r.e.a.l. math 1010 intermediate algebra supplemental student … · 2019. 8. 6. · r.e.a.l. math...

Real-world Explorations. Active Learning.

R.E.A.L. Math 1010 Intermediate Algebra

Supplemental Student Packet

Fall 2017

Developmental Mathematics Program

Index of Contents S1 Equivalence

S2 Mathematical Properties -- Inverse and Identity

Lab 1: Linear Patterns & Paired Data

S3 Systems of Equations

S4 Inequalities

S5 Add Like Things

S6 Exponent Review

S7 Polynomial Multiplication

S8 Polynomial Division

S9 Rational Exponents

S10 Simplify, Add & Subtract Radical Expressions

S11 Multiply Radical Expressions

S12 Divide Radical Expressions

S13 Composition of Functions

Lab 3: Introduction to Inverse Operations

S14 Exponential Growth and Decay

S15 Intro to Logarithms

Lab 4: Exponential and Logarithmic Graphs

S16 Factoring to Simplify Rational Expressions plus extra practice

S17 Multiply and Divide Rational Expressions

S18 Add and Subtract Rational Expressions

S19 Rational Equations Classwork

S20 Complex Fractions Classwork

Lab 5: Applications of Rational Expressions -- Swimming Pool

Lab 6: Expressions vs. Equations

S21 Graphing Functions using Transformations

S22 Complete the Square

S23 Zero Product Property and Solving Quadratic Eqns

S24 Complex Numbers

R.E.A.L.Math1010SupplementalActivity Name:___________________________S1EquivalencePartI.

ObjectiveI:Numericalexpressionsthathavethesamenumericalvalue(thesameanswers)areequivalentnumericalexpressions.

1.Findasmanyexpressionsasyoucanthatareequivalenttothenumber4.Howmanyarethere?

2.Writeasmanyexpressionsasyoucanthatareequivalentto(7+4)+5

3.DON’TCALCULATE.Whichofthefollowingnumberexpressionshasthesameansweras:

367+68·214·1966+814·45

a) 367+68·1966·214+814·45

b) 214·68·1966+367+814·45

c) 45·814+367+214·68·1966

d) 68+367·214·1966+814·45

e) 1966·214·68+45·814+367

f) 367+68·214·814+1966·45

1

4.Withoutcalculating,insertthesymbol=or≠betweenthenumberexpressions.Givethereason(s)foryourchoice.

a) (208+59)·61·48 208+59·61·48

b) (415·58)·(232÷29) 415·58·232÷29

c) (151+36)+75 151+(36+75)

d) 862–354 354–862

e) x·y·z z·x·y

PartII.

ObjectiveII:AlgebraicexpressionsthatareequivalentforALLvaluesofthevariableareequivalentalgebraicexpressions.Thisiscalledanalgebraicidentity.

1.Answerthefollowingquestionsforeachalgebraicstatement.

• Arethereanyvaluesforwhichthisalgebraicstatementistrue?Whichvalues?• Arethereanyvaluesforwhichthisalgebraicstatementisnottrue?Whichvalues?• Isthisstatementanalgebraicidentity?

a)𝑥 + 𝑥 = 𝑥$

b)4𝑥 + 12 = 7𝑥 + 50

c)10𝑥 + 40 = 10𝑥 + 50

2

2. Completethefollowingtable.

x Letx=1 Letx=2 Letx=5 Letx=19 Letx=382x+5x3x+4x12x-5x9x–2x

a) Whatdoyounoticeinthetable?

b) Aretheexpressionsinthetableequivalent?

3. Notallalgebraicexpressionsarealgebraicidentities,butcanbeequivalentforoneormorevaluesofthevariable.Findthevalueofthevariablethatwouldmakeeachpairofalgebraicexpressionsnumericallyequivalent.

a) 3𝑥 + 2and5𝑥 + 3

b) 4𝑥 + 12and7𝑥 + 3

NOTE:Eachpairofexpressionsisnumericallyequivalentforaspecificvalueofthevariable,buttheyarenotalgebraicallyequivalent.

PartIII.Themeaningoftheequalsignasrelational.

1. Ifinagivensituationy=x+4andy=3x–7canwesayx+4=3x–7foratleastonevalueofthevariable?Whyorwhynot?

2. IfA=BandB=C,doesA=C?Explain.

3. Giveadefinitioninyourownwordsofwhatthe=signmeans.

3

R.E.A.L.Math1010SupplementalActivity Name:___________________________S2InverseandIdentityProperties

1. Statethenumberthatistheidentityforthefollowingoperationsandgiveanexample.a) Addition

b) Subtraction

c) Multiplication

d) Division

2. Representthenumber1intwodifferentways,withoutusingthenumeral“1”

3. Fillintheblanks:

a) 𝑚 ∙ _____ = 1 b) 𝑚 + ____ = 0

4. Writethemultiplicativeinverseforeachofthefollowing:

a) 2 b) 7 c)++,

d) -+.

Inmathematics,anIdentityisanumberthatwhenadded,subtracted,multipliedordividedwithanynumber(n)theidentityallowsthatnumbertostaythesame.Thisisanimportanttoolformaintainingequivalenceinalgebraicexpressions.

Inthisactivity,wearegoingtofocusontheidentityformultiplication,thenumeralone.Wecall1theMultiplicativeIdentity.Weknowthatanynumbermultipliedby1equalsitself.

Thevaluesthatmakethesestatementstruearecalledinverses.Exercise3ademonstratesthemultiplicativeinverseand3bdemonstratestheadditiveinverse.Inthisactivity,wearegoingtofocusonthemultiplicativeinverse.

5

5. Whenyoumultiplymultiplicativeinversestogether,whatistheoutcome?Whatdowecallthatnumber?

6. Inthefollowingexpressions,findthenumbersorvariablesthataremultiplicativeinversesofeachother.Explainwhatmakesthemmultiplicativeinverses.

a),∙/0∙,

b)123024

10. Letx=1,y=2,andz=3ineachofthefollowingexpressions,123034

and1204.Whatdoesyour

outcomedemonstrate?

Fractionsvs.RationalExpressions

SimplifyingRationalExpressions

AfractionthathasvariablesinitiscalledaRationalExpression.Weusethesameprinciplestoadd,subtract,multiplyanddividerationalexpressionsasweusetoaddfractions.

Whensimplifyingrationalexpressions,welookformultiplicativeinversepairsthatmakeupthemultiplicativeidentityandapplythemultiplicativeidentityproperty.Forexample:

123034

canbewrittenshowingthemultiplicativeinversepairof33likethis12

04∙ 33

Themultiplicativeinversepairof33= 1.

Therefore1204∙ 1 = 12

04,bythemultiplicativeidentityproperty.

6

TheBigOne

11. Is22amultiplicativeinversepairin

26+26,

?Whyorwhynot?Canyouproveyouranswer?

12. Simplifythefollowingrationalexpressionsbyfollowingtheexamples:1- Factorasneeded2- Writethemultiplicativeinversepairseparatelyandwritea“BigOne”aroundeachone3- Writethemultiplicativeinversepairasthemultiplicativeidentity(1)4- Writetheequivalentexpressionwithoutthe1.

Example1:723893/3

= /3(,28/)/3

= /3/3∙ 2𝑥 + 3 = 1 ∙ 2𝑥 + 3 = 2𝑥 + 3

a),22>

b) 1467+?48-

Toemphasizethevalueofmultiplicativeinversepairs,a“BigOne”isdrawnaroundthemultiplicativeinversepair.

(26+)(28/)(26,)(28/)

=(26+)(26,)

7

R.E.A.L.Math1010SupplementalActivity Name:___________________________Lab1LinearPatternsinPairedData(AdaptedfromMIAInstructorResources)

Variablesariseinmanycommonmeasurements.Yourheightisonemeasurementthathasprobablybeenrecordedfrequentlyfromthedayyouwereborn.Inthisproject,youareaskedtopairupandmakethefollowingbodymeasurements:height(h);armspan(a),thedistancebetweenthetipsofyourtwomiddlefingerswitharmsoutstretched;femur(f)fromthecenterofthekneecaptotheboneontheoutsideofthehip.Forconsistency,measurethelengthsininches.1. Gatherthedatafor15peopleinyourclass,andrecorditinthefollowingtable:

Student Height(h)

ArmSpan(a)

Femur(f)

9

PredictingHeightfromBoneLengthAnanthropologiststudieshumanphysicaltraits,placeoforigin,socialstructure,andculture.Anthropologistsareoftensearchingfortheremainsofpeoplewholivedmanyyearsago.Aforensicscientiststudiestheevidencefromacrimesceneinordertohelpsolveacrime.Bothofthesegroupsofscientistsusevariouscharacteristicsandmeasurementsofthehumanskeletalremainstohelpdeterminephysicaltraitssuchasheight,aswellasracialandgenderdifferences.

Intheaverageperson,thereisastrongrelationshipbetweenheightandthelengthoftwomajorarmbones(thehumerousandtheradius),aswellasthelengthofthetwomajorlegbones(thefemurandthetibia).

Anthropologistsandforensicscientistscancloselyestimateaperson’sheightfromthelengthofjustoneofthesemajorbones.

2. Ifyouwanttopredictheightfromthelengthofthefemur,whichvariableshouldrepresenttheindependentvariable?Explain.

3. Useyoucalculatortomakeascatterplotofthedatafromyourtableshowingtherelationshipbetweenheightandlengthoffemur.Whatistheequationoftheregressionlineforthedata?

4. Usetheequationoftheregressionlinein#3topredicttheheightofapersonwhosefemurmeasures17inches.

5. Anthropologistshavedevelopedthefollowingformulastopredicttheheightofamaleorfemale,wherehrepresentstheheightininchesandLrepresentsthelengthofthefemurininches.Doestheformulaaccuratelypredictyourheightfromthelengthofyourfemur?Ifnot,whatcouldaccountforthedifference?Male:ℎ = 1.888𝐿 + 32.010Female:ℎ = 1.945𝐿 + 28.670

InterestingNote:Thedevelopmentoftheseandotherformulasusedtoestimateaperson’sheightbasedonbonelengthisbasedontheworkofDr.MildredTrotter(1899-1991)inskeletalbiology.Herresearchalsoledtodiscoveriesaboutthegrowth,racialandgenderdifferences,andagingofthehumanskeleton.

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R.E.A.L.Math1010SupplementalActivity Name:___________________________S3Systemsof2equationswith2variables.WatchtheproductreviewvideoaboutOruKayakshttps://www.youtube.com/watch?v=4CGclIZBlRAorthepromotionalvideoonthispagehttps://www.orukayak.comThebusinessownersofOruKayaksappearedontheTVshowSharkTankseekinganinvestmentof$500,000dollarsfora12%equitystake.Duringtheepisode,thefollowinginformationwasshared.

ü Thecompanysold473kayaksduringtheirfirst2monthsofbusinessthroughaKickstartercampaign.

ü Thekayakshavebeensellingfor$1100ü Thekayakscost$505tobuildü Atthepointbeingontheshow,9monthsaftertheKickstartercampaign,theyhavesold1228

kayaks.1.Assumingthesaleshadalineargrowthrate,writeafunctionthatrepresentsthenumberofkayakssoldasafunctionofthenumberofmonthssincethecompanystarted.2.Usethefunctiontoestimatethenumberofkayakssoldduringthe12monthsfollowingtheSharkTankepisode.Doesthisprojectionseemreasonable?Whyorwhynot?Usinganon-linearmodelananalystcalculatedamorereasonableprojectionthatthecompanyshouldsell3153kayaksinthenextyear.OntheSharkTankepisode,thebusinessownersprojectedsalesfortheirsecondyearofbusinesstobeat4milliondollars.Theygottheirinvestmentof$500,000fromaSharkTankinvestor.Sincethattime,thepriceoftheoriginalkayakwasraisedto$1600andtheydevelopedanothermodelforbeginnerpaddlersthatsellsfor$1200.Wewanttofindouthowmanyofeachmodelneedstobesoldtoreachprojectedsalesof4milliondollars.Wheneverwewanttofindtwounknownvalues(variables)weneed2equationsaboutthosevalues.3.Letx=thenumberofunitssoldfor$1200andy=thenumberofunitssoldfor$1600.Writetwoequationswithxandy,a)onerepresentingthetotalunitssoldbasedontheanalyst’sprojectionb)onerepresentinghowmanyofeachpricedkayakshouldbesoldtomeettheprojectedamountofsales.Beforewesolvethissystemofequationsforxandy,weneedtoreviewafewskills.

11

4. Tosolveasystemofequationswewanttofindthevaluesoftheorderedpair(x,y)thatareasolutionforbothequations.Forexample,theorderedpair(6,10)isasolutionforthefirstequationbelow,butnotforthesecondequation.Therefore,(6,10)isnotasolutionofthesystem.𝑦 = 𝑥 + 4𝑥 + 3𝑦 = 8

Tofindthesolutiontoasystemofequations,solveforonevariableatatimebycreatingoneequationwithonevariableandsolveforthatvariable.Therearetwowaystoalgebraicallysolvesystemsoftwoequationsintwovariables--substitutionandelimination.Bothmethodsuseequivalenceasatool.

a) Solvetheabovesystemusingthesubstitutionmethod.Thendrawagraphthatrepresentsthesystemandit’ssolution.

5. Solvethissystemofequationsusingelimination.Thendrawagraphthatrepresentsthesystemanditssolution.

2𝑥 − 3𝑦 = 12𝑥 + 3𝑦 = 9

12

7. UseeithersubstitutionoreliminationtosolvetheOruKayaksystemofequations.Howmanyofeachmodelofkayaksoldwouldproducetheprojectedoutcomes.

