### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Coll Alg MATH 1050

The U

GPA 3.95

### View Full Document

## 23

## 0

## Popular in Course

## Popular in Mathematics (M)

This 80 page Class Notes was uploaded by Miss Noel Mertz on Monday October 26, 2015. The Class Notes belongs to MATH 1050 at University of Utah taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/229941/math-1050-university-of-utah in Mathematics (M) at University of Utah.

## Reviews for Coll Alg

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/26/15

53 i m gr g 77 2 vgas gm 533M654 rgawgfxw em fun 0 Kidm 2362 f mw E FfCi r2 2 s if Wwa j i39 29 quot w altar m awaw a Wm gxmkrpwkuiiwzg MMMWWWWMM hhhhhh M 6ka5 tuba Via ail W i f t AS QP grim Sawit g g i 2 Peewww why 536391 iw i g 5L ng 5 1 1 MWS D Ex i 510 any r1 vv iltz s 4 a4 m4e Lanny A 2 39X wt V3 ZX35 2 2 Ws sfazJ 4w www gg Wt a vi ti MWSO lt3 3 MW 33 EX fv 6 Lo 14 gtth 2A Wumjtw xj L w M quotL 9 44 404 03 slap2 a HUM3L p 1 W Lag3 W ww g Wm waxpm 4 3 3m m kg wx wwwmv 36 WNW m5 ap kn TACO 32 me rm a g im x mg mr E muw 3 brim Wm M i EuxgiW 3 W t 2 193 3 i3 W K r a Qb ji i mq 31 W15 if Ci XLva a 3 5 3 M MWwwmruw wwwmm gtM HWWMWWM ggggg awryaz ens c gg L30 lt34 gmuj j V515 A a W 51 7 iv 6A5quot wigvi 4 LL 13x 3 A l v e r5 G MM CES QUAi RmQ 2quotquot W L if 15335 Q E 55mg 5 5 2 gtE i i x9 f gg g ri 5 3 6 23a Km 3 1 3 X 3 x 2 g i Xi 2 0 E 5 2 436x 5L 155 ii 395 s15 Sft r 5W a Carma x fifo j X g k X g E35 35 45K Wer mng mm4 me w XmX mum b i 3 ME q w m O m W 9Q u E QVV KWNX 4 V xme Nov hmwwwv W3 w a W 59 93 Ww mix awwa LO XNK 472 3 3va m HA 33 w ms aquot guqcv wj 5 gt Fig p WW w L r f i M W vw39 V t gigr if Ef e5 that 723 9 fquot X 39gd lm s a 1135 Q a S g 73 i 13 f 539 z e k K F rA i 519 ma abmg as 44 33 Miamm 3 i x5 x 5M 36 22 yaw 4C53lt14 Lgt E E e dag ag 3 E g A g E 3 E XMWMMW E r 4 Ma 31 x wa ea E mS Af 28f 37 A a WM 1 E g i v Nixix xmw 5 gs g 1 Ci X3541 W Xi ix r1361 5 C513 5 543 if r gmwk a WWW rvia Rux imx WW 3 ffx 341 203 5 w 548 515 3va at N f x A f I x J j 5 1 lt9 j vs Na 1 ng m 5M5 i 3958 1C 52 k f M 231 424 MM 2m mg m 155 56gt 35 445 Q c g gmx j k y k vmiiy aw j wim i gfK E3 g cw 9 ixs wd fZ wi k r k 1 t E35 2 im rZ mVS lt3 4 gage X a 5 a g WW qx s a amx izf X ngig f V v LV or iHX Vf 3 st 3 Cgf g x 1551 M x MR 055 32 m Km m e v mm 2 b m 95 X J 5 a x if Md xv V M VMNW x h w x i m 3 C 39 po39 1 39 gougw mw 5w M pHh MMA KW M g g 0M ngtc a cd Lof oinltO K E r X f f A 1 g g M i g 1 5 J i quot K X s L wwwwwmw m y f i l E a M zquot I r z l s E g 1 k g g L i Q i 2 W o Po ngyvg F1quot L g x r 3W Ck A k a ha A 60 gt W N Mn x1a as a we i Z i 2 Y 2R 3 Ci Satax hm ow gt0 6 54quot 33 a g SC K lt9 Cm CM Wm 9 Mk 4 m