unit 4 exponents and exponential...
TRANSCRIPT
Unit4ExponentsandExponentialFunctions
TestDate:________________
Name:___________________________________________________________________________________Bytheendofthisunit,youwillbeableto…
• Multiplyanddividemonomialsusingpropertiesofexponents• Simplifyexpressionscontainingexponents• Differentiatetheoutcomebetweenanegativesigninthebaseorinthepowerofanexpression
withexponents• Understandtherelationshipbetweenrationalexponentsandnthroots• UsethePowerPropertyofEqualitytosolveexponentialequations• Distinguishbetweenalinearandexponentialfunctionintheequation,table,andgraph• Describethedomainandrangeforanexponentialfunction• Graphanexponentialgrowth/decayfunction
2
TableofContentsMultiplicationPropertiesofExponents.......................................................................................................................................3
DivisionPropertiesofExponents...................................................................................................................................................6
SquareRootsasExponents............................................................................................................................................................10
nthRoots..................................................................................................................................................................................................11
RationalExponents............................................................................................................................................................................12
SolvingExponentialEquations.....................................................................................................................................................13
ExponentialFunctions......................................................................................................................................................................15
IdentifyingExponentialBehavior...........................................................................................................................................16
ExponentialGrowthvs.Decay......................................................................................................................................................17
ExponentialFunctionsPractice....................................................................................................................................................18
Summarize:GraphsofExponentialFunctions.......................................................................................................................20
ExponentialGrowthandDecay....................................................................................................................................................21
ExponentialGrowth......................................................................................................................................................................21
CompoundInterest.......................................................................................................................................................................22
ExponentialDecay.........................................................................................................................................................................22
3
4.1MultiplicationPropertiesofExponentsAmonomialisanexpressionwith____________________________________________________connectedonlybymultiplicationanddivision.
• No________________________________________
• No________________________inthedenominatorAconstantisamonomialwhichisa_______________________________________.Examples:Monomial NotaMonomial
Trythis!Expandandevaluatethefollowing:
1. 2" ∙ 2$
2. 4& ∙ 4"
3. 𝑥( ∙ 𝑥Whatdoyounotice?ProductofPowersProperty:Examples:
1. 5" ∙ 5& 2.𝑎(𝑎,)(𝑎&) 3.𝑥𝑦 ∙ 𝑥𝑦
4. (6𝑛&)(2𝑛1) 5. 6𝑐𝑑( 5𝑐(𝑑" 6.(−4𝑥𝑦"𝑧&)(−6𝑥(𝑦"𝑧)
4
Trythis!Expandandevaluatethefollowing:
1. 3" $
2. 2( "
3. 𝑥$ &Whatdoyounotice?PowerofaPowerProperty:Examples:
1. 2& " 2. 3& $ (
3. 𝑥( , 4. 𝑥" & "Trythis!Expandandevaluatethefollowing:
1. 𝑥𝑦 &
2. 2𝑧 $Whatdoyounotice?PowerofaProductProperty:Examples:
1. 𝑥𝑦$ , 2. −3𝑝(𝑡, $
3. 4𝑎$𝑏:𝑐 " 4. −4𝑥"𝑦(𝑧; &CHALLENGE:
1. Simplify 5𝑥𝑦& −3𝑥"𝑦" & " 2. Simplify −3𝑥( $ 𝑥"𝑦& ( ,
5
WarmUp:𝑏& ∗ 𝑏>> =_____________ 2( & =_____________ −2𝑥𝑦" ( =_______________
Reminder:
WhenMULTIPLYINGpowerswiththesamebase,ADDtheexponents.
Whenraisingapowertoapower,MULTIPLYtheexponents.
Whenthere’salotgoingon,followtheorderofoperations:
• P:Takecareofanythinginsideparentheses.Startwiththeinnermostsetofparentheses.• E:Takecareofexponents.Raiseeverythinginsideparenthesestothepower!• M:Multiplyeverythingtogether.
o Combineliketermso Addexponents
Examples
1. 2𝑎& $ 𝑎& &
2. 𝑐& " −3𝑐( "
3. 5x"𝑦 " 2𝑥𝑦&𝑧 &(4𝑥𝑦𝑧)
4. 2𝑥"𝑦 & ( 3𝑦 "
6
DivisionPropertiesofExponents1. QuotientofPowersProperty
ExpandandSimplify:"A
"B= CB
CD=
Inwords:Todividetwopowerswiththesamebase,____________________theexponents.Insymbols:Foranynonzeronumbera,andanyintegersmandp,Examples:
1. EFF
EG 2.
