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## Unit 2 for precal

by: Emmaline Murphy

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34

# Unit 2 for precal MAC 1140

Emmaline Murphy
FSU

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includes sections 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 9.2, 9.3, 9.4
COURSE
Precalculus Algebra
PROF.
David Ekrut
TYPE
Bundle
PAGES
34
WORDS
CONCEPTS
precal
KARMA
75 ?

## Popular in Mathematics (M)

This 34 page Bundle was uploaded by Emmaline Murphy on Tuesday March 1, 2016. The Bundle belongs to MAC 1140 at Florida State University taught by David Ekrut in Spring 2016. Since its upload, it has received 32 views. For similar materials see Precalculus Algebra in Mathematics (M) at Florida State University.

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Date Created: 03/01/16
L. S'ecttolr.2ExpoNENTIAt FrnrcrroNs DefinitionL.L. An exponerntialfunetionisa janctiono,fhefann f (r): a* w.lreaei.a reanumber witha > 0and,aI 0. 2.Tns cnAprr oFTHE EXpoNENTTAL FUNCTToN (t)-f(") ot whereo > , {=Ax a> I A )=0 [+ (z) : a* where 0< s< 1 "f(r} x otaLl o Y= fl-,4 Section 4.2 3.PnopsnTtps .f*) a'wherc >1 f(*): o'wher0< a< 1 Domain fi, (-r",r") R (-no, eo) Range Wr (0, o.,) ff* (o,*) Increasing N/A lrl Decreasi:og tl N/A Intercept(s)Y- tnt ft, I y -rn+ (0, t) N0 - x-lht No *d* t,t| Asymptote(s) H A-Y=o 14 A. \=CI Section 4. MoncRAPHTNc Match the folequati:3*theirph:: (C)e ((D)yz-* (E)y:3*T tF)y:3*-' (G)u:3r-* (H)y:1-3' F c .-x- | -b* )= 2 Y= L-, A n l-X \= 3' )= c ft I _x r/= ? Y= t- (5) :3x- I Y: -3- (7) Section5LogarithmandExponentiEquations 1. TrrFoRM au aP Ithe equation can be resoithaitisintheformau: a'thenwe can changehitou,:'t). 4 "* Example l.lFi,nilsuc11ro116{z-*} \V= 2 ' 8r] r (t -x)= n A _) )a J.3 4+ 21 An A 2222 22 4te* x) = 3 *4x* * ?r *4Y = -5 A=S - + Example 1.2Findalr suc71ro"3-2*::) a" )x :c -l o *t tfb-?x =t* 3 'A',v -l I -1 -4 -AX.-, " r/f, d., X,e Sect4.5 2. TneFoRM a,ulf Itheequatiocan brewrittsothaitis the forau: brthenwe... (1takethe natural logof botsides, (2usetheproperlna": u,lnat (3Solvefor tvariabby (a) multiply out (b)add/subtrsotaltermswitthe variabare oone siand other terms othe othside (c) facout thvariable (ddivide (eusethepropertof logarittorewrittomatcthe solution Example 2.1Fi.nal* sucth,aB,-1 42-* - (1)2 (2).