FACTS YOU MUST KNOW COLD FOR THE REGENTS EXAM 2 review sheet_1.pdf · use your calculator or you...

ALGEBRA II (COMMON CORE)

FACTS YOU MUST KNOW COLD FOR THE REGENTS EXAM

©NYSMathematicsRegentsPreparation||Createdby:TrevorClark

©NYSMathematicsRegentsPreparation||RevisedSpring2017||www.nysmathregentsprep.com

AlgebraII[CommonCore]RegentsExamStudyGuide

FACTORINGTheOrderofFactoring:

GreatestCommonFactor(GCF)

DifferenceofTwoPerfectSquares(DOTS)

Trinomial(TRI)

“AC”Method/EarmuffMethod(AC)

QuadraticFormula(QF)GCF: !" + !$ = !(" + $)DOTS: )* −,* = () + ,)() − ,)TRI: )* − ) + - = () + *)() − .)AC(a≠1): *)* + /0) + /1 )* + /0) + .- () + /*)() + .) () + /*

* )() +.*)

() + -)(*) + .)QF:

Ifallelsefailstofindtherootstoaquadratic,usetheQuadraticFormula:

) = −" ± √"* − 4!$*!

ALGEBRA&FUNCTIONS

OTHERFORMSOFFACTORINGFactorbyGrouping:

). + *)* − .) − -

)*() + *) − .() + *) ()* − .)() + *)FactoringPerfectCubesbySOAP:S–“Same”asthesigninthemiddleoftheoriginalexpression”O–“Opposite”signAP–“AlwaysPositive”

). − 1()). − (*).

() − *)()* + *) + 4)

PerfectCubeFactor SOAPFactor

DIVIDINGPOLYNOMIALSDivisionAlgorithm:567689:85676;<= = >?<@69:@ + A9B!6:89=

5676;<= LongDivisionofPolynomials: SyntheticDivisionofPolynomials:(*)* + C) + -) ÷ () + *) (). + -)* + C) − -) ÷ () + 4)

THEREMAINDERTHEOREMWhenthepolynomialE())isdividedbyabinomialin

theformof() − !),theremainderequalsE(!).4)* + *) − 0() − /)

E(/) = 4(/)* + *(/) − 0 ⇒ /

Theremainderis1!

THEFACTORTHEOREMIfE(!) = HforpolynomialE()),thenabinomialintheformof() − !)mustbeafactorofthepolynomial.

)4 + -). + C)* − -) − 1() + 4)

E(−4) = (−4)4 + -(−4). + C(−4)* − -(−4) − 1E(−4) = *0-+ (−.14) + //* − (−*4) − 1

E(−4) = HTheremainderiszero,therefore() + 4)isafactor!


QUADRATIC:Aquadraticequationisapolynomialequationwithadegreeoftwo(2).

THESTANDARDFORMOFAQUADRTATICEQUATION

Thestandardformofaquadraticisintheformof!)* + ") + $ = H,

wherea,b,andcareconstantswherea≠0.

THEDISCRIMINANTThediscriminantisapartofthequadraticformulawhichallowsmathematicianstoanticipatethenature,orkindsofrootsaparticularquadraticequationwillhave.

"* − 4!$wherea,b,andcareconstants

THEPARTSOFAQUADRATICRoot/Zero/X-Intercept:apointonaquadraticwhereI(J) = 0.ItisapointwherethequadraticintersectstheJ − LJMN.TurningPoint(Vertex):thepointonaquadraticwherethedirectionofthefunctionchanges.AxisofSymmetry:alineofsymmetryintheformofJ = O,whereOisaconstant.ThevalueofOisthesamevalueastheJvalueoftheturningpoint.Focus:apointwhichlies“inside”theparabolaontheaxisofsymmetry.Directrix:alinethatisperpendiculartotheaxisofsymmetry&lies“outside”theparabola.

THESUMOFTHEROOTSOFAQUADRATIC

SumoftheRoots:=/ + =* = −"!

where!and"areconstantsfromaquadraticequationintheformof!)* + ") + $ = H.

THEPRODUCTOFTHEROOTSOFAQUADRATIC

ProductoftheRoots:=/ ⋅ =* = $!

where!and$areconstantsfromaquadraticequationintheformof!)* + ") + $ = H.


