Algebra2Unit4A:Exponentials
Ms.Talhami 1
Algebra2Unit4A:Exponentials
Name_________________
Algebra2Unit4A:Exponentials
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
Algebra2Unit4A:Exponentials
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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.
Algebra2Unit4A:Exponentials
Ms.Talhami 4
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
Algebra2Unit4A:Exponentials
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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...
Algebra2Unit4A:Exponentials
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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:
Algebra2Unit4A:Exponentials
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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 .
Algebra2Unit4A:Exponentials
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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
Algebra2Unit4A:Exponentials
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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= ?
Algebra2Unit4A:Exponentials
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
Algebra2Unit4A:Exponentials
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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
Algebra2Unit4A:Exponentials
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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
Algebra2Unit4A:Exponentials
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
Algebra2Unit4A:Exponentials
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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
Algebra2Unit4A:Exponentials
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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
Algebra2Unit4A:Exponentials
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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.
Algebra2Unit4A:Exponentials
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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.
Algebra2Unit4A:Exponentials
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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±
Algebra2Unit4A:Exponentials
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
Algebra2Unit4A:Exponentials
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.
Algebra2Unit4A:Exponentials
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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
Algebra2Unit4A:Exponentials
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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.
Algebra2Unit4A:Exponentials
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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.
Algebra2Unit4A:Exponentials
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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 =
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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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Ms.Talhami 26
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.
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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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Ms.Talhami 28
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.
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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.
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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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Ms.Talhami 32
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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Ms.Talhami 33
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.