*14 tr@ilW lIt ,.4n1 (s)tn2 (rf^ 3 ={2_x} (44 XIN 8- ln #; Qtn4* xtnl (5)Lr^4 *xl n4 fxlh4 (b),n48 xlnS+xi n4-1 ns z etn* 1nn +1n3 *Itn3 + .r )\n 3 + J{ih4 : f; t*",4 + in? +rn4) f + tn3 x(th3 atrn4 x(\h? +t4q)= ln4'+ \n? \---'re*_*_*-_4 *(w; :* --1L'-*:/". {;"ri-il;"\$) \n3+\n\$ Sect4.5 Example 2Fi,nallsucthat e2'-3 *x-? a Ve: I lM =1 Wi? :lrj+ * ail lx*t= ll",t {' I lvr{ tff lr,frf 3.TnpFoRMlog,u:logou: =*f= 1n.'".ft*\$= Ifthe equacan re*rittenhe folog*'loo, thwe cachange rhis to?).p6ptArlr.,,h ly ? U > a Exampl3.1.i,nalri3tog,-:logq2lX 76 \orj -) qq*, =lDq o(4 t Q5)"* -) 0^,;)u/^= x=! e>r'* :) i x=la5''' ,4[]! Example 3Fi,natrrbtr:::r-r"f lrf , :6_S=r12 dh I** Y = le *1 twV*,t*&tug .sDl(*r * -l:- t2 / nH- A sotuti*ro4 x"^"k*l?: 0 -); Soltx-h nrr (x+4'Xx-3 r o X= -1 x=S Sect4.5 4.TlteFoRMlagou:b LL:&b.equation can betten iformog'u,:,thewe can chathis to Example 4.Find",fP *: -Z 3A _l <\ togxG;=)i -> i: i* = = (a") \-/ ^a ie = o-e')-' xd= a- ,/t y;,. tr*f (# =)-' c x,- J '= !A'* Gu) Example 4.Fi,ndi,logrr(\$)-2 r, -"x 3e b;)r = (*.e- )* = Jc{ -' 3*-#, S# *)-- H=b x+; r lb= fi-ta (Vy#r x-4 todtqrx+a) 't\$-*{L}3- tfrrtqt'4 )f-+t=lddi .*) \01 +a) - t![s#i#) = tCx = t X1-;= tUX *\$*< ),t* =. *""{* : tog+(r+a) irqo{x;-4)- *-*::-}i"Fr i t1.- 1;'A- L. Sec:rr4.7 GeowrH AND DBcan FORMULA: If Arows ordecays exponentiallthenitsatisftheequation A- Pn o* o ,4rows 'n 0 K7 o ,decays if KLA ? (t) , Yor* th,eple L.P{t):800e0'02'.tehowamanydayswilli,papulati,onaysobeys function the reac1800? (1)_ng A r'rq\- ' '0.08 P&)= Aooeu'u-( = l8o0 h1 -ffi] \''or2 80(] = r,9)n4 <)-1a1- - * '' 0.02 , a 0at , q ttt_ln9 = +++€r : ln 4 '*'o.o2lr.4 ln4 = O O?\ = \nI tv- l^* '')o.olng o.oa ().o 2 o.o J Example L.2.A culture - grows erponent'iIf800 bactera,rer-esei,nitiand,therare2lA0 aft1rt hour,howyllenwi,bepreseni,t?tcultur*fte3hours? Ptc) , ?o*o* t = I {hsar o.rp*#W Po , 1ao kt PCr) = ).1#O P cr): bfr; . \ 24aa erfl*) P(O - fcry , ffiu :: J4#{) EE) -ffi lne,El = ln t3) Fcr). f,cnd-;l:::., [{rl* \n3 , ,/ P(*)'*#tJet?Jti{"\*l rl # SooetnS*= In,tl lL-- ,,,cJ, : Section 2 Example L.3.ad,iI-31i,arad,i,oarnaterithad,ecaaccord'ito(t): Aoe-o'*zt, wherisheinitialoantprzsenandA(t)is tamountpresenat timet(indays)Whati,shhalf-lifeodinI-31? u*- (,) *(r)= As- o fi e=i #*, -0.BBb r r ,, (o) rd) o : i =) lne ,lnlZL. -#r,2 \/ -0.83 I : -0.b3U ln n 3+=E 4,fr'e - a.fr!