Algebr /Trig Fil Em Study Guide (Sprig Semester) Mrs. Duphy YOUR FINAL IS THURSDAY, MAY 4 th from 10:30 to 1:15 Iformtio About the Fil Em The fil em is cumultive for secod semester, coverig Chpters, 3, 9, d 7.1-7.3 All problems will be multiple choice with oe correct swer out of 4. There will be NON-CALCULATOR prt d CALCULATOR prt. The study guide below hs ll of the topics we hve covered d you will be workig selected review problems. The prctice problems my ot cover every cocept, so good strtegy is to study those topics you do ot immeditely kow. Ech dy we will hve wrm-up review d set of problems to work. You will tur i pcket with ll prctice problems o the dy of the fil em. It will cout s QUIZ grde (completio/effort). The fil em ccouts for 0% of your totl semester verge. While your grde is ot just umeric computtio, your verge does ply huge prt i determiig your grde. You MUST be preset o the specified dy d time. There will NOT be y mke-ups! BRING YOUR GRAPHING CALCULATOR!!! Chpter : Polyomil & Rtiol Fuctios Sectio.1: Qudrtic fuctios d models Covert qudrtic fuctios of the form f ( ) b c ito stdrd form f ( ) ( h) k by completig the squre. Fid the zeros of qudrtic fuctio by fctorig or by usig the qudrtic formul. hk, d grph y Grph prbol from fuctio i stdrd form. It's esy: plot the verte from this "ew origi". Also be ble plot the is of symmetry h d the zeros. Crete qudrtic fuctio from give iformtio, such s the verte d poit through which it psses or from its zeros d its mimum or miimum. Fid the miimum or mimum of qudrtic fuctio by fidig its verte. The -coordite of the verte is where the mi/m occurs d the y-coordite is the vlue of the mi/m. b Kow how to fid the verte of qudrtic fuctio i the form f ( ) b c s h d k f ( h) (i.e. just substitute h ito the fuctio to fid k). Solve word problems ivolvig qudrtic fuctios. Sectio.: Polyomil fuctios of higher degree 1 Sketch the grph of polyomil fuctio of the form f ( ) 1 L 1 0, where is the ledig coefficiet. o Use the ledig coefficiet test to determie the "til behvior" of the grph. o Fid the zeros by fctorig d determie their multiplicity (i.e. sigle, double, etc.). The grph crosses the -is t zeros of odd multiplicity. The grph touches, the leves the -is t zeros of eve multiplicity. o Clculte dditiol poits t vlues of o ech side of the zeros. o Sketch smooth curve betwee the poits. Uderstd tht if = is zero of f( ), the is fctor of f( ) d,0 is -itercept. Be fmilir with the Itermedite Vlue Theorem i the followig wys: o Polyomils re cotiuous, meig o itervl b, they tke o every vlue betwee f( ) d f( b ). o If f( ) 0 d f( b) 0 or vice-vers, the the fuctio hd to cross through zero somewhere betwee d b.
