly10feet.Thedepthdecreasesexponentiallyas

showninthegraphbelow.Theengineersmeasurethedepthafter1hourtobe64feetandafter4hourstobe 28 feet. Develop an exponential equation in ( )xy a b= to predict the depth as a function of hoursdraining.Roundatothenearestintegerandbtothenearesthundredth.Then,graphthehorizontalline

10y = andfinditsintersectiontodeterminethetime,tothenearesttenthofanhour,whenthereservoirwillreachadepthof10feet.

WaterDep

th(ft)

Time(hrs)

(a) Representthe information inthisproblemastwocoordinatepoints.

(b)Determine a linear function in the form that passes through these two

points.Don't roundthe linearparameters (mandb).

(c) Determineanexponentialfunctionoftheform that passes through these two

points.Roundbtothenearesthundredthandatothenearesttenth.

(d)Whichmodel predicts a larger population offishintheyear2000?Justifyyourwork.


Ms.Talhami 18

THE METHOD OF COMMON BASES COMMONCOREALGEBRAII

There are very few algebraic techniques that do not involve technology to solve equations that containexponentialexpressions.Inthislessonwewilllookatoneofthefew,knownasTheMethodofCommonBases.Exercise#1:Solveeachofthefollowingsimpleexponentialequationsbywritingeachsideoftheequationusingacommonbase.

(a) 2 16x = (b)3 27x = (c) 1525

x = (d)16 4x =

Ineachofthesecases,eventhe last,morechallengingone,wecouldmanipulatetheright-handsideoftheequationsothatitsharedacommonbasewiththeleft-handsideoftheequation.Wecanexploitthisfactbymanipulatingbothsidessothattheyhaveacommonbase.First,though,weneedtoreviewanexponentlaw.

Exercise#2:Simplifyeachofthefollowingexponentialexpressions.

(a) ( )32 x (b) ( )423

x (c) ( )3 715

x−− (d) ( )2134x−−

Exercise#3:Solveeachofthefollowingequationsbyfindingacommonbaseforeachside.

(a)8 32x = (b) 2 19 27x+ = (c) ( )41125 25x

x−

=

Exercise#4:Whichofthefollowingrepresentsthesolutionsettotheequation

2 32 64x − = ?

(1){ }3± (3){ }11±

(2){ }0, 3 (4){ }35±


Ms.Talhami 19

Thistechniquecanbeusedinanysituationwhereallbasesinvolvedcanbewrittenwithacommonbase.Inapractical sense, this is rather rare. Yet, these types of algebraicmanipulations help us see the structure inexponentialexpressions.Trytotacklethenext,morechallenging,problem.

Exercise#5:Twoexponentialcurves,524

xy

+= and

2 112

x

y+

⎛ ⎞= ⎜ ⎟⎝ ⎠areshownbelow.TheyintersectatpointA.A

rectanglehasonevertexattheoriginandtheotheratAasshown.Wewanttofinditsarea.(a) Fundamentally,whatdoweneedtoknowabout

arectangletofinditsarea?(b) HowwouldknowingthecoordinatesofpointA

helpusfindthearea?(c) FindtheareaoftherectanglealgebraicallyusingtheMethodofCommonBases.Showyourworkcarefully.

Exercise#6:Atwhatxcoordinatewillthegraphof 25x ay −= intersectthegraphof3 11

125

x

y+

⎛ ⎞= ⎜ ⎟⎝ ⎠?Showthe

workthatleadstoyourchoice.

(1) 5 13ax −= (3) 2 1

5ax − +=

(2) 2 311ax −= (4) 5 3

2ax +=

y

x

A


Ms.Talhami 20

THE METHOD OF COMMON BASES COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. SolveeachofthefollowingexponentialequationsusingtheMethodofCommonBases.Eachequationwill

resultinalinearequationwithonesolution.Checkyouranswers.

