Chapter 2 Algebra and Functions Review代数与函数

Chapter2Algebraandfunctionsreview1.Operationsonalgebraicexpressions•Youwillneedbeabletoapplythebasicoperationsofarithmetic-addition,subtraction,multiplicationanddivision-toalgebraicexpression.Forexample:yxyxxyzyzxx3152322210542.Factoring•Thetypesoffactoringincludedonthemathematicssectionare:Differenceoftwosquares:bababa22Findingcommonfactors,asin:yxxx4222Factoringquadratics:124322xxxx3.Exponents•Definitionyyxxxxppppppaaaaabbaba210443111111ThreePointstoRemember•1.Whenmultiplyingexpressionswiththesamebase,addtheexponents:752aaaaaaaaaanmnmaaa32511ttttttttttThisrulealsoholdsforexponentsthatarenotpositiveintegers:•2.Whendividingexpressionswiththesamebase,subtractexponents:235rrrrrrrrrrrnmnmaaa3.Whenanumberraisedtoanexponentisraisedtoasecondexponent,multiplytheexponents:186363nnnmnnmaa4.Evaluatingexpressionswithexponentsandroots•Youwillneedtoknowhowtoevaluateexpressionsinvolvingexponentsandroots.Forinstance:?,832yiswhatyIf?,6432xofvaluetheiswhatxIf5.Solvingequations•Mostoftheequationsthatyouwillneedtosolvearelinearequations.Equationsthatarenotlinearcanusuallybesolvedbyfactoringorbyinspection.1)Workingwith“Unsolvable”Equations?22,5baofvaluetheiswhatbaIf2)Solvingforonevariableintermsofanother?,3zandyoftermsinxiswhatzyxIf3)Solvingequationsinvolvingradicalexpressions29145x6.Absolutevalue•Theabsolutevalueofanumberisitsdistancefromzeroonthenumberline.TheabsolutevalueofthenumberisdenotedwwYoumaybeaskedtoworkwithexpressionsandsolveequationsthatinvolveabsolutevalue.Forexample:107t11672x7.Directtranslationintomathematicalexpressions•Manywordsrequireyoutotranslatetheverbaldescriptionofamathematicalfactorrelationshipintomathematicalterms.Alwaysreadthewordproblemcarefullyanddouble-checkthatyouhavetranslateditexactly.Forexample:isynumberaodividedisnbydividedtotranslatesthanlessxtotranslatesknumberathanlesstotranslatesbydecreasedynumberatotranslatesxquantitythetimesint2020556064320205560643ynxkyx8.Inequalities•Aninequalityisastatementthatonequantityisgreaterthanorlessthananother.•Inequalitiesareshownusingfoursymbols:toequalorthanlessthanlesstoequalorthangreaterthangreater1592x9.Systemsoflinearequationsandinequalities•Youmaybeaskedtosolvesystemsoftwoormorelinearequationsorinequalities.Forexample:Forwhatvaluesofaandbarethefollowingequationsbothtrue?18312babaSubstitutionandElimination10.Solvingquadraticequationsbyfactoring•Youmaybeaskedtosolvequadraticequationsthatcanbefactored.(Youwillnotbeexpecttoknowthequadraticformula)Forinstance:Forwhatvaluesof?420102xxisxxx712225201322xx11.Rationalequationsandinequalities•Arationalalgebraicexpressionisthequotientoftwoexpressions.Anexampleis223yyYoumaybeaskedtosolveequationsorinequalitiesinvolvingsuchexpressions.Forexample:Forwhatvalueofxisthefollowingequationtrue?3213xx12.Directandinversevariation•Thequantitiesxandyaredirectlyproportionalifforsomeconstantk.kxyForexample:xandyaredirectlyproportional.Whenthevaluexis10,yisequalto-5.Ifx=3,whatisy?Thequantitiesxandyareinverselyproportionalifforsomeconstantk.xkyForexample:Ifxy=4,showthatxandyareinverslyproportional.•Example1ThenumberofjuicepacksthatRamblingPinesDayCampbuyseachdayisdirectlyproportionaltothenumberofchildrenatthecampthatday.Ifthecampbought150juiceboxesyesterdayfor60children,howmanyjuiceboxeswilltheybuytodayfor52children?Example2Ifcanddareinverselyproportionalandc=10whend=6,whatisdwhenc=30?13.Wordproblems•Example1Thepriceofasweaterwentup20%sincelastyear.Iflastyear'spricewasx,whatisthisyear'spriceintermsofx?Example2Oneyearago,anaveragerestaurantmealcost＄12.Today,theaveragerestaurantmealcost＄15.Bywhatpercenthasthecostoftheaveragerestaurantmealincreased?•Example3Theaverageheight(arithmeticmean)of4membersofa6-personvolleyballteamis175cm.Whatdoestheaverageheightincentimetersoftheother2playershavetobeiftheaverageheightoftheentireteamequals180cm?•Example4Acartravelingatanaveragerateof55kilometersoerhourmadeatripin6hours.Ifithadtraveledatanaveragerateof50kilometersperhour,thetripwouldhavetakenhowmanyminuteslonger?14.Functions•1)FunctionnotationandevaluationAfunctioncanbethoughtofasaruleorformulathattellshowtoassociatetheelementsinoneset(thedomain)withtheelementsinanotherset(therange).Forexample,thesquaringfunctioncanbethoughtofastheruletakethesquareofxorastherule.2xThesquaringfunctioncanbewrittrninfunctionnotationas2xxf2,13gthenxxgIfx2)Domainandrange•Thedomainofafunctionisthesetofallthevaluesforwhichthefunctionisdefined.•Therangeofafunctionisthesetofallthevaluesthataretheoutput,orresults,ofapplyingthefunction.ExampleWhatarethedomainandrangeofxxf13)Usingnewdefinitions•Forfunctions,especiallythoseinvolvingmorethanonevariable,aspecialsymbolissometimesintroducedanddefined.•Thesesymbolsgenerallyhaveunusuallookingsigns,soyouwon'tconfusethemwithstandardmathematicalsymbols.•Youmaybeaskedtocomparetwovalues,eachofwhichrequirestheusesofthesymbol.•Youmaybeaskedtheevaluateanexpressionthaninvolv