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.