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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
4
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
8
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.
10
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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Worksheet by Kuta Software LLC
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
16
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Worksheet by Kuta Software LLC
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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Worksheet by Kuta Software LLC
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.
Whatisthenameoftheexponentrulethatthisdemonstrates?
Howwouldyourgroupexpressthisrulewithwordsonly(nosymbols)?
Asagroup,determinehowtowritethisruleinsymbolicformusingvariablestoshowthatthisrulecanbeusedforbasesandexponentsofanynumber.
PartC:ExponentiatingExponents
1. Asagroup,determinehowtowrite(23)4withouttheuseofexponents.
2. Nowexpresstheresultwithoneexponentonly.
Whatisthenameoftheexponentrulethatthisdemonstrates?
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
28
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
34
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.
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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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Worksheet by Kuta Software LLC
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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Worksheet by Kuta Software LLC
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
44
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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Worksheet by Kuta Software LLC
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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Worksheet by Kuta Software LLC
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
50
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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Worksheet by Kuta Software LLC
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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Worksheet by Kuta Software LLC
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.
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.
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Adding and Subtracting Radicals
54
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.
7. Determine the perimeter of the triangle shown. Simplify as much as possible.
57
8. Determine the perimeter of the triangle shown. Simplify as much as possible.
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?
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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
68
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
76
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.
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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?
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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.)
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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
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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
7. 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 = ( )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
14. 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 = 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
94
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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Worksheet by Kuta Software LLC
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
102
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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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Worksheet by Kuta Software LLC-2-
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
106
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Worksheet by Kuta Software LLC-1-
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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Worksheet by Kuta Software LLC-2-
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
110
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Worksheet by Kuta Software LLC
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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Worksheet by Kuta Software LLC
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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Worksheet by Kuta Software LLC
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
120
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
124
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
9. Graph all of the following functions on
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
13. Graph all of the following functions on
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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Worksheet by Kuta Software LLC
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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Worksheet by Kuta Software LLC
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