gt LN f3 PAGN X a 39 k7 a 6va 933 R Li 105 X27445 as yap 1 A in 1quot g ml hadNa g Xg ad V 2 1 me a 9 iii 9 01410 I 72 5 a POWWQWQ 9 104 459 k mm X NJWVJ 5 ij 4 MN m w wz ia we n 0 ww 4amp0 A If2 w 09 4390 145 hr 0 W X 5 gawk m mvf W mewi Maw 34 w 395 M 49 m5 112 35 9 av C X P 3VLX W WWJW 430370 35 E 22 quotmg W EX ems 9 00 WE 96 z 3 3 so am CBS x A 3 m j x may S j x m 4 g 4 y 3 3 4 HMS WAR 5 3 9 it mix 63quot mix3V mm 392 3 WWW 44 35Wer R 3365 W fwdv quotWmm 3 K 4w M3 Gk 45 gt PIW 5 r i 1 amp give med 47 23 4LMV x k w r i V f s endow 05 new N V WC ac MM 470 6 9M z xi M03 pgy o SW WY mg ml 2916 we Wm smog X k Ag Q 23 age M7 GW S39NM 9x3 0lt quotZx 352gtlt 1 W037 GE 23 New 3 11 gt13folquot3x47 quotr X S tQ K DV RWquot bwism 39 I A Egt X39s 3 L ZX Sgw mkc divum 40 ampW c z WV 584 0 g y2gt 29x 4 x W N 1330 K x I 39V YC7M a j I w V 6v y F X 5 40 w X gt C85 V3 S a X Hkivgk 34 L u 7 s LibSva 47 53km Rad Li gxg ig L 2 U 5 Skatti m 4393 K t q g 0 a M quot Mrva MW m M89360 2 CW 0 f vewa z x ch 3 CPSb gig 19V 435 a Mix f 92 gt f rqi Lg I Pmy s 9 WEN 4353 WWWWM wvwm 6 a Lia jail 14 10 CM aged 3 c a W W Pr Ema PM C35 way Cc c l gt tc iaed a 451 39 Hail 3 61 r L 911 a oLfc4 Lt ac icnci bill acwcfi a 27CDL iHa 4119CBKCampgt ac waif rrbclv ba ix ma chair awwyumwm MM 21mg 3 x L g m ux 61312 tad w at W 39 CAWV wgww o4 WWW Mumw 1 M S W 32cm m5 N cw mavng V a I V Wx W my g a f Lr Nr W VJ MO 2 2 3 3 QL 239 L C v 3 9amp3 wage a QM 72 vi L EX L EM523i LR Shiv5M i xlwqx zg VHQW mg 0 3i 1 L EB M Haj g 393 53 La vi 397 i I g 1 Z l 1 33 30 m2 3 3r 3 i A 394 W P L4 K3Wgt1 i 90 3 312 W quot3 193 is LEV vio 3913 b 413 37quot 3 2 Li L 5 r K L u 1 w Af rmR K 4 Cg 1 Wt g 1 J 93 no 8313 L H 11 32 M r 5 is N by l 0 v 3 Em C 20 E lz it 31 4 33 30 Li W lt 1 5 j 3 x v i quot 3 1 g F W j a gag g Egg i M 3amp9 39 jig 356 4M2 wj 33 VD is if b i 537 Ji ga g 3amp3 gt5 17 5 A 5 37x t ES w f amp 3959 5 A 9 gm x39 3 i3 quot 29gt 3 gt Wamwz 4 a 1 fig v v x F I n I 23 r RMMWWMW Wa V xng MR 2 93 m 32339 VX g bk gswk awg m A x A y M by L 1 f L L I lt1 16 a 2 mm 1 6 J 1 E 5 h k 3 6 3 x if n a Tm w i gt I I 9 xv3 wv m 1 2 i xji 39Wi w Vwa I f 3lt Qx ng LI w ka wga 3r 2 S J quot5 haze m 6 3 i Jag g WU 4 if X X a w Q6 WHO 5 f 5 Li H n g 5 5 4 gigs am wg f jjggw g x c MA i X fww g may 6 m w L Q 2 i w lt9 6 quot73 sa ggi i i g5 ngcWi EZ 3 5i z E E gt 1 W2 5 fbif xx gs5 lt3 Wm W a C z W 5 ir rb fx i Wm W 547 2 2 r w E i lt gx i De mw m Na Q x e 9 Wwi Mairr s 6dr 5 ng 0 1 9 g gng 13 Add ECOGJ Om quot m WQ W2 5 1amp5 1 Aw 39 a mail 4 34 Q 53 c 513 3 25 C Veg K V g N r x a 39 2 ridz 3MW i Q gi x v J i y 39 gt w i ii X Z y y i k iii Vg gba L wxa y L 51m Maw ie i 21 gg WM WWV MaN E Q 4 f 19 V1 EagiiN m x mf a 51 