x x x a b) Math 233B Intermediate Algebra Fall 2012 Final Exam Study Guide

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1 Mth B Iteredite Alger Fll 0 Fil E Stud Guide The fil e is o Thursd, Deceer th fro :00p :00p. You re llowed scietific clcultor d 4" 6" ide crd for otes. O our ide crd e sure to write foruls ou eeded for of the proles listed. I will ot provide ou with foruls o the e. The Fil E is coprehesive. For the Fil E, ou will eed to e le to:. Fctor usig geerl fctorig procedures. 5.7 * Fctor poloil fctorig out the GCF of ll the ters. * Fctor poloil usig coitio of fctorig ethods. * For ters: - the differece of squres ( )( ) - the su of cues ( )( ) - the differece of cues ( )( ) * For ters: c - with ledig coefficiet : fid two uers tht dd to d c. Just use these uers. - with ledig coefficiet ot : fid two uers tht dd to d c. Replce the ter usig the uers ou foud d fctor groupig. * For 4 ters: - fctor groupig - group d fctor ters, the fctor reiig ters. Use fctorig procedures to solve pplictio proles There re severl tpes: use the Pthgore Theore to fid ukow side of right trigle; the equtio is give d ou eed to kow how to use it to swer the questio sked; give the re of rectgle fid its diesios of uifor order.. Fid ll uers for which rtiol epressio is defied, ultipl d divide two rtiol epressios, dd or sutrct two rtiol epressios with the se or differet deoitors, d siplif. Reeer: ou c ot cler our frctios whe it is epressio. 6. & Siplif cople frctios 6. - get it ito oe frctio over oe frctio, the flip d ultipl the otto frctio to the top frctio. 5. Solve rtiol equtio. Reeer ou hve to check the proposed solutio i the origil equtio to eliite vlues tht would ke epressio udefied Solve pplictio ivolvig rtiol equtios, like distce proles, work proles, Evlute rdicl epressios. Negtive uers uder rdicls with eve idices re ot rel, ut egtive uers uder rdicls with odd idices re egtive. We use solute vlues to sigif tht vriles re ot egtive uder eve roots Chge rdicl epressios to epoetil epressios, d vice-vers, to evlute or siplif, while pplig the rules of epoets to rtiol d egtive epoets. 7. * * product rule: * * power rule: ( ) * 0 * zero epoet rule: * dd/sutrct with se se: * quotiet rule: * egtive epoet rule: * epded power rule: or ( ) * frctio rised to egtive epoet: o rule! d

2 9. Siplif squre root epressios usig the product rule d the quotiet rule for rdicls. 7. * product rule for rdicls: * quotiet rule for rdicls: 0. Add or sutrct rdicl epressios. Reeer: 5, ut c t Multipl two squre roots usig the distriutive propert or the FOIL ethod. If possile, siplif squre roots tht pper i the product Siplif quotiet ivolvig squre roots. Rtiolize deoitors. 7.5 * For deoitors with ter: get rid of the root i the otto ultiplig top d otto wht the deoitor eeds to coe out of the rdicl. * For deoitors with ters: get rid of the root i the otto ultiplig top d otto the deoitor s cojugte (the se ioil ut with opposite sigs i the iddle). * For uertors d deoitors with differet idices, chge to epoetil for d coie epoets for fctors of the se se.. Solvig rdicl equtios 7.6 * For oe rdicl: get the rdicl loe o oe side of the equl sig, rise oth sides to the power of the ide, d solve the reiig equtio. * For two rdicls: get ech rdicl to ech side of the equl sig, rise oth sides to the power of the ide, d solve the reiig equtio. * For two rdicls d o-rdicl: get oe rdicl o oe side d the other rdicl d o-rdicl to the other side of the equl sig, rise oth sides to the power of the ide. You will still hve rdicl d eed to repet the process. 4. Solve proles ivolvig rdicls usig the Pthgore Theore, d solve for specific vriles i foruls ivolvig rdicls Perfor opertios ivolvig cople uers. This icludes covertig rdicls to cople uers, ddig, sutrctig, ultiplig, d dividig cople uers, d Siplif powers of i Solve qudrtic equtios usig the squre root propert. Solve qudrtic equtios copletig the squre. 8. * The Squre Root Propert: If, the * To solve + + c = 0 copletig the squre: ) set up the equtio so tht the vrile ters re o the left of the equl sig, i stdrd for, d the costt ter is o the right. Bsicll, get it ito the for c. ) divide, so the coefficiet of is. ) coplete the squre tkig oe-hlf the coefficiet of the -ter, squrig it, d ddig this qutit to oth sides of the equtio. Bsicll, dd to oth sides. 4) fctor the Perfect Squre Trioil o the left side of the equtio d siplif the right side. Reeer, it lws fctors ito 5) use the priciple of squre roots. Tke the squre root of oth sides. 6) solve the reiig equtio. 7. Solve qudrtic equtios usig the qudrtic forul. 8. * For + + c = 0, ou c solve for usig The Qudrtic Forul. 4c 8. Solve pplictio proles usig vrious techiques for solvig qudrtic equtios. 8. * There re 4 ws tht we kow how to solve qudrtic equtio..) Fctorig,.) Squre Root Propert,.) Copletig the Squre, 4.) Usig the Qudrtic forul You use of these 4 ethods to solve qudrtic equtio otied fro pplictio prole. I.e.: fidig the tie it tkes,t, for projectile to e certi height, h (t). 9. Solve equtios tht c e de ito qudrtic equtios usig u-sustitutios. 8.4 * With the equtio writte i descedig order, let u the iddle vrile or ioil, the u the first vrile or ioil. Rewrite the equtio usig u to solve, the replce u with wht u=, so ou c solve wht ou were origill sked to solve.

