ImportantMath125Definitions/Formulas/PropertiesExponentRules(Chapter3)

Let𝑚&𝑛beintegers,and𝑎&𝑏realnumbers.

ProductProperty 𝑎%𝑎& = 𝑎%(&

QuotientProperty 𝑎%

𝑎& = 𝑎%)&

PowertoaPower 𝑎% & = 𝑎%&

ProducttoaPower 𝑎𝑏 & = 𝑎&𝑏&

QuotienttoaPower *+

&= *,

+,for𝑏 ≠ 0

ZeroExponent 𝑎/ = 1for𝑎 ≠ 0

NegativeExponent 𝑎)& = 1*,& 1

*2,= 𝑎&for𝑎 ≠ 0

Multiplying/FactoringBinomials(Chapters3&4)

SquareofaBinomial/PerfectSquareTrinomial 𝑎 ± 𝑏 4 = 𝑎4 ± 2𝑎𝑏 + 𝑏4

MultiplyingConjugates/DifferenceofSquares 𝑎 + 𝑏 𝑎 − 𝑏 = 𝑎4 − 𝑏4

Sum/DifferenceofCubes 𝑎8 ± 𝑏8 = (𝑎 ± 𝑏)(𝑎4 ∓ 𝑎𝑏 + 𝑏4)*

*Thetrinomial𝑎4 ∓ 𝑎𝑏 + 𝑏4isprimeandcannotbefactoredanyfurther

PythagoreanTheorem(Chapter4) Foraright(90°)triangle:

𝑎4 + 𝑏4 = 𝑐4

FormulasRelatingDistance,Rate,&Time(Chapter4)

𝑑 = 𝑟𝑡 ⟹ 𝑡 =𝑑𝑟

FormulaForCombinedWork(Chapter4)Let𝑡1bethetimeittakesforthefirstpersonorgrouptocompleteaparticulartask.Let𝑡4bethetimeittakesforthesecondpersonorgrouptocompletethistask.If𝑡isthetimeittakeswhentheyworktogetheronthetask,then

1𝑡1+1𝑡4=1𝑡

LinearEquations(Chapter6)

SlopeFormula 𝑚 =𝑦4 − 𝑦1𝑥4 − 𝑥1

ParallelSlope 𝑚1 = 𝑚4

PerpendicularSlope 𝑚1 = −1𝑚4

Slope-InterceptForm* 𝑦 = 𝑚𝑥 + 𝑏

Point-SlopeForm 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

StandardForm 𝑎𝑥 + 𝑏𝑦 = 𝑐

HorizontalLine 𝑥 = 𝑘

VerticalLine 𝑦 = 𝑘

*Infunctionnotation,wecanwriteslope-interceptformas𝑓(𝑥) = 𝑚𝑥 + 𝑏.

PropertiesofRadicals(Chapter8)Let𝑚&𝑛bepositiveintegerswhere𝑛 > 1,and𝑎&𝑏benon-negativerealnumbers.

NthRootofaPerfectPower 𝑎&, = 𝑎

RationalExponents 𝑎%/& = 𝑎%, or 𝑎, %

ProductRule 𝑎, 𝑏, = 𝑎𝑏,

QuotientRule *,

+, = *+

, for𝑏 ≠ 0

Distance&MidpointFormulas(Chapter8)

DistanceFormula 𝑑 = 𝑥4 − 𝑥1 4 + 𝑦4 − 𝑦1 4

MidpointFormula 𝑀 =𝑥1 + 𝑥42 ,

𝑦1 + 𝑦42

SquareRootProperty(Chapter9)Isolatethesquaretermtoonesideoftheequation,thenapplythesquareroottobothsides:

𝑥4 = 𝑏 ⟹ 𝑥4 = 𝑏 ⟹𝑥 = ± 𝑏

𝑎𝑥 + 𝑐 4 = 𝑏 ⟹ 𝑎𝑥 + 𝑐 4 = 𝑏 ⟹ 𝑎𝑥 + 𝑐 = ± 𝑏

QuadraticFormula(Chapter9)If𝑎𝑥4 + 𝑏𝑥 + 𝑐 = 0for𝑎 ≠ 0,then

𝑥 =−𝑏 ± 𝑏4 − 4𝑎𝑐

2𝑎

ParabolasinStandardForm(Chapter9)

StandardForm 𝑓 𝑥 = 𝑎𝑥4 + 𝑏𝑥 + 𝑐,𝑎 ≠ 0

AxisofSymmetry 𝑥 = −𝑏2𝑎

Vertex −𝑏2𝑎 , 𝑓 −

𝑏2𝑎

y-intercept Evaluate𝑓 0 = 𝑐

x-intercept(s) Solve𝑓 𝑥 = 0

Weusalltheaboveinformationtographaparabola.

