3.2D Review: Graphing ANY Rational Function
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1 College Algebra F Ln0v1i8o wkauft\a_ zswokfvtewtahrueq ^LTLSCM.e j QAIlWlj `rliigwhrtzsi vreeasuenrvwekdr. 3.D Review: Graphing ANY Rational Function 1) Review: Finding Asmptotes of Rational Functions: Name Date Period Given V() = p() d() in lowest terms, where the leading term of p is an and the leading term of d is b m (polnomial p has degree n and polnomial d has degree m). I. If, there is a horizontal asmptote at = 0, the -ais. II. If, there is a horizontal asmptote at = a b. III. If, the graph has no horizontal asmptote. NOTE: The graph of a rational function will never "cross" the asmptote, but it is possible for a graph to cross the asmptote. **To find where a function crosses its horizontal asmptote, set the function equal to and solve. ) To find an Oblique or Non-linear asmptote: Consider Option III. If n > m, the graph has no horizontal asmptote. The function will have an oblique or non-linear asmptote, determined b the after division. (disregard the remainder.) Oblique: Degree of numerator is more than of denominator Non-linear: Degree is larger b or more. g Gg0E1B8T mksuwtrao QSDohfqtcwuaSrSe^ KLFLgCR.t i LAklzlR srbilg\hrtesz NrseqseCrivpe^di.U ` RMuaCdTe_ ]wiiqtkhw hi\nmffinniihtued raflygmelbhras XB. -1-
2 Graph the function. 3) b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: \ TJ0p1T8Q FKUuGtQaN ks\obfttowtajrqet NLaLJCq.L U mapl_l\ VraiqgRhVtWsA rr_emsoerryvxepdm.n k SMVadre_ sweiktzhm OIFnDfCiBnniYtKeK FA]l[g[embUrVaO B. -
3 ) f () = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: L e^0f1l8a [KPuotaq usooofdtfwzawr[eg clflqcv.^ q haclhlz UrTiKg^hItse ZrseGsmeKrevSe^d].N L amajdqe^ NwEiQtWhv EIdnvfdiJneiSteS [AflSgweibErjak LW. -3-
4 5) g() = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: W Nr0t1e8j KVultFav TSPoefetowDacr_eP `LsLqC.a P maslnlv rjiigmhatbs\ r_estegrzv[eedk.v U vmaacd]e owsietth IQnffBiinniOt]e_ satlcgee^bhrtal rg. --
5 ) v() = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: j HC0F1e8X VKOuStia isqodfqtmwlasroee qlqltc`.m D hadlplt [rfixghhitosv grreiseerrrvtezdq.s Z emaagdeen fwlixtohd \IenRfKiHnLi`tTec sawlzg[ewbsrwac mw. -5-
6 7) Graph: f () = - b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: H of0n1u8r okbuhteaj VSUoVfRtqw_agrUeb LXLJC\.I w qavllo srcilgphptdsv XrMesbeervvgeUdH.s F QMkamdGeB IwMiXtqho airnsfxijnqi_t`eq OASlgPe_bKrOav _T. --
7 8) Graph: f () = 5-1 b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: H jo0r18z AKUuXtoa sszo\fttywcafrle\ ^LnLLCU.P K VAMlXli r[iwglhyt`sn jreeysze^revzetdw.i K ^MoaOdZeM iwyijtwh qientffijn]ietqeb vacldg^eobkrjab Fd. -7-
8 9) Graph: f () = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: P zj0^1l8i WKuLtaaD qssovfdtpwyazreel slzlycm.q ^ UAmlalb zrziugphwttsb arvezsoezrov]eqdj.` T MRakdFe` BwEi[tGh\ kilnafficnxipt`ev \AolOgXeabMrCag ep. -8-
9 10) Graph: f () = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: s w^0w1w8c Kgustual TSozfjtkwMasreq alllco.d ] da]l_l` ErViCgjhjtQsZ \ree[sgezrnvkeido.a N jmja^diei ^wjiftfhn LInJfpirnUiMtTep SA^lYg^etbprZaV H_. -9-
10 11) Graph: f () = Oblique or Non-Linear Asmptote? (Show long division to find it...) b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: K cp0c1i8r gkwu\tva] GSXoMfMtYwUa]r\ef RL`LeCf.m n faulzlz sr]itg`h_tosk `raeyskewrgvte[dz.k r cmzajdneh ^w]iitvhq OInDffi\nniHtbeM EAulRgMefb_rOaB XI. -10-