* Example 1.4.Fi,ntheezponenti,al N(t),th,asati,sthecond:i,ti,ons o.83 If(0):16andI[(6):3. functi,on, N(t) Nor** No= lb = u Ff k+' S), Ibe. N[p)' tJod(n =-fi ; K'-rn , \$id lb T -Frc : I -Tvl+ f,," A NCt), lloe dne {b tn* ' wlc : [n,* 2. LocrsrrMoosl or Gaow:rn K: \fi*B Anthe*oaHofgrowthithelogistic modofgrowthwitcarrying capacity P(t): b>0, c>0 I+ e-u' 1. SpcuoN 4.6Inrnnpsr CoutpouNDED nTIMEs ppn'YEAR The Future Value,A,is amountinaccounat theendofgiventimeperiodofan account. The PresentValue orPrinciple, Pisthe amountinitialdeposited. The Annual rate of intereor nominal rater,isthe rate for fullearin decimalform. n ithenumber of timeser yeatheaccountiscompounded, i.e. theumberof timesperyeartheinteresis calculated and atoetheaccount. tisthenumber ofyearstheaccountis held. FORMULA forA: zI\nf A= P(t+ n) Example L.L. Findthe amountthatresul.from\$350-U,pes!gJ12% *nmpoqnded *.t""rr,aftea peri,of9 years. W T nrl fi= ? A ?(tn+r..)nu v=*50 = fl'* 0^ l;a tef [tl .3w(.1 + Ht- 4 6{ - 3bo(' n +- ry)uo @sso,(r.T) 350 (d) o+q fl'J Section6 -\--*-.-**-*'*\ 2 Example 1.2Ifan'i.naestmen8Tompoundedsemiannually, how much should youdeposWt tohaae\$2532n20years? Pralgr* [&.#*t"'lP. A 'J\$S#" r'ao f , # sff A--(t *[] ** P= assa { r*?} h'& , P=A(eufl]* ?,A53a(\ + o\$ y+o t; 2,0 flrn*, Example 1.3How lory,gittokW totirli,ii,s,nuesat10%,inte,rest compoundemonthly? X3 h= ?Cttt)n* f=iJ 3F, ytt+ +.,]** ln"1= \n (l+ H)'" ; :Zr ' !lf= r-- lh3 : *g=([ +aL"]tz-L_.",,t .=,ffil _H) ?,? 2.IxrsRosrCorqtruuousrCoupouuoen Anaccounthaiscontimrouscornpoundedithevalutheprevioformula approaches wne-)oo- FORMULA for A: fr= ? *'+ Exarnple 2.L.note wi,l*8,450t maturi,3 yearHow muchisthe note warth worth no,ssurnconti,nucnrnpoundat,7.3%? A runtt) = ?en+ Pr Lrlhflt's Wor+tn ? *-'* A = 8450 {*ute- ?, A -> Y,#"S?t5 fi+lae-oGl#-# L"3 o a-lq = frleoe section4-6 3 A= PuY* Example 2.2.How rn&nyyearwi,ll i,tforan,i,nuestmo/\$10,00 togrouto S40,000?Assumea ratefi,ntereof9% compounded continuously. {}ffi t fl= 4A,&?# 40ff1= tCI7trt7, 4 go-aq{' : , Y; lorffi = h4 0#qt ln* ffi""--#f'fl q ' lry{= Gt*: b:'? f,).*q 0 "#''t j.d,q* = L- l-n-3 # ---ipif}4" Example 2.3.Wh,ati,nteresternm,pound,cttntinwously, wi,tl take an'inaestment o/\$10,000to\$40,000in5years? 3. ErpBcrrvpRaro Theeflective raofinteresrefecti.uetesimplecompoundedonce)interest ratethawouldgivethesarneeturnasangivenaccount. Example 3.L.Fi,nd the efferate ointerefor5.5% mmpoundedquarterly. flt;4 fw{ = I {**f*'}o* : Ct*W)- d t ; CI 05bi 1. Spclloru 9.2 Tnn PlRnror,a Definition L.l.A parabola isthe setaf allpoi,ntsP : (r,g), thatare equ,id,istant .froma fisedpoi,ntcalled tlrecus and afined li,ne called directrix. 