ParabolasinVertexForm(Chapter9)Letℎ&𝑘berealnumbers.

VertexForm 𝑓 𝑥 = 𝑎(𝑥 − ℎ)4 + 𝑘,𝑎 ≠ 0

Vertex (ℎ, 𝑘)

StepstoGraph

1. Graph𝑦 = 𝑎𝑥42. Move𝑦 = 𝑎𝑥4

ℎunitshorizontally&𝑘unitsvertically.

GraphsofParentFunctions(Chapter10)

GraphingFunctionsusingShifting&Reflecting(Chapter10)Wegeneralizegraphingfunctionsbasedusingshiftingandreflectioingstheaboveparentfunctions.

GeneralizingTrandfomations:𝒚 = 𝒂𝒇(𝒙 − 𝒉) + 𝒌Togeneralize,weletℎ&𝑘berealnumbers(likevertexformofaparabola).

Ø Identifytheparentfunction𝑓(𝑥)andmakeatableofitskeypoints.Ø Graph𝑦 = 𝑎𝑓(𝑥).

o If𝑎 = 1,thenthegraphopensupward.o If𝑎 = −1,thenthegraphopensdownward.

Ø Shiftthegraphof𝑦 = 𝑎𝑓(𝑥)horizontallybyhunits,thenverticallybykunits.Ø Theoriginof𝑓 𝑥 isnowshiftedtoitsnewcenter(ℎ, 𝑘).

o Thispointisavertexiftheparentfunctionisasquareorabsolutevaluefunction.

Variation(Chapter10)

Wecreatenewvariationsusingthe3propertiesabove.

AbsoluteValueEquations(Chapter11)

Isolatetheabsolutevaluetermtooneside,thenapplyoneofthefourpropertiesbelow:

EquationsInvolvingAbsoluteValuesLet𝑎isapositiverealnumberand𝑥beanyalgebraicexpression.

• If 𝑥 = 𝑎,then𝑥 = 𝑎or𝑥 = −𝑎.• If 𝑥 = 0,then𝑥 = 0.• If 𝑥 = −𝑎,then𝑥 =∅(nosolution;emptyset)• If 𝑥 = 𝑦 ,then𝑥 = 𝑦or𝑥 = −𝑦

AbsoluteValueInequalities(Chapter11)Isolatetheabsolutevaluetermtooneside,thenapplyoneofthetwopropertiesbelow:

InequalitiesInvolvingAbsoluteValuesLet𝑎isapositiverealnumberand𝑥beanyalgebraicexpression.

• If 𝑥 < 𝑎,then−𝑎 < 𝑥 < 𝑎o <canbeinterchangedwith≤.

• If 𝑥 > 𝑎,then𝑥 < −𝑎or𝑥 > 𝑎o >canbeinterchangedwith≥.

AlgebraofFunctions(Chapter12)

FunctionOperationsLet𝑓(𝑥)and𝑔(𝑥)bedefined,then

• 𝑓 + 𝑔 𝑥 = 𝑓 𝑥 + 𝑔(𝑥)• 𝑓 − 𝑔 𝑥 = 𝑓 𝑥 − 𝑔 𝑥 • 𝑓 ∙ 𝑔 𝑥 = 𝑓 𝑥 𝑔(𝑥)

• YZ

𝑥 = Y [Z([)

for𝑔(𝑥) ≠ 0

FunctionComposition(Chapter12)Let𝑓(𝑥),𝑔 𝑥 , &𝑓(𝑔 𝑥 )bedefined,then

𝑓 ∘ 𝑔 𝑥 = 𝑓(𝑔 𝑥 )

One-to-OneFunctions(Chapter12)

Afunctionisone-to-oneifeach𝑦correspondstoone𝑥.Ø Theabovetellsusaone-to-onefunctionneverhasrepeating𝑦-values.

TheHorizontalLineTestIfeveryhorizontallineintersectsthegraphofafunctionatmostonce,thenthefunctionisone-to-one.

InverseFunctions(Chapter12)

TheInverseFunctionLet𝑓(𝑥)beaone-to-onefunction.Then,

• If𝑓 𝑥 = 𝑦,then𝑥 = 𝑓)1(𝑦).o Thistellsusthatifthepoint 𝑥, 𝑦 isonthegraphof𝑓,then

𝑦, 𝑥 isonthegraphof𝑓)1.Thatistosay,thegraphof𝑓)1isfoundbyreflectingthepoints𝑓abouttheline𝑦 = 𝑥.