HDIJ
HIK
3. LJ
LK
4. MANFOPMJNDP
2. PowerofaQuotientProperty
ExpandandSimplify:&$
&= E
Q
"=
Inwords:Tofindthepowerofaquotient,findthepowerofthenumeratorandthedenominator.Insymbols:Foranyrealnumbersaandbnotequaltozero,andanyintegerm,Examples:
1. &(
$ 2.
&PD
1
"
3. &RB
$
&
4. "SK
&RD
"
7
3. ZeroExponentPropertyExpandandSimplify: UsetheQuotientofPowerProperty:&J
&J= &J
&J=
Inwords:Azeroexponentisanynonzeronumberraisedtothezeropower.Itisalwaysequalto1.Insymbols:Foranynonzeronumbera,Examples:
1. TE
U 2.
RJS0
RD
3. "RDSRKS
U
4. "TAEO
TK
4. NegativeExponentProperty
ExpandandSimplify: UsetheQuotientofPowersProperty:EK
EJ= EK
EJ=
Inwords:Fora(anotzero)andn(anynumber),𝑎WXand𝑎Xarereciprocals.
Insymbols:Foranynonzeronumberaandanyintegern,Examples:
1. 2W$ 2. >YZB
3. >&ZK
4. 𝑥W>U
5. XZDPB
LZK
6.
[ZD\RK
\SZJ
8
Directions: Simplify each of the following.
1. 𝟖𝟖
𝟖𝟒 2. 𝒂
𝟒𝒃𝟔
𝒂𝒃𝟑 3. 𝟓𝒄𝟐𝒅𝟑
W𝟒𝒄𝟐𝒅
4. 𝟒𝒇𝟑𝒈𝟑𝒉𝟔
𝟑 5. W$RK
"$RJ 6. ,\J
1PjLD
"
7. 𝒙𝟑(𝒚W𝟓)(𝒙W𝟖) 8. &1
W" 9. ""LDmK
>>LKmZD
10. 𝟔𝒇Z𝟐𝒈𝟑𝒉𝟓
𝟓𝟒𝒇Z𝟐𝒈Z𝟓𝒉𝟑 11. W>"C
ZFnJRZB
"CZDnRJ 12. ("o
ZKT)ZD
(oKTB
9
Directions: Simplify each of the following.
1. 𝒎𝟓𝒏𝒑𝒎𝟒𝒑
2. 𝟓𝒄𝟐𝒅𝟑
W𝟒𝒄𝟐𝒅 3. 𝟖𝒚𝟕𝒛𝟔
𝟒𝒚𝟔𝒛𝟓
4. 𝟒𝒇𝟑𝒈𝟑𝒉𝟔
𝟑 5. W$RK
"$RJ 6. ,\J
1PjLD
"
7. 𝒙𝟑(𝒚W𝟓)(𝒙W𝟖) 8. &1
W" 9. ""LDmK
>>LKmZD
10. 𝟔𝒇Z𝟐𝒈𝟑𝒉𝟓
𝟓𝟒𝒇Z𝟐𝒈Z𝟓𝒉𝟑 11. W>"C
ZFnJRZB
"CZDnRJ 12. N
ZKXZJ
NBXD ZF
13. 𝒋Z𝟏𝒌𝟑
Z𝟒
𝒋𝟑𝒌𝟑 14. ("o
ZKT)ZD
(oKTB 15. "RDSKx
&RBSxZK
W"
10
SquareRootsasExponentsDoNow:Useyourcalculatortoevaluatethefollowing.16= (16)
FK =
(100)
FK = 100 =
Whatdoyounotice?____________________________________________________________________________________________________________Whyisthishappening?
Checkitout: 𝑏FK"=________________________________
Examples:Writeeachexpressioninradicalform,orwriteeachradicalinexponentialform.Example1: 25
FK
Example2: 18
Example3:5𝑥FK
Example4: 8𝑝
Example5: 49FK
Example6: 22
Example7: 7𝑤FK
Example8:2 𝑥
Definition:
CalculatorTutorial#1Useparenthesestoevaluateexpressionsinvolvingrationalexponentsonagraphingcalculator.Forexample,tofind125
FD,press
125[^][(]1[÷]3[)][ENTER].