Sectio.3: Polyomil divisio Divide polyomils usig log divisio d the divisio lgorithm: the divisor, Q ( ) is the quotiet, d R ( ) is the remider. f ( ) R( ) Q ( ), where D ( ) is D( ) D( ) Kow how to perform log divisio, prticulrly with divisor of the form b c. f( ) Be especilly good t sythetic divisio of the form c. Determie if give vlue is zero of polyomil fuctio by dividig d checkig the remider. f( ) Kow the meig of the Remider Theorem: the vlue of f() c is the remider of c. Fidig such vlue is clled sythetic substitutio (s opposed to direct substitutio). Kow the meig of the Fctor Theorem: If f( ) hs zero c, the c is fctor. The coverse is lso true. Sectio.4: Comple umbers Kow the defiitio of the imgiry umber i: i 1 ( umber tht whe squred produces egtive rel umber). Uderstd the hierrchy of comple umbers bi, prticulrly tht y umber c be epressed i comple form: purely rel: 0i; purely imgiry: 0 bi. Kow how to perform ll rithmetic opertios o comple umbers (,,, ). Kow the defiitio of comple cojugte: bi d bi d the property tht bi bi b Simplify rdicl epressios ivolvig comple umbers. Solve qudrtic equtios hvig comple roots (fuctios with comple zeros). Sectio.5: Zeros of polyomil fuctios Fudmetl Theorem of Algebr: All polyomil fuctios with comple coefficiets hve t lest oe comple zero. Lier fctor theorem: All polyomil fuctios with comple coefficiets hve ectly lier fctors c i, where c i is comple umber (purely rel, purely imgiry, or comple). Rtiol zero theorem: If f( ) hs iteger coefficiets, the every rtiol zero will hve the form p q, where p is fctor of the costt term d q is fctor of the ledig coefficiet. Cojugte root theorem: If f( ) hs rel coefficiets, the y comple zeros will occur i cojugte pirs. Descrtes' rule of sigs: If f( ) hs rel coefficiets, the: o The umber of positive rel zeros = umber of sig chges of f( ) (or less by eve iteger). o The umber of egtive rel zeros = umber of sig chges of f( ) (or less by eve iteger). Upper d lower boud theorem: If f( ) hs rel coefficiets ( 0) d f( ) is divided by c, the o If c > 0 d ech umber i the resultig coefficiets is positive or zero, the c is upper boud of ll rel zeros. o If c < 0 d the resultig coefficiets re ltertely siged, the c is lower boud for ll rel zeros. o While this rule is true, it is usully more helpful to lyze the behvior of the grph of f( ) to help fid good guesses for zeros.
Fid zeros of polyomil fuctios usig the followig procedure d tips: o Before doig ythig sophisticted: If f( ) is triomil or if it is cubic, look to see if it fctors first! If y zeros re give, divide them out first to geerte depressed polyomil fuctio tht is qudrtic. Fid the zeros of the qudrtic fuctio by fctorig or the qudrtic formul. o If f( ) is ot fctorble d o zeros re give, list ll possible rtiol zeros. o Apply Descrtes' rule of sigs to potetilly elimite prt of the list of rtiol zeros. o Strt checkig rtiol zeros usig sythetic divisio, begiig with 1 d 1. If either of these re zeros, plot the poits f (1), f ( 1), d f (0) o grph ( simple sketch will do) d sketch the til behvior o your grph. Thik bout where potetil -is crossigs could occur d use this to pick the et rtiol zero to test. If the degree is higher th 3 d you fid zero, mke sure to check it gi with the depressed coefficiets to see if it is double zero! o Cotiue the process util the depressed fuctio is cubic d fctorble by groupig or is qudrtic. Use the clcultor to fid ll rtiol zeros first, the use sythetic divisio to get the depressed fuctio. Chpter 3: Epoetil & Logrithmic Fuctios Sectio 3.1: Epoetil fuctios d their grphs Kow the defiitio of epoetil fuctio: f ( ) with > 0 d 1. Uderstd the structure d behvior of the grph of f ( ) d f ( ). Use the oe-to-oe property of epoetil fuctios ( y = y) to solve simple epoetil equtios (i.e. mke the bses the sme d equte epoets). Grph epoetil fuctios d trsform them usig L/R/U/D shifts d - & y-is reflectios. Kow the turl bse e to 16 deciml plces (.7188188459045...). r Be ble to solve compoud iterest problems usig the formuls A P1 for ul iterest rte rt r compouded times per yer for t yers d A Pe for cotiuous compoudig. t Sectio 3.: Logrithmic fuctios d their grphs Kow the defiitio of the bse- logrithm s the iverse of bse- epoetil fuctio: If y, the log y. Tht is, the logrithm is the epoet to which ws rised to get y. Recogize the ottio "log" s meig the bse-10 logrithm d "l" s the bse-e logrithm. Covert epoetil form ito logrithmic form d vice vers. Solve simple logrithm equtios usig the oe-to-oe property of logrithms ( log log y = y). Grph logrithmic fuctios d trsform them usig L/R/U/D shifts d - & y-is reflectios. Fid the domi of logrithmic fuctio usig the property tht the iput must be positive.