8. AsmallpotterycompanyspecializesinalargevasethatsellsonEtsyfor$100.Thetotalcostindollars,𝐶 𝑥 ,ofproducingxvasesismodeledby𝐶 𝑥 = 25𝑥 +500.a) Whatisthepracticalmeaningoftheslopeandinterceptofthecostfunction?

b) Writearevenuefunctionthatrepresentstheamountofmoneycollectedindollars,R,fromthesaleofx vases.

c) Acompanywillbreakevenwhenitsrevenueexactlyequalsitscost.Determinethebreak-evenpointonthevasesgraphicallyandalgebraically.

13

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Worksheet by Kuta Software LLC

1010 REAL Classwork

S3 Solving Systems of 2 Equations

Name___________________________________ ID: 1

Date________________ Period____

Solve each system by elimination.

1) −10x + 9y = 1520x − 6y = 30

2) −3x − 3y = −24−6x − 6y = 12

3) −8

5x −

9

5y = 5

−5x − 5y = 10

4) −10

3x +

4

3y = −4

7x − 9y = −4

5) −18x + 5y = 289x + 7y = 5

6) 8x − 10y = −6−16x + 20y = 12

7) Mei's school is selling tickets to the annual dance competition. On the first day of ticket sales theschool sold 1 senior citizen ticket and 12 child tickets for a total of $146. The school took in$108 on the second day by selling 3 senior citizen tickets and 6 child tickets. What is the priceeach of one senior citizen ticket and one child ticket?

8) The senior classes at High School A and High School B planned separate trips to the water park.The senior class at High School A rented and filled 7 vans and 9 buses with 560 students. HighSchool B rented and filled 14 vans and 1 bus with 168 students. Every van had the same numberof students in it as did the buses. Find the number of students in each van and in each bus.

14

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Answers to S3 Solving Systems of 2 Equations (ID: 1)1) (3, 5) 2) no solution 3) (7, −9) 4) (2, 2)5) (−1, 2) 6) Infinite number of solutions7) senior citizen ticket: $14, child ticket: $11 8) Van: 8, Bus: 56

15

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CE 1010 Classwork

S4 Solving Multi-Step and Compound Inequalities

Name___________________________________ ID: 1

Date________________ Period____

Solve each inequality and graph its solution. Then give the solution in interval notation.

1) 82 > −4 + 2(5x + 3)0 1 2 3 4 5 6 7 8 9 10

2) −96 ≤ −2(−7b − 8)−14 −12 −10 −8 −6

Solve each inequality and graph its solution.

3) 9 < −7

3k +

7

3 −

3

2

−5 −4 −3 −2 −1 0 1 2 3 4 5

4) −3 > b − 7

2− 1

−5 −4 −3 −2 −1 0 1 2 3 4 5

Solve each compound inequality and graph its solution. Then give the solution in intervalnotation.

5) −3 ≤ −5b + 7 ≤ 2

−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7

6) −45 ≤ −3 − 7m ≤ 18

−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7

7) 29 ≥ −8x − 3 > −27

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

8) −21

20 ≤

9

4p −

3

2 <

9

4

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

9) −41

16≤ −

7

4m +

1

2< −

29

12

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

10) −53

12 <

5

3r −

11

4≤ −

19

36

−4 −3 −2 −1 0 1 2 3 4 5 6 7 8

17

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Answers to S4 Solving Multi-Step and Compound Inequalities (ID: 1)1) x < 8 :

0 2 4 6 8 10

2) b ≥ −8 :−14 −12 −10 −8 −6

3) k < −7

2 :

−4 −2 0 2 4

4) b < 3

2 :

−4 −2 0 2 4

5) 1 ≤ b ≤ 2 :−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7

6) −3 ≤ m ≤ 6 :−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7

7) −4 ≤ x < 3 :−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 8)

1

5≤ p <

5

3 :

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

9) 5

3< m ≤

7

4 :

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

10) −1 < r ≤ 4

3 :

−4 −3 −2 −1 0 1 2 3 4 5 6 7 8

18

+

R.E.A.L.Math1010SupplementalActivity Name:___________________________S5AddLikeThings

1. Discussinyourgroupwhythisistrue

butthisisnot.

2. Whenyouaddwholenumberstogethersuchas438+56,whycan’tyouaddthe5tothe4?

3. Whydowelineupthedecimalpointwhenweadd43.056+2.4?

4. Whydowehavetofindacommondenominatortoadd!"+ $

%?

Thereisoneanswertoallofthesequestions.Thatis:wecanonly“addlikethings.”Understandingthe“addlikethings”principlemakesalgebramucheasiertolearn.Whenweaddlikethingswearemaintainingequivalence.

+ =5

=5

19

5. CirclethefollowingitemsthatarelikethingsandcanbeaddedandExplainwhyorwhynotforeachitem.Addorsubtractthosethatarelikethings.

a) 2𝑥$ + 5𝑥$ b) 𝑥) + 𝑥*

c)+"+ +

%d) 5𝑖 + 13𝑖

e) 𝑥𝑦! + 𝑥!𝑦 f) 2𝑥 + 𝑥

g)!+"+ +

"h) 7 + 4𝑖

i) 𝑥!) + 4 𝑥!) j) 2++3!

− "++3!

6. a)Whyare!"+ $

%notlikethings?

b) Howdowemake!"+ $

%likethings?Whycanwedothat?Whichmathematicalproperties

makethatpossible?

c) Whatnamedowegivetotwofractionsthatlookdifferent,buthavethesamevalue?Giveanexample.

20

7. Whatwouldbethecommondenominatorofthefollowingpairsofrationalexpressions.

a) 5+67and "

276

b) !+35

and 2+95

8. UsingtheMultiplicativeInverseandIdentityProperties,converteachfractiontoitsequivalentfractionandaddthetwofractions.Explainthepropertiesusedtoconverteachfraction.

5+67and "

276

!+35

and 2+95

9. Aexpressionthathasrealnumbercoefficientsandnon-negativeexponentsiscalledapolynomialexpression.Labelthefollowingexpressionswithallofthefollowingwordsthatapplytoeach:

Polynomial,Monomial,Binomial,Trinomial

a)𝑥2𝑦" _______________________________________________________

b) 𝑥2 + 𝑦" _______________________________________________________

c) 𝑥2 + 𝑦" − 𝑧_____________________________________________________

d) 𝑥2 + 𝑥𝑦" + 𝑦𝑧 − 4________________________________________________

21

AddingandSubtractingPolynomialExpressionsandPolynomialFunctions

Polynomialexpressionsthathavejustonevariablecanalsobefunctions.Theexpression𝑥2 + 2𝑥" − 7𝑥! + 𝑥 − 5canalsobeafunctionrepresentedas𝑝 𝑥 = 𝑥2 + 2𝑥" − 7𝑥! + 𝑥 − 5.Functionsarelikethingsthatcanbeaddedtogether.

7. Considerahomeownerwhoneedstoimprovethesoilforgrowingflowersinheryard.Sheidentifiestwosoilamendmentsneededtoturnherexistingdirtintotheperfectsoilforgrowingbeautifulflowers.Sheneedsequalamountsofeachamendmentwhicharesoldbythecubicyard.Peatmossis$32percubicyardandcompostedmanureis$49percubicyard.Thetotalcostfortheamendments,basedonthenumberofcubicyardsneeded,canbecalculatedtwodifferentways.

a) Whatarethetwodifferentwaystocalculatethetotalcost?

b) If𝑓 𝑥 = 32𝑥and𝑔 𝑥 = 49𝑥,whatis(𝑓 + 𝑔)(𝑥)?

8. Theprofitearnedfromthesaleofaproductorserviceiscalculatedbysubtractingthecosttoproducetheproductorservicefromthetotalrevenue,moneycollectedbysellingtheproductorservice.Revenueandcostarefunctionsofthenumberofproductorservicessold.Afarmthatmakesandsellsbarsofgoatmilksoap.Thefunctionrepresentingthecosttomakeabarofsoapis𝑐 𝑥 = 3.25𝑥 + 125.Therevenuefunctionis𝑟 𝑥 = 6.75𝑥.

a) Writetheequationoffunctionsthatrepresenttheprofitbasedonmakingandsellingxbarsofsoap.

b) Whatistheprofitfunction?

9. Becausewecanaddlikethings,wecanadd(orsubtract)polynomialfunctions.Forthefollowingfunctionsfind(𝑓 + 𝑔)(𝑥)and(𝑔 − 𝑓)(𝑥):a) 𝑓 𝑥 = 12𝑥! + 3𝑥 − 5b) 𝑔 𝑥 = 7𝑥 + 9

22

R.E.A.L.Math1010SupplementalActivity Name:______________________S6ExponentReview

Ingroups,followtheinstructionbelowbywritingonalargewhiteboardorsharedposter.

Definition:Inanexponentialexpressionthenumberbeingraisedtoapoweriscalledthebaseandthepoweriscalledtheexponent.

1. For3"whichnumberisthebase?_______Whichnumberistheexponent?________

PartA:MultiplyingwithExponents

1. Write23withoutanexponent.2. Write24withoutanexponent.3. Withouttheuseofexponents,represent23x24.4. Nowrepresentthisproductwiththeuseofonlyoneexponent.

Whatisthenameoftheexponentrulethatthisdemonstrates?

Howwouldyourgroupexpressthisrulewithwordsonly(nosymbols)?

Asagroup,writethisruleinasymbolicformusingvariablestoshowthatthisrulecanbeusedinforbasesandexponentsofanynumber.

Usethisruletosolvethefollowingequation:2"×2% = 2'

23

PartB:DividingwithExponents

1. Write27withoutanexponent.2. Write24withoutanexponent.3. Withouttheuseofexponents,represent27dividedby24.4. Nowrepresentthisquotientwrittenwithoneexponent.


Howwouldyourgroupexpressthisrulewithwordsonly(nosymbols)?

Asagroup,determinehowtowritethisruleinsymbolicformusingvariablestoshowthatthisrulecanbeusedforbasesandexponentsofanynumber.

PartC:ExponentiatingExponents

1. Asagroup,determinehowtowrite(23)4withouttheuseofexponents.

2. Nowexpresstheresultwithoneexponentonly.


Asagroup,determinehowyouwouldexpressthisrulewithwordsonly(nosymbols).

Asagroup,determinehowtowritethisruleinsymbolicformusingvariablestoshowthatthisrulecanbeusedforbasesandexponentsofanynumber.

24

PartD:ZeroExponents

1. Whatisthevalueofanynumberdividedbyitself?

2. Withoutusingexponentrules,whatisthevalueof"(

"(?Whatisthevalueof

)*+

)*+?

3. Nowapplythequotientruletosimplify)*+

)*+.Therefore)

*+

)*+=c____–____=c___=____

4. Itmakessense,then,thatif)*+

)*+=1and

)*+

)*+=c0,then________=_________

5. Wouldthisbetrueforanybaseraisedtothezeropower?Why?

6. Insmallgroupsfillintheblanks.

Writetheexponentialexpressionwithoutexponentsandsimplify.Ifnecessaryleaveanswersinfractionform.

Simplifyeachexpressionusingthequotientruleforexponents

Example,-

,= ,',',',

,= ./

,= 8 21

2= 212. = 23 = 8

21

2,= ___________________ = _______ = ______

21

2,= ___________________ = _______ = ______

21

23= ___________________ = _______ = ______

21

23= ___________________ = _______ = ______

21

21= ___________________ = _______ = ______

21

21= ___________________ = _______ = ______

21

2"= ___________________ = _______ = ______

21

2"= ___________________ = _______ = ______

21

2/= ___________________ = _______ = ______

21

2/= ___________________ = _______ = ______

21

2%= ___________________ = _______ = ______

21

2%= ___________________ = _______ = ______

21

25= ___________________ = _______ = ______

21

25= ___________________ = _______ = ______

25

Part E: DerivingtheMeaningofNegativeExponents

Insmallgroups,fillintheblanksandmakealistofthefollowingonyourwhiteboardorsharedposter.

25=2x2x2x2x2=32

24=______________=16

23=______________=_____

22=____________=_____

21=_______

20=_______

2-1=

2-2= ._____'_____

= .,+=.

1

2-3= .

= =

2-4= .

= =

Asagroup,discussthepatternsthatyousee.

Howwouldthesepatternsbedifferentifthebasewere.,insteadof2?

Whichoftheaboveexponentsalwaysproducesthesamenumberregardlessofwhatthebaseis?

Withyourgroup,definethemeaningofanegativeexponent.

26

UnderstandingNegativesandExponents

Insmallgroups,expressthefollowingwithoutexponents.Compareandcontrastwhathappenswhenthenegativesignisindifferentpositions.Bepreparedtoexplaintotheclass.

-32=

(-3)2=

3-2=

(-3)-2=

-3-2=

Asagroup,discussthefollowingquestionsandbepreparedtojustifyyouranswerstotheclass.

AlwaysTrue,SometimesTrue,orFalse:Anegativeexponentwillchangethesignofitsbase.

AlwaysTrue,SometimesTrue,orFalse:Anegativeexponentmeanstotakeareciprocal.

AlwaysTrue,SometimesTrue,orFalse:Anegativeexponenttellshowmanytimestodividebyitsbase.

-(1/2)2=

(-1/2)2=

(1/2)-2=

(-1/2)-2=

-(1/2)-2=

27

R.E.A.L.Math1010SupplementalActivity Name:___________________________S7MultiplyingPolynomialExpressions

1. Whatistheareaofthefollowingfigure?Nameatleast2waystodeterminethearea.

Removeifleftinequivalencelab2. Usingtheindicateddimensionsofeachrectangle,writeexpressionsrepresentingtheareasofthetworectanglesinsideeachofthefiguresbelow.

3. WriteanexpressionrepresentingtheareaofthisrectangleintermsofLengthtimesWidthusingtheindicateddimensionsoftherectangle.

