MCR3U–Unit4:ExponentialRelations–Lesson4 Date:___________Learninggoal:Icangraphandidentifykeypropertiesofexponentialfunctions.Icandistinguishbetweenalinear,quadratic,andexponentialfunctionwhengivenagraph,tableofvalues,orequation.

IntroductiontoExponentialFunctionsExponentialfunctionsarecurvesthatincreaseordecreasethroughtheirdomains.Theyhavethebasicform! ! = !! , ! ≠ 0.COMPARINGGRAPHSExample1:Createatableofvaluesforeachfunction,andgraphthembothonthesameaxis.

a)! ! = 2! b) ! ! = 3!

c)ℎ ! = !!!

! ! !

-2

-1

0

1

2

! ! !

-2

-1

0

1

2

! ℎ !

-2

-1

0

1

2

Property !(!) !(!) !(!)Domain Range !-intercept !-intercept HorizontalAsymptote Increasing/Decreasing? As! ↓

Note:Theequationℎ(!) = !!!isequivalenttoℎ(!) = 2!!.During

transformationslastunitwesawthatwhenthe!valueofafunctionisnegativeitcausesa________________acrossthe!-axis.Thatisexactlywhatwehavedonehere!

COMPARINGLINEAR,QUADRATIC,ANDEXPOENTIALFUNCTIONSThereare3wayswecancomparelinear,quadratic,andexponentialfunctions:

1. Equations

Usetheequationsofthefollowingrelationstodeterminewhethertheyarelinear,quadratic,orexponential.

a) ! ! = 3! + 1 b) ! ! = !! − 2 c) ! ! = 3(2!)

2. TablesofValues

Usethetableofvaluestoconfirmwhethertherelationsarelinear,quadratic,orexponential.a) ! ! = 3! + 1 b) ! ! = !! − 2 c) ! ! = 3(2!)

3. Graphs

Usethegraphstorecognizewhethertherelationsarelinear,quadratic,orexponential(growthordecay).

x y 1stdiff. x y 1st

diff.2nddiff. x y 1st

diff.2nddiff. Ratio

-3 -3 -3 -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3

SuccessCriteriaforDeterminingBetweenLinear,Quadratic,andExponentialFunctions

Functions Equations TableofValues GraphsLinear

Quadratic

Exponential

SUMMARY

HW:IntroductiontoExponentialsWorksheet

IntroductiontoExponentialsWorksheet

! ! = 2! ! ! = 52

!

! ! = 0.8!

KeyPoints:(,),(,), (,),(,)

KeyPoints:(,),(,), (,),(,)

KeyPoints:(,),(,), (,),(,)

! ! = 10! ! ! = 3! ! ! = 12

!

KeyPoints:(,),(,), (,),(,)

KeyPoints:(,),(,), (,),(,)

KeyPoints:(,),(,), (,),(,)

! ! = 23

! ! ! = 3

2!

! ! = 1.1!

KeyPoints:(,),(,),

KeyPoints:(,),(,),

KeyPoints:(,),(,),

(,),(,) (,),(,) (,),(,)! ! = 4! ! ! = 7

3! ! ! = 1

3!

KeyPoints:(,),(,), (,),(,)

KeyPoints:(,),(,), (,),(,)

KeyPoints:(,),(,), (,),(,)

! ! = 1! ! ! = 0.3! ! ! = 34

!

KeyPoints:(,),(,),(,),(,),(,)

KeyPoints:(,),(,), (,),(,)

KeyPoints:(,),(,), (,),(,)

! ! = 110

!

! ! = 1.4! ! ! = 14

!

KeyPoints:(,),(,), (,),(,)

KeyPoints:(,),(,), (,),(,)

KeyPoints:(,),(,), (,),(,)

MCR3U–Unit4:ExponentialRelations–Lesson5 Date:___________Learninggoal:Icanapplytransformationstoexponentialfunctionsandsketchtheirgraphs.

TransformationsofExponentialFunctionsExponentialFunctionsoftheform! ! = !!canbetransformedusingthesamealgorithmasourother

functionswesawlastunit,wherepoint(!.!)on! ! = !!mapsontothepoint(!! ! + !,!" + !)on! ! = ! ∙ !!(!!!) + !.

! ! = ! ∙ !!(!!!) + !Example1:Eachfunction!(!)istransformedtoresultinthefunction!(!).Foreach:

i) writeanequationfor!(!)usingfunctionnotation.ii) writeanequationfor!(!)given!(!).

a)Transformationsapplyto! ! = 5!

• Verticalstretchbyafactorof3• Horizontalstretchbyafactorof4• Horizontaltranslation5unitright• Verticaltranslation1unitup

b)Transformationsapplyto! ! = !!!

• Verticalreflectionacrossthe!-axis• Horizontalstretchbyafactorof

!!