(a) 2 53 9x− = (b) 3 72 16x+ = (c) 4 5 15 125x− =

(d) 2 18 4x x+= (e) ( )3 22 1216 1296

xx

−− = (f) ( )

315 151 312525

x x+ −=

2. Algebraicallydeterminetheintersectionpointofthetwoexponentialfunctionsshownbelow.Recallthat

mostsystemsofequationsaresolvedbysubstitution.

1 2 38 and 4x xy y− −= = 3. Algebraicallydeterminethezeroesoftheexponentialfunction ( ) 2 92 32xf x −= − .Recallthatthereasonit

isknownasazeroisbecausetheoutputiszero.


Ms.Talhami 21

APPLICATIONS4. Onehundredmustberaisedtowhatpowerinordertobeequaltoamillioncubed?Solvethisproblemusing

theMethodofCommonBases.Showthealgebrayoudotofindyoursolution.

5. Theexponentialfunction251 10

25

x

y−

⎛ ⎞= −⎜ ⎟⎝ ⎠isshowngraphedalongwiththehorizontalline 115y = .Their

intersectionpointis ( ),115a .UsetheMethodofCommonBasestofindthevalueofa.Showyourwork.REASONING6 TheMethodofCommonBasesworksbecauseexponentialfunctionsareone-to-one,i.e.iftheoutputsare

thesame,thentheinputsmustalsobethesame.Thisiswhatallowsustosaythatif 32 2x = ,thenxmustbeequalto3.Butitdoesn'talwaysworkoutsoeasily.

If 2 25x = ,canwesaythatxmustbe5?Coulditbeanythingelse?Whydoesthisnotworkoutaseasilyas

theexponentialcase?

x


Ms.Talhami 22

EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY COMMONCOREALGEBRAII

Exponentialfunctionsareveryimportantinmodelingavarietyofrealworldphenomenabecausecertainthingseitherincreaseordecreasebyfixedpercentagesovergivenunitsoftime.YoulookedatthisinCommonCoreAlgebraIandinthislessonwewillreviewmuchofwhatyousaw.Exercise#1:Supposethatyoudepositmoneyintoasavingsaccountthatreceives5%interestperyearontheamountofmoneythatisintheaccountforthatyear.Assumethatyoudeposit$400intotheaccountinitially.ThethinkingprocessfromExercise#1canbegeneralizedtoanysituationwhereaquantityisincreasedbyafixedpercentageoverafixedintervaloftime.Thispatternissummarizedbelow:

Exercise#2:WhichofthefollowinggivesthesavingsSinanaccountif$250wasinvestedataninterestrateof3%peryear?

(1) ( )250 4 tS = (3) ( )1.03 250tS = +

(2) ( )250 1.03 tS = (4) ( )250 1.3 tS =

INCREASINGEXPONENTIALMODELS

IfquantityQisknowntoincreasebyafixedpercentagep,indecimalform,thenQcanbemodeledby

where representstheamountofQpresentat andtrepresentstime.

(a) Howmuchwillthesavingsaccountincreasebyoverthecourseoftheyear?

(b)Howmuchmoneyisintheaccountattheendoftheyear?

(c) By what single number could you havemultipliedthe$400byinordertocalculateyouranswerinpart(b)?

(d)Usingyouranswerfrompart(c),determinetheamountofmoneyintheaccountafter2and10years. Round all answers to the nearest centwhenneeded.

(e) Giveanequationfortheamountinthesavingsaccount as a function of the number ofyearssincethe$400wasinvested.

(f) Usinga tableonyour calculatordetermine, tothe nearest year, how long itwill take for theinitial investment of $400 to double. Provideevidencetosupportyouranswer.