11 College Algebra P KK0Z1l8j ukbuftsaw YSgoEfHtFwtaYr[ek GLNLNCF.w O EA^lwlX kroijgmhqtvsb MrteVseeArYvXe^da. 3.D Review: Graphing ANY Rational Function 1) Review: Finding Asmptotes of Rational Functions: Name Date Period Given V() = p() d() in lowest terms, where the leading term of p is an and the leading term of d is b m (polnomial p has degree n and polnomial d has degree m). I. If, there is a horizontal asmptote at = 0, the -ais. II. If, there is a horizontal asmptote at = a b. III. If, the graph has no horizontal asmptote. NOTE: The graph of a rational function will never "cross" the asmptote, but it is possible for a graph to cross the asmptote. **To find where a function crosses its horizontal asmptote, set the function equal to and solve. ) To find an Oblique or Non-linear asmptote: Consider Option III. If n > m, the graph has no horizontal asmptote. The function will have an oblique or non-linear asmptote, determined b the after division. (disregard the remainder.) Oblique: Degree of numerator is more than of denominator Non-linear: Degree is larger b or more. quotiend polnomial, one, two, factored, -intercept, denominator, zero, -intercepts, numerator, zero, oblique/ horizontal, degree, numerator, denominator, cross, interval _ vp0a1f8g _KYuSt_aW wsvodfdtbwjalrleg NL[LrCS.e \ ^AclZlj wrkifghathsm rnelssedravseodg.a L HMharddeZ hwkietwhv kijn[fmiqnxijtcen EAqltgsekbrvai Bb. -1-
12 Graph the function. 3) b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: l ce0c1n8\ \KLuOtvaa MSZoDfTtHwZaZrPef \LuLbCQ.r B CAqlSlB MrQiwgfhRtXsA IrkeSsoeirrv^ecdm.L D qmoaldhe] aw[ictihd si\nofmi[n]ictneb ZANlggeObArMaj im. -
13 ) f () = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: b me0g1h8b ukmuwtnan ts`oyfjtmwdalrked qldlsco.t ] oablhls urhixgghstsl crvedsfegrsvteodm.z o RMda\dZeS pwsi`tvha CIjnefniwnJiCtieo `AwlQgqeabtrla _e. -3-
14 5) g() = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: l b0g1p8a JKuBtXaa sseohfktfwmaqrlej CLsLgCs.z q IAIlulo ^rpiegjhtxsr qrje[s\eerlv_ecdv.q [ lmdacdyeg bwfivtrhu RI`nifPirnPiJtleS ]A`lYgLesbIrEai _n. --
15 ) v() = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: N n`0h1b8f RKeuBtNaL js^oefjtbwxabrgem ulvlca.] M garlsl ZrgiWgDhatEsg rme`s^eqrevsede.k J ZMRasdOew bweiet[hp mihnwfgilnhieteh TAzlug`eJb_rTaI EI. -5-
16 7) Graph: f () = - b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: z XZ0m1L8l BKju\tTab rskozftt\wlarroeq wlqlicp.a ] AflHlr Nr_iRgghgttsu \rfetsaeirevtedd[.f Y bmdaxdkev YwziatuhJ oifnifhinnwintpes \AhljgoedbXrYaP nb. --
17 8) Graph: f () = 5-1 b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: r to0p1^8n tksu\tdam BShopf[tvwnabr^e_ QLPLTCK.^ P paaltll Kr^iNgohkt_sT krxemsbeur_veeudy.j W BMJaWdter `waimtqhv DIgnaf]ihnei[tpeE EA`lEgPeGb[rJau Dn. -7-
18 9) Graph: f () = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: K AI0^1v8O DKsuatLaI Sqo^fStmw[adrPes QLTLlCD.o O CAIlclT OrDiIgVhOtNs srqensuegrivkegda.q h gm[akdneh fwsigthhl eifn\f_ipnui_tles hapllgeelb`rsar R\. -8-
19 10) Graph: f () = b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: I L\0A1p8p hkluoteai psjozfltzwmawrqeo LZLhC].A e AAhlplQ jrsiagnhgtds] DrkeIsPe`rEv_eUdo.k w IMvaUdnem Sw^iJtYhT iionffi]nji]tuee AWlRgXeQbLrAaB \F. -9-
20 11) Graph: f () = Oblique or Non-Linear Asmptote? (Show long division to find it...) b. What are the equations of the vertical asmptotes? c. Find the -intercept: e. Horizontal, Oblique, or Non-Linear Asmptote? Show how ou know: M WI0P1B8v dkpujtma^ ZSLoLfitfwna\rwez PLoLkCh.c A maplulz _rwiagohatysd Rr_easeecrcvIeFdB.Y hmaadmeo Wwi_tShS ciannfgivn_iwtoet fawljgpebberzao TC. -10-
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