3 0. Grph qudrtic equtios usig the is of setr, the verte, d the -itercepts. Also, kow how to shift the grphs. 8.5 * The is of setr is the verticl lie tht goes dow the iddle of the prol, through the verte. Sice the is of setr is lie the equtio of this lie is. * The verte is the iiu poit for, or the iu poit for. Sice the verte is poit, its coordites re verte:, f ( ). Or, verte: (h,k) whe i the for f ( ) ( h) k The verte is used to solve pplictio proles tht require ou to solve for the iiu or iu vlues, i.e.: highest height of projectile, iu re of rectgle, iiu cost, etc... *The -itercepts re the poits where the grph crosses the -is (where =0). Ech tie ou set equtio equl to zero d solve for ou re fidig the -itercepts. To deterie the uer of -itercepts qudrtic equtio hs, clculte the discriit. Discriit 4c If 4c > 0 If 4c = 0 Solutios to + + c = 0 The equtio hs two uequl REAL solutios. If 4c is perfect squre, the solutios re RATIONAL uers.) If 4c is NOT perfect squre, the solutios re IRRATIONAL CONJUGATES. The equtio hs ol oe REAL solutio. It would e doule root d if,, d c re rtiol, it would e RATIONAL uer. Grph of = + + c Two -itercepts. Oe -itercept. The equtio hs NO REAL solutios. It hs two IMAGINARY solutios. No -itercepts. If 4c < 0 The would e COMPLEX CONJUGATES. * Usig trsltios (shiftig the grph) is es w grph qudrtic equtio puttig it ito the for f ( ) ( h) k, with verte: ( h, k), d deteries whether the grph will e rrower or wider th the origil grph. *The greter the solute vlue of, the rrower the grph*. Solve qudrtic, poloil, d rtiol iequlities i oe vrile. 8.6 * Fid the zeros of the iequlit d the vlues tht ke the fuctio udefied d set the s oudr regios o uer lie. Test ech regio. If the stteet is true, shde tht regio. Write the shded regio i itervl ottio.. Fid the coposite of two fuctios. 9. ( f g)( ) f ( g( )) es first write the f-fuctio, ut replce ll s with ig lk ( ). Iside the ig lk ( ) write the g-fuctio d siplif.. Deterie whether fuctio is oe-to-oe. 9. * A fuctio is -to- if it psses the horizotl d the verticl lie test. (Drw horizotl or verticl lie through the grph. If it itersects the lie ore th oce, it fils the test.) -to-: Or,,, ot -to-: Or,, 5 * Fuctios with odd ledig epoets re -to-. Eple: f ( ), f ( ), f ( ), d hlf the fuctios with eve ledig epoets re -to-. Eple: ( ), 0 4. Fid the iverse of -to- fuctio. 9. Steps to fid the iverse fuctio:. Replce f () with.. Iterchge the vriles d.. Solve the equtio for. 4. Replce with f ( ). This is the iverse of the fuctio, () f, f ( ), 0 f. 5. Verif f () d ( f ) re iverses ( f f )( ) f ( f ( )) d ( f f )( ) f ( f ( ))