Ms.Talhami 23

Decreasingexponentialsaredevelopedinthesameway,buthavethepercentsubtracted,ratherthanadded,tothebaseof100%.Justremember,youareultimatelymultiplyingbythepercentoftheoriginalthatyouwillhaveafterthetimeperiodelapses.Exercise#3:Statethemultiplier(base)youwouldneedtomultiplybyinordertodecreaseaquantitybythegivenpercentlisted.(a)10% (b)2% (c)25% (d)0.5%Exercise#4:Ifthepopulationofatownisdecreasingby4%peryearandstartedwith12,500residents,whichof the following is its projectedpopulation in 10 years? Show the exponentialmodel youuse to solve thisproblem. (1)9,230 (3)18,503 (2)76 (4)8,310Exercise#5:ThestockpriceofWindpowerIncisincreasingatarateof4%perweek.Itsinitialvaluewas$20pershare. Ontheotherhand,thestockpriceinGerbilEnergyiscrashing(losingvalue)atarateof11%perweek.Ifitspricewas$120persharewhenWindpowerwasat$20,afterhowmanyweekswillthestockpricesbethesame?Modelbothstockpricesusingexponentialfunctions.Then,findwhenthestockpriceswillbeequalgraphically.Drawawelllabeledgraphtojustifyyoursolution.

DECREASINGEXPONENTIALMODELS

IfquantityQisknowntodecreasebyafixedpercentagep,indecimalform,thenQcanbemodeledby

where representstheamountofQpresentat andtrepresentstime.


Ms.Talhami 24

EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY COMMONCOREALGEBRAIIHOMEWORK

APPLICATIONS

1. If$130isinvestedinasavingsaccountthatearns4%interestperyear,whichofthefollowingisclosesttotheamountintheaccountattheendof10years?

(1)$218 (3)$168 (2)$192 (4)$3242. Apopulationof50fruitfliesisincreasingatarateof6%perday.Whichofthefollowingisclosesttothe

numberofdaysitwilltakeforthefruitflypopulationtodouble? (1)18 (3)12 (2)6 (4)283. Ifaradioactivesubstanceisquicklydecayingatarateof13%perhourapproximatelyhowmuchofa200

poundsampleremainsafteroneday?

(1)7.1pounds (3)25.6pounds (2)2.3pounds (4)15.6pounds4. Apopulationofllamasstrandedonadesertislandisdecreasingduetoafoodshortageby6%peryear.If

thepopulationofllamasstartedoutat350,howmanyareleftontheisland10yearslater?

(1)257 (3)102 (2)58 (4)1895. Whichofthefollowingequationswouldmodelapopulationwithaninitialsizeof625thatisgrowingatan

annualrateof8.5%? (1) ( )625 8.5 tP = (3) 1.085 625tP = +

(2) ( )625 1.085 tP = (4) 28.5 625P t= + 6. Theaccelerationofanobjectfallingthroughtheairwilldecreaseatarateof15%persecondduetoair

resistance. If the initial acceleration due to gravity is 9.8meters per second per second,which of thefollowingequationsbestmodelstheaccelerationtsecondsaftertheobjectbeginsfalling?

(1) 215 9.8a t= − (3) ( )9.8 1.15 ta =

(2) 9.815

at

= (4) ( )9.8 0.85 ta =


Ms.Talhami 25

7. RedHook has a population of 6,200 people and is growing at a rate of 8% per year. Rhinebeck has apopulationof8,750andisgrowingatarateof6%peryear.Inhowmanyyears,tothenearestyear,willRedHookhaveagreaterpopulationthanRhinebeck?Showtheequationorinequalityyouaresolvingandsolveitgraphically.

8. Awarmglassofwater,initiallyat120degreesFahrenheit,isplacedinarefrigeratorat34degreesFahrenheit

anditstemperatureisseentodecreaseaccordingtotheexponentialfunction

( ) ( )86 0.83 34hT h = +

REASONING9. Percentscombineinstrangewaysthatdon'tseemtomakesenseatfirst.Itwouldseemthatifapopulation

growsby5%peryearfor10years,thenitshouldgrowintotalby50%overadecade.Butthisisn'ttrue.Startwithapopulationof100.Ifitgrowsat5%peryearfor10years,whatisitspopulationafter10years?Whatpercentgrowthdoesthisrepresent?

(a) Verify that the temperature starts at 120degreesFahrenheitbyevaluating .

(b)Using your calculator, sketch a graph of Tbelow for all values of h on the interval

.Besuretolabelyoury-axisandy-intercept.