FUNCTION:Afunctionisarelationthatconsistsofasetoforderedpairsinwhicheachvalueof)isconnectedtoauniquevalueof,basedontheruleofthefunction.Foreach)value,thereisoneandonlyonecorrespondingvalueof,.Afunctionalsopassestheverticallinetest.DOMAIN:Thelargestsetofelementsavailablefortheindependentvariable,thefirstmemberoftheorderedpair()).

RESTRICTIONSONDOMAIN:1. Fraction:Thedenominatorcannotbezero.

Settheentiredenominatorequaltozeroandsolve.

E()) = ) − 4) + . ; ) ≠ −.

2. Radical:Theradicandcannotbenegative.Settheradicandgreaterthanorequaltozero

andsolve.E()) = √) − 0; ) ≥ 0

3. RadicalintheDenominator:Theradicalcannotbenegativeandthedenominator

cannotbezero.Settheradicandgreaterthanzeroandsolve.

E()) = /√) + C

; ) > −CRANGE:Thesetofelementsforthedependentvariable,thesecondmemberoftheorderedpair(,).

COMPOSITIONFUNCTIONS:Onefunctionissubstitutedintoanotherinplaceofthevariable.Thiscaninvolvenumericsubstitutionsorsubstitutionsofanalgebraicexpressioninthefunctionintheplaceofthevariable.

NOTATION:E(U()))orE ∘ U())Alwaysreadfromrighttoleftwhenusingthisnotation.

Example1:IfI(J) = J + 9andX(J) = 2J + 3,findI(X(3))

X(3) = 2(3) + 3 ⟹ 6 + 3 = ]I(9) = (9) + 9 = /1

Example2:IfI(J) = J + 5andX(J) = 3J + 4,findI ∘ X(J)

X(J) = J + 5I(J + 5) = 3(J + 5) + 4 ⇒ 3J + 15 + 4 = 3J + 19

INVERSEFUNCTIONS:Theinverseofafunctionisthereflectionofthefunctionovertheline, = ).Onlyaone-to-onefunctionhasaninversefunction.

NOTATION:E())isthefunctionEa/())istheinverse

ONE-TO-ONEFUNCTIONAone-to-onefunctionmustbeafunction,wherewhentheorderedpairsareexamined,thearenorepeating)valuesor,values.One-to-onefunctionsalsopassboththehorizontalandverticallinetests.

ONTOFUNCTIONAll)valuesandall,valuesareused.


ENDBEHAVIORTheendbehaviorofagraphisdefinedaswhatdirectionthefunctionisheadingattheendsofthegraph.Theendbehaviorcanbedeterminedbythefollowing:

1. Thedegreeofthefunction2. Theleadingcoefficientofthefunction

NOTATION:As) → ±∞,E()) → ±∞

Thisnotationisreadas.“As)approachespositive/negativeinfinity,,approaches

positive/negativeinfinity.”(*NOTE*:InAlgebra2,thesearetheonlytwonotationsyoushouldknow)

MULTIPLICITYMultiplicityisdefinedashowmanytimesaparticularnumberisazeroforagivenpolynomial.Inotherwords,it’stheamountoftimesarootrepeatsitselfgiventhe

featuresofthefunction.


COMPLEXNUMBERSTheimaginaryunit,6,isthenumberwhosesquareisnegativeone.

√−/ = 6 ⇔ 6* = −/TosolveforavalueofM,youcanuseyourcalculatororyoucanusetheM −clock!Example:Solvefor6CTosolve,startatthetop(Mf)andcountaroundtheclockateachquarterinterval,andstopwhenyoureachMg.Theansweris– 6.

LOGARITHMSi9 = j ⇔ klmi j = 9

Anexponentandalogarithmareinversesofeachother!

PropertiesofLogarithmsklm"(B ⋅ :) = klm" B + klm" :klm"(B:) = klm" B − klm" :klm" B= = = klm" B

klm" " = /klm" / = H

PropertiesofNaturalLogarithms

kn(!") = kn ! + kn "kn o!"p = kn! − kn"

kn !" = " kn!kn / = Hkn 9 = /

LogarithmicFormExponentialForm

Theinverseof, = 9)is, = kn)

PROPERTIESOFEXPONENTS&RADICALS

)aB = /)B

():)B = ):⋅B

(),): = ):

,:

)H = /

)B ⋅ ): = )Bq:

)B): = )Ba:

(),): = ): ⋅ ,:

)r= = √)r=

√)! = )/!

√!:: = !

√!": = √!: ⋅ √":

s!": = √!:

√":

RATIONALEXPRESSIONS&EQUATIONSToaddorsubtractrationalexpressions,youneedtofindacommondenominator!

