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NAME DATE SECTION
IntrotoExponentialFunctions–Day2
RepresentingExponentialGrowth
MathTalk:ExponentRules
Rewriteeachexpressionasapowerof2.
2! ⋅ 2!
2! ⋅ 2
2!" ÷ 2!
2! ÷ 2
WhatDoes𝒙𝟎Mean?1. Completethetable.Takeadvantageofanypatternsyounotice.
𝑥 4 3 2 1 0 3! 81 27
2. Herearesomeequations.Findthesolutiontoeachequationusingwhatyouknowaboutexponentrules.Bepreparedtoexplainyourreasoning.
a. 9? ⋅ 9! = 9!
b. !!"
!?= 9!"
3. Whatisthevalueof5!?Whatabout2!?
MultiplyingMicrobes1. Inabiologylab,500bacteriareproducebysplitting.Everyhour,onthehour,each
bacteriumsplitsintotwobacteria.
a. Writeanexpressiontoshowhowtofindthenumberofbacteriaaftereachhourlistedinthetable.
b. Writeanequationrelating𝑛,thenumberofbacteria,to𝑡,thenumberofhours.
c. Useyourequationtofind𝑛when𝑡is0.Whatdoesthisvalueof𝑛meaninthissituation?
2. Inadifferentbiologylab,apopulationofsingle-cellparasitesalsoreproduceshourly.Anequationwhichgivesthenumberofparasites,𝑝,after𝑡hoursis𝑝 = 100 ⋅ 3! .Explainwhatthenumbers100and3meaninthissituation.
hour numberofbacteria
0 500
1
2
3
6
t
GraphingtheMicrobes1. Referbacktoyourworkinthetableoftheprevioustask.Usethatinformationandthe
givencoordinateplanestographthefollowing:
a.Graph(𝑡,𝑛)when𝑡is0,1,2,3,and4. b.Graph(𝑡,𝑝)when𝑡is0,1,2,3,and4. (Ifyougetstuck,youcancreateatable.)
2. Onthegraphof𝑛,wherecanyouseeeachnumberthatappearsintheequation?
3. Onthegraphof𝑝,wherecanyouseeeachnumberthatappearsintheequation?
UnderstandingDecay
NoticeandWonder:TwoTables
Whatdoyounotice?Whatdoyouwonder?
TableA TableB
𝑥 𝑦0 21 3
12
2 53 6
12
4 8
What'sLeft?1. HereisonewaytothinkabouthowmuchDiegohasleftafterspending!
!of$100.
Explaineachstep.
– Step1:100− !!⋅ 100
– Step2:100 1− !!
– Step3:100 ⋅ !!
– Step4:!!⋅ 100
2. Apersonmakes$1,800permonth,but!!ofthatamountgoestoherrent.Whattwo
numberscanyoumultiplytofindouthowmuchshehasafterpayingherrent?
3. Writeanexpressionthatonlyusesmultiplicationandthatisequivalentto𝑥reducedby!
!of𝑥.
𝑥 𝑦0 21 32 9
2
3 274
4 818
ValueofaVehicle
Everyyearafteranewcarispurchased,itloses!!ofitsvalue.Let’ssaythatanewcarcosts
$18,000.
1. Abuyerworriesthatthecarwillbeworthnothinginthreeyears.Doyouagree?Explainyourreasoning.
2. Writeanexpressiontoshowhowtofindthevalueofthecarforeachyearlistedinthetable.
year valueofcar(dollars)
0 18,000
1
2
3
6
𝑡
3. Writeanequationrelatingthevalueofthecarindollars,𝑣,tothenumberofyears,𝑡.
4. Useyourequationtofind𝑣when𝑡is0.Whatdoesthisvalueof𝑣meaninthissituation?
5. Adifferentcarlosesvalueatadifferentrate.Thevalueofthisdifferentcarindollars,
𝑑,after𝑡yearscanberepresentedbytheequation𝑑 = 10, 000 ⋅ !!
!.Explainwhatthe
numbers10,000and!!meaninthissituation.
Day2Summary
Inrelationshipswherethechangeisexponential,aquantityisrepeatedlymultipliedbythesameamount.Themultiplieriscalledthegrowthfactor.
Supposeapopulationofcellsstartsat500andtripleseveryday.Thenumberofcellseachdaycanbecalculatedasfollows:
numberofdays numberofcells0 5001 1,500(or500 ⋅ 3)2 4,500(or500 ⋅ 3 ⋅ 3,or500 ⋅ 3!)3 13,500(or500 ⋅ 3 ⋅ 3 ⋅ 3,or500 ⋅ 3!)𝑑 500 ⋅ 3!
Wecanseethatthenumberofcells(𝑝)ischangingexponentially,andthat𝑝canbefoundbymultiplying500by3asmanytimesasthenumberofdays(𝑑)sincethe500cellswereobserved.Thegrowthfactoris3.Tomodelthissituation,wecanwritethisequation:𝑝 = 500 ⋅ 3! .
Theequationcanbeusedtofindthepopulationonanyday,includingday0,whenthepopulationwasfirstmeasured.Onday0,thepopulationis500 ⋅ 3!.Since3! = 1,thisis500 ⋅ 1or500.
Hereisagraphofthedailycellpopulation.Thepoint(0,500)onthegraphmeansthatonday0,thepopulationstartsat500.
Eachpointis3timeshigheronthegraphthanthepreviouspoint.(1,1500)is3timeshigherthan(0,500),and(2,4500)is3timeshigherthan(1,1500).
Sometimesaquantitygrowsbythesamefactoratregularintervals.Forexample,apopulationdoubleseveryyear.Sometimesaquantitydecreasesbythesamefactoratregularintervals.Forexample,acarmightloseonethirdofitsvalueeveryyear.
Let'slookatasituationwherethequantitydecreasesbythesamefactoratregularintervals.Supposeabacteriapopulationstartsat100,000and!
!ofthepopulationdieseach
day.Thepopulationonedaylateris100, 000− !!⋅ 100, 000,whichcanbewritten
as100, 000 1− !!.Thepopulationafteronedayis!
!of100,000or75,000.Thepopulation
aftertwodaysis!!⋅ 75, 000.Herearesomefurthervaluesforthebacteriapopulation
numberofdays bacteriapopulation0 100,0001 75,000(or100, 000 ⋅ !
!)
2 56,250(or100, 000 ⋅ !!⋅ !!,or100, 000 ⋅ !
!
!)
3 about42,188(or100, 000 ⋅ !!⋅ !!⋅ !!,or100, 000 ⋅ !
!
!)
Ingeneral,𝑑daysafterthebacteriapopulationwas100,000,thepopulation𝑝isgivenby
theequation:𝑝 = 100, 000 ⋅ !!
!,withonefactorof!
!foreachday.
Situationswithquantitiesthatdecreaseexponentiallyaredescribedwithexponentialdecay.Themultiplier(!
!inthiscase)isstillcalledthegrowthfactor,thoughsometimes
peoplecallitthedecayfactorinstead.