2.EeuauoNS,a)0 Equation Vertex IFocus IDirectrix :4a(r (v - k)' - lt) (v-k)':-4atr-tt\ X= h+a :4a(y (* - h)' - k) .VTV PIY (*-h)':-a"(y-k) X= K+r'l 4+& +5 Example2.1.Sketch thep,,'Y':;'r1t., (x n)'. <a61-4 (h, a)- Q,o) x2 --4a (v) (\>o 4ctt=-C=j \$=-* Example2.2.Find, the eofthearabod*rn"abebw. U-a)'= 4(t+3 j v&r.!ex: c \,e (V-Q'= 4a0-t, 6r-l^- 4a(x+t -4 -+c?,0 (o-;)a= 4a(-A+4 L'"hcn Ca)' = 4 6L,Ct) ",1:-. 0 /:4a-i- tzi - Example2.3.Fi,nd equatiofthearabouitlfocus1,-zand ueratr (11). 'PtvedYtx (x-h)" , ",(-h)n = - +a(V- o kt,,ri({} (x-D{= : * la cvv - r) ****f-f _+I_ !l[Ltuu It5.t,,(h,Y- =(t,-2) a=3) Example 2.Findtheequatiotheparabowi,th (-r, (1-2). focus -2)anduerter at 4a{x- h) .i (x rt ii (x*!) ((u:t ---? Section 3 Example2.5Fi,nd,equat'ofonearabaw'ituerter1-2)anddi,rectri,r ,-rr. I I (9 -v)' = -a(x*h) I (v+r)'= +r) I -rbCx i Example 2.6.ndtheequatiofn,earabowi,foeus-1,-Z)and,;i,rectri,r U:3. I Dir v€l'ttr=(-l,i) - I (x -a A -h)*= - 4a (v -r) t.J\ (x+r)a= _t}(y-i) parabola r2:6ale2.7Fi,nd,focus,i,reanduerter of wi,theequat'ion h r 4 q-h):=2 +alv-?, r..\ Y{r1 ry{a,o) \J / (x- -o)- *-+:- 0)^ = b (v 4et - b _ f - Drrr frcw.r, (o , ?) 7 L cirr-- Y=- ,Z u =37 Example2.8Fi,ntlfacus,,irectrii'a\d thefiarabotaheequation 2(A+2)2:*+3 ( - 2) vrv*Lx y+ 3 2, #(vt2)r= * tq, -2) Fcr,.,ts (y+ 1, (o +*") -# v\vecrY = * {= x ,= *a Example 2.9.,nd,focus,,i,rectaer-terthearabola wi,equat'ion =o() (Y+Z)2--2r*3 fl::tr -v\^ -- - 1 a (x-r,) ^r=rr,*l I C./ 'ir"I\.x 6t *e)'. _Jy+?,(y*;)r_ *,{y_i) v{t?(x - (A u\ -4a -;" 7 ,-d) = ff: i 1.Sscrrolr.4PnoppRTrESop LocaruTnMrc FuNcrroNs (1)Recallhe definitianlogarithm: logr : yitandonly7aP: r (2)Specialogarithmsushoulduicklrecognizand/oevaluate: 0\ rog,r":, ["] log,a:1 I Xo*', x0 x l(b) I rlnx I ,";:ro*roz I : lnX I,u,tor:to""J I L' EXPONTNITIATIN\ft Example 1,.LSoluforA,lny: r- 41n2. -4 lne : I t Iny x- 1ln2 : ln )4 E. ln lb w 0 Y -4lnL y= (r" ,(,, I Y ln tv \/ = .a- 4rvl J_x\ tx = .0, . = -lre' Y= lae (3)Sincf(*): a'andS(*):logorare one-to-one: =7 (u) o" a?ifandonlylu : u ,tr. (b)l"su : trsgifand ontyu,: EX p0Nfn liINTg :ln(r Example 1.2.Solaforg:lng - 16). ^ tny -ln(x-l*) O =& clurn&tn : ftk,"W) :) Y ' x'lh x> lb Section 2 (4Sinfe*) anallLd)lograrinverofeach other: (a)Ioo,u'tl (b)ar"s"u ExamplL.3.ualuate ezusi,ngoperoetponents. 