4. Combiningallthreeoftheexpressionsfrom#2and#3intoasingleequation,showtheequivalentrelationshipbetweentheareasoftherectanglesin#2andtheareaoftherectanglein#3.Thisrepresentsthedistributivepropertyofmultiplicationoveraddition.

4

5

x

3x

6

3x

x

3x

6

29

5. ThesquareparkinglotatLaCasitaMexicanRestaurantisgoingtobeenlargedsotherewillbeanadditional30ft.ofparkingspaceinthefrontofthelotandanadditional30ft.ofparkingspaceonthesideofthelot,asshowninthefigurebelow.Labelthefiguretorepresentthedimensionsoftheoriginallotandtheadditionalspace.Writeanexpressionintermsofxthatcanbeusedtorepresenttheareaofthenewparkinglotintermsoflengthtimeswidthusingthedimensionsofthenewparkinglot.

Dimensions:L=___________W=____________

Area=LxW=_______________________

6. Multiplytheexpressionfrom#5byapplyingthedistributivepropertytwice.Thatis,eachofthetermsinthefirstbinomialaredistributedoverthesecondbinomial.Thisexpressionrepresentstheareaofthenewparkinglot.

7. Fillintheareasofeachpartoftheparkinglotthatcorrespondtotheexpressionyouhavejustwritten.

8. Writeanequationshowingtherelationshipbetweenthedimensionsofthenewparkinglotandtheareaofthenewparkinglot.Writethedimensionsasasinglebinomialraisedtoapower.

Street

30

9. Kekauwasaskedtomultiply(𝑥 − 8)&andcameupwithananswerof𝑥& − 64.a) Isthatcorrect?

b) Useanareamodelliketheoneinexercise#5toverifyyourconclusion.

c) Useanotherareamodeltocreateanidentity(orformula)formultiplying(𝑎 + 𝑏)&.

10. Anotherterminologyformultiplying(𝑎 + 𝑏)&is“squaringabinomial”because(𝑎 + 𝑏)isabinomialanditisbeingsquared.Usetheformulayoudevelopedin#10tosquarethefollowingbinomials

a) (𝑥 + 3)& b) (𝑦 − 7)&

c) (𝑎 + 3𝑏)& d) (5𝑦 − 6)&

e) (2𝑥 − 1)&

31

11..Createanareamodel,likethefigurein#5,torepresenteachexpression.Showthedimensionsandareaofeachindividualrectangleintheareamodelandwritetheexpressionsthatcorrespondtoit.

Example:(x+2)(x+3)

𝑥& + 2𝑥 + 3𝑥 + 6𝑥& + 5𝑥 + 6

a) (x+4)(x+7) b) (x+3)(x+1)

Anareamodelisnotdirectlyapplicabletoallpolynomialmultiplicationproblems.However,atablecanbeusedinasimilarwaytorepresentthedistributiveprocessusedinmultiplyingpolynomials,eventhoughitdoesnotspecificallyrepresentarea.

12. Fillinthetabletoidentifythepartialproductsof(x+2)(x-5).Thenwritetheproductinstandardform.Explainwhyanareamodelwon’tworkforthisproduct.

x 2

x

3

x2 2x

3x 6

x - 5

x+2

32

13. Useatabletomultiplythefollowingpolynomialproducts.

a) (𝑥 + 7)(𝑥 − 4) b) (𝑥 − 6)(𝑥 + 6)

c) (𝑥& + 4)(𝑥 − 9) d) (𝑥 − 3)(𝑥& − 6𝑥 + 9)

e) (𝑥 + 2)(𝑥& + 4𝑥 + 9) f) (3𝑥 + 5)&

33

a. Multiply these polynomials using the tabular method.

(2𝑎𝑎 + 5)(𝑎𝑎2 + 5𝑎𝑎 + 1)

b. How can you use the expression in part (a) to quickly multiply 25 ⋅ 151?

Exploratory Challenge

1. Does 2 1 2 2

2= (𝑎𝑎2 + 5𝑎𝑎 + 1)? Justify your answer.

Tabular DivisionOpening Exercise

Name:___________________________R.E.A.L.Math1010SupplementalActivityS8Division of Polynomials

This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

https://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US35

2. Describe the process you used to determine your answer to Exercise 1.

3. Reverse the tabular method of multiplication to find the quotient: 2 2 10

2.

𝒙𝒙

𝟐𝟐𝒙𝒙𝟐𝟐 −𝟐𝟐

𝒙𝒙 −

4. Test your conjectures. Create your own table and use the reverse tabular method to find the quotient.

𝑎𝑎 + 4𝑎𝑎3 + 3𝑎𝑎2 + 4𝑎𝑎 + 2𝑎𝑎2 + 1

36

5. Test your conjectures. Use the reverse tabular method to find the quotient.

3𝑎𝑎 − 2𝑎𝑎 + 6𝑎𝑎3 − 4𝑎𝑎2 − 24𝑎𝑎 + 16𝑎𝑎2 + 4

6. What is the quotient of11

? What is the quotient of 11

?

37

Problem Set

Use the reverse tabular method to solve these division problems.

1. 2 2 1 1

2 3

2. 3 12 11 2 2 2

3 2 1

3. 2 22

4. 2 2 122 2 3

5. 1 2

6. 2

7. 2 2 2 1

1

8. 2 2 2 2 1

1

9. Use the results of Problems 7 and 8 to predict the quotient of

2 2 2 2 2 11

.

Explain your prediction. Then check your prediction using the reverse tabular method.

10. Use the results of Problems 7–9 above to predict the quotient of

2 2 2 2 11

. Explain your prediction.

Then check your prediction using the reverse tabular method.

11. Make and test a conjecture about the quotient of

2 2 2 2 12 1

. Explain your reasoning.

38

Polynomial Long Division

Opening Exercise

1. Use the reverse tabular method to determine the quotient 2𝑥𝑥3+11𝑥𝑥2+7𝑥𝑥+10

𝑥𝑥+5.

2. Use your work from Exercise 1 to write the polynomial 2𝑥𝑥3 + 11𝑥𝑥2 + 7𝑥𝑥 + 10 in factored form, and then multiply

the factors to check your work above.

39

Example 1

If 𝑥𝑥 = 10, then the division 1573 ÷ 13 can be represented using polynomial division.

3753 23 ++++ xxxx

Example 2

Use the long division algorithm for polynomials to evaluate

2𝑥𝑥3 − 4𝑥𝑥2 + 2

2𝑥𝑥 − 2

40

Exercises 1–6

Use the long division algorithm to determine the quotient. For each problem, check your work by using the reverse tabular method.

1. 𝑥𝑥2+6𝑥𝑥+9𝑥𝑥+3

2. 7𝑥𝑥3−8𝑥𝑥2−13𝑥𝑥+2

7𝑥𝑥−1

3. 𝑥𝑥3−27 𝑥𝑥−3

4. 2𝑥𝑥4+14𝑥𝑥3+𝑥𝑥2−21𝑥𝑥−6

2𝑥𝑥2−3

5. 5𝑥𝑥4−6𝑥𝑥2+1

𝑥𝑥2−1

6. 𝑥𝑥6+4𝑥𝑥4−4𝑥𝑥−1

𝑥𝑥3−1

41

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1010 REAL Classwork

Polynomial Divison With Remainders

Name___________________________________

Date________________ Period____

Divide.

1) (8n2 + 15n + 10) ÷ (n + 1) 2) (n4 − 3n3 + 7n2 − 3n − 10) ÷ (n − 1)

3) (n4 − 12n3 + 32n2 + 31n − 39) ÷ (n − 6) 4) (m4 − 6m3 − 2m2 + 11m − 10) ÷ (m − 1)

5) (a4 − 6a3 + 12a2 − 12a + 16) ÷ (a − 3) 6) (12v2 + 44v + 25) ÷ (6v + 4)

7) (12x3 + 4x2 − 52x − 56) ÷ (6x + 8) 8) (2a3 − a2 − 9a − 41) ÷ (2a − 7)

9) (4m4 − 21m3 − 53m2 − 11m − 8) ÷ (4m + 7) 10) (6x2 − 47x + 38) ÷ (x − 7)

42

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Answers to Polynomial Divison With Remainders

1) 8n + 7 +3

n + 12) n3 − 2n2 + 5n + 2 −

8

n − 13) n3 − 6n2 − 4n + 7 +

3

n − 6

4) m3 − 5m2 − 7m + 4 −6

m − 15) a3 − 3a2 + 3a − 3 +

7

a − 36) 2v + 6 +

1

6v + 4

7) 2x2 − 2x − 6 −4

3x + 48) a2 + 3a + 6 +

1

2a − 79) m3 − 7m2 − m − 1 −

1

4m + 7

10) 6x − 5 +3

x − 7

43

Thepropertiesofexponentscanbeexpandedtoincluderationalexponents.

Forexample,given43,theexponent3meansthereare3factorsof4.

1. Assumingtheuseofanexponentdoesn’tchangefromonescenariotoanother,whatdoesthismeanabout41/2

Example1:Write41/2withoutanexponent.Inthecaseof41/2,startbyfactoring4:4®2x2,andthentakehalfofthosefactors.41/2=2

Example2:Evaluate91/2.Takinghalfofthefactorsof9meanstakingoneofitstwofactorsof3.91/2=3.

2. Find271/3bylistingthefactorsof27andtaking1/3ofthosefactorslisted.

3. Usethesamemethodtofind272/3.

Youmayhavenoticedthatevaluating91/2issimilartoevaluating 9.Itisthesame.

NotethattakingthesquarerootofanumberistheinverseoperationofsquaringanumberANDthesamethingastakinghalfofthefactorsofthesquarednumber.

( 9)$ = 9, 9$ = 9

AND (91/2)2=9,(92)1/2=9

Ingeneral,sincesquaringandsquarerootingareinverseoperations,( 𝑎)2= 𝑎𝑖𝑓𝑎 ≥ 0.

4. Explainwhy 𝑎 = 𝑛𝑜𝑡𝑎𝑟𝑒𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟ifaisnegative.

Letarepresentanonnegativerealnumber,symbolicallywrittenasa𝑎 ≥ 0.Theprincipalsquarerootofa,denotedby√𝑎,isdefinedasthenonnegativenumberthat,whensquared,producesa.

Name:___________________________R.E.A.L.Math1010SupplementalActivityS9 Rational Exponents

45

5. Provealgebraicallythat 𝑎 = 𝑎7/$.Inotherwords,solvefortheexponentmif𝑎9 = 𝑎, 𝑓𝑜𝑟𝑎 ≥ 0.

6. Evaluatethefollowing.a. 361/2 b. -491/2 c. (-49)1/2 d. 01/2

7. Ifthevolumeofacubeis64cubicinches,determinethelengthofonesideofthecube.

8. Tofindtheexponent,m,that’sequivalenttotakingthecuberootofanumber,solve𝑎9 = 𝑎; , 𝑓𝑜𝑟𝑎𝑛𝑦𝑛𝑢𝑚𝑏𝑒𝑟𝑎

9. Evaluatethefollowing.a. 8; b. 125; c. −1000; d. 0;

e. −8; f. 100; d. − 27; e. − −125;

(nearesttenth)

10. Calculateeachofthefollowing,andthenverifyyouranswerusingyourcalculator.a. 81C b. 321/5 c. −32E d. -2251/4(calculator)

11. Trytocompute −81C onyourcalculator.Whathappensandwhy?

Ingeneral,√𝑎F = 𝑎7/G,thenthrootofa.Thenumbera,calledtheradicand,mustbenonnegativeifn,calledtheindex,iseven.

Thecuberootofanyrealnumbera,denotedby√𝑎; ,isdefinedasthenumberthatwhencubed,givesa.

46

12. YachtsthatcompeteintheAmerica’sCupmustsatisfytheInternationalAmerica’sCupClassrulethatrequires𝐿 + 1.25 𝑆 − 9.8 𝐷; ≤ 16.296meters.

Where𝐿representstheyacht’slengthinmeters,Srepresentstheratedsailarea,insquaremeters,andDrepresentsthewaterdisplacement,incubicmeters.

a. Isayachtwithlength21.85meters,sailarea305.5squaremeters,anddisplacement21.85cubicmeterseligibletocompete?Explain.

b. Explainwhytheunitsofyournumericalanswerinpart(a)aremeters.

Thepropertiesofexponentscanbeexpandedtoincluderationalexponentswherethenumeratorisdifferentfromone.Forexample:8$/N = 8$∙(7/N)

= (8$)7/N= 8$; = 64; = 4

Inthisexample,thecuberootwastakenafterthesquaringwasdone.Anequivalentanswercanbefoundbytakingthecuberootof8first,thensquaringtheresult.

13. Computeeachofthefollowing.Showeachstepofthecomputation.Thenverifytheanswerusingyourcalculator.

a. 253/2 b. (-8)2/3 c. 324/5 d. -163/4

e. 2432/5 f. (-16)3/4 g. -253/2 h. 4QN/$

14. Compute72/3onyourcalculator,andexplainhowyoumightreversetheoperationtochecktheanswer.

Ingeneral,𝑎R/S = √𝑎RT 𝑜𝑟𝑎R/S = (√𝑎

T )R,where𝑎 ≥ 0ifqisevenandpandqareintegers.

47

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1010 REAL Classwork

Rational Exponents

Name___________________________________ ID: 1

Date________________ Period____

Simplify.

1) (n4)1

2 2) (64x4)1

2

3) (r6)4

3 4) (100000r5)3

5

5) (16x6)1

2 6) (81r2)1

2

7) (x8)−5

4 8) (36n4)1

2

9) (x4)3

2 10) (64k2)3

2

Write each expression in radical form.

11) (2x)−3

4 12) x2

5

13) n3

4 14) n3

2

Write each expression in exponential form.