j W accgmrix ek ki G i xr k g xLQQ a w e x 6 29631 32 41xe 5 quot3 gquot H K g 1 g T I quot3 5 3s m v a m a j 52x15 2 5 gr Lwlt 138 15 i 5 A 5 i a 3 g L x Z quot5 3 539 mg was W 4953 a fr 343 54 1 W W i H x j quot W39 sf giggzi x quot s 393 a D J W 39 a 112 a Q 2Lvam aw L1 2 cg I f f x 55 dLQ Q x it gm 2 g quot 39 gquot lt5 xquot A m 14 yr Wye V 52 gk fa ML gxmg g g swagin wfdgm iangewdw i 39 7 gas a xix iv31 3 m quot M a i mi ngw i 5 9quot gig893 N 3r i 63 Ii 4 11 5464 g 7 3quot k t 3 14 x fugig im ig My Tana x W g x if M akl L55 2 of wena vWWst 3 9 39 9 r ff so Rf g W a wwzf 3v in 2 w m39i 39 55 Wmv f 33 ME we 24w EL 7 gaggaasii Mg g I 3 5 E rw sob 9 w m w aw M 501 UWSK C P mdsgi a Liwai 3M KW 5 ixm 55 iwgm W W V ok a A 63 M x If V g 55L j G A 35 M 1 Eggsth IL 5 S we 39 63X 3 gt quotE S V C N4 m WK W m a M g E mcrfmm my E A J v m Gav wxwmn N w x N MMO 942 g 2 1 I m akaQ 365553 a kcvx 3 gii syai v ifg 9 quot wv g g Vdii i 3 an 35811 ziwj gig 5i gfu if j kw m wws 40548 E rm vr 3 1 fw m ka 4amp3 45 252 L1 45 39 39 r k 2i i gquot i 7 SET ifch 1 44kg xxoiz 9U SW M quotquot v N n a n N y q v Elm W amp SWQ viM gad a pz hf Mug Um i5 mi 2 5 223awmva gt if i n UAR Mgt 39 313 b I 2 5 1 2 gwww IWWK a Er 1 mixck w ma 5 g5 132 55 gig 0 42145 33 M1 9g NS 2439quot S j fLk bxcgy 3 Cm17 x is 3 k A A quot 546 L g gg xvfng quotgm kamj 5y 5 A gram 92 3st 2 H a g f 7ng 39 4a 2 IV 6 a If I ifer i r iwmw Q32 4 461 Q 5 2 53 45 Kafka mVg w f W A 3 r p m 5 25 5 3531 ww s w Wm qs a 3 g x E 921 2quot 2 ti f 3 j Z D W MI I viiy 2 45 g 29 is ff agv i 3 vi WI a 55 a waif 4 43 S W 133 4 5M5 O4 4 F 41 wwm 16 w 16 3 r a 42 L r AL 6 Q3 Q I If M 3951 WK 3 5 346 Su 9 i Xavfi QL J 3 3i I iyb Pa wg 52ch Q Zi l wwi z 53 xii 35 Sumx aQ 7quot Rff 9 u 5 MWWWWuWM y Qk X g W 2 3398 QWQJV 5Y3 kg f gw gratis 6 g a A u LEW K ca tc c l f gt m if A A A 4 x lt 2 A gr 8 yg g a a 2 2 Ci X If f 5 E w Wx 4494 i i m i 2 ia L243 mf 3k3mm L 9 g f is 9 xx H a a x 32gt 5 mg W 3am a m w iwk W W 313 n i 391 3 5 m x x N a y 1 x E Z f w wi fv fk Sig fw n ca 3 3 a a Sm 3 f WW quotWWMM lm cazwimwilt r quot B 5quot 5 j 3W V10 3ampVm z g mb if 4 W x ML 85w 5quot 8 Ci gnaw A 95 Cw 2m Agar ag Qy m 32 me wwwwkw F F x j J V quotgyV Li MWa4 c p S39WML gumx 0 g M ibR i kwv c S mutin 5A E h gt Maggi W SW noquot i 3 3 K 0 f SVLXWX CA 5543 L by M E 2 qu a n wk m quot 5 W x 11 I iii 4 3 4 x k 39 22 393 n mm WM 2 a ea miota m s amm A SSOQ aimt a leg yaM Lam amen MM 131QVEFK I S ki f fir Gf2o gt r af d 5 Qaigquot m gtd 3 25a mg n S39W V i I w r f O 3ng aw vwf W1 1 W WL 3 x A ka a S g zw mm 41 ca W V VMWMNMMM gt TIC A r 0 IiA iEVVLQ ZBC 2 4C 9 1 W MM 3 Rwy 5 a 93 9 35 b 3 big 9qu k by gm 2 gm 3 Q to Q 8 M 9 gfgcj Lia3 1mm SC I