• Verticaltranslation7unitsdown

SuccessCriteriaforGraphingSquareRootFunctions• Labelyourscale• Labelyourequation• 5keypointsareclearlymarkedwithadot• Labelanyintercepts• Labelandsketchanyasymptotes

Example2:Sketchthefollowingtransformedfunctionsonthegridsbelow(usesuccesscriteria).Firstwritetheexponentialfunctionthatistobetransformed.Listthetransformationsinorderonthebasefunction ! ! = !!andthenewmapping.Statethedomainandrange.

a) Basefunction:! ! = (!!)! b)Basefunction:! ! = (2)!

! ! = −! ! + 1 − 2 ! ! = ! 2! − 6 + 5

Description:______________________ Description:______________________

________________________________ ________________________________

________________________________ ________________________________

________________________________ ________________________________

________________________________ ________________________________

!,! → (__________, ___________) !,! → (__________, ___________)

D=________________________________ D=________________________________R=________________________________ R=________________________________HW:TransformingExponentialFunctionsWorksheet

TransformingExponentialFunctionsWorksheet

1. Sketchthefollowingtransformedfunctionsonagrid(usesuccesscriteria).Firstwritetheexponentialfunctionthatistobetransformed.Listthetransformationsinorderonthebasefunction ! ! = !!andthenewmapping.Statethedomainandrange.**CheckyouranswersonDesmos

a)! ! = −! ! + 5 − 4,! ! = 2! b)! ! = !(−! + 3)+ 2,! ! = 2!c)! ! = 3! !

! ! − 1 + 5,! ! = (!!)! d)! ! = !

! ! −! − 2,! ! = 3!

e)! ! = −2![3! + 6]+ 1,! ! = (!!)! f)! ! = ! 9− 3! + 2,! ! = 3!

2. Eachfunction!(!)istransformedtoresultinthefunction!(!).Foreach:iii) writeanequationfor!(!)usingfunctionnotation.iv) writeanequationfor!(!)given!(!).

a)Transformationsapplyto! ! = 3!

• Verticalreflectionacrossthex-axis• Verticalstretchbyafactorof5• Horizontaltranslation6unitsleft• Verticaltranslation3unitsdown

b)Transformationsapplyto! ! = !!!

• Verticalstretchbyafactorof!!

• Horizontalstretchbyafactorof3• Horizontaltranslation6unitsright• Verticaltranslation2unitsdown

MCR3U–Unit4:ExponentialRelations–Lesson6 Date:___________Learninggoal:Icandeterminemorethanoneequationwhengivenanexponentialgraph.

SimilarExponentialFunctions

Example1:Identify3pointsanddeterminetheequationofeachgraph.

a) b) c)

DIFFERENTBASES

Someinterestingthingshappenwithtransformationsofexponentialfunctions.Unlikeourotherparents,manytransformationsofbaseexponentialfunctionsarenotunique.

DifferentBases

Function Transformations Domain Range

! ! = 8!

! ! = 2!!

Weknowthat2! = 8sowecanalsosaythat! ! = 2!! = 2! ! = 8! = ! !

Therefore,wecannotdistinguishbetweenagraphof! ! = 2!thathasbeenstretchedhorizontallyby!!andagraphof! ! = 8!.

STRETCHS&SHIFTS

Function Transformations Domain Range

! ! = 3!

! ! = 3 3!

! ! = 3!!!

Weknowthat 3 3! = so! ! =Thereforewecannottellthedifferencebetween:

Example2:Determineanotherequationthatisthesamegraphas! ! = !! 3!

REFLECITONSANDRECIPROCALS

Function Transformations Domain Range

! ! = 3!

! ! = 13

!

! ! = 1! ! = 1

3!

! ! = 3!!

Weknowthat

!!

! = and

!!! = so! ! = ! ! = ! ! =

Thereforewecannottellthedifferencebetween:

Example3:Determinetwoequationsforthegraphsbelow.a) b) c)

HW:SimilarExponentialFunctionsWorksheet

SimilarExponentialFunctionsWorksheet

1. Statethedomain&range,andthenmatcheachfunctiontoitsgraph.

i)! = 2! + 1 ii)! = −2! + 1 iii)! = −2! − 1 iv)! = 2! − 1v)! = 2!! + 1vi)! = 2!! − 1 vii)! = −2!! + 1 viii)! = −2!! − 1

2. Given! ! = !!

!,writeanequationfor! ! = ! −! andℎ ! = !

! ! ,thengraphf,!andℎ.

Writetheequationofanotherexponentialfunctiononadifferentbasethatisequivalentto!orℎ.Communicateyourdiscoveriesarticulatelyandattempttostateasmanypropertiesaboutthesefunctionsasyoucan.

3. Given! ! = 2!,writethetransformedfunctionandgrapheach.Writetheequationofanotherexponentialfunctiononadifferentbasethatisequivalentto!ℎ,! and!.Communicateyourdiscoveriesarticulatelyandattempttostateasmanypropertiesaboutthesefunctionsasyoucan.

a) ! ! = 2! ! b) ℎ ! = ! ! + 1 c) ! ! = !