2tn\t lnT ln4i+ t*S In 1b i*# O = O = U 'O : lb.fi s 10 (5Operations: : (a)Lag"(mn1om,*Iogon \_. lllff"[ll:",ii*;_,.s"n Example LEualuaezb{tus'ithaboueoperti,es. * +ln3 ,,tn tl^ ,e) r1tna9* q b oln4 =8- =e Example LErpand, asasuchsdt"ff .4 +octoK *7 t +t'4'rfl,s rn l*G*)r \$h{v-,f}- rn(ti:") tnft') W = *S lr"i = iln.y + * t"V- l J lhtr.r L Sec,4.4n 3 Example1-.Rewri,tesinglelogarita*(" 1)+2log(3+2) 31ogc -r - - (gx+il^ OryT ExampleL.7.f logaa:andlogaSb, thenlo--t9 (azab lol,r\B = loqq2'b'3 (babz (c2(+ b) =loql(J.3)' (d,) @( a+bz =toiqte+lo4q3 7 ''l%lA + &1,01+ = cL + Zb- 4 ExampleL.8Sotrfory:,.y:lnr_41n, : Iny _ ln 2 +lny lny= l"Go) D0 l'-{6t-uu e{/ X (x >o) 4 j = rt" Section4 4 :W. (6)ChangeofbaseormulaIfb>0 andb+ 0,thenlogr Inparticular, logor:*:;: H;. Example L.9Rewrifeog35withbase usi,changeofbasformula. lWtS = + chw'qp, bttw bpsq 'tg't, ln ',ffi 'A) I +r*{.P*' {\ {"): losorisonldefinedora} 0,a* 1,andr> 0. Example \.LO.Forthefollowi,ngrlnll,U>0 andylL. Tnteor [alse? (a)tos*t:o tntg Xo . I r"*ny:tosr.loss P F6{ l-tO r togx + wqy 9ror,":'# trelSO hpq t 104Y (d)tos_:,logr VCt\|e/ /yrcga -- ! roqx - loqy : (elosoS-3 -3 tfUt"f,., (fr"s-r?5)':3 €ASgt t Y^ronr {,(,al [o5ti,,t+ a\8, f,t,Lt\$t ,vf'(.) + b vf* \dlrr't 1.SpcuoN 4.3LocanITHMIC FuNcuoNs Definition.A.The losarithofruith, respertlr.e baisdefi,ned, by g:logo* iand,ontya1 X = OX Example L.t.Rewri,tei: ro,s{l,logarithm. loqrrTL Example 1.2.Reuritelogr(r +): 3asan erponent. (ilu= x+t Example L.3.logo7: C qo = I Example 1.4.logoo: I a'-- a 2.LocanrruMs v.s.ExPoNENTS (1)al"s"' X (2)logo': X Example 2.!.Eaaluatelogtrqlffi)a - I (2) q= W* sew_fr Example 2.2,Si.mpt (0.S)t"sg{i+=) t + [ 5#*\$tq# Section 4.3 3.Evar,uarrLocARrrHMs Example3.1.Eualuaeaclogarithm. Nthord,ofoperations. fu)a+bg24 4+ )' 10fl^l 0 lol e \,n^*,1= = b (b)los2nn * (i)Log24a \ocl +1= 2,+4 to to1^?n)r ' i,q,f ra = (\$-rig,1+1 B (j)(togrqa t toq, S : loga x : 3 '' '[,.0"q^et''(t d14 -tb o! A)+- (d,tas,a.( rcg ,+ *)-, a+) (*)tosr)= lota4-'' loqrb ( togr a),{ * r ?= 0 to4 *Q-*) = -'& ZIL I ta5efiq = 0ql1 .1 logr= -' ^'\ J 1 1 wlr?") \oq a 7 '"''4'tiug* *t 2") lofl+ 4 E = _t I -t =B (rnt g^| -t - Ios.4 a ; I (9/ +. 