gt V Mng j r lt A in f if ggui w gt ngr 55 i lt37 Witva j W gm w a mum v a 2 j g xiiquotk 39 f r quot La 7 fV n gt E a m Rwy 9 N CA 213 VVgxwr quot 3 m Wavg mb Q muw jk 4 4 by v j w 5 p Q 3 21M Qm t 52m W lt g t MAB7 as g quot a V A 433 1wa as w ww 5 quotjC m ngVwa c 5 5k 2 7 112 5 i 5 J 5 am 29 hi 3 Q a i 611 L wan 34in 3 2 6 5 L 13 0x 5 LJi M J Smuggsm Wm WM x 52 Law 1895 3 1m 3 aw Ei agx o mwvisxfabg vf 335m 5 E L f K g 0quot K if 5 CW 53 0 F x is is 5i 2 iii r a 1 g 4 inxf xz 5 I 45 i g V 3 1 9 A 5 W 4 5 65 r SA a 55min quotW L 7 gar 5quot RC i QSY39 quotQirlg n An Cara 3 a 39 X r W i V K 3 VK Citf gir 43721 as V P Y A4 r1 39 65 D gt6 x Vgm gv gf ax Mb i Sf m 2 ma a 3 a ESRJW OZD ESQ i V g 35 r Smx alLQi3f war n 4 Hi i 1 j 3 7gt SA 813 1Q39gff 1Vz r f39 igfr amp 1 6m5 B WNWM 5539 39 Avg S pg g 5 quotgt1 quotgm1 3 E4 39quot1 2 W K i 2 r 352 1 a f CA A EL uquot h C Ex L Lease m3 w who Swwxm W 6amp1 a r 4 5 39m vdr f 1vaxe i QLWLLVKQ 4 664 W 562 33 am3 3 3 9 52m rkwvix va igv rmwiex 43quot g39gi 2 m 2amp1 quot 39 3 7 s a Egg3 as 3 gt m 3 W mm 5 9 1 4mm a w 3 km g E X g 4 22 sapx 3 kg 5 w cj 100 I s I L LW E a g 00 21 V111 x 3 739 a 2 39s E i E g E E a E iin smurw wv Jquot Wquot k Si quot 1 37 wa i5 3 imnm i 3 725 5 WM Q r v W E n SQ gt iquot 3 r MN We WW 31125 2313quot C PK 54 lri i W v gt O gt g a may as a i V va gag j 3mm 2 13 a WC g amf Egg 11b 3 30 A E ivxg grmoi agM gr9943 D654 Y 3 a wn w 29 W15 C 9 2 196353 cave 6w MAR I 66 My 3151 2 x4 amp awashw V vain W page wagMW 35ch a QWKS39Z Igt v cisKWV 2 xfyw qd gawy mi b 3 2 S a L7 cf W i i c SWQE 5 3ri 2 2gm WW I N I X kw15 S 6 59 77 R og 3 x w t V I 4 g gt5 i Si fiv a 1 LX 3 u 7 K x 9 L I L I 2 v i i 1 i fv 2633 w av Ex 2 Sk zin 41v 47 z 59 a 3 39 c 39 quot a x fS be i w quotc TVC ROJ L fagg r 39 Jigth gymd amp a y 3 a L iv v g f a cam Lb x cx g aw t viai is ac 13 ggxn ic l gi r i wigtr 4 g 3 iw k 2v 414 O 3 U CK 2 3 5368113 54 gt 3 10055 x 7 eiaccikwllzizjrf gm h v2 p 1000 H iWCHij m 3 f w g lgt 17 m a a gaga4 2 WAXLi k Cfm wfacamf g g 5 O f i n f 1 2 2 2 3 a Ctr N ma a 5 i 4 jag5 p Q fmdgy 1meswxwaQ um rc mam if 5 t big WWW IL 323 ffLE 5 fk 574 i gawk V ngg g 5 a I wa w ruin MW wig FM 5 1 i 3A 1 Pay r Lx j km vie ML 5 4c u k i m a 21 WE x H 4 t g 930 E S 51 4 ymS5r wm i V j 3 1 S 1 pixUCQCK 1t myp mny C ivxEv uugm3 quot g3 6114 gm LR 1 11 Cu 7 L 4 ALn V 3 quot Hm M 3 Leggmm a gm ng TiW em mj MW e f i w agax 2 X351 i I x70 a7 v xi J x 5 t Q4 39 Lacks 0K t A lt w H k iC W4 axMeg o m 3 M 1 H 13 i iS 5 tquot Y v 38 gwem are ed sg 41X 1 X X W i fw fw gv Q9 5m W 8 L Lizg ULd a I 00333 O a2lt 33330 aquot G 4 50 ELMw a 3quot i a all 6 3 V J gt4 x VXI 62 53 33quot 39 gt4 quot339 a 12 gt52 47 5113 wa gggagg