15) ( 35a)5 16) ( 3

7x)4

17) 4r 18) ( 6k)5

19) ( 4x)7 20)

1

10n

48

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Answers to Rational Exponents (ID: 1)1) n2 2) 8x2 3) r8 4) 1000r3

5) 4x3 6) 9r7)

1

x108) 6n2

9) x6 10) 512k311)

1

( 42x)3

12) ( 5x)2

13) ( 4n)3 14) ( n)3

15) (5a)5

3 16) (7x)4

3

17) r1

4 18) (6k)5

2 19) x7

4 20) (10n)−1

2

49

Asquarerootradicalissimplified,orinitssimplestformwhentheradicandhasnosquarefactors.

Acuberootradicalissimplified,orinitssimplestformwhentheradicandhasnocubedfactors.

Afourthrootradicalissimplifiedwhentheradicandhasnofactorsthatareraisedtothe4thpower,andsoon.

8isnotaperfectsquareso 8can’tbesimplifiedintoaninteger.However,8hasaperfectsquarefactor(4)thatCANbesimplifiedintoaninteger.

Forexample: 8 = 4 ∙ 2 = 4 ∙ 2 = 2 2

Use your calculator to verify that 𝟖 = 𝟐 𝟐.

Note that 2 2 is EXACTLY equal to 8 so it’s more accurate than using a calculator to get a decimal approximation.

When simplifying radicals, it’s helpful to easily recognize numbers that are perfect squares and perfect cubes. The most common perfect square factors you will use when simplifying square roots are: 4, 8, 16, and 25. The most common perfect cube factors are 8 and 27 (and 125 for simplifying really large radicands.)

Example 1: Simplify 150 .

150 = 25 ∙ 6 = 𝟐𝟓 ∙ 6 = 𝟓 6

Notice that there are other ways of factoring 150 but the idea is to rewrite 150 using the largest, perfect square factor of that number. A calculator can be helpful for dividing radicands by 4, 8, 16, or 25 to see if the number is divisible by these perfect square factors.

There are no perfect square factors left under the radical so 5 6 is simplified.

Example 2: Simplify 32/ .

32/ = 8 ∙ 4/ = 𝟖𝟑 ∙ 4/ = 𝟐 4/

Notice that there are other ways of factoring 32 (like 16∙ 2)but the idea is to rewrite the number using a perfect cube factor -in this case, 8. When simplifying a cube root, start by seeing if the number is divisible by 8 or 27

There are no perfect cube factors left under the radical so 2 4/ is simplified.

1. Explain why x2, x4, x6, x8, or any other variable with even exponents are perfect square factors.How might this idea help you simplify 𝑥3 ?

2. Explain why x3, x6, x9, or any other variables with exponents that are multiples of 3 are perfectcube factors. How might this idea help simplify 𝑥4?

Name:___________________________R.E.A.L.Math1010SupplementalActivityS10 Simply, Add, Subtract Radicals

51

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1010 REAL Classwork

Simplifying Single Radicals

Name___________________________________

Date________________ Period____

Simplify.

1) 48 2) 125

3) 27 4) 32

5) −5 36m2 6) 5 36x

7) 3 45n 8) − 125x

9) −33−320p6 10) 5

4120x

11) −4112v 12) −2

3189x9

13) 33−24m8n2 14) −6

3−500xy3

15) 2480m7n5 16) 8

3135x6y7

17) −6 72r3 18) − 243b5

19) 8 72n3 20) 7332n3

52

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Answers to Simplifying Single Radicals (ID: 1)

1) 4 3 2) 5 5 3) 3 3 4) 4 25) −30m 6) 30 x 7) 9 5n 8) −5 5x9) 12p2

35 10) 5

4120x 11) −2

47v 12) −6x3

37

13) −6m2 33m2n2 14) 30y

34x 15) 4mn

45m3n 16) 24x2y2

35y

17) −36r 2r 18) −9b2 3b 19) 48n 2n 20) 14n34

53

Classwork

Exercises 1–5

Simplify each expression as much as possible.

1. √32 = 2. √45 =

3. √300 =

4. The triangle shown below has a perimeter of 6.5√2 units. Make a conjecture about how this answer was reached.

5. The sides of a triangle are 4√3, √12, and √75. Make a conjecture about how to determine the perimeter of this

triangle.


https://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

Adding and Subtracting Radicals

54

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US




Exercise 6

6. Circle the expressions that can be simplified using the distributive property. Be prepared to explain your choices.

8.3√2 + 7.9√2

√13 − √6

−15√5 + √45

11√7 − 6√7 + 3√2

19√2 + 2√8

4 + √11

√7 + 2√10

√12 − √75

√32 + √2

6√13 + √26

Example 1

Explain how the expression 8.3√2 + 7.9√2 can be simplified using the distributive property.

Explain how the expression 11√7 − 6√7 + 3√2 can be simplified using the distributive property.

55





Example 2

Explain how the expression 19√2 + 2√8 can be simplified using the distributive property.

Example 3

Can the expression √7 + 2√10 be simplified using the distributive property?

To determine if an expression can be simplified, you must first simplify each of the terms within the expression. Then,

apply the distributive property, or other properties as needed, to simplify the expression.

56





Problem Set

Express each answer in simplified radical form.

1. 18√5 − 12√5 = 2. √24 + 4√54 =

3. 2√7 + 4√63 =

4. What is the perimeter of the triangle shown below?

5. Determine the perimeter of the triangle shown. Simplify as much as possible.

6. Determine the perimeter of the rectangle shown. Simplify as much as possible.


57






9. Determine the perimeter of the shaded triangle. Write your answers in simplest radical form, and then approximateto the nearest tenth.

58





Workwithapartnertocompletethefirstpartofthislesson.

Findtheareaoftherectanglebelow.

3

4

Youprobablygot12.J

Let’smakeittougher.Write3,4and12asradicals.Hint:3is 9 .

Redothequestionsusingradicals.Labelthemissinglengthandsolveforthearea.Whatdoyounotice?

9

Trythisone.Comeupwithanareaasaradical,ratherthanabiglongmessydecimal.

2

11

Checkyouranswerto 2 × 11 usingyourcalculator.

Haveyouseenenoughtomakearule?Whatis a × b ?

Willthisrulestillapplyifthevalueofaand/orbisnegative?

A=

A=

A=

Name:___________________________R.E.A.L.Math1010SupplementalActivityS11 Multiplying Radicals

59

Workwithapartnertofindtheareaofeachofthefollowingrectangles.Pleaseexpressyouranswerasaradical(exactvalue–nodecimals).Youaregoingtohavetofigureoutwhattodowiththecoefficients.

20

√6

60

Let’smakeittougher.Usingwhatyoulearnedonthepreviouspage,trytoanswerthefollowing.Youcan(andshould)useyourcalculatortocheckyouranswers,butyouranswersshouldnothaveanydecimalsinthem.Theyshouldallstillhaveradicals.

Question ExactValueAnswer(radicals)

CalculatorCheck

2 3 × 4 5

( )3 2 4 3 2 5-

( )( )2 7 3 3 4 2 5 5- -

61

MultiplyingRadicals

a b´ =

Ruleinyourownwords:

NumericalExample:

a b c d´ =

Ruleinyourownwords:

NumericalExample:

( )a b c d e f+ =

Ruleinyourownwords:

NumericalExample:

( )( )a b c d e f g h+ + =

Ruleinyourownwords:

NumericalExample:

62

Classwork

Exercises 1–5

Simplify as much as possible.

1. √172 =

2. √510 =

3. √4𝑥4 =

4. Complete parts (a) through (c).

a. Compare the value of √36 to the value of √9 × √4.

Name:___________________________R.E.A.L.Math1010SupplementalActivityS12 Dividing Radicals & Rationalizing Denominators


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b. Make a conjecture about the validity of the following statement: For nonnegative real numbers 𝑎 and 𝑏,

√𝑎𝑏 = √𝑎 ∙ √𝑏. Explain.

c. Does your conjecture hold true for 𝑎 = −4 and 𝑏 = −9?

5. Complete parts (a) through (c).

a. Compare the value of √100

25 to the value of

√100

√25.

b. Make a conjecture about the validity of the following statement: For nonnegative real numbers 𝑎 and 𝑏, when

𝑏 ≠ 0, √𝑎

𝑏=

√𝑎

√𝑏. Explain.

c. Does your conjecture hold true for 𝑎 = −100 and 𝑏 = −25?

64





Exercises 6–16

Simplify each expression as much as possible, and rationalize denominators when applicable.

6. √72 = 7. √17

25=

8. √32𝑥 = 9. √1

3=

10. √54𝑥2 = 11.√36

√18=

65





12. √4

𝑥4= 13.

4𝑥

√64𝑥2=

14.5

√𝑥7= 15. √

𝑥5

2=

16.√18𝑥

3√𝑥5=

66





Problem Set

Express each number in its simplest radical form.

1. √6 ⋅ √60 = 2. √108 =

3. Pablo found the length of the hypotenuse of a right triangle to be √45. Can the length be simplified? Explain.

4. √12𝑥4 =

5. Sarahi found the distance between two points on a coordinate plane to be √74. Can this answer be simplified?

Explain.

6. √16𝑥3 =

7.√27

√3=

8. Nazem and Joffrey are arguing about who got the right answer. Nazem says the answer is1

√3, and Joffrey says the

answer is √3

3. Show and explain that their answers are equivalent.

9. √5

8=

10. Determine the area of a square with side length 2√7 in.

67





Name:________________________R.E.A.L.Math1010SupplementalActivityS13CompositionofFunctions

JCPenneyishavingastorewidesaleof30%offselectfamilyapparel,shoes,accessories,fine&fashionjewelry.Additionally,ifyoupaywithyourJCPenneycreditcard,yougetanextra20%discountonthosepurchases.

1. Representeachdiscount,separately,asafunctionofx,wherexrepresentstheregularpriceoftheitem.

Thefunctionthatcalculatestheamountofa30%discountontheregularprice,x,is:

f(x)=_______________________

Thefunctionthatcalculatestheamountofa20%discountontheregularprice,x,is:

g(x)=_______________________

2. Createfunctionsthatcalculatethesaleprice,S(x)andT(x)afteradiscountisappliedtotheregularprice,wherexrepresentstheregularpriceoftheitem.

Thesalepriceaftera30%discountofanitemthatcostsxdollarsisrepresentedas:

S(x)=_____________________

Thesalepriceaftera20%discountofanitemthatcostsxdollarsisrepresentedas:

T(x)=____________________

3. DoestheJCPenneysalemeanyouget50%offanysalemerchandise?Explainwhyorwhynot.

4. Explaininwordshowyouwouldcalculatethecombineddiscountsforanitemthatisregularlypriced$100andwhatyourfinalsalepricewouldbeifyouuseaJCPenneycreditcard.HowcouldthisprocessberepresentedwithfunctionnotationusingS(x)andT(x)from#2?

69

5. FindS(100)forthefunctionin#2.S(100)=______________

6. FindT[S(100)]forthefunctionsin#2.T[S(100)]=_________________

Compareanddiscussyouranswersto#4and#6.

7. Youreceivethiscouponinthemailanddecidetouseittobuyanewcoat.

a) Writethefunctionf(x)thatrepresentshowmuchyouwouldpayforacoatusingthe$10offcoupon.Letxrepresenttheretailprice.

b) Writethefunctiong(x)thatrepresentshowmuchyouwouldpayusingtheextra15%offcoupon.Letxrepresenttheretailprice.

c) Whenyouarriveatthestore,youfindthatallcoatsare30%off.Writethefunctionh(x)thatrepresentsthesalepriceforacoatbeforeapplyingthecoupon.

d) Writethefunctionnotationandthefunctionequationthatrepresentshowmuchyou’dpayifyoupurchasedthecoatwiththe30%discountandtheadditional15%discountcoupon.

70

e) Writethefunctionnotationandthefunctionequationthatrepresentshowmuchyou’dpayifyoupurchasedthecoatwiththe30%discountandtheadditional$10offcoupon.

f) Thecoatyouwanttopurchasehasaretailpriceof$100.Whichisthebetterdiscountcoupontouse?Whatwillbethefinalsalepriceforthecoat?

ThisprocessoffindingthefunctionofafunctioniscalledtheCompositionofFunctions.Itcanbethoughtofastheprocessofapplyingtwofunctionstoasingleinputvalue,wheretheoutputofthefirstfunctionbecomestheinputofthesecondfunction.

fromhttps://en.wikipedia.org/wiki/Funct1

71

8. Usethetwotablestofindthecompositions:

x -4 -3 -2 -1 0f(x) 3 2 5 8 11

x 2 4 6 8 10g(x) -1 -3 -5 -7 -9

9. Considerthefollowingfunctionalrelationshipsassociatedwithacityleaguesoccerteam.

Function Input OutputF Gameswoninaseason,w Averagenumberoffanspergame,F(w)

W Averagenumberofdaysofpractice,d Gameswoninaseason,W(d)

P Numberofrainydays,r Averagenumberofdaysofpractice,P(r)

Whatdoestheexpression𝐹(𝑊(𝑑))represent?Usethephrase“asafunctionof”inyouranswer.

10. Asmallstoneisthrownintostillwaterandcreatesacircularwave.Theradiusrofthewaterwaveincreasesattherateof2cmpersecond.

a) Findanexpressionfortheradiusrintermsoftimet(inseconds)afterthestonewasthrown.

b) IfAistheareaofthewaterwave,writethecomposition𝐴[𝑟(𝑡)].

a) 𝑔[𝑓(−3)]=

b) 𝑔[𝑓(−1)]=

c) 𝑓[𝑔(2)] =

d) 𝑓[𝑔(4)] =

72

c) Whatisthemeaningof𝐴[𝑟(𝑡)]?

c) FindtheareaAofthewaterwaveafter60seconds.