(c) Afterhowmanyhourswillthetemperaturebeat50degreesFahrenheit?Stateyouranswertothenearesthundredthofanhour.Illustrateyouransweronthegraphyourdrewin(b).


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MINDFUL MANIPULATION OF PERCENTS COMMONCOREALGEBRAII

Percentsandphenomenathatgrowataconstantpercentratecanbechallenging,tosaytheleast.Thisisduetothefactthat,unlikelinearphenomena,thegrowthrateindicatesaconstantmultipliereffectinsteadofaconstantadditiveeffect(linear).Becauseconstantpercentgrowthissocommonineverydaylife(nottomentioninscience,business,andotherfields),it'sgoodtobeabletomindfullymanipulatepercents.Exercise#1:Apopulationofwombatsisgrowingataconstantpercentrate.IfthepopulationonJanuary1stis1027andayearlateris1079,whatisitsyearlypercentgrowthratetothenearesttenthofapercent?Exercise#2:Nowlet'strytodeterminewhatthepercentgrowthinwombatpopulationwillbeoveradecadeoftime.WewillassumethattheroundedpercentincreasefoundinExercise#1continuesforthenextdecade.

Exercise#3:Let’sstickwithourwombatsfromExercise#1.Assumingtheirgrowthrateisconstantovertime,whatistheirmonthlygrowthratetothenearesttenthofapercent?Assumeaconstantsizedmonth.Exercise#4:Ifapopulationwasgrowingataconstantrateof22%every5years,thenwhatisitspercentgrowthrateoverat2yeartimespan?Roundtothenearesttenthofapercent.

(a) After 10 years, whatwill we havemultipliedthe original population by, rounded to thenearesthundredth?Showthecalculation.

(b)Usingyouranswerfrom(a),whatisthedecadepercentgrowthrate?

(a) First,giveanexpressionthatwillcalculatethesingle year (or yearly) percent growth ratebased on the fact that the population grew22%in5years.

(b)Now use this expression to calculate thepercentgrowthover2years.


Ms.Talhami 27

Exercise#5:Worldoilreserves(theamountofoilunusedintheground)aredepletingataconstant2%peryear.Wewouldliketodeterminewhatthepercentdeclinewillbeoverthenext20yearsbasedonthis2%yearlydecline.

Exercise#6:Aradioactivesubstance’shalf-lifeistheamountoftimeneededforhalf(or50%)ofthesubstancetodecay.Let’ssaywehavearadioactivesubstancewithahalf-lifeof20years.(a)Whatpercentofthesubstancewouldberadioactiveafter40years?(b)Whatpercentofthesubstancewouldberadioactiveafteronly10years?Roundtothenearesttenthofa

percent.(c)Whatpercentofthesubstancewouldberadioactiveafteronly5years?Roundtothenearesttenthofa

percent.

(a) Writeandevaluateanexpressionforwhatwewouldmultiplytheinitialamountofoilbyafter20years.

(b)Use your answer to (a) to determine thepercent decline after 20 years. Be careful!Roundtothenearestpercent.


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MINDFUL MANIPULATION OF PERCENTS COMMONCOREALGEBRAIIHOMEWORK

APPLICATIONS

1. Aquantityisgrowingataconstant3%yearlyrate.Whichofthefollowingwouldbeitspercentgrowthafter

15years? (1)45% (3)56% (2)52% (4)63%2. Ifacreditcardcompanycharges13.5%yearlyinterest,whichofthefollowingcalculationswouldbeusedin

theprocessofcalculatingthemonthlyinterestrate?

(1) 0.13512

(3) ( )121.135

(2)1.13512

(4) ( )1121.135

3. Thecountydebtisgrowingatanannualrateof3.5%.Whatpercentrateisitgrowingatper2years?Per5

years?Perdecade?Showthecalculationsthatleadtoeachanswer.Roundeachtothenearesttenthofapercent.