g v wgmsz fv g fg a 39v W a E E 3 73 3 35 2 Ev ivft 61 359 239 1 a 510 My 6M CL Egg 1 g i m 9 ng caf w Cir vii Si i EKLN 5X 3 iw x OH w I m 2amp9 MS N we EQCNWU x C A VHS mgm n 2y Kw W X I u X x 3 D W JQWMVY T JG J m W g Mm xxx 6 X7 1 i 6 KM A39 gt 9a mwgax g gyv ej x at age X 359 Wak 42an airwng j quot 3 4 a Q x b quot E I d 5amp3 Mg X g 5 v Q A X 5 g QWX grit 4 3 yijc 4 3 W 5ng va 3x5 W L Evde My mewLaw a Sl o zv 0 M 52L ha 105 Wage3765 4e pr wxa a S73 1 3f 3 at b m 1 m is 3 gkgjqw kEi gti xwf ir 6ij X ms 3 WM quotxix0U Q 7 X W Mng 1 5 Sunad st rms aWLh 331158 L92 WVWv WS 4 35 SOKGC C avx faxKL m M 35 9 C bf 03 33quot gm em a A Bew k 015 M 54237 e 1 x 4 m ltNm dlt 229 7 279 X96 sir Ean Shag 5 is bade waZkNJJ C a 03 Wk 53 3 a g 2 msh advMg W V 39 n 2391 b ygw miv f x r g3 Mada a 9523 ydvkg w 551 i 33121110 m mm 03 13an 32 mi r L 523 3 E KS 0 Vt Cap q x 39 I 3901 If 5 WW 58 a GM 138 g4 L 3lt Vii gs W 5 311 WN 66gt 9 ltsltm gm 0 2quot f 7 Xi ak r 7 m 4 lt vi 5 5 so we gr Wu K fu 9 a m quot M 39 x v fNS AiLQ 9quot E g if r LiQMV t kq V 3 m vaM L f v 4c 63396 5Gn 6quot g v g V 0 quot5 39a Z 3 2mm 5 2 2 Mg ami gi quot K 3quot 4 i X i Egg vi12 2 Q V 7 3 gigWE i E N i gig ng39g Xx ag l EM w s W 93 We mam w Ma mg 1 Q 39 m 5 e mrif i SIM a 1 mg az w 4 yd5 44 9am ampamp 339 MW K AW W m M05 u lt x K w m 1K a 5133 n a y 12 mw am WNW prx rm WS AQTKQW T1 6 i nwm3m 3amp3 mm Wmnwm my Wm M m USwa h 23 a gymmxrfx gfrzmm Em mm Q33 63 gt0 r ffixkii ivzzkEai9ig iampamp143aww iwa xp x x wg 333 3mm 3 x i x I fiwag EAf 559 fl I i 3 39 s i i 3 f 1 an m xxs an Lumzw SE 9km E 5 LWCS 6 2 2 V A 3 E e v xe 9 gym 501m 5 2 V9 336 i z k f m f a g 5p y a ml WWW WW a Law sgswm lt5 X 9 2 if W3 Z 393 M3 3x my 3 2 r x s i E g y 1 gig t ff39f vx 9 m Ly g g xiiav Cmg iz 2 lt g amp amp a g WWM W Nm xwnMV a 2zgt Qamp gagg m Vm 9 s X 33 W Ex 2 3395ng x 32 3 gxnggg 339 Exagggq 2 5343 SAW x2 17 S 254 t 8 gVUOS Q LEpa m 3 g i J 4 539 iii 51 Lard3 f Kg 14 33 W EM Rag MA M bm 23 q am Lag magiquot 35 3quot L gWquot amp 4210st Dgcbm gc 55 3 MW ab 445 2 aw gmwg agggf avg am Mia Wyxg R way 99m vm 5 ew W G 9U3 amp EVKC RGV Uwgzkmx W39gv n z 3 2amp3 51 1 MW 4r gm iigg 5kn 341 9L f WW5 ij V k 5552 43am six 43741 5 a Xviig 3r 451 A x CV xl amp C gt gykxhg g L x 39W X C 3 Ex m Q39M iwm 5 ECM KPQS VEW g4 inqx 3 gg We x W 5K1 U i g 3 1 1 y Hvix Z X i g39a X3 gt322 f i 1 mg 3 39 A K3 3 k X as 35 E Summary of Part 2 of the course Solving systems of equations An equation in 1 cuts out a curve in the plane e g straight line parabola a linear equation in 1 y2 cuts out a plane in 3 space Solving systems of equations corresponds to nding points of intersection 742 and p580 There are two different zqiproaches 1 If