Ntes 3.1: Ply Fucs Chapter 3.1: Plymial Fuctis I Algebra I ad Algebra II, yu ecutered sme very famus plymial fuctis. I this secti, yu will meet may ther members f the plymial family, what sets them apart frm ther families f fuctis, ad will lear hw t ucver deep secrets abut may family members just by lkig at their equati. Frmal Defiiti f a Plymial Fucti A plymial fucti is a fucti whse dmai is