Finding&VerifyingFunctions(Chapter12)

StepstoFind𝒇)𝟏(𝒙)Let𝑓(𝑥)beaone-to-onefunction.

1. Replace𝑓 𝑥 with𝑦.2. Interchange𝑥and𝑦.3. Solvefor𝑦.4. Replace𝑦with𝑓(𝑥).

VerifyingInversesTwoone-to-onefunctions,𝑓(𝑥)and𝑔 𝑥 ,areinversesifandonlyif

𝑓 𝑔 𝑥 = 𝑥&𝑔 𝑓 𝑥 = 𝑥.

Logarithms(Chapter12)

For𝑥 > 0,𝑏 > 0,and𝑏 ≠ 1,thelogarithmfunctionofbase𝒃of𝒙ifdefinedby𝑦 = log+𝑥

where𝑏d = 𝑥.

ExponentialForm𝑏d = 𝑥

LogarithmicForm𝑦 = log+𝑥

Tosolvealogarithmicequation,wemakesureitiswritteninlogarithmicform,theconvertittoexponentialform.

ie,loge(𝑥 + 1) =14⟹ 9

gh = 𝑥 + 1 ⟹ 𝑥 + 1 = 9 ⟹ 𝑥 + 1 = 3 ⟹ 𝑥 = 2

PropertiesofLogarithms(Chapter12)

PropertiesofLogarithmsForanyrealnumbers𝑥, 𝑦, &𝑏 > 0with𝑏≠ 1,andanyrealnumberr:

1. ProductRule:log+ 𝑥 ∙ 𝑦 = log+ 𝑥 + log+ 𝑦

2. QuotientRule:log+[d= log+ 𝑦 − log+ 𝑦

3. PowerRule:log+ 𝑥j = 𝑟 log+ 𝑥

Note:IfweuseProperties(1-3)insolvingalogarithmicequation,weMUSTcheckoursolutions.

4. SpecialProperties:a. log+ 1 = 0b. log+ 𝑏 = 1c. log+𝑏[ = 𝑥d. 𝑏klmn[ = 𝑥

5. ChangeofBase:log+ 𝑥 =klm[klm +

𝑜𝑟 kp [kp +

CommonLogarithmlog1/𝑥 = log 𝑥log 10[ = 𝑥10klm [ = 𝑥

Naturallogarithmlogq𝑥 = ln 𝑥ln 𝑒[ = 𝑥𝑒kp [ = 𝑥

SolvingExponentialEquations(Chapter12)Ø Ifthebothsidesoftheexponentialequationcanbewrittenintermsofthesamebase,weusethe

propertybelow:

Uniquenessof𝒃𝒙Foranyrealnumbers𝑥, 𝑦, &𝑏,with𝑏 > 0&𝑏≠ 1,

if𝑏[ = 𝑏d ,then𝑥 = 𝑦.

ie,2[ = 1t⟹ 2[ = 1

4u⟹ 2[ = 2)8 ⟹𝑥 = −3

27[ ∙ 3[(1 = 94 ⟹ 38 [ ∙ 3[(1 = 34 4 ⟹ 38[ ∙ 3[(1 = 3w ⟹ 3w[(1 = 3w ⟹ 4𝑥 + 1 = 4 ⟹𝑥 =34

Ø Ifthebothsidesoftheexponentialequationcannotbewrittenintermsofthesamebase,weusethepropertybelow:

LogarithmicPropertyofEqualityForanypositiverealnumbers𝑥, 𝑦, &𝑏with𝑏≠ 1,

if𝑥 = 𝑦,thenlog+ 𝑥 = log+ 𝑦.

ie,2[ = 5 ⟹ log 2[ = log 5 ⟹ 𝑥 log 2 = log 5 ⟹𝑥 = klmyklm 4

18𝑒[(1 = 5 ⟹ 𝑒[(1 = 15 ⟹ ln 𝑒[(1 = ln 15 ⟹ 𝑥 + 1 = ln 15 ⟹𝑥 = −1 + ln 15.

Disclaimer:Theformulasheetismeantasastudyaidandnotacomprehensivelistorreviewofalltopicsinthiscourse.Youarestillrequiredtolookoveroldexams,quizzes,homework,classwork,andnotesforallwehavecoveredthissemester.Furthermore,theseformulasneedtobecommittedtomemorytosuccessfullycompleteproblemsontheFinalExam.