one of the equations isnquott linear then you must use substitution 7414 24 If all m equations in n variabl y 2 are linear then solve using ele mentary row operations ERG s ie 7381 a write the augmented matrix b use ERO s to write the augmented matrix in echelon form c rewrite the sv stem of equations using the variables 4 and use back substitution There are three possibilities i no solutions ii unique solution iii in nitely many solutions Matrices Addition subtraction multiplication of matrices Make sure you know the con ditions on the dimension of the matrices for these operations to be well de ned 82 We can only 39dividequot by a matrix if it has an inverse To nd out whether an n x n matrix A has an inverse and to compute the inverse you should 843 4 augment A by the n x n identity matrix In N use EROquots to write the matrix AIn in row reduced echelon form 63 if the resulting matrix is of the form I A then you ve found the inverse Otherwise A has no inverse An application but by no means the only reason for introducing inverses is to solve systems of n linear equations in n unknowns Write the system as AX B and trv to compute A as above The tem has a unique solution 1 1 if and only if A exists and the solution is X A Partial fractions To write the partial fraction decomposition for a rational function 27 1 if degree numerator gt degree denomenator use long division giving a poly nomial plus a rational function f x gx with degree f lt degree 94 2 factor 917 into linear and quadratic terms no complex numbers 34 for every linear term Ax8 include Mir 3 if there is a quadratic term Cu2 D41 E include 011qu 4 If there is a linear term squared Ax By include Kiwi ikuim 44 add the terms from 3 to give something like em quotI quotI quotI 3a l 5172 4a 3 20 a b 1 gm Set this equal to f x gx giving you a system of equations 3a b r coe icient of x2 in f 4a 3 20 coe i ent of a in f a b c constant term from f Solve the system Six steps to drawing graphs of rational functions Given a rational function rx f x gx with integer coe icients draw 7 2 Mar as follows 1 Is Mar common function shifted re ected stretched 2 Plot y intercepts 34 a Plot XiiIltCI39CCpts i4e nd zeros of f x by factoring b Plot vertical asymptotes ie nd zeros of gx 26 44 Look at behaviour to far right and far left ie nd the horizontal slant asymptotes 2 6 a if degree f lt degree 9 then 7 0 is asymptote A 3 if degree f 2 degree g then y LnE is asymptote ie leading coe of f divided by leading coe o g A n if degree f gt degree g then use long division to write 39rx qx rmnwindzi39r xrgx so that q x is a 39slantquot asymptote Use cunning and plot at least one point between and one point beyond all zeros