4& Section 4.3 4.Two SPECTALLocARTTHMS Def(1)fheCommon Logarithm:ogr: Iogror fheNatural Logaritlnr:Log.r @) -Y Example4.L. ,t!: 7 lnq, = Exampre 4.2.ros1000-: l1qtol000d= @q(o')* = l,j,o,r'* =vw Example 4.8.ehst: 5I {Et = 5.Tsp cRApH oF THELocARrrHMrc FUNCTToN (t)J(") lo8,zwherea> I l-\$r.X It rstrO (z) : lo8*rwhereo o#t1';1,= the, t*,'r/f #{""fr= a-x f(r) < < Section 6.noppRTIps f(r) logowhere>1 r):los^rwhere0<o<1 Domain K] (a,oq) rr* b,d Range W Cro, s#) ffi,(-oo, o") Increasing rl N/A Decreasing N/A t.l Intercept(sx-lhT (r,o) (t,o) X- \i"tf n0 Y- thf no Y-thr AsS.mptote(s) V.A X=O V A. X=0 4.3 5 Section Exercis6.1.Matehthfuncti,on,tseph. Sls fr 7l (A) s:logrr {B)y:1og3(-r) (C) s - -Logrr (D)A:-logu(-r) C. 3' {. CT, Y--t%, Y= Ialri i-xi \$) !1 t+. 3, J= l- t0ga 1= lo6J3 L-x) (6) D g= -tc\$ell t% I 9= er- (7) (s) - {(r) GIru.SS r cn( Y="+ry) \$= {t-"Yi G I rts( Y - crxrs 3' Sectionf.3 7. DolrarN Example 7.1. Find tlLe rlorno.irof f (.r): ln(.l 3er) - Example 7.2 . Fin,rl, th,rl,omain al : .1 3log.,(.r2 I G) - ) : Example 7.3. Fi,n,dthe d,om.a,in o.fQ) iog(l:2 - +) '? ntlfi6r fb (-q, hr,,o lr,*niu, ? Mgtx' &xtt \s X-aYtt r.€iro) oFy \.[_z F.ltnor Aillt is Y -q,xls b2+ c?? olz = bQ = A2- (1. n.rrns Er,r,rpsp Definitiont-1.. ellipsisthesetofal,loi.nPs, {r,y},suehho.t the sofn thedi.stancestweetwopo'i,ncalled,efociof thelli,psconstant. The rnajor axisthe ari,s througfoci,. The rninor axastheo.ri,sraugth,centererpend,i,etothemajor *ri,s. The verticeofheelli,pse theointwhere thelli,'intersectsmajararis. 2.EeuarroNs (*-hl)'-ik)- k)':,1|(*-hh)z ,(y-k)'_,, n -(a-V---;;-^ o' -(y-a'-- Axisr x- qx ts Y-ax15 a)b, relatibetween 6r-- q'- c2 b2, a2 - c2 a,b,antc Center V) Ch, E) Ch, Foci (vt *C, k) (h, Y+ A Ih 'At K) Ch, v:4 Vertices (h+o, k) (-v\ + e) ,V (h- a, K) (h r K-q) Sketch Atov' ,--l L, rnr AV r) filho r v, Section Example2.L.Fi,ni|,uati,ofthcnni,epheibelow. Fo6: ;{ A') ,) x' Y- f -F 1 x 1--. 7l V: -b'o a' \ .. _)< A=4 lt= 2 XZ , Y2 : + tn6\\0r 4 rh, o(rs 2.2Fi,nthequati.anthcani,caphed' Example 1 (x -h)' cy- k)= lo 0r' - *t L\3,-z (x -3)' (q+e, 1o2' I qe a,=3 b r l rnlh0 y : (x -3)= ct--A-i )-*-- S -*-- I A I u-sT (Y+2/ l--t : I I _ __3 4tt-1.1 bo= o3- c* b'=b2* 1' Sect9.3 aO, 3t0* lb Example 2.Writeusilowecnsr)thformuforg'nthe elwithcenter .u,u""""s"m* ""aucen"""?r,|'ti"'**'fu' o = 3b = | =Tr.{,";#=o,u'*,*! Example 2.Findthequatioftheelli.pse watth and, and,=; '' 3U(l-#} foci {-1,2) {3,2) - rr'!+'.\^,^-.. ,*l t)e+ (v*rte u* - q tri,',,'f = -q s-=l - , b^=5 q- b*= 4 Example2.5. "#i"* # c '--lb U^2*U dl= 4 x-0 elli,pse2/,i,r2r., t\2oci,,rtimajoraris,dm,inor aforhe ceffr;"'l(t, 4)lq## . +1)' {- 2)2 16 -0 1-I"o = t = 2 3:= .l 1t-''r-r(\-t,K1(r) .