Alger Iportt Thigs to Kow Chpters 8. Chpter - Qudrtic fuctios: The stdrd for of qudrtic fuctio is f ( ) c, where 0. c This c lso e writte s (if did equl zero, we would e left with The grph of qudrtic fuctio is prol: the prol opes up if 0 the prol opes dow if 0 c, which is lie) Fctorig qudrtic fuctios: Eples Es eples ( ): c c e fctored ito... 6 r s, where rs c d r s c p r q s Slightl tougher eples ( ):... choose p, q, r d s so tht whe ou FOIL the two fctors ou ed up with the origil qudrtic fuctio (with, d c s coefficiets) Fctorig the Differece of Two Squres... 8 9 9 Fctorig Perfect-Squre Trioils... 8 6 ( 4)( 4) 4... 4 ( )( ) ( Zero-Product Propert If pq 0 the 0 9 ) p or q 0... 0 This is iportt s it tells us the vlues of tht solve the equtio. 0 or 0 Copletig the Squre Strtig with, dd, which ields or... 0 0 = This c lws e fctored ito = Verte For c A qudrtic fuctio i stdrd for [ The verte of the prol is t the poit (h,k) d the is of setr is the lie =h ] c e epressed i verte for: h k Qudrtic Forul If c 0 d 0 the the solutios to the equtio re give 4c Ais of Setr The is of setr of prol is give FWB Jur, 04 v Alger Topics Pge of 7
Discriit The discriit of qudrtic equtio [ ++c=0] is the epressio -4c If -4c >0, the the equtio hs rel solutios (the grph of the prol itersects the -is i plces) If -4c =0, the the equtio hs rel solutio (the grph of the prol itersects the -is i plce) If -4c <0, the the equtio hs 0 rel solutios (the grph of the prol does ot itersect the -is) Igir Nuers The igir uer Thus, r is defied s: r r i r... 7 i 7 i d i 6 i 6 4i Cople Nuers + i is cople uer where is the rel prt d is the igir prt The cojugte of + i is - i Cople uers c e grphed o the cople ple, which hs es: rel d igir. The solute vlue of igir uer is i 4 i 4 Igir is 4+i Rel is Addig d sutrctig cople uers: coie the rel prts d igir prts seprtel...(+i) + (6+i) = 8+8i (4+i) (+8i) = - i Multiplig cople uers: use the FOIL ethod...(+i)(6+i) = +0i+8i+i = = +8i+(-) = +8i- = -+8i Dividig cople uers: ultipl oe (the cojugte of the deoitor divided itself) d solve i i 6 i 6 i 6 i 6 i 0i 8i i 8i 7 8i 6 0i 0i i 6 6 Qudrtic Iequlities Fid the vlues of tht ke the iequlit true... +6+8<0 (+)(+4)<0-4<<- Hit: sketch the fuctio to see where the prol lies FWB Jur, 04 v Alger Topics Pge of 7
Chpter 6 - Epoetil d Logrithic Fuctios I epoetil fuctio, the se is costt d the epoet is vrile.... If epoetil fuctio is i the for = the it represets - epoetil growth if >... = 4 () - epoetil dec if <... = 6 (/) A sptote is lie tht grph pproches ut does ot rech s its - or -vlues ecoe ver lrge or ver sll. Copoud Iterest t r.0 A ( t ) P... A $00 $64. 70 where P is the pricipl out, r is the ul iterest rte, is the uer of ties iterest is copouded per er, d t is the tie i ers ()(0) $00 ivested t %, copouded othl, is worth how uch fter 0 ers? Logriths The followig fors of the se reltioship re equivlet: = log =...0 = 000 log 0 000 = Reeer: A logrith is epoet of the se. A positive uer c e se, ut 0 d e re used ost ofte. Oe-to-Oe Propert of Epoets: If =, the =... = 6 = 4 = 4 Properties of Logriths Product Propert: log () = log + log...log 8 = log ( 4) =log + log 4 = + = Quotiet Propert: log (/) = log log...log (6/) = log 6 log = 4- = Power Propert: log p = p log...log 4 = log 4 = = 6 Epoetil-Logrithic Iverse Propert: log log d... log 7 log 7 7 Oe-to-Oe Propert: If log = log, the =...log 4 = log 4 64 = 64 Chge of Bse Forul: log log 6 4 log... log 6 log log Nturl Logriths A turl logrith is logrith with se e d is revited l. l is the se thig s log e e is irrtiol uer (like π); its pproite vlue is.788. The l fuctio o clcultor will clculte turl logriths....l =.609 Cotiuous Copoudig A rt (.0)(0) ( t) Pe... A $00e $64. 87 Where P is the pricipl out, r is the ul iterest rte d t is the tie i ers $00 ivested t % is worth how uch fter 0 ers if copouded cotiuousl? Applictios of Logriths Bse 0 logriths (log o the clcultor) re used to clculte ph vlues i Cheistr, erthquke gitudes (usig the Richter scle), soud power levels (usig deciels db) d other thigs. Nturl logriths (l o the clcultor) re used to clculte copoud iterest, rdioctive dec, popultio chges, d other thigs. FWB Jur, 04 v Alger Topics Pge of 7