and vertical asymptotes 64 Join dots with smooth curve 2 gym a L 9 r k v N MW NQQ w Lava b 1M Avw zav 7 a W Ed39g 544 p warm 31 5 Lx was a m 39 Maya 53 wws a 339 8 gt43 i A L k gi Qg Reg WWz M wig ag 7 99 3f bm g V a Vina 3a m3 gag 6L Elf Aj rxk 63L NVA gs w A b aw3 if bx V 1mg 4 S J 5x2 Eaka 5 ti Wigquot 663 I Saw W a Xx 7 3 5X49 7 5 isxni gt e gtltlt 221 X g E Ea 8 x3 W Way s mars 4mm RAQ 393 mamas ack5 w w i 22 6 39w39z Mfwggs m a xngg 3 5 wa 5 20m r i iewag 5mgij assam Mote j akmzi 4km f yk mxs 33 4447 JVQ 452 5315 Cg rg 0 3mg mg lama lt2 mgwo a Mm g da s m i KMLi ggyvag39 x a 55 3 24 mm A r39 41 53 44mg 3 5 CiXv a 39soif i W WW 3 2 3436236 23 44 3 sH N L 2 39 9310i v3a ii 1 a 9 mm W M w 5 W 3 r LvK AQ fawi we 2w g tmga m a g 6m a 56 a WkgkW 41325 4953 Md3 Lab S39 ac wag2 wk Mme S a f 5 vw Jms m V WuN WW V w 2 1 5 gr N Ex aw 458 g 45x 93 Swikg ij 4C 5 awn M 233 a w vazw gwama 2 i i gt54 L53 5 2L x20 828 423 5 w w 733 wag 3amp3 M 3 gt395 c nv w lt3 lt acts 5 gxl gQQ wamp udbxgw BMf K gt 2x f Swg gf 3 Q Lgt J wm mg s xzq 30 gigg 3x32i4 gt5 395 5 H3 0379 W i 2 33x 3 g ng wvi 33 E a w5 JUii f Cw 39 x A w 391 g r Yawvapkkgw sirquot QM gm 3 1 lt3 3 g i xc 133 5 W2 33 Z Cm f3 l ir g li f an 5 a A aw ws 05w 3 41 quotM a UK 6 5g 3 ng aw 2x110 ags fear gages mad a m 51C 0 Wit 4 M Cm r S V Cmp p f i x C3 v Exam 5 13 3 agaiia y a fax Pg A5e Lagsir 4 M 32 futif turf E xa v kgzm walk 4sz W W way Math1010 Equation Solving Strategies 1 Linear Equations Simplify both sides ofthe equation isolate the variable 3x7215xi7 3X 61 5X 7 3X 5 5X7 52X7 22 1x Example 2 Quadratic Equations a If it has an X term in it then move everything on one side of the equation with the other side zero Try to factor it 1 If it factors set each factor equal to zero and solve 2 If it doesn39t factor use the quadratic formula ie For ax2bxc0 x ibiv sb274ac 2a b If it doesn39t have an X term in it then isolate the x2 and take the square root of both sides Rememberto put the plus or minus sign on the answer or you can complete the square Examples 3x213x10 3x213X7100 3X72x50 a 1 3X 20 or x50 3X2 or x5 x2 or x5 3 2x2x750 a2b1c5 x ilivl174275 a 2 29 x71v140 4 x1i 4 2x27500 2x250 b x225 x5 Note For quadratic equations the quadratic formula method will ALWAYS work So ifyou39d like you can always use that method 3 HigherOrder Polynomial Equations Get everything on one side of the equation with the other side zero Factor and then set each factor equal to zero and solve This is basically a more intense version ofthe 2a1 type 10x45x315x2 10x475x3715x20 5x22x27x730 Example 5x22xi3x10 5x20 or 2X 30 or x10 x0 or x2 or x1 2 4 Rational Equations Find the least common denominator LCD Multiply both sides