- \l'r,a (_l,{r)e(-t,"p) \__ t' 1{r {ut' Il,l;fi* )ac- \,2.,,rre) 1. SecrroN 9.4Tnp HrrppasolA. Definition 1-1.A hyperberlads the seof allaints,P : (*,y),sachthatthe di,ffereof thed'istancesneenhropo'inkcallethefociofthe-fuA 'constant. Peuu'otn" The centeri,thepointhalfwaybetweenthe twfoci. S The transverseaxis i,theo*ithroughthe ?frconjugate axis i,theari,s throughthecenterperpendi,cutorthetransuerseff:is. Theveriices ofhehyperbola athepoi,ntwherethe hyperbola'intersectsrans- ). xx uerseam,s. \$ tv') "/ 2.treuArroNs (-( f 0,d\ \$,0) tf0Oxlsetsg 0 ro) [-G,r) (4rr (s (a (* Y- \ t*-=n)' _ t,k)'_ , a,2)'_ b2-): t Transverse fo D,-b Axis Po"ralir \ Jo 86Yd.\\{l 0n{ur{ii-{" x- axr5 v- oxls ti relatiobetween 'ctf1s a, b,andc ba = c,a* aZ b'= c?*a" ,(. 0 Center (h,k) ch, r; Foci c, K) (n rrc) Ch: Ve-rtices (nt e, K) ('n, K:4 Y-K" Lu ^) Asymptotes Ltl,At\ Y-K= J ot-+ J Y- K: - 1.r'-nzr ;1 Sketch T*CIlxlt o- Y=K IF\)r' V-Kfitr I {,i-ot#,(x-h) c- {.[n5 v-[to o Section 9.4 Example 2.1Find the.atiafthennicraphbelow. {oy}n', &,a) Y x' | 01 az-@-- be k,trt= b k'0 a!,3b a2 ba a:b t ua xa , u =l 3to* 3ro V-l =l ea lo: O. I01 4z uety { (4,10) = Tt= l@_ lb , 100 tty;,H 1b loz= Tt, -*wb lb - tA-\V "b+l4' 7b bb toT =-Ab v* Yf = 5b - o"= I (v -r)' Example2.2.Findheequatiatthe mni,e,pbelow.{orm Ujn)' =l j- 6- b^ fw;t -t)z (v-p)a =) U I yong-v{ ' a*2 1 b- l [ge =) (x-l)' cv-a) 2 {IXl5V =-i* T,-', 7=l I I u, t9-)a- i Xrl "1 T) be-l w vatu 8r c" Jhe ve rtl(ps. Frnd Verlil:-t rntdrornt ryeen Ceh-ler * (,, 2) & ''' Ch,r.) - -) se E' qb' =l (9'-(lE)', =) qab I =) tr_* i 4b' tot- .tr- W\) (x) =, dqtb2- rt11-gal'i"/ Ser:9.4n o (usi,ng center0)occus@fife anTaerter at).mula thhyperbowi,th {orrn-(h,r)= (oo) b2. c^- a' yx \, ! -Da,bb-io:'?0 b2^=l -7# *=t* ^^g =7 Y'= tbct *#) Example 2.Fi,nd, th,e equati,oneuithl'at(-2,2and,,2and foci 'ol-?'+-0115. torrni'v'| (qz'-rff Y=1. b^* ; (.j= bi i_ C'2aA22 -Filrv:d--r" I tr ba= Q-4 ' = LX--t) -i)' .1__j 'rf**<j oxts '). 5r ' '_ cY-u) - , :) *q - 3**l r ll Srnl. foci"erti'ces'uers"ny' - o*'and' -Tx#:ymptoforthconic:"'' X2 -*'2n*", VltltN' (o,o) *-*:' A I o:*,y':t'fo,rn'l0rt[rue sriir 4t={r\, I I =z*rqr,sutrrr 't.J.? ruL' ,'"(rPt-'x--' ,ryX */ | \'\ d 4= ca veft -(rfr,0)&5afa,o) ./_,4=Example 2.6. Fthcentefoc'i,r"ttransuerse an'is, ankandate Y=- rl-aXhe asympforese conic -@*?Y:fr Qefi1tY=(h'k)' (l''p) \, (+ /- , 5fl q4=Bt4=r{A (tt,ay\ rJ 6r,r)-- (it}'ai-))'-, t ,;i:,ffi;:qY_' \ (n:c,k); ":: ?:,,:\\rr-:G l+ '"",4- .',r.cL(-t,;,,,F..