4. Apopulationof llamas isgrowingataconstantyearly rateof6%. Atwhat rate is the llamapopulation

growingpermonth?Pleaseassumeallmonthsareequallysizedandthatthereare12oftheseperyear.Roundtothenearesttenthofapercent.


Ms.Talhami 29

5. Shanaistryingtoincreasethenumberofcaloriessheburnsby5%perday.Bywhatpercentisshetryingtoincreaseperweek?Roundtothenearesttenthofapercent.

6. Ifabankaccountdoublesinsizeevery5years,thenbywhatpercentdoesitgrowafteronly3years?Round

tothenearesttenthofapercent.Hint:Firstwriteanexpressionthatwouldcalculateitsgrowthrateafterasingleyear.

7. Anobject’sspeeddecreasesby5%foreachminutethatitisslowingdown.Whichofthefollowingisclosest

tothepercentthatitsspeedwilldecreaseoverhalf-anhour? (1)21% (3)48% (2)79% (4)150%8. Overthelast10years,thepriceofcornhasdecreasedby25%perbushel. (a) Assumingasteadypercentdecrease,bywhatpercentdoesitdecreaseeachyear?Roundtothenearest

tenthofapercent. (b) Assumingthispercentcontinues,bywhatpercentwillthepriceofcorndecreasebyafter50years?Show

thecalculationthatleadstoyouranswer..Roundtothenearestpercent.


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COMPOUND INTEREST COMMONCOREALGEBRAII

Intheworldsofinvestmentanddebt,interestisaddedontoaprincipalinwhatisknownascompoundinterest.Thepercentrateistypicallygivenonayearlybasis,butcouldbeappliedmorethanonceayear.Thisisknownasthecompoundingfrequency.Let'stakea lookatatypicalproblemtounderstandhowthecompoundingfrequencychangeshowinterestisapplied.Exercise#1: A person invests $500 in an account thatearnsanominalyearlyinterestrateof4%.

So,thepatternisfairlystraightforward.Forashortercompoundingperiod,wegettoapplytheinterestmoreoften,butatalowerrate.Exercise#2:Howmuchwould$1000investedatanominal2%yearlyrate,compoundedmonthly,beworthin20years?Showthecalculationsthatleadtoyouranswer.(1)$1485.95 (3)$1033.87(2)$1491.33 (4)$1045.32Thispatternisformalizedinaclassicformulafromeconomicsthatwewilllookatinthenextexercise.Exercise#3:Foraninvestmentwiththefollowingparameters,writeaformulafortheamounttheinvestmentisworth,A,aftert-years. P=amountinitiallyinvested r=nominalyearlyrate n=numberofcompoundsperyear

(a) Howmuchwouldthisinvestmentbeworthin10years if thecompoundingfrequencywasonceperyear?Showthecalculationyouuse.

(b) If,ontheotherhand,the interestwasappliedfour times per year (known as quarterlycompounding),whywoulditnotmakesensetomultiplyby1.04eachquarter?

(c) Ifyouweretoldthataninvestmentearned4%per year, how much would you assume wasearnedperquarter?Why?

(d)Usingyouranswerfrompart(c),calculatehowmuchthe investmentwouldbeworthafter10years of quarterly compounding? Show yourcalculation.


Ms.Talhami 31

TherateinExercise#1wasreferredtoasnominal(innameonly).It'sknownasthis,becauseyoueffectivelyearnmorethanthisrateifthecompoundingperiodismorethanonceperyear.Becauseofthis,bankersrefertotheeffectiverate,ortherateyouwouldreceiveifcompoundedjustonceperyear.Let'sinvestigatethis.Exercise#4:Aninvestmentwithanominalrateof5%iscompoundedatdifferentfrequencies.Givetheeffectiveyearly rate,accurate to twodecimalplaces, foreachof the followingcompounding frequencies.Showyourcalculation.(a)Quarterly (b)Monthly (c)DailyWecouldcompoundatsmallerandsmallerfrequencyintervals,eventuallycompoundingallmomentsoftime.InourformulafromExercise#3,wewouldbelettingnapproachinfinity.Interestinglyenough,thisgivesrisetocontinuous compounding and theuseof thenatural basee in the famous continuous compound interestformula.Exercise #5: A person invests $350 in a bank account that promises a nominal rate of 2% continuouslycompounded.