/H*)* +

0.)⟹

33 ⋅

/H*)* +

0.) ⋅

2J2J ⟹

.H-)* +

/H)-)* =

.H + /H)-)*

Tomultiplyrationalexpressions,factorfirst,reduce,andthenmultiplythrough.Todividerationalexpressions,flipthesecondfraction,factor,reduce,andthenmultiplythrough. =ComplexFractions:MultiplyeachfractionbytheLCD,cancelwhat’scommon.&simplify.


SPECIALRIGHTTRIANGLES

TRIGONOMETRY&

TRIGONOMETRICFUNCTIONS

RADIANSTochangefromdegreestoradians,multiplyby t/1H.DEGREESTochangefromradianstodegrees,multiplyby/1Ht .ARCLENGTHOFACIRCLE

; = = ⋅ uwhereNisthelengthofthesector,visthelengthoftheradius,andwisananglein

radians

THEUNITCIRCLE THEUNITCIRCLE–EXACTVALUESRememberthesefacts&thetablebelow!

$<;u = );6:u = ,

@!:u = ,) =

;6:u$<;u

;6:u = ,

SPECIALRIGHTTRIANGLES–EXACTVALUESRememberthetablebelow!

TRIGONOMETRICFUNCTIONSsin w = {||{NM}~

ℎÄ|{}~ÅÇN~

cosw = LÖÜLO~Å}ℎÄ|{}~ÅÇN~

tan w = {||{NM}~LÖÜLO~Å}

csc w = ℎÄ|{}~ÅÇN~{||{NM}~

sec w = ℎÄ|{}~ÅÇN~LÖÜLO~Å}

cotw = LÖÜLO~Å}{||{NM}~

RECIPROCALFUNCTIONS

sin w = 1csc w cos w =

1sec w tan w =

1cot w

csc w = 1sin w sec w =

1cosw cot w =

1tan w


TRIGONOMETRICGRAPHSSTANDARDFORMSOFTRIGONOMETRICGRAPHSTHETANGENTGRAPH

, = ä ãån(i() − ç)) + 5

, = äélã(i() − ç)) + 5

Amplitude(A):èê |íLJMìÇì −íMÅMìÇì|Frequency(B):Thenumbercyclesthegraphcompletesin2îradians.HorizontalShift(C):Themovementofafunctionleftorright.Thesignusedintheequationisoppositethedirectionofthefunction.VerticalShift(D):Themovementofafunctionupordown.Thesignusedintheequationisthesamedirectionofthefunction.Period:Thehorizontallengthtocompleteonecompletecycle.Theformulatocomputethishorizontaldistanceisêïñ ,whereóisthefrequencyofthefunction.SketchPoint:tellsyouwhereandhowoftentoplotpoints.Theformulaisòôöõúùû .

THE TANGENT CURVE Remember!

Thetangentfunctionisundefinedatt*

and.t* radians,whichiswhythefunctionisdiscontinuousatthoselocationsinthegraphshown.

THEPYTHAGOREANIDENTITIESãån* u + élã*u = /

tan2θ+1=sec2θ

/ + él†*u = éãé*u

INVERSENOTATIONS• Theinverseof, = ãån )is, = ãåna/ )or

, = °¢éãån())• Theinverseof, = élã )is, = élãa/ )or

, = °¢éélã())• Theinverseof, = †°n )is, = †°na/ )or

, = °¢é†°n())

THEQUADRANTS&TRIGONOMETRICRELATIONSHIPS

(1,0)

(0,1)

(-1,0)

(0,-1)

xy

(x , y)1 (cos , sin )

Quadrant IQuadrant II

Quadrant III Quadrant IV

ALLSTUDENTS

TAKE CALCULUS

QUADRANTI:AlltrigonometricfunctionsarepositiveQUADRANTII:OnlysineispositiveQUADRANTIII:OnlytangentispositiveQUADRANTIV:Onlycosineispositive


FORMULAS REMEMBER!

SIGMANOTATION:SigmaNotationisusedtowriteaseriesinashorthandform.Itisusedtorepresentthesumofanumberoftermshavingacommonform.Thediagrambelowshowsthepartsofasigmanotation(otherwiseknownasasummation).