11. Forthefunction𝑓(𝑥) = 3𝑥 − 6,findthefollowing.

a) 𝑓(4)

b) 𝑓(𝑎)

c) 𝑓(SAM)

d) 𝑓(∎)

e) 𝑓(𝑟(𝑠))

12. Forthefunctions𝑓 𝑥 = 𝑥= − 4𝑥 + 3and𝑔 𝑥 = 3𝑥 − 7,findthefollowing.

a) (𝑓 ∘ 𝑔)(𝑥)

b) (𝑔 ∘ 𝑓)(𝑥)

c) (𝑓 ∘ 𝑔)(−3)

d) (𝑔 ∘ 𝑓)(2)

Thenotationforthecompositionoffunctionsfandgcanbewrittenas𝑓[𝑔(𝑥)]oras(𝑓 ∘ 𝑔)(𝑥).Itishelpfultoalwaysconvertthenotation(𝑓 ∘ 𝑔)(𝑥)to𝑓[𝑔(𝑥)]asthefirststeptocreatingacompositefunction.

73

AdditionalPractice:

1. Airescapesfromaballoonattheconstantrateof100cm3persecond.Whatistherateofchangeoftheradiusoftheballoon(supposedtobeasphere)whenr=10cm?

2. Startingfrom50meters,theradiusrofacircularoilspillincreasesattherateof0.5meters/second.a) Expresstheradiusrasafunctionoftime.b) TheareaAofacircularshapeisgivenbyA=πr2.Findthecompositefunction(𝐴 ∘ 𝑟)(𝑡)andexplainitsmeaning.c) Howlongwillittaketheareatobelarger10,000m2?

3. Fuelisbeingpumpedintoastoragetank.Thevolume,V,ofthefuelinthetankdependsonthedepth,d,accordingtotheformula𝑉 𝑑 = 4(3𝑑= + 5)Cwheredismeasuredinmeters.Supposethedepth,d,ofthefueldependsontime,t,measuredinhoursaccordingtotheformula𝑑 𝑡 = D

C𝑡 − 5.Usefunctioncompositiontowritethevolumeofthefuelinthetank

asafunctionoftime,V(t).

74

4. Laurawillgoforarunduringherlunchbreakifthetemperatureisbetween60degrees,and80degreesFahrenheit.Theaveragespeed,S,inmilesperhour,atwhichLaurarunsisdependentonthetemperature,t,indegreesFahrenheit,atthestartofherrunandcanbemodeledbythefunctionS(t)=6+0.1(80-t).Thedistance,D,inmiles,thatshecanrunin30minutesgiventhatheraveragespeedisxmilesperhourcanbemodeledbythefunctionD(x)=0.5x.WritethealgebraicexpressionthatmodelsthedistancethatLaurarunsin30minutesgiventhatitistdegreesFahrenheitoutsideatthestartofherrun.

5. Anais,amathteacher,noticedthattheaveragegradeonanexam,G,giventhatnstudentswatchtherelatedvideocanbegivenbythefunctionG(n)=50+1.5n.Shealsonoticedthatthenumberofstudents,S,whowatchanmminutevideocanbemodeledbythefunctionS(m)=30-m. Writethealgebraicexpressionthatgivestheaveragegradeontheexamiftherelatedvideoismminutes.

6. TheHobbitsarebuildingawatchtowersotheycanpreparetobattleincasetrollsdecidetoattackthem.OneHobbitwillalwaysbeonthelookoutandtheHobbitswillprepareforbattleassoonasthetrollsarevisible.Thetime,T,inminutes,thattheHobbitshavetopreparefortheattackoftrollskmetersawayisgivenbythefunctionT(k)=k/80Thevisibility,V,inmeters,thattheHobbitshavefromanmmeterwatchtowerisgivenbythefunctionV(m)=50m.WritethealgebraicexpressionthatmodelstheamountoftimetheHobbitshavetoprepareforatrollattackgiventhatthewatchtowerismmeterstall.

75

R.E.A.L.Math1010LabActivity Name:___________________________Lab3:IntrotoInverseOperations

Inverseoperationsareoppositeoperationsthatundoeachother.

1. Writedownsomeexamplesofreallife“operations”thathaveinverse:

Eg.Tyinganduntyingashoelace

a.

b.

2. Notalloperationshaveinverses.Writedownsomeexampleofreallife“operations”thatdonothaveinverses.(Cannotbeundone.)

a.

b.

3. Additionandsubtractionareinverseoperations.Writesomeexamples:

a. 𝑥 + 14 − 14 = 𝑥

b.

c.

4. Multiplicationanddivisionareinverseoperations.Writesomeexamples:

a. 𝑥 ÷ 14 ∙ 14 = 𝑥

b.

c.

5. Whatistheinverseoperationofsquaring?Showanexample.

6. Isthefollowingatruestatement?Whyorwhynot?𝑥) = ( 𝑥))

77

Solvinganalgebraicequationistheprocessofapplyinginverseoperationsto“undo”whatishappeningtox.

7. Explaininwordstheinverseoperationsyouwouldapplytosolvethefollowingequations.Foreachstep,explainwhatoperationneedstobeundoneandhowyouwouldundoit.

a.𝑥 − 79 = 2

b.(𝑥 − 4)) = 25

c.

(𝑥 + 3)12 = 2

8. Let𝑓 𝑥 = 5𝑥 − 1.Findthefunction𝑔(𝑥)sothat𝑓 𝑔 𝑥 = 𝑥.

9. Whatcouldwesayaboutthefunctions𝑓(𝑥)and𝑔(𝑥)?

Step1:

Step2:

Step1:

Step2:

Step1:

Step2:

78

1. AccordingtotheU.S.CensusBureau,in2012thecityofGeorgetown,Texas,asuburbofAustinwasoneofthefastestgrowingcitiesintheentirenation.In2012,thepopulationofGeorgetownwas52,303.

a) Assumingthatthepopulationincreasesataconstantpercentrateof3%,determinethepopulationofGeorgetown(inthousands)in2013.

b) DeterminethepopulationofGeorgetown(inthousands)in2014.

c) Dividethepopulationin2013bythepopulationin2012andrecordthisratio.

d) Dividethepopulationin2014bythepopulationin2013andrecordthisratio.

e) Whatdoyounoticeabouttheratiosinpartscandd?Whatdotheseratiosrepresent?

Linearfunctionsrepresentquantitiesthatchangeataconstantaveragerate(slope).ExponentialfunctionsrepresentquantitiesthatchangeataconstantPERCENTrate.

Populationgrowth,salesandadvertisingtrends,compoundinterest,spreadofdisease,andconcentrationofadruginthebloodareexamplesofquantitiesthatincreaseordecreaseataconstantpercentrate.

2. a)Lettrepresentthenumberofyearssince2012(t=0correspondsto2012).Usetheresultsfromproblem1tocompletethefollowingtable.

t,years(since2012) 0 1 2 3 4 5

P,population(inthousands)

Onceyouknowthegrowthfactor,b,andtheinitialvalue,a,youcanwritetheexponentialequation.Inthissituation,theinitialvalueisthepopulationinthousandsin2012(t=0),andthegrowthfactorisb=1.03.

b) Writetheexponentialequation,𝑃 = 𝑎 ∙ 𝑏& ,forthepopulationof Georgetown,TX.

Name:___________________________R.E.A.L.Math1010SupplementalActivityS14 Exponential Growth and Decay

79

Example2:Recallthatapercentincreasecanbedeterminedindifferentways.

Method1:Acommonmethodforcalculatinganamountafterapercentincreaseistodeterminetheamountofincreaseandaddtotheoriginalamount.

Forexample,wepayanincreasedpercentforcommoditiesinUtahduetoanaverage6.5%salestax.

Todeterminetheincreasedcostofapairof$30jeansduetosalestax,wemightfirstdeterminetheamountofthe6.5%increase,thenaddittothebaseamountof$30.

30(.065)=$1.95(amountoftax)

1.95+30=$31.95

Method2:Anotherwaytodetermineanamountafterapercentincreaseistofirstdeterminethetotalpercentaftertheincrease.

Forexample,the$30priceofthejeanscouldbeconsidered100%ofthebaseamount,sothepercentwewouldpayafterthe6.5%taxincreasewouldbe106.5%.

Thenwewoulddetermine106.5%of30.

30(1.065)=$31.95

Inthiscase,1.065isconsideredtheGROWTHFACTORbecausewemultiplybythisnumbertogettheamountthatresultsfroma6.5%rateincrease.

Thesamereasoningcanbeappliedforpercentdecreases.

Tocalculateanamountafterapercentdecrease,wecanfirstdeterminetheresultingpercentafterthedecrease.

Forexample,a$40coatmightbeonsalefor20%off.$40wouldbe100%oftheprice,sothesalepriceAFTERthe20%decreasewouldbe80%oftheoriginalprice.Thendetermine80%of$40todeterminetheamountyouwouldpayforthecoatonsale.

40(.80)=$32

Inthiscase,.80couldbeconsideredtheDECAYFACTORbecausewemultiplybythisnumbertogettheamountthatresultsfroma20%decrease.

3. Determineifthesalepriceofthecoatdeterminedintheexampleaboveisthesamesalepricefoundbycalculating20%of40andthensubtractingthatamountfrom40.

80

4. a)UseMethod2fromExample2tocalculatetheamountyouwouldpayfora$65onlineorderafterpayinganadditional15%chargeforshipping.

b) WhatistheGROWTHFACTORyouusedinpart(a)?

5. a)Comparedtotraditionalincandescentlightbulbs,energy-efficientlightbulbssuchashalogenincandescents,compactfluorescentlamps(CFLs),andlightemittingdiodes(LEDs)useabout25%-80%lessenergy.

UseMethod2fromExample2tocalculatetheamountyouwouldpayforyourelectricbillifyoucoulddecreaseyourpaymentby25%byupgradingtoenergy-efficientlightbulbs.(Assumethatpreviouselectricbillswere$150.)

b) WhatistheDECAYFACTORyouusedinpart(a)?

c) Whatistherelationshipbetweenthedecayfactorandthepercentdecrease?

6. Whatisthegrowthfactorforagrowthrateof8%?

7. WhatisthegrowthRATEforagrowthfactorof1.054?

8. a)Completethefollowingtable.

t Calculationforpopulation(inthousands)

ExponentialForm P(t),Populationinthousands

0 52.3 52.3(1.03)/

1 (52.3)1.03 52.3(1.03)0

2 (52.3)(1.03)(1.03)

3

b) Usethepatterninthetableinpart(a)tohelpyouwritetheequationforP(t),thepopulationofGeorgetown(inthousands),usingt,thenumberofyearssince2012,astheinputvalue.Howdoesyourresultcomparewiththeequationobtainedinproblem#2b?

81

c) DeterminethegrowthfactorforthefunctionP(t).

d) DeterminethegrowthRATEofthepopulationofGeorgetownwrittenasapercent.

e) DetermineP(8).WhatisthepracticalmeaningofthevalueyoufoundforP(8)?

f) Graphthefunctionwithyourgraphingcalculator.UsethewindowXmin=0,Xmax=100,Ymin=0,Ymax=1000.DetermineP(0)fromyourgraph.WhatisthegraphicalandpracticalmeaningofP(0)?

g) UseyourgraphtopredictthepopulationofGeorgetown,Texas,in2022.Thenusethegraphtodeterminetheyearthatthepopulationwillreach75,000,assumingitcontinuestogrowatthesamerate.RememberthatP(t)isthenumberofthousandsofpeople.

h) PredictthepopulationofGeorgetownin2035?Doyouthinkthisisanaccurateprediction?Whyorwhynot?

9. Assumingthegrowthrateremainsconstant,howlongwillittakethepopulationofGeorgetown,Texas,todoubleits2012population?Explainhowyoureachedthisconclusion.

10. Determinethegrowthfactorandthegrowthrateofthefunctiondefinedby𝑓 𝑥 =250(1.7)𝑥.

11. Youareworkingatawastewatertreatmentfacility.Youarepresentlytreatingwatercontaminatedwith18micrograms(𝜇𝑔)ofpollutantperliter.Yourprocessisdesignedtoremove20%ofthepollutantduringeachtreatment.Yourgoalistoreducethepollutanttolessthan3microgramsperliter.

a) Whatpercentofthepollutantpresentatthestartofatreatmentremainsattheendofthetreatment?

82

b) Theconcentrationofpollutantsis18microgramsperliteratthestartofthefirsttreatment.Usetheresultofpart(a)todeterminetheconcentrationattheendofthefirsttreatment.

c) Completethefollowingtable.Roundtheresultstothenearesttenth.

n,NumberofTreatments 0

C(n)ConcentrationofPollutantatendofnthtreatment

d) Writeanequationfortheconcentration,C(n),ofthepollutantasafunctionofthenumberoftreatments,n.

12. Determinethedecayfactorandthedecayrateofthefunctiondefinedby 𝑓 𝑥 = 123(0.43)8 .

13. Ifthedecayrateofafunctionis5%,determinethedecayfactor.

14. Ifthedecayrateis2.5%,whatisthedecayfactor?

Congratulations!Youhaveinherited$20,000!Yourgrandparentssuggestthatyouusehalfoftheinheritancetostartaretirementfund.Yourgrandfatherclaimsthataninvestmentof$10,000couldgrowtooverhalfamilliondollarsbythetimeyouretire.Youareintriguedbythisstatementanddecidetoinvestigatewhetherthiscanhappen.

15. a)Supposethe$10,000isdepositedinabankat3.5%annualinterest.Useagrowthfactortofindtheamountintheaccountafteroneyear.

b) Usingtheamountintheaccountafteroneyearasthestartingamount, computetheamountintheaccountafterthe2ndyear.(Usethegrowth factoragain.)