Chpter 7 - Poloil Fuctios The stdrd for of poloil is writte with the... 4 + -7 + epoets i descedig order of degree. To dd or sutrct poloils, coie like ters....( + 7 ) + (4 +7) = 6 + 4 + 4 ( + 7 ) (4 +7) = + 0 0 To ultipl poloils, distriute ll ters....( )( + )( + ) = ( )( + 6 + ) = ( + 6 + ) ( + 6 + )= + 6 + 8 = + To divide poloils, either use log divisio...divide + 0 + 8 + 4 ( ) + 0 + 8 0 6-4 + 8-4 + 8 0 or sthetic divisio... -0 8 6-8 -4 0 Fctorig the su d differece of two cues + = ( + )( + )...( + 8) = ( + )( + 4) = ( )( + + )...( 64) = ( 4)( + 4 + 6) The Fctor Theore ( r) is fctor of the poloil epressio tht...( ) is fctor of + 0 + 8 defies the fuctio P if d ol if P(r) = 0. ecuse () + () 0() + 8 = 0 The Reider Theore If the poloil epressio tht defies the fuctio P is...fid P() if P() = + 7 + + divided, the the reider is the uer P(). Usig sthetic divisio: Note: It is ofte esier to use sthetic divisio th 7 to plug i the vlue d evlute the fuctio. 0 8 4 7 87 46 Therefore, P() = 46 Multiplicit refers to root occurrig ore th oce...the fctors of + + 9 re ( + ), ( ) d ( ). The roots re d (with ultiplicit ). FWB Jur, 04 v Alger Topics Pge 4 of 7
Grphig Poloil Fuctios Locl Miu d Miiu f() is locl iu if there is itervl roud such tht f() > f() for ll vlues of i the itervl, where f() is locl iiu if there is itervl roud such tht f() < f() for ll vlues of i the itervl, where Icresig d Decresig Fuctios The fuctio f is icresig over itervl if, for ever < i the itervl, f( ) < f( ) The fuctio f is decresig over itervl if, for ever < i the itervl, f( ) > f( ) This grph of = f() - hs locl iu t = 4 - hs locl iiu t = - is icresig for < < 4 d < < - is decresig for 4 < < - right ed ehvior: rise - left ed ehvior: fll The Rtiol Root Theore Let P e poloil with iteger coefficiets i stdrd for. If p/q (i lowest ters) is root of P() = 0, the - p is fctor of the costt ter of P - q is fctor of the ledig coefficiet of P Cople Cojugte Root Theore If P is poloil fuctio with rel-uer coefficiets d + i (where 0) is root of P() = 0, the - i is lso root Of P() = 0. FWB Jur, 04 v Alger Topics Pge of 7
Chpter 8 Rtiol Fuctios d Rdicl Fuctios The vrile vries directl s if there is ozero costt... = 4 k such tht = k. This equtio is clled direct-vritio equtio d the uer k is clled the costt of vritio. Two vriles, d, hve iverse-vritio reltioship... = 4/ If there is ozero uer k such tht = k or = k/. The costt of vritio is k. Joit Vritio: If = kz, the vries joitl s d z, d the... = 4z costt of vritio is k. Whe ore th oe tpe of vritio occurs i the se... = (4)/z equtio, the equtio represets coied vritio. 6 4 4 8 4 6 A rtiol fuctio is the quotiet of two poloils.... (Equtio ) (Equtio ) (Equtio ) (Equtio 4) Rel uers for which rtiol fuctio is ot defied...i Eq., the deoitor c re clled ecluded vlues (vlues of tht ke the e fctored ito (-)(+). Thus, = d deoitor equl zero). = - re ecluded vlues. Verticl Asptotes If ( ) is fctor of the deoitor of rtiol fuctio...eq. hs verticl sptote t = ut ot fctor of its uertor, the = is verticl d = -. sptote of the grph of the fuctio. Horizotl Asptotes Let R() = P/Q e rtiol fuctio, where P d Q re poloils. ) If the degree of P is less th the degree of Q, the = 0...Eq. hs horizotl sptote t = 0. is the equtio of the horizotl sptote of the grph of R. ) If the degree of P equls the degree of Q d d re the...eq. hs horizotl sptote t = ¾. ledig coefficiets of P d Q, the = / is the equtio of the horizotl sptote of the grph of R. ) If the degree of P is greter th the degree of Q, the the...eq. hs o horizotl sptote. grph of R hs o horizotl sptote. Hole i the Grph If is fctor of the uertor d the deoitor of...i Eq. 4, (+) is fctor of oth the uertor rtiol fuctio, the there is hole i the grph of the fuctio d the deoitor, so there is hole i the t = uless = is verticl sptote. grph t = -. FWB Jur, 04 v Alger Topics Pge 6 of 7
FWB Jur, 04 v Alger Topics Pge 7 of 7 Other Useful Thigs Order of Opertios Pretheses, Epoets, Multiplictio/Divisio, Additio/Sutrctio The doi of fuctio is ll possile -vlues (or ll vlues of the idepedet vrile). The rge of fuctio is ll possile -vlues (or ll vlues of the depedet vrile). Properties of Epoets Product of Powers:... = 7 = 8 = 6 Quotiet of Powers:... 7 8 8 Power of Power:... 6 6 8 4 Power of Product:... 6 9 4 Power of Quotiet:... 6. 4 9 Rtiol Epoets ( rtiol es uer tht e epressed s frctio)...... Lier Fuctios The stdrd for of lier fuctio is = +,... = + 4 where is the slope of the lie d is the -itercept. ru rise Slope