SEQUENCES&SERIES DEFINITIONSSequence:alistoftermsorelementsinorder.ThetermsareidentifiedusingpositiveintegersassubscriptsofL:Lè, Lê, L§, …L¶.Thetermsinasequencecanformapatternortheycanberandom.Series:thesumofallthetermsofasequence.ExplicitFormula:Ifspecifictermsarenotgiven,aformula,sometimescalledanexplicitformula,isgiven.Itcanbeusedbysubstitutingthenumberofthetermdesiredintotheformulafor"Å".RecursiveFormula:Inarecursiveformula,thefirstterminasequenceisgivenandsubsequenttermsaredefinedbythetermbeforeit.IfL¶isthetermwearelookingfor,L¶aè,whichisthetermbeforeL¶,mustbeused.

CommonDifference(8):!* − !/ CommonRatio(=):!*!/

Example:Evaluate(3(*) − 2) + (3(.) − 2) + (3(4) − 2) + (3(0) − 2)

(4) + (7) + (10) + (13) = .4

SUMOFFINITESEQUENCESArithmeticSeriesFormula:

©: =:(!/ + !:)

* where:isthenumberoftermsinthesum,!/isthefirstterm,and!:isthe:@™term

inthesumGeometricSeriesFormula:

©: =!/(/ − =:)/ − =

where=isthecommonratioand= ≠ /,:isthenumberoftermsinthesum,!/isthe

firstterm.


SETNOTATIONINPROBABILITY

STATISTICS&PROBABILITY THENORMALDISTRIBUTIONCURVE

Z-SCORESAz-scorerepresentsthenumberofstandarddeviationsagivenvalue)fallsfromthemean,´.Formula:

¨ = :?B"9= −B9!:;@!:8!=88976!@6<: =

) − ´≠

where)isthevaluebeingexamined,´isthepopulationmean,and≠isthe

populationstandarddeviation.Notes:• Anegativez-scorerepresentsa

valuelessthanthemean.• Az-scoreofzerorepresentsthe

mean• Apositivez-scorerepresentsa

valuegreaterthanthemean.

TYPESOFSTATISTICALSTUDIESSurvey:usedtogatherlargequantitiesoffactsoropinions.Surveysareusuallyaskedintheformofaquestion.Forexample,“DoyoulikeAlgebra,Geometry,orneither?”wouldbeasurveyquestion.ObservationalStudy:theobserverdoesnothaveanyinteractionwiththesubjectsandjustexaminestheresultsofanactivity.Forexample,thelocationastowheretheSunrisesandsetsoneachdaythroughouttheyear.ControlledExperiment:twogroupsarestudiedwhileanexperimentisperformedwithoneofthembutnottheother.Forexample,testingiforangejuicehasaneffectinpreventingthe“commoncold”withagroupof100people,where50peoplewilldrinkorangejuiceandtheother50willnotdrinkthejuice.Thestatisticianwillthenanalyzethedataofthecontrolgroupandtheexperimentalgroup.

CONFIDENCEINTERVALSAConfidenceintervalisarangeorintervalofvaluesusedtoestimatethetruevalueofapopulationparameter.Theformulatocalculatetheconfidenceintervalisgivenby:

where≠isaknownvalue,)isthemean,and¨changesvalue

dependingontheconfidencelevel.INDEPENDENT&DEPENDENTEVENTSOFPROBABILITYIndependentEvent:Twoeventsareindependentifonehappening(ornothappening)hasnothingtodowhetherornottheotherhappens(ordoesn’thappen).DependentEvent:Twoeventsaredependentiftheoutcomeoroccurrenceofthefirstaffectstheoutcomeoroccurrenceofthesecondsothattheprobabilityischanged.

COMPLEMENTOFEVENTSINPROBABILITYTheprobabilityofthecomplementofaneventisoneminustheprobabilityoftheevent:Æ(äç) = / − Æ(ä)

MUTUALLYEXCLUSIVEEVENTSINPROBABILITY1) IfAandBaremutuallyexclusive

events,Æ(ä<=i) = Æ(ä) + Æ(i)2) IfeventsAandBareNOTmutually

exclusive,

CONDITIONALPROBABILITYTheconditionalprobabilityofaneventB,inrelationtoeventA,istheprobabilitythateventBwilloccurgiventheknowledgethataneventAhasalreadyoccurred.

NOTATION:Æ(i|ä)Readas“theprobabilityofBgivenA”

FACTS YOU MUST KNOW COLD FOR THE REGENTS EXAM 2 review sheet_1.pdf · use your calculator or you...

Documents

Transcript of FACTS YOU MUST KNOW COLD FOR THE REGENTS EXAM 2 review sheet_1.pdf · use your calculator or you...