Sectio 3.3: Properties of logrithms Kow the four properties of logrithms s well s you kow your me! logb log l o Chge of bse: log (the lst form we will use i Clculus 1) log log l b o Product: log y log log y Quotiet: log log log y o Power: log log Be ble to epd logrithmic epressio ito sum d/or differece of idividul logrithms d to codese such epressio bck to sigle logrithm. Sectio 3.4: Epoetil d logrithmic equtios Solve epoetil equtios oe of two wys: o Usig the oe-to-oe property d equtig epoets o Isoltig the epoetil fuctio d tkig the logrithm of both sides of the equtio to elimite the epoetil. Solve logrithmic equtios oe of two wys: o Usig the oe-to-oe property d equtig the rgumets (iputs) of the logrithms. o Isoltig the logrithm d epoetitig both sides of the equtio to elimite the logrithm. Fid the time it tkes to double, triple, etc. d ivestmet of P dollrs t ul iterest rte of r% compouded times per yer by solvig for time (see formuls bove i sectio 3.1). Solve other pplictio problems ivolvig epoetil or logrithmic fuctios. Sectio 3.5: Epoetil d logrithmic models b Be ble to work with epoetil growth d decy models y e i word problem, especilly those ivolvig popultio growth or decy. Kow how to work with logistic growth or decy models of the form y k 1 be ; prticulrly be ble to solve for whe give the other prmeters i the cotet of problem. Chpter 9: Sequeces, Series, Ad Probbility Sectio 9.1: Sequeces d series Uderstd the differece betwee sequece (list of umbers i prticulr order) d series (the sum of the umbers i sequece). Kow the structure of sigm ottio of the form summtio, m is the upper limit of summtio, d k is the summd. Be ble to write the k th term of series d put it ito sigm ottio. m k 1 k, where k is the ide, 1 is the lower limit of Sectio 9.: Arithmetic sequeces d prtil sums Kow the defiitio of rithmetic sequece: hs commo differece betwee successive terms. d, or more geerlly Uderstd the structure of the th term of rithmetic sequece: 1 1 d m d be ble to fid the th term give either the first three terms or y two terms. m Be ble to fid the umber of terms i give rithmetic sequece by solvig for. y
Uderstd the structure of the prtil sum of rithmetic series: the prtil sum. You will hve to solve for whe 1 d re kow. S 1 d be ble to fid Sectio 9.3: Geometric sequeces d series Kow the defiitio of geometric sequece: hs commo rtio betwee successive terms. Uderstd the structure of the th 1 term of geometric sequece: r, or more geerlly 1 m m d be ble to fid the th term give either the first three terms or y two terms. r Be ble to fid the umber of terms i give rithmetic sequece by solvig for usig logrithms, or by kowig powers of the bse (ofte power of or 3). 1 Uderstd the structure of the prtil sum of geometric series: r S 1 r d be ble to fid the prtil sum. 1 Fid the sum of ifiite geometric series: S 1 r whe it eists (i.e. whe r 1). Solve word problems ivolvig geometric sequeces d series by idetifyig the type of sequece. Sectio 9.5: The biomil theorem k k k k0 Uderstd the structure of the biomil theorem: y C y d how to use it to epd give biomil. Some thigs to remember: o There re lwys + 1 terms i biomil epsio of degree (otice the lower limit of summtio is zero, so there re 0 + 1 = + 1 terms). o The coefficiets fit ito the ptter of Pscl's trigle, which hs verticl symmetry. o Alwys write the coefficiets first, lwys begiig with 1 d d edig with d 1. 4 5 o d y re geeric vribles; the epsio my be somethig like 3 4y 6 o If the epsio is y ( y)., just mke the sigs lterte + +, etc. Be ble to fid the coefficiet o give term of epsio or to fid the k th term of epsio. Sectio 9.6: Coutig priciples Kow the meig of the fctoril opertio:! ( 1) ( ) L 31. Uderstd the fudmetl coutig priciples d be ble to solve "how my wys?" problems. Uderstd the meig of permuttio ("rrgemet") d how to solve "how my wys c you! rrge?" type of problems. Kow the formul for permuttios: Pr r! Be ble to fid the umber of permuttios of somethig with ideticl elemets, such s the letters i the word RACECAR. Uderstd the meig of combitio ("group" without regrd for rrgemet of the elemets) d how to solve "how my groups?" type of problems (e.g. committees, crd hds, etc.). Kow the formul! Pr for combitios: Cr. r! r! r!