83

c) Writeanexponentialequationusingthestartingvalueintheaccountandthegrowthfactor.

Thisisanexampleofcompoundedinterestbecauseduringthe2ndyearandanyyearstofollow,interestwillbeearnedofftheinterestthatwasearnedpreviously.Inthisexample,theinterestiscompoundedannuallybutinterestedcanbecompoundedatotherfixedintervals.Oftenitiscompoundedquarterly(4timesayear),monthly(12timesayear),ordaily(365timesayear).

Ifinterestiscompounded,thecurrentbalanceisgivenbytheformula

𝐴 = 𝑃(1 +𝑟𝑛)

=&

whereAisthecurrentbalance,orcompoundamountintheaccount,Pistheprinciple(theoriginalamountdeposited),ristheannualinterestrate(indecimalform)nisthenumberoftimesperyearthatinterestiscompounded,andtisthetimeinyearsthemoneyhasbeeninvested.Thegivenformulaiscalledthecompoundinterestformula.

16. Howdoesthecompoundinterestformulacomparetotheequationyoufoundinproblem#15c?

17. Determinehowmuchmoneyyouwouldhaveinyouraccountifyouinvestedyour$10,000atthesameannualinterestrateof3.5%buttheinterestiscompoundeddailyinsteadofyearly.Howdoesthisamountcomparetotheamountyouwouldhaveafter2yearsofcompoundingannually(problem#15b)?

Compoundingmoreoftenatthesameratewillyieldagreateramountoveragiventime.Infact,youcouldcalculatetheamountwheninterestiscompoundedeveryhour,minute,orevensecond.However,compoundingmorefrequentlythaneveryhourdoesn’tincreasethebalanceverymuchbecausethegrowthfactordoesn’tchangemuchasngetslargerandlarger.

84

Soforcompoundingthatoccursmoreoftenthandailycompounding,bankswilluseadifferentformulaforwhatiscalled“continuous”compounding:

𝐴 = 𝑃𝑒?&whereAisthecurrentamount,orbalance,intheaccount;Pistheprincipal;ristheannualinterestrate(annualpercentageindecimalform)tisthetimeinyearsthatyourmoneyhasbeeninvested;andeisthebaseofthecontinuouslycompoundedexponentialfunction(eiscalledEuler’sconstantandisapproximately2.718)

18. Determinehowmuchmoneyyouwouldhaveinyouraccountafter2yearsifyouinvestedyour$10,000atthesameannualinterestrateof3.5%compoundedcontinuously.Howdoesthiscomparetotheamountsin#15b(compoundedannually)and#17(compoundeddaily)?

19. Historically,investmentsinthestockmarkethaveyieldedanaveragerateof11.7%peryear.Supposeyouinvest$10000inanaccountat11%annualinterestratethatcompoundscontinuously.

a) Usetheappropriateformulatodeterminethebalanceafter35years.

b) Whatisthebalanceafter40years?

c) Yourgrandfatherclaimedthat$10,000couldgrowtomorethanhalfamilliondollarsbythetimeyouretire(in40years).Isyourgrandfathercorrectinhisclaim?

85

AGeneralFormulaforContinuousGrowth

Consideringtheequationforcontinuousgrowthofmoneyinabankaccount(𝐴 = 𝑃𝑒?&),amoregeneralformofthisequationisusedforothertypesofcontinuousgrowth.Therearemanyothersituationsinwhichgrowthoccurscontinuouslyandnotjustyearly,monthly,ordaily.Populationgrowthisagoodexampleofcontinuousgrowthbecausebabiesareborneverysecondofeveryday,notjustonamonthlyoryearlyschedule.

Whereas𝐴 = 𝑃𝑒?& isusedwhendealingwiththecontinuousgrowthofmoney,amoregeneralformulaforcontinuousgrowthordecayis:𝑦 = 𝑎𝑒A&whereAhasbeenreplacedwithy,theoutput;Phasbeenreplacedwitha,theinitialvalue;Andrhasbeenreplacedwithk,thecontinuousgrowthordecayrate.

Thesecontinuousgrowthordecayequationsarecloselyrelatedtobasicexponentialequations.

Forexample:Theequation𝑦 = 42 1.23 & canbewrittenasthecontinuousgrowthequation𝑦 = 42𝑒.B/C&.Thedifferencebetweenthetwoequationsisthatthefirstreflectsthegrowthatarateof23%overanumberofyears,t,inwhichthegrowthincreasesonceeachyear.(Iftrepresentedhoursinsteadofyearsthenthepercentgrowthwouldoccuronceperhour,etc.)Thesecondone,thecontinuousgrowthequation,reflectscontinuous,ongoinggrowththroughouttheyearsatacontinuousgrowthrateof20.7%.Bothequationswillgivethesameapproximatevalueforyforthesamenumberofyears.Noticethattherateforcontinuousgrowthisslightlylessthentherateforannualgrowth.Thecontinuousgrowthrate,k,thatcorrespondstoagrowthfactor,b,canbefoundbysolvingforkintheequation𝑏 = 𝑒A .Becausethevalueofk=.207representsacontinuousgrowthrate,kisapositivenumber.

Theequation𝑦 = 35 . 97 & ,hasadecayrateof1-.97=.03or3%;thisequationcanbewrittenasthecontinuousexponentialdecayequation,𝑦 = 35𝑒F./G& .Noticethatkisnegativewhenrepresentingarateofdecay.

Identifythegivenexponentialfunctionsasincreasingordecreasing.Ineachcase,givetheinitialvalueandrateofincreaseordecrease.

a) 𝑅 𝑡 = 25 1.098 & c) 𝑓(𝑥) = 95.2𝑒F./K8

b) 𝑆 = 3025 0.72 & d) 𝐵 = 0.59𝑒./CN8

86

R.E.A.L.Math1010SupplementalActivity Name:___________________________S15IntroductiontoLogarithms

1. Completethetablefor𝑓(𝑥) = 10(

x f(x)0

1

2

3

4

2. Withacalculator,completethetablefor𝑔(𝑥) = log 𝑥

x g(x)1

10

100

1000

10000

3.a) Whatdoyounoticeaboutthetwotables?

b) Whatistherelationshipbetweenthetwoequations?

c) Alogarithmicfunctionis_____________________________________________.

4. Whatdowecallthepositionofthenumber10intheequationinexercise1?Whatishappeningtothe10inexercise1?

87

5. Whatdoes10havetodowiththeequationinexercise2?

6. Writetheinverseofthefollowingfunctionsandequations.

a) 𝑓 𝑥 = 2(

b) 𝑦 = log/ 𝑥

c) 𝑦 = 7(

d) 𝑓(𝑥) = log12 𝑥

e) 𝑦 = log3 𝑥

f) 𝑦 = 𝑏(

7.Lookatthetableinexercise2.a) Describetherelationshipbetweenthebase(10),thex,andtheyintheequation.(It

mighthelptowritein10asthebaseforthisexercise.)

b) Canyouwriteageneralrulethatexplainswhatalogarithmis?

8. Useyourgeneralruleofalogarithmtocalculatetheselogswithoutusingacalculator.

NOTE:Alllogarithmshaveabase.Thebaseiswrittenasasubscriptlikethis:𝑦 = log/ 𝑥andthisequationisread“yequalslogbase3ofx.”Whennobaseisindicatedweunderstandittobebase10.Itiscalledthe“CommonLogarithm”andisthebasethattheLOGbuttononthecalculatoruses.Exponentialandlogarithmicfunctionsthatareinversesofeachotherhavethesamebase.

88

a) log/ 9

b) log2 8

c) log7 64

d) log: 5

9. Usingthelogsandyouranswersfromexercise8,writeeachloganditsanswerinitsequivalentexponentialformat.Forexample:

a) log/ 9 = 2 and 32 = 9

b)

c)

d)

e)

f)

g)

h)

10. Dothetwoformatsusedinexercise9representinverserelationshipsorthesamerelationship?Why?

e) log= 1

f) log212

g) log100

h) log: 5/

89

11. Let’sdevelopasecond“definition”ofalogarithm.a) Inthelogarithmicformof9a,whatisthevalueofthelogarithm?

b) Intheexponentialformof9a,whatisthevalueoftheexponent?

c) Doesthisrelationshipholdtrueforallthelogsandexponentsinexercise9?

d) Therefore,wecanalsosayalogarithmis_________________________

Thiscan’tbeemphasizedenough:Alogarithmisanexponent.

Thefollowingexpressiondescribestheexponentonthebaseof2thatgives8.log2 8

Thelogexpressionisequalto3because2/ = 8

12. Whatistheexponentthateachexpressionisdescribing?

a) log1> 100

b) log? 64

c) log/ 27

d) log2 16

90

R.E.A.L. Math 1010 Supplemental Activity Name: ___________________________

Lab 4 Exponential and Logarithmic Graphs

(Adapted from MIA Instructor Resources)

1. Make a table of values for each of the

following functions and graph all of

them on the coordinate plane to the

right.

a. (x)f = 2x

b. (x)g = ex

c. (x)h = 4x

2. As x increases, what happens to y ?

3. As x decreases, what happens to y ?

4. Will the value of y ever be equal to 0? Why or why not?

5. State the domain and range of each of the functions.

6. How are the graphs

a. Similar?

b. Different?

91




right.

a. (x)f = ( )21 x

b. (x)g = ( )41 x

c. (x) )h = ( 54 x

8. As x increases, what happens to y ?

9. As x decreases, what happens to y ?

10. Will the value of y ever be equal to 0? Why or why not?

11. State the domain and range of each of the functions.

12. How are the graphs

a. Similar?

b. Different?

13. What point do all the graphs (#1 and #7) have in common? Explain why.

92




right.

a. (x)f = x

b. (x)g = 2x

c. (x) xh = log2

15. Fold your paper along the line . What do you observe about the graphs of the(x)f = xother two functions?

16. Write the inverse function for y = 4x

17. Write the inverse function for (x) xh = log3

93

Name:___________________________R.E.A.L.Math1010SupplementalActivityS16FactoringtoSimplifyRationalExpressions

Abasketballplayerneedstoknowmultipleskillssuchasdribbling,passing,andshootinginordertoplayabasketballgame.Similarly,therearemanyskillsneededtoplaythegameofalgebra–solvingequationsforunknowns.Factoringisoneofthoseskills.

ReviewofFactoring.

Thenumber15canbefactoredintothenumbers3times5.Thisfactorizationcouldberepresentedasanarrayofitems,suchas3rowsofdeskswith5desksineachrow,orasthedimensionsofarectangle3feetby5feet.Therectanglebelowis3unitswideby5unitstallwithanoverallareaof15units.

Similarly,apolynomialisanalgebraicrepresentationofsinglenumber,dependingonwhatvalueisgiventothevariable.Forexample,𝑥" + 𝑥$ − 𝑥 − 4isthenumber6when𝑥 = 2.Thispropertyimpliesthatpolynomialscanbefactored.Somepolynomialscanbefactored.Andtheycanalsoberepresentedinarectangle.

Consider,𝑥$ + 5𝑥 + 6.Letthisshapehavedimensionsof𝑥unitswideand1unittall.

Thatmeansthisshapehasthedimensionsof𝑥 ⋅ 𝑥or𝑥$

Andthisshapehasthedimensionof1by1.

95

Therefore,thefollowingrepresents𝑥$ + 5𝑥 + 6

Andcanbearrangedinthefollowingrectangletoshowthedimensionsoftherectangleasthefactors 𝑥 + 2 and 𝑥 + 3 .

Labelthedimensionsofthisrectangle.

Furtherinvestigationoffactoringwithvisualscanbeexploredseparatelyfromthisexercise.Seefinalpageforapaperalgebratilespattern.

96

FactoringPractice:

1. 4𝑧$ − 6𝑦𝑧

2. 𝑚$ − 64

3. 𝑧$ + 9𝑧 + 14

4. 𝑥$ + 8𝑥 + 12

5. 𝑦$ − 5𝑥 − 6

Thefollowingisasummaryofhowtoapproachafactoringexercise.

1. Always,firstlookforacommonfactorandfactoritout.2. Counttheterms:

• Twotermsmaybethesumordifferenceofsquares.Thesumcannotbefactored,thedifferenceisfactoredas𝑎$ − 𝑏$ = (𝑎 + 𝑏)(𝑎 − 𝑏).

• Threetermsmaybeaperfectsquaretrinomialsuchthat𝑎$ ± 2𝑎𝑏 + 𝑏$ =(𝑎 ± 𝑏)$,orfactorusingtrialanderror.

• Fourtermsmaybefactoredbygrouping.3. Alwaysfactorcompletely.

97

SimplifyingRationalExpressions

Arationalexpressionissimplifiedwhentherearenomultiplicativeidentitypairswithintheexpression.

6. Followthesestepsexactlytosimplifythefollowingrationalexpressions:1- factorthenumeratorsanddenominators2- writethemultiplicativeinversepairseparatelyandwritea“BigOne”aroundeachone3- writethemultiplicativeinversepairasthemultiplicativeidentity(1)4- writetheequivalentexpressionwithoutthe1.

Example1: :;<=:>?@:;>A:<?B

= (:>$)(:<A)(:>=)(:>$)

= (:>$)(:>$)

∙ :<A:>=

= 1 ∙ :<A:>=

= :<A:>=

a):;<A:<?$:;<=:<D

b)E;>$==E<E;

c)$:;>$::;>?

d)$F;>$F>GF;>HF<$B

e) G:I>?D:;

$:I>$:;>@:

98

PaperAlgebraTiles

99

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Name___________________________________

2) 5v3 − 4v2 + 25v − 20

4) 18x3 + 6x2 − 15x − 5

6) r4 − 4r2

8) v2 + 12v + 27

10) 6x2 − 18x − 168

12) x2 − 5x + 4

14) b2 + 2b + 1

16) x2 − 12x + 20

18) m4 − 25

20) 3m4 − 5m2 − 28

22) 27x7 + 6x5 + 21x3

24) 6x4 + 31x2 + 18

26) 125x3 − 8

1010 REAL Classwork

S16 Factoring ReviewFactor each completely.