11
nthRootsUseyourcalculatortoevaluatethefollowing.6&= 216D = 2, = 64j =Whatdoyounotice?______________________________________________
______________________________________________Weknowthatif8" = 64,then 64 = 8.Similarly,if2$ = 16,then 16B = 2.Definition:Foranyrealnumbersaandbandanypositiveintegern,if𝑎X = 𝑏,thenaisannthrootofb.Examples:Evaluate.Example1: 27D
Example2: 32J
Example3: 64D
Example4: 10,000B
Likesquareroots,nthrootscanberepresentedbyrationalexponents.Definition(Part2):Examples:Usethenthrootdefinitiontoconvertformsandevaluate.Example1: 125
FD
Example2: 1296FB
Example3: 27FD
Example4: 256FB
CalculatorTutorial#2Touseexponents,pressthecaretsymbol(^)toraiseanumbertoapower.CalculatorTutorial#3Tofindnthroots,enteryournumbern,thenpress[MATH]andchoose√� .(5)
12
RationalExponentsSimplifytheseexpressionsusingMultiplicationProperties:
36FK&=________________
32$
FJ =________________
Simplifytheseexpressionsusingthenthrootdefinition:
36FK&=________________
32$
FJ =________________
Definition: Examples:Convertformsandevaluatethefollowingexpressions.Example1: 8
KD
Example2: 64KD
Example3: 36DK
Example4: 27KD
Example5: 256JB
Example6: 81JK
Example7: 7𝑤DK
Example8:2 𝑥J &
ChallengeProblems:
1. −8KD
2. 81 WJB
3. 𝑥"𝑦$ WFK
13
SolvingExponentialEquationsWarmUp:Answerthefollowingquestions.
1. 2towhatpoweris32? 2. 6towhatpoweris216?
3. 5towhatpoweris625?
Findasolutiontothefollowingequations.
4. 2R = 32 5. 6R = 216 6. 5R = 625ThePowerPropertyofEqualityAslongasbisarealnumbergreaterthanzeroandnotequalto1,then𝑏R = 𝑏Sifandonlyif𝑥 = 𝑦.Examples:
1. If5R = 5&,then𝑥 = 3. 3.If𝑛 = >",then4X = 4
FK.
2. 7R = 343 4.3&R�> = 81
Thispropertyhelpsuswhensolvingmorecomplicatedexponentialequations(likeexample4).AnotherExample:25RW> = 5
14
Examples:Solveeachequationforx.1. 2&R = 512
2. 3"R = 9R�>
3. 36RW> = 6
4. 2$R = 32R�>
5. 16R = >"
6. >&,
R�>= 216
7. >"1
R= 81
8. 25R = >>"(
1. Thesunprotectionfactor(SPF)ofasunscreenindicateshowwellitprotectsyoufromthesun’s
harmfulrays.SunscreenwithanSPFof𝑓absorbsabout𝑝percentoftheUV-Brays,where𝑝 = 50𝑓U.".FindtheSPFthatabsorbs100%ofUV-Brays.
2. Thepopulationpofaculturethatbeginswith40bacteriaanddoublesevery8hoursismodeledby𝑝 = 40 2
�G,wheretistimeinhours.Findtif𝑝 = 20,480.
15
ExponentialFunctionsThezombiesarehere…Eachnight,everyzombiewillinfectanewperson…Howmanynightsdoyouthinkitwilltaketoinfectthewholeroom?_______Writeafunctionthatrepresentsthisscenario:AnexponentialfunctionhastheformThefollowingrestrictionsapply:1.__________________2.___________________3._____________________Note:Thebaseisa___________________.Theexponentisa_________________________.Directions:Useyourtableabovetographthefunction.
1. Whatisthey-interceptofthefunction?Whatdoesitrepresentinthisscenario?
2. Whatisthedomainofthefunction?
3. Whatistherangeofthefunction?Summarize:Howdoyoufindthey-intercept?Howdoyoufindthedomainandrange?
Night #ofzombies
16
IdentifyingExponentialBehaviorUpuntilnow,wehavebeenworkingwithlinearfunctions.Thegraphofalinearfunctionis_______________,andalinearfunctionhasa_________________________________________.Thereare2methodswecanusetodeterminewhetherafunctionislinearvs.exponential:1. Graphing
Example:Graphthedatainthetable.Determinewhethertherelationshipislinearorexponential.
x y
-2 9
-1 3
0 1
1 13
2. Lookingforaconstantratio
Example:Exponentialfunctionshaveconstantratiosinsteadofaconstantrateofchange.Thismeansthatifthex-valuesareatregularintervalsandthey-valuesdifferbyacommonfactor,thedataisprobablyexponential.Inthisexample,theconstantratiois_______.Summarize:Howcanyoudeterminewhetherafunctionislinearorexponential?
17
ExponentialGrowthvs.DecayAfterthezombieoutbreak,ourclassisnowfullofzombies.Theschooladministrationfiguresoutwhat’sgoingonandsendsPrincipalWaynetoclearourclassofthezombieepidemic.PrincipalWaynecancureonehalfoftheremainingzombieseachdaywithavaccinecreatedinMr.Benters’BiologyLab.Whenwillourentireclassbecured?_________________________Writeafunctionthatrepresentsthisscenario:Useyourtabletographthefunctionbelow.