CONTINUOUSCOMPOUNDINTEREST

Foraninitialprincipal,P,compoundedcontinuouslyatanominalyearlyrateofr,theinvestmentwouldbeworthanamountAgivenby:

(a) Write an equation for the amount thisinvestmentwouldbeworthaftert-years.

(b)Howmuchwouldtheinvestmentbeworthafter20years?

(c) Algebraicallydeterminethetimeitwilltakeforthe investment to reach $400. Round to thenearesttenthofayear.

(d)What is the effective annual rate for thisinvestment?Roundtothenearesthundredthofapercent.


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COMPOUND INTEREST COMMONCOREALGEBRAIIHOMEWORK

APPLICATIONS1. Thevalueofaninitialinvestmentof$400at3%nominalinterestcompoundedquarterlycanbemodeled

usingwhichofthefollowingequations,wheretisthenumberofyearssincetheinvestmentwasmade? (1) ( )4400 1.0075 tA= (3) ( )4400 1.03 tA= (2) ( )400 1.0075 tA= (4) ( )4400 1.0303 tA= 2. Whichof the following represents thevalueof an investmentwithaprincipalof$1500withanominal

interestrateof2.5%compoundedmonthlyafter5years? (1)$1,697.11 (3)$4,178.22 (2)$1,699.50 (4)$5,168.71

3. Francoinvests$4,500inanaccountthatearnsa3.8%nominalinterestratecompoundedcontinuously.Ifhewithdrawstheprofitfromtheinvestmentafter5years,howmuchhasheearnedonhisinvestment?

(1)$858.92 (3)$922.50 (2)$912.59 (4)$941.62

4. Aninvestmentthatreturnsanominal4.2%yearlyrate,butiscompoundedquarterly,hasaneffectiveyearlyrateclosestto

(1)4.21% (3)4.27% (2)4.24% (4)4.32%

5. Ifaninvestment'svaluecanbemodeledwith12.027325 1

12

t

A ⎛ ⎞= +⎜ ⎟⎝ ⎠thenwhichofthefollowingdescribes

theinvestment? (1)Theinvestmenthasanominalrateof27%compoundedevery12years. (2)Theinvestmenthasanominalrateof2.7%compoundedever12years. (3)Theinvestmenthasanominalrateof27%compounded12timesperyear. (4)Theinvestmenthasanominalrateof2.7%compounded12timesperyear.


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6. Aninvestmentof$500ismadeat2.8%nominalinterestcompoundedquarterly.REASONING

7. Theformula 1ntrA P

n⎛ ⎞= +⎜ ⎟⎝ ⎠

canberearrangedusingpropertiesofexponentsas 1tnrA P

n⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

.Explain

whattheterm 1nr

n⎛ ⎞+⎜ ⎟⎝ ⎠

helpstocalculate.

8. The formula rtA Pe= calculates the amount an investment earning a nominal rate of r compounded

continuouslyisworth.Showthattheamountoftimeittakesfortheinvestmenttodoubleinvalueisgiven

bytheexpression ln 2r

.

(a) WriteanequationthatmodelstheamountAthe investment is worth t-years after theprincipalhasbeeninvested.

(b)Howmuch is the investmentworth after 10years?

(c) Algebraicallydeterminethenumberofyearsitwill take for the investment to be reach aworth of $800. Round to the nearesthundredth.

(d)Whydoes itmakemoresensetoroundyouranswerin(c)tothenearestquarter?Statethefinalanswerroundedtothenearestquarter.

Algebra 2 Unit 4A: Exponentials Algebra 2 Unit 4A: Exponentials · 2018-11-29 · Algebra 2 Unit...

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Transcript of Algebra 2 Unit 4A: Exponentials Algebra 2 Unit 4A: Exponentials · 2018-11-29 · Algebra 2 Unit...