1) 24x3 + 56x2 − 3x − 7

3) 5k3 + 8k2 − 25k − 40 5)

x2 + 14x + 49

7) 5n2 + 30n + 25

9) a3 + 13a2 + 36a

11) n3 − 9n2 + 8n

13) 4 p3 + 64 p2 + 252 p

15) x3 − 36x

17) x4 − 5x2

19) x4 − 4x2 − 5

21) 25u6 − 65u4 + 36u2 23)

30u6 + 117u4 + 105u2 25)

250 − 128u3

27) 250a3 − 2

28) −81a3 + 24

100

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Answers to Factoring Review (ID: 1)1) (8x2 − 1)(3x + 7) 2) (v2 + 5)(5v − 4) 3) (k2 − 5)(5k + 8) 4) (6x2 − 5)(3x + 1)5) (x + 7)2 6) r2(r − 2)(r + 2) 7) 5(n + 5)(n + 1) 8) (v + 9)(v + 3)9) a(a + 9)(a + 4) 10) 6(x − 7)(x + 4) 11) n(n − 8)(n − 1) 12) (x − 1)(x − 4)13) 4p(p + 9)(p + 7) 14) (b + 1)2 15) x(x − 6)(x + 6) 16) (x − 10)(x − 2)17) x2(x2 − 5) 18) (m2 + 5)(m2 − 5) 19) (x2 − 5)(x2 + 1)20) (3m2 + 7)(m − 2)(m + 2) 21) u2(5u2 − 9)(5u2 − 4) 22) 3x3(9x4 + 2x2 + 7)23) 3u2(5u2 + 7)(2u2 + 5) 24) (3x2 + 2)(2x2 + 9) 25) 2(5 − 4u)(25 + 20u + 16u2)26) (5x − 2)(25x2 + 10x + 4) 27) 2(5a − 1)(25a2 + 5a + 1) 28) 3(−3a + 2)(9a2 + 6a + 4)

101

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Worksheet by Kuta Software LLC-1-

Simplify each expression by reducing BEFORE multiplying.

1) 9

7 ⋅

3

82)

12

3 ⋅

12

10

3) 4

6 ⋅

7

34)

6

6

7

12

5)

3

4

3

10

6)

6

11

10

6

Simplify each expression by reducing BEFORE multiplying. Then state any values that willmake any part of the expression undefined- these values must be excluded.

7) 3

13k÷

9

15k8)

15x14

⋅ 13

18

9) 14

15x3 ÷2

1810)

2v3

19v ⋅

9

2

Name:___________________________R.E.A.L.Math1010SupplementalActivityS17 Multiply and Divide Rational Expressions Classwork

103

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11) 12

11÷

11m3

17m3 12) 18k2

13k3 ⋅11k3

7

Simplify each expression by factoring first, reducing, then multiplying. Then state any value(s)that will make make any part of the expression undefined- these values must be excluded.

13) 6

5x2 ÷x + 4

5x3 + 20x2 14) 10x − 10

10÷

9

2x

15) a2 + 11a + 10

a − 5 ⋅

1

a + 1016)

m + 9

4m + 36÷

1

4m2 + 4m

17) b2 − b − 56

b − 2 ⋅

b − 2

b2 + 9b + 1418)

1

x + 9÷

x + 2

x2 + 17x + 72

19) 5x + 1

20x + 4÷

1

4x20) (3p − 3) ⋅ 7

21 − 18p − 3p2

21) 5

35k − 10÷

5k2

42k − 1222) (5x2 − 38x + 21) ÷

5x2 + 12x − 9

x + 8

23) 4r

14r + 7 ⋅

14r2 − 9r − 8

7r − 824) 5x ⋅ 3x

3x2 + 24x

104

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Answers to Multiplying/Dividing Rational Expressions (ID: 1)

1) 27

562)

24

53)

14

94)

12

7

5) 5

26)

18

557)

5

13; {0} 8)

65x84

; None

9) 42

5x3 ; {0} 10) 9v2

19; {0} 11)

204

121; {0} 12)

198k2

91; {0}

13) 6; {0, −4}14)

2x(x − 1)9

; {0} 15) a + 1

a − 5; {5, −10}

16) m(m + 1); {−9, 0, −1}17)

b − 8

b + 2; {2, −7, −2} 18)

x + 8

x + 2; {−9, −8, −2}

19) x; {− 1

5, 0} 20) −

7

7 + p; {1, −7} 21)

6

5k2 ; {2

7, 0}

22) (x + 8)(x − 7)

x + 3; {−8,

3

5, −3} 23)

4r7

; {− 1

2,

8

7} 24)5xx + 8

; {0, −8}

105

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1) 3

2 +

2

32)

5

12 +

1

2

3) 3

5 +

4

154)

6

7 +

4

5

5) 8

15 −

1

36)

11

12 −

3

8

7) 6

7 −

2

68)

3

4 −

2

3

Simplify each expression using the same procedures used to simplify the fractions above.

9) x + 3

x2 − x − 6 +

x + 1

x2 − x − 610)

6

9b2 − 54b −

5b − 1

9b2 − 54b

11) a − 6

2a − 2 −

a + 3

2a − 212)

x + 3

x2 − x − 2 +

x + 1

x2 − x − 2

Name:___________________________R.E.A.L.Math1010SupplementalActivityS18 Add and Subtract Rational Expressions ClassworkSimplify each expression without a calculator.

107

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13) 4

4x2 − 12x −

3

214) 5r − 4

4r + 20

15) 3k

3k − 15 −

4k3

16) 6

x − 2 +

5xx − 3

17) 2r − 2

3r2 + 18r +

4

2r18)

3

n + 5 +

6nn + 3

19) 8

x − 5 −

7

x + 720)

6

2 +

3

3m2 − 8m − 3

21) 3bb − 5

−3b

3b + 722)

3k − 7

3k2 + 5k − 28 +

7

4

23) 2x + x − 3

x2 − 3x − 1824)

4

b + 3 +

5

5b + 15

108

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Answers to Adding and Subtracting Rational Expressions (ID: 1)

1) 13

62)

11

123)

13

154)

58

35

5) 1

56)

13

247)

11

218)

1

12

9) 2

x − 310)

7 − 5b9b2 − 54b

11) −9

2a − 212)

2x + 4

x2 − x − 2

13) 2 − 3x2 + 9x

2x(x − 3) 14) 5r2 + 25r − 1

r + 515)

23k − 4k2

3(k − 5) 16) −4x − 18 + 5x2

(x − 3)(x − 2)17)

8r + 34

3r(r + 6) 18) 33n + 9 + 6n2

(n + 5)(n + 3) 19) x + 91

(x − 5)(x + 7) 20) 9m2 − 24m − 6(m − 3)(3m + 1)

21) 6b2 + 36b

(b − 5)(3b + 7) 22) 32 + 7k4(k + 4) 23)

2x3 − 6x2 − 35x − 3(x − 6)(x + 3)

24) 5

b + 3

109

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Solve each equation. Remember to check for extraneous solutions.

1) 3

4n −n + 1

n =

1

2n2) 1 =

3r + 3

2r +

4r − 12

r

3) 4

a =a − 4

a −

1

a4)

1

p −

3p + 4

p2 = p + 2

p2

5) 1

4p =

1

p2 +1

2p6)

4p − 1

6p2 − 5

6p2 =p + 5

3p2

7) a − 3

5a2 + 1

5a2 =1

a8)

4k − 20

3k2 = 1

3k +

1

3k2

9) 1 =7v + 49

8v + 1 −

1

8v + 110)

4

5k + 3 =

1

4 −

1

20k + 12

11) 4

n2 + 8n =

1

n2 + 8n +

1

n12)

k − 3

4k + 8 = 1 +

k − 1

k + 2

13) 6n + 30

n2 − 7n + 1 =

n2 − 36

n2 − 7n14)

8

a =

7a2 − 35a − 98

a2 + 3a +

8

a2 + 3a

15) v − 6

v +

1

v + 7 =v + 3

v + 716)

8

5v =v + 4

5v+ v − 4

Name:___________________________R.E.A.L.Math1010SupplementalActivityS19 Solve Rational Equations

111

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Answers to S19 Solve Rational Equations (ID: 1)

1) {− 3

4} 2) {7

3} 3) {9} 4) {−2}

5) {−4} 6) {8}7) {− 1

2} 8) {7}

9) {47}10) {14

5 } 11) {−5} 12) {−1}

13) {66}14) {57

7, −2} 15) {−42}

16) {4, −1

5}

112

Review: Rational Expressions and Rational Equations

1)If the rational expression

!"#$%

&'#( represents the area of a rectangle and

&")$*

') represents

the length of the rectangle, what rational expression represents the width?

2) In the given problem +,($.+),($01)

÷ ?($01)($04)

= ,($04)($0+)

, what must be the factors that are represented by the question mark?

3) Explain how to subtract rational expressions with different denominators.

4) What are two possible LCDs which could be used for the sum 1".6

+ ,6."

?

5) If one form of the correct answer of a sum of two rational expressions is .!8.!

, what would be an equivalent answer using the denominator 9 − 𝑘 ?

6) Give a simple reduced fraction that is equivalent to the complex fraction

<=0

<%

<<=0

<>?

.

7) Explain the difference between adding rational expressions and solving rational equations.

8) Which of the following complex fractions is equivalent to

@.A>+.#=

?

A) .@0A>.+0#=

B) [email protected]>.+.#=

C) .@0A>+.#=

D) @0A>+0#=

9) Without multiplying by the least common denominator and solving, explain why the

rational equation "

".+= +

".+has no solution. (Hint: Examine both numerators and

denominators carefully.)

10) True or false? The LCD of the two fractions @B and

@C is 𝑎𝑐 if the greatest common factor

of a and c is not 1. Explain your answer.

113

11) True or false? If a fraction with denominator 𝑥 + 9 must be written as an equivalentfraction with denominator (𝑥 + 9)&then the original fraction must be multiplied by 𝑥 + 9in both the numerator and the denominator.

12) True or false? If a fraction with denominator must be written as an equivalent fraction with denominator (𝑥 + 1)+ then the original fraction must be multiplied by in both the numerator and the denominator.

13) If (7x - 6)(4x - 3) is the LCD of two fractions, is 6 − 7𝑥 3 − 4𝑥 also acceptable as an LCD?Why or why not?

Solve the equation algebraically. Round the answer to three decimal places whenever necessary. 14) &

"0++ 10 = 15

15) M.@"@0O.&"

= 15.5

16) +4"

= 4 − @"

17) 1"0@

= 2

Multiply or divide as indicated. Write the answer in lowest terms. 18) 4𝑥& − 25

𝑥& − 4 ÷2𝑥 − 5𝑥 − 2

19) 𝑘& + 9𝑘 + 20𝑘& + 11𝑘 + 28 ∙

𝑘& + 7𝑘𝑘& + 3𝑘 − 10

20) &U?.+U.!+U?0&U.@

∙ +U?0@MU.,

U?0+U.@6

Add or subtract. Write the answer in lowest terms. 21) 4

𝑦& − 3𝑦 + 2 +6

𝑦& − 1

22) 𝑥𝑥& − 16 −

4𝑥& + 5𝑥 + 4

23) 57 − 𝑦 +

6𝑦 − 7

24) 6𝑥 + 4 +

1𝑥

114

25) Suppose that the concentration of a particular drug in the bloodstream, measured in

milligrams per liter, can be modeled by the function 𝐶 = @+U+U?0&.4

where t is the numberof minutes after injection of the drug. What is the concentration of the drug 10 minutes after injection. Round the answer to three decimal places whenever necessary.

26) A formula for electric circuits is @X= @

X<+ @

X? . If 𝑅@ = 10 ohms and 𝑅& = 13 ohms, find R.

Round the answer to three decimal places.

27) A formula for the focal length of a lens is f = BZZ0B

Calculate f (the focal length) for a = 5 cm and b = 20 cm. Round the answer to one decimal place.

Simplify the complex fractions. 28) 1

7𝑥 −14𝑥

18𝑥 +

12𝑥

29) 9 + 3𝑥𝑥4 +

112

30) 53𝑥 − 1 − 55

3𝑥 − 1 + 5

115

Answer Key

1) !"$?

1'(2) (y - 3)(y + 3)3) Answers will vary. On e possible explanation: First, find the LCD and rewrite each expression

with the LCD. Subtract the numerators (the LCD is the denominator of the difference). Last,write the rational expression in lowest terms.

4) Answers will vary. One possible solution: x - 8, 8 - x.5) !

!.86) &,

47) When adding rational expressions, we use the least common denominator to write an

expression equivalent to the sum of the given expressions. We do not clear fractions whenadding rational expressions. When solving rational equations, we use the least commondenominator to clear fractions and then proceed to find the value(s) of the variable for which theequation is true.