1. Whatisthey-intercept?Whatdoesthatrepresentinthisscenario?
2. Whatisthedomain?
3. Whatistherange?
Aslightlymorerealisticbiologyexample:Acertainbacteriapopulationdoublesinsizeevery20minutes.Beginningwith10cellsinaculture,thepopulationcanberepresentedbythefunction𝐵 = 10 2 C ,where𝐵isthenumberofbacteriacellsand𝑡isthetimein20minuteincrements.Howmanybacteriacellswilltherebeafter2hours?
Day #ofzombies
18
ExponentialFunctionsPracticeCreateatableandgraphthefunction.Youwillneedtochoosewhichvaluestouseinyourtable.Identifythey-intercept,domain,andrangeofeachfunction.Alsoidentifywhetherthefunctionrepresentsexponentialgrowthordecay.USEPENCIL!1. 𝑦 = 2R
Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________2. 𝑦 = 2R − 1
Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________3. 𝑦 = 2R + 3
Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________ClassDiscussion:
x y
x y
x y
19
4.𝑦 = >"
RW>
Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________
4. 𝑦 = >"
R�"
Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________
5. 𝑦 = >"
RW"+ 6
Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________ClassDiscussion:
x y
x y
x y
20
Summarize:GraphsofExponentialFunctionsExponentialGrowthFunctions ExponentialDecayFunctionsEquation:Domain:Range:Intercepts:Endbehavior:Sketchofgraph:
Equation:Domain:Range:Intercepts:Endbehavior:Sketchofgraph:
21
ExponentialGrowthandDecayExponentialGrowthThenumberofonlineblogshasrapidlyincreasedinthelast15years.Infact,thenumberofblogsincreasedatamonthlyrateofabout13.7%over21months,startingwith1.1millionblogsinNovember2003.Theaveragenumberofblogspermonthfrom2003-2005canbemodeledbytheequation𝑦 = 1.1 1 + 0.137 Cor𝑦 = 1.1 1.137 CwhereyrepresentsthetotalnumberofblogsinmillionsandtisthenumberofmonthssinceNovember2003.Labelthediagrambelowwithwhateachvariableorconstantrepresents.
𝑦 = 1.1 1 + 0.137 CIngeneral,theequationforexponentialgrowthisasfollows:
𝑦 = 𝑎 1 + 𝑟 CExample1:Theprizeforaradiostationcontestbeginswitha$100giftcard.Onceaday,anameisannounced.Thepersonhas15minutestocallortheprizeincreasesby2.5%forthenextday.
a. Writeanequationtorepresenttheamountofthegiftcardindollarsaftertdayswithnowinners.
b. Howmuchwillthegiftcardbeworthifnoonewinsafter10days?Example2:Acollege’stuitionhasrisen5%eachyearsince2000.Ifthetuitionin2000was$10,850,writeanequationfortheamountofthetuitiontyearsafter2000.Predictthecostoftuitionforthiscollegein2020.
CalculatorTutorial#4Whensolvingexponentialequations,youwilloftenencounter“unfriendly”decimals.Ifyouroundthesebeforeyourfinalanswer,youmaygetaslightlyincorrectanswer.Onyourcalculator,usethe[2nd][(-)]keystoget[Ans],yourEXACTpreviousanswer.
22
CompoundInterestCompoundinterestisaspecialkindofexponentialgrowth.Itisinterestearnedorpaidbothontheinitialinvestmentandpreviouslyearnedinterest.Ingeneral,theequationforcompoundinterestisasfollows:
𝐴 = 𝑃 1 +𝑟𝑛
XC
Example3:Maria’sparentsinvested$14,000at6%peryearcompoundedmonthly.Howmuchmoneywilltherebeintheaccountafter10years?Example4:Determinetheamountofaninvestmentif$300isinvestedataninterestrateof3.5%compoundedeveryothermonthfor22years.ExponentialDecayIngeneral,theequationforexponentialdecayisasfollows:
𝑦 = 𝑎 1 − 𝑟 CExample5:Afullyinflatedchild’sraftforapoolislosing6.6%ofitsaireveryday.Theraftoriginallycontained4500cubicinchesofair.
a. Writeanequationtorepresentthelossofair.
b. Estimatetheamountofairintheraftafter7days.Example6:ThepopulationofCampbellCounty,Kentuckyhasbeendecreasingatanaveragerateofabout0.3%peryear.In2000,itspopulationas88,647.Writeanequationtorepresentthepopulationsince2000.Ifthetrendcontinues,predictthepopulationin2018.