*j_.,-"#,\'r/ tI vfrkx(l'"r1c,ki' L"' ei X--t t\ryf;y-K =I|cx-r) :7 y-l = t,l-(x+ r) Sect9.4 4 Example 2.7. Findtheeenterfoc'i,,t'ices, transuerse a*i,s, asisand,e the symptpetforshrco*:"c \o 1a '' f n -t) , .62 fl=2fi 2@+L)'-(*#2)2:76. (e,'r"-t'-r 14'' \ I ) //b3*tF--4 4-T tL?- rr: lu r.nwr,:. cY-Y) 2 CJ- l|' \l/ 7 C^,--'=-Jb , *J. - fuYrY1 - pa "r , , / {Lt+t}- .\-' j T " (e,-lt e{'") c-6r'\5 rvvfcl:(n,*"L,l = \ t l4\ vu+ Cr, #a) = {Qt,- 1:-11{;) 'x-)zt y-K -i,.f:ni Synp Z, #. 5 T-Gxtt 3.DrscnrurNANr ^ tj: t ir_ *; 4jyfiip:{y+ f The equationfa conic(parabolellipsor hyperbola) maye writtennthe form Ar2+ Bry * Cy'+ Dr * Eg *F :0 where A,B,C,D,E, andF arereal numbersWe may determinwhich conithe aboveformulaifortry examinithediscriminant Discriminant bu- lAC type oequation (ordegenerate) Discriminant Pc r&boie Discrimina<t0 (ov crrcte) E"ltrps* Discrimina>t0 Hyps r\ol w 4 e- q 4 (t{tttr Gn M**ci, # Secti9.4 4. Corvtpr,errNTHE Seuanp To changean equatioofa conifromthe form Ar2 + Bry * Cy'+ Dr * Ey *F : 0 intoan equation theforrndiscussed earliemust cornpletethesquare. Steps: (1) Group the terwith rtogether, the terms constanto the othside. with gtogether, anmove the (2) Factocoefficiofrz and coefficiofgrout oeach Soup. (3)To completthesquareof Mr) {r2* we add (M l2)2ince 12+ Mr +(M/42 : (r* Ml2)2. (4) Keep the equatibalancedbyaddingequivalenvaluestothe otherside. Keep in mind tvaluefactoreoutin ste2. Example 4.I.Descri,thegraphof 4r2+9y'- 16r- 18y: 11 That i,findthetypeofgraphand whereappli,cable uerfoci,,,irectra,sllrrlp- totesetc. a-'1 '{ &: s il t'. 4x'+ ttpr -* 4y': it G -i '{;' ,-* iSY= co,l 5 C''lE 6'! *-- 1, 1{xs - 4'x ++) + f jy+ I J * lt +tb+7 'd t({:i)' r q('i -l)* = 3A q,ki.(3,tltr, 3b V{ irt (ht 3'o 4b F0ri : tht C,V).(2, j r) {x -2)a + r1 , (v:)) ^ *t t q "t {l tt p5€, {;, r) Sec,tion 9-4 .-1G)5-t) Example.2Descrthe4rzy, B* 4yr44ac E rv> o + * hyye{bolA, =] lxr+Bx*yr*4y' =4 lLv'+?x r\$) - Uz4y+4) +t-+ =4 4(l:)14- L\d2= 4 (x-h)' (yr, t.) 4 + '|'1(,*7 b' (N)' (Y-2)' , t 4:t V.trl',(Atq,z) ? a) =(-t!l,lu),'il nVrjvbo\{4 Ct r ),,rX- J'i, q2=t -T +lY; (ht:,2 frr=o I toci= Exampt43.Descrtheeaof (0n-X\l Lr-1!S,z) ,:a'-8u-4v*t2 ?M#,V{X1, t I 2- q tryrr,t',v)'=+a (x-t') AL-4 0\'(r)'l1+0x-0,= y \y + \4v+ty'(t,e) 4a=6 @a Q=A [-l orv

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