8) A9) Since the denominators are the same, the numerators must be the same. Then ; but this

value of x makes the denominators 0, so the equation has no solution. 10) False11) True12) False13) Yes. Explanations will vary.14) -2.615) 3.87516) 917) 118) &"04

"0&

19) 88.&

20) &U0+U0@

21) @O$.6($.@)($0@)($.&)

22) "?.+"0@,(".1)("01)("0@)

23)@

M.$

24) M"01"("01)

25) 0.4 mg/liter

26) 5.652 ohms

27) 4.0 cm

28) − ,+4

29) +,"

30) &.+"+"

116

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1)

5

3

3

2

2)

5

9

1

4

3)

1

4 −

1

8

2

5

4)

3

16 −

1

2

9

5)

25

9 −

1

3

1

5 −

9

4

6)

3

25 +

6

5

1

5 +

5

9

7)

1

x5x + 25

x2

8)

4

x2

x

9)

1

2

2

3 −

2

x

10) x

1

5 −

5

x2

11)

16

3 −

4

25

x4

+ 16

5

12)

12

u +u2

5

3

u2 −u4

13)

4

3u − 9 −

4

u − 3

4

u − 3 +u2

9

14)

4x3

+ 4

2x − 5

2x − 5

16 −

9

2x − 5

Name:___________________________R.E.A.L.Math1010SupplementalActivityS20 Complex Numbers Classwork

Simplify each expression.

117

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Answers to Complex Fractions (ID: 1)

1) 10

92)

20

93)

5

164) −

5

144

5) −440

3696)

297

1707)

x5x + 25

8) 2

9) 3x

4x − 1210)

5x3

x2 − 2511)

1552

75x + 96012)

240u + 4u4

60 − 5u3

13) −24

36 + u3 − 3u2 14) 128x2 − 320x + 192

12x2 − 60x − 357

118

Name:___________________________R.E.A.L.Math1010SupplementalActivitySwimmingPoolTask

1. DavidandAlice,whoareneighbors,arefillinganinflatableswimmingpoolinAlice’sbackyard.Tospeedthingsup,theyusehosesconnectedtobothoftheirhouses.Whilewaitingforthepooltofinishinflating,Davidnoticeshecanfill3smallbucketsin4minutes.Alicenoticesshecanfill2smallbucketsin5minutes.Iftheyusethehosesfromboththeirhouses,whatisthecombinedrateforfillingthepool?

2. Inanotherneighborhood,SusanaandEmilyarealsousinghosesfromboththeirhosestofilltheirswimmingpool.Susananoticesthehoseconnectedtoherhousewillfill3bucketsintminutes.IttakesEmily2minuteslongerthanSusanatofillonly2buckets.Iftheyusethehosesfromboththeirhouses,whatisthecombinedrateforfillingthepool?

3. Explainhowfindingthefirstrate(DavidandAlice)issimilartofindingthesumofthesecondrate(SusanaandEmily).

4. Modelanauthenticsituationwithalgebraicrationalexpressions.Youmaywanttoincludeoperationsofaddition,subtraction,multiplication,orsimplifying.

119

—Lab 5

SolveRationalEquationsFollowthesestepstohelpyousolveeachproblem.

1. _____________________________________________________________________

2. _____________________________________________________________________

3. _____________________________________________________________________

4. _____________________________________________________________________

1. xx851 -

=+

2. 23

4312=+

n

3. 21

12

-=

+ xx

4. 4

321

21

2 -=

-+

+ aaa

R.E.A.L.Math1010SupplementalActivity Name:___________________________Lab6RationalExpressionsvs.RationalEquations

121

5. 6

723

12 -+-

=-

+++

xxx

xx

xx

6. 25

1051

51

2 -=

-+

+ aaa

Add&SubtractRationalExpressionsFollowthesestepstosimplifyeachexpression.

1. _____________________________________________________________________

2. _____________________________________________________________________

3. _____________________________________________________________________

4. _____________________________________________________________________

1. !"+ 5

2. !%&+ '

(

122

3. %")!

+ !"+%

4. !,)%

+ !,+%

5. ")!")'

+ ""+%

6. !,+-

+ !-+,

7. Howissolvingarationalequationdifferentfromsimplifyingarationalexpression?

123

R.E.A.L. Math 1010 Supplemental Activity Name: ___________________________

S21 Graphing Transformations of Functions

1. Use a table of values to graph all of

the following functions on the

coordinate plane to the right. Clearly

label each graph. Also list the vertex

of each function.

a. (x)f = x2

Vertex:

b. (x)g = x2 + 3Vertex:

c. (x)h = x2 3Vertex:

2. How does the graph of compare to the graph of ? How do their vertices(x)g (x)fcompare? Be specific.

How does the graph of compare to the graph of How do their vertices(x)h (x)?fcompare? Be specific.

3. Make a generalization about the transformation of a function’s graph when a constant is

added to or subtracted from the term.x2

125

4. Graph all of the following functions on

the coordinate plane to the right.

Clearly label each graph. Also list the

vertex of each function.

a. (x)f = x2

Vertex:

b. (x) x )g = ( 3 2

Vertex:

c. (x) x )h = ( + 3 2

Vertex:

5. How does the graph of compare to the graph of ? How do their vertices(x)g (x)fcompare? Be specific.

How does the graph of compare to the graph of How do their vertices(x)h (x)?fcompare? Be specific.

6. Make a generalization about the transformation of a function’s graph when a constant is

added to or subtracted from before being squared.x

126

7. Graph both of the following functions on

the coordinate plane to the right. Clearly

label each graph.

a. (x)f = x2

b. (x) x )g = ( 2 2 4

8. How does the graph of compare to(x)gthe graph of Be specific.(x)?f


the coordinate system to the right.

Clearly label each graph.

a. (x)f = x

b. (x)g = x + 2

c. (x)h = x 2

10. Are these quadratic, exponential, or

linear functions?

11. How does the graph of compare to the graph of Be specific.(x)g (x)?f

How does the graph of compare to the graph of Be specific.(x)h (x)?f

12. Do your generalizations made in #3 still hold true for these functions?

127


the coordinate plane.

a.

b.

c.

14. Are these quadratic, exponential, or linear

functions?

15. Do your previous generalizations hold true for these functions?

128

Parent Functions Graphs: Every function can be classified as a member of a “family.” The

“parent” of a function family is the most basic representation of the family. Below are graphs of

some basic parent functions.

In this course we are studying Constant, Linear, Absolute Value, Quadratic, Exponential and

Logarithmic. Each can be transformed.

129

14. Describe each transformation of the following functions from their parent function.

Parent Function Function Description of transformation from parent function

(x)f = x (x)f = x 2 Vertical shift down two units

(x) x 0)p = ( 1 2 7

(x)h = √x + 5

(x)g = x| 2| + 7

(x)f = √x 3+ 2

(x) 8m = x3 1

(x) 3q = 2x 1

(x) og xw = l + 1

(x) x 1)k = ( + 1 3 + 2

(x)z = 2x 5

(x) og (x 2)r = l + 2 6

(x) 8d = x| | 1

130

R.E.A.L.Math1010SupplementalActivity Name:___________________________S22CompletetheSquare

1. Discusswithyourgrouppossiblemathematicalrelationshipsbetweenthefollowing:

162. Whatsinglewordcouldbeusedtodescribebothitems?

3. Inthecontextofthisexploration:a) Whatisthemathematicalprocessusedtoobtaintheitemontheleft?

b) Showthenumericalrepresentationofthisconceptusingtheitemontheright.

4. Thegraypartofthefollowingfigurehasarea𝑥" + 10𝑥

a) Fillinthequestionmarks.

b) Whatistheareaofthewhitesquare?

c) Whatistheareaofthelargesquarecontainingtheentirefigure?

d) Iftheareaofthegraypartis39,whatistheareaofthetotalfigure?Useyouranswertodeterminethevalueofx?

e) Howwouldyourepresentthedimensionsofthefigure?Writeinsimplestform.

f) Writeanalgebraicstatementshowingtheequivalencebetweenthearea(representedbyatrinomial)andthedimensionsofthisfigure.

5. Thegraypartofthefollowingfigurehasarea𝑥" + 20𝑥

131

a) Fillinthequestionmarks.

b) Whatistheareaofthewhitesquare?

c) Whatistheareaofthelargesquarecontainingtheentirefigure?

d) Iftheareaofthegraypartis125,whatistheareaofthetotalfigure?Useyouranswertodeterminethevalueofx?

e) Howwouldyourepresentthedimensionsofthefigure?Writeinsimplestform.

f) Writeanalgebraicstatementshowingtheequivalencebetweenthearea(representedbyatrinomial)andthedimensionsofthisfigure.

6. Thetrinomialsrepresentingtheareasofeachofthefiguresareperfectsquaresbecausetheyareequivalenttothesquareofanexpression(binomials).Writethreemorealgebraicstatementsthatshowtheequivalencebetweenaperfectsquaretrinomialandthesquareofabinomial.

____________________________________________________

____________________________________________________

____________________________________________________

7. Giventhefollowingpairsoffigures:

132

a) Redraweachpairintoasinglesquareandlabelthedimensions.b) Determinetheareaofthespacethatneedstobefilledinordertocompletethesquare.c) Writetheexpressionfortheperfectsquaretrinomialanditsequivalentbinomialsquared.

𝑥" + 12𝑥

𝑥" + 8𝑥

8. Withorwithoutusinganareamodel,completethesquareforthefollowing:

a) 𝑥" + 6𝑥

b) 𝑥" + 14𝑥

c) 𝑥" + 7𝑥

𝑥"12x

𝑥"8x

133

9. Howwouldyouapplytheprincipleofinversetosolvethefollowingequations?

(𝑥 + 5)" = 8 and (𝑥 − 7)" = 64

a) Explaintheprocessinwords:

b) Solve(𝑥 + 5)" = 8

c) Solve(𝑥 − 7)" = 64

ThisprocessisreferredtoasapplyingtheSquareRootPropertytosolvequadraticequations.

10. Combinetheprocessesofcompletingthesquare,factoringtheperfectsquaretrinomialintoabinomialsquared,andusingtheSquareRootPropertytosolvethefollowingequations.

a) 𝑥" + 20𝑥 + 40 = 0 b) 𝑥" + 8𝑥 + 6 = 0

c) 𝑥" − 5𝑥 − 8 = 0 d) 𝑥" − 9𝑥 − 10 = 0

134

R.E.A.L.Math1010SupplementalActivity Name:___________________________S23ZeroProductPropertyandSolvingQuadraticEquationsbyFactoring

StudentTask1:1. Ineachofthefollowingequations,thevariablesrepresentrealnumbers.Assumingeachequationistrue,whatcanyouconcludeaboutthevaluesofthevariables?Explaineachstepinyourreasoning.

a. 2𝑥 + 3 = 0

b. 7𝑥 = 0

c. 7(𝑦 − 5) = 0

d. 𝑎𝑏 = 0

StudentTask2:2. TheZeroProductProperty(ZPP)statesthatiftheproductoftwonumbersiszero,thenatleastoneofthenumbersiszero.(Insymbols,ifab=0,thena=0orb=0.)Wecanusethispropertywhenwesolveequationswhereaproductis0.Foreachequationbelow,usetheZPPtofindallsolutions.Explaineachstepinyourreasoning.

a. 𝑥(13 − 4𝑥) = 0

b. 7(𝑦 + 12) = 0

c. (𝑥 − 19)(𝑥 + 3) = 0

d. (𝑦 − 6)(3𝑥 − 4) = 0

135

StudentTask3:3. FactorandsolvethefollowingquadraticequationsusingtheZeroProductProperty

a. 𝑥3 + 7𝑥 + 12 = 0

b. 2𝑥3 − 9𝑥 + 4 = 0

c. 𝑥3 + 11𝑥 + 20 = 2

d. 𝑥3 + 5𝑥 − 30 = −6

136

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Solve each equation using the square root property. Simplify radicals with imaginary numbers.

1) n2 = −52 2) k2 = −1

3) v2 = −14 4) n2 − 3 = −9

5) a2 − 5 = −8 6) v2 − 4 = −14

Simplify. Give solutions in standard, a + bi, form.

7) (−8i) + (−2 − 5i) 8) (−i) − (4i)

9) (7 − 7i) − (6i) 10) (5 + 6i) + (4i)

11) (−8 − 7i) + (1 + 6i) 12) (3 − 8i) + (−5 − 5i)

13) (4 + 7i) + (4i) − (6i) 14) (−3 − 8i) + (6 + 5i)

15) (8 − 8i)(−1 − 4i) 16) (−4 + 7i)2

17) (8 + 7i)2 18) (5 − 4i)(6 − 3i)

19) −7(−8i)(6 + i) 20) (−7 + 3i)(8 + 6i)

Solve each equation with the quadratic formula. Simplify all radicals as needed and givesolutions as reduced fractions when applicable. Then indicate the number of x-intercepts thatwill be found on the graph of the quadratic.

21) 6x2 − 10x = −9 22) 9a2 + 4 = −4a

23) 2n2 = −9 24) 3p2 − 2p = −7

25) −5p2 = 3 26) 9x2 + 10 = x

Name:___________________________R.E.A.L. Math 1010 Supplemental ActivityS24 Complex Numbers Classwork

137

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Answers to Complex Numbers (ID: 1)

1) {2i 13 , −2i 13} 2) {i, −i} 3) {i 14 , −i 14} 4) {i 6 , −i 6}5) {i 3 , −i 3} 6) {i 10 , −i 10} 7) −2 − 13i 8) −5i9) 7 − 13i 10) 5 + 10i 11) −7 − i 12) −2 − 13i13) 4 + 5i 14) 3 − 3i 15) −40 − 24i 16) −33 − 56i17) 15 + 112i 18) 18 − 39i 19) −56 + 336i 20) −74 − 18i

21) {5 + i 29

6,

5 − i 29

6} No x-intercepts 22) {

−2 + 4i 2

9, −2 − 4i 2

9} No x-intercepts

23) {3i 2

2, −

3i 2

2} No x-intercepts 24) {

11

3±

2 5

3i} (standard form) no x-intercepts

25) {−i 15

5, i 15

5} no x-intercepts

26) {1 + i 359

18,

1 − i 359

18} no x-intercepts when solutions are imaginary

138

r.e.a.l. math 1010 intermediate algebra supplemental student … · 2019. 8. 6. · r.e.a.l. math...

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