4 5. Grph epoetil fuctios Solve pplictio proles ivolvig epoetil fuctios. 9. * These proles will e like the oe s fro the hoework. Be prepred to clculte popultio growth, dec, Copoud r t iterest forul: A P( ), d so o. 7. Covert fro epoetil for to logrithic for. 9. Defiitio: If log, the 8. Grph logrithic fuctios. 9. * log is rell, so we teporril look t, crete the poits (,) d the grph the poits (,). This is the grph of log. 9. Use the properties of logriths to epd or siplif logrithic epressios. 9.4 * product rule: log log log * idetit: log * quotiet rule: log log log * power rule: log 0. Solve epoetil d logrithic equtios. 9.6 * If, the. Ad if, the. * If, thelog log. Ad, if log log, the.. Solve pplictio proles ivolvig logriths. 9.6 * Use the defiitio of logriths to solve for the vrile s the epoet.. Use the properties of turl logriths to siplif. 9.7 * zero power: 0 log log * idetit d power: log * Nturl epoetil fuctio: f ( ) * Nturl logriths: l e e e. log If l, the * l e * l e * l l * l l l * l l l * e l. Use the chge of se forul to clculte logriths. 9.7 log * Chge of se forul: log log 4. Solve equtios d pplictio proles ivolvig turl logriths. 9.7 * Popultio growth, Cotiuousl copouded iterest forul: A rt Pe, Rdioctive dec: A kt A o e, d so o. Repetig fro ove! O our ide crd e sure to write foruls ou eeded for of the proles listed ove. I will ot provide ou with foruls o the e. Prctice Proles for the Fil To stud for the fil do the followig proles AND look t the proles tht were o the es (ut ol those siilr to those i this hdout). The swer to the proles listed elow, (For those of ou who hve the Chpter Test Prep Video cd tht ce with the ook, ou c view it to see soeoe workig out ech of the proles tht re i the Chpter Tests.)

5 Mth B Prctice Fil E. Fctor. 0. Fctor Solve for Solve for. ( 7) ( 8) 5. Fid the issig legths usig the Pthgore Theore, c, d the fctorig Divide Sutrct Siplif Solve for

6 0. Solve for c. c. The wid is lowig t verge of 0 iles per hour. Ridig with the wid, icclist c ccle 75 iles i the se out of tie it tkes to ccle 5 iles gist the wid. Wht is the cclist s verge rte i cl ir? Let = With the wid Rte Distce tie dist rte Agist the wid. Siplif. ( 5 5 ). Multipl. 4 4 ( ) 4. Siplif Siplif

7 6. Multipl d siplif Multipl d siplif. ( 5) 8. Rtiolize the deoitor Rtiolize the deoitor Solve for Multipl. ( 9 i )( 5i). Solve for. Epress irrtiol swers i siplified rdicl for: 6. Solve usig the pproprite u-sustitutio: The equtio for the height of ll throw ito the ir is h ( t) 6t 48t 00, where h(t) is the height of the ll fter t secods..) Estite the tie it tkes for the ll to hit the groud..) Estite the height of the ll d secods it tkes to rech its iu height.

8 For the qudrtic fuctios give elow, ) Epress the verte s ordered pir. ) Is the verte iu or iiu poit? c) Write the equtio of the is of setr. d) Fid the -itercept d epress it s ordered pir. e) Fid the -itercepts d epress the s ordered pirs. f) Grph the fuctio. 5. f ( ) ( ) 4 6. f ( ) Solve the iequlit, epress the swer i set-uilder ottio, itervl ottio, d grph it o uer lie. 8. Let f ( ) 4 8 d let g ( ) 5, fid ( g f )( ). 9. Let f ( ), fid ( ) f.

9 0. Grph f 4 ( ).. Grph ( ) log. f 4. If ou ivested $6000 t 4.8% copouded othl, how log will it tke to doule our oe?. S the epected popultio of tow which presetl hs 000 residets c e pproited the forul (.). Fid the epected popultio of the tow i 50 ers. 4. Solve for. log ( ) 5. If ou ivested $,50 t rte of 5% copouded cotiuousl, how ers will it tke to oti $6,5.5?

10 Aswers:. ( )(5 7). ( )(4 6 9) 4., 5 5. Leg: 5, hpoteuse: 7. ( )( ) 0. c., 4 ( ) 6. ( ) 9. No solutio iles per hour i Hits the groud 4.4. sec 6 M 6 ft fter.5 sec (, ) (, ) f ( ) oths or 4.5 ers. Aout,46. people t ers