! ! !

d) ! ! = ! ! − 2

4. Matchtheequationofthefunctionsfromthelisttotheappropriategraph.

a) ( ) 341

+⎟⎠

⎞⎜⎝

⎛−=− x

xf

b) 341

+⎟⎠

⎞⎜⎝

⎛=x

y

c) ( ) 345

+⎟⎠

⎞⎜⎝

⎛−=−x

xg

d) ( ) 3452 +⎟⎠

⎞⎜⎝

⎛=x

xh

SimilarExponentialFunctionsWorksheetSolutions

1.Statethedomain&range,andthenmatcheachfunctiontoitsgraph.

i)! = 2! + 1 Domain: ! = ! ∨ ! ∈ ℜ Range:! = ! ∨ ! > 1,! ∈ ℜ

ii)! = −2! + 1 Domain: ! = ! ∨ ! ∈ ℜ Range:! = ! ∨ ! < 1,! ∈ ℜ

iii)! = −2! − 1 Domain: ! = ! ∨ ! ∈ ℜ Range:! = ! ∨ ! < −1,! ∈ ℜ

iv)! = 2! − 1 Domain: ! = ! ∨ ! ∈ ℜ Range:! = ! ∨ ! > −1,! ∈ ℜ

v)! = 2!! + 1 Domain: ! = ! ∨ ! ∈ ℜ Range:! = ! ∨ ! > 1,! ∈ ℜ

vi)! = 2!! − 1 Domain: ! = ! ∨ ! ∈ ℜ Range:! = ! ∨ ! > −1,! ∈ ℜ

vii)! = −2!! + 1 Domain: ! = ! ∨ ! ∈ ℜ Range:! = ! ∨ ! < 1,! ∈ ℜ

viii)! = −2!! − 1 Domain: ! = ! ∨ ! ∈ ℜ Range:! = ! ∨ ! < −1,! ∈ ℜ

vi) vii) iii) v)

viii) ii) iv) i)

2.

3.

4.a)ii b)iv c)i d)iii

MCR3U–Unit4:ExponentialRelations–Lesson7 Date:___________Learninggoal:Icancreateanequationtomodelexponentialgrowthanddecay.Icansolveforanunknowninanexponentialapplication.

ApplicationsofGrowthandDecayExponentialgrowthordecayoccurswhenquantitiesincreaseordecreaseatarateproportionaltotheinitialquantitypresent.Thisgrowthordecayoccursinsavingsaccounts,thesizeofpopulations,appreciation,depreciation,andwithradioactivechemicals.Example1:ThepopulationofGuelphisexpectedtogrowby3%peryear.Thepopulationwas96000in1996.

a)Findanequationtomodelthepopulation.

b)Whatwouldyouexpectthepopulationtobein2018?c)Howlongwouldittakeforthepopulationtoeach234000?

ExponentialFunctionsExponentialgrowthanddecayproblemscanbemodelledusingtheformula: !(!) = !(!)!.

• !(!)isthefinalamount• Where!istheinitialvalue• Where!isthegrowth/decayfactor

• Thebaseiscalledthe“growthfactor”when! > 1• Thebaseiscalledthe“decayfactor”when0 < ! < 1• Thegrowthordecayrateis|! − 1|• Thefunctionneithergrowsnordecayswhen! = 1

• Where!isthenumberofgrow/decayperiods

Example2:Acarcosts$24,000.Avirtualcostassociatedwiththetime-valueofthecariscalleddepreciation.Thiscardepreciatesanaverageof18%peryear.

a)Modelthissituationwithanequation

b)Whatistheapproximatevalueafter31months?Example3:Abacteriapopulationdoublesevery20minutes.

a)Writeanequationforapopulationthatstartswith100bacteria.

b)Howmanybacteriawillyouhaveafter2hours?

Example4:Ryanhasbeensavingforhiscollegetuitionfor4years.Heput$5,550inasavingsaccount4yearsago(withoutaddingtoit)andnowhas$6492.72.Calculatetheannualgrowthrateasapercenttotwodecimals.

Half-lifeistheamountoftimerequiredforanamounttodiminishbyhalftheinitialvalue.Example5:A200gsampleofradioactivepolonium-210hasahalf-lifeof138days.

a)Writeanequationforthemassremainingaftertdays.

b)Determinethemassleftafter5years,tothenearestthousandth.

c)Howlongagowasthesample800g?

HW:Pg.80#9-13,19,20,23,pg.95#12,14,17,20

Half-life

Half-lifeproblemscanbemodelledusingtheformula: !(!) =!0 !12!!ℎ

• !(!)isthefinalmass• Where!!istheinitialmass• Where!istime• Where!ishalf-life