ofthe equation by the LCD which will get rid ofthe fractions Then it turns into either a linear or quadratic equation and you solve those using the previously stated methods 7 2 7x2 x1 x57x1 LCD x1x5 x1x5 2x1x57x2x1x5 x1 x5 x1 x572x1x2x5 x572X72x27X10 7x3x27x10 0x28x7 0x7x1 0x7 or 0x1 x7 or x1 Example Note You MUST check your answers for rational equations to make sure they don39t give you a zero in the denominator ofthe original equation lfa solution makes the denominator go to zero then you throw out that solution 5 Radical Equations Isolate the radical expression Ifthere are two radical expressions isolate one ofthem either one Then raise both sides of the equation to the powerthat will undo the radical So if it39s a square root then you would square both sids If it39s a cube root then cube both sides Ifthere is still a radical expression left in the equation repeat this process until there are no radical expressions left Then the equation will either be a linear or quadratic equation and you can solve appropriately v alm v 32vm2 foi63v 3x9 x763 V39xi6 3 V 9x9 x7696yquotx76x9 Example x36v x76x9 Note lfyour original equation has an even root in it then you MUST check your solution by plugging it into the original equation to see if it really works If it doesn39t work then throw it out 6 Exponential Equations Isolate the exponential expression on one side of the equation Then rewrite the equation as its corresponding logarithmic equation using the de nition of Logarithm Then the equation will either be a linear or quadratic equation and you can solve appropriately Example 27382 5 50 400 2738quot 5 50 2736quot 5 400273e2quot 5 50400273e2quot 5 7 Logarithmic Equations Isolate the logarithmic expression on one side ofthe equation Ifthere are several logarithmicterms you39ll need to use logarithmic properties to condense them into one logarithmic expression rst Then rewrite the equation as its corresponding exponential form using the de nition of Logarithm The equation will most likely either be a linear or quadratic equation to nish solving Example 210g4x7310g4x716 210g4x710g4x7163 log4x2710g4x7163 2 10g4 x x716 2 3 x x716 43x716x2 64X76416x2 0x2764x1024 0x732x732 x7320 x32 Note For logarithmic equations you MUST check your solution by plugging it into the original equation to see ifit is in the domain of the logarithm We can only take the log of positive numbers so ifour proposed solution makes us try to take the log ofa negative number or zero then we must throw out that solution

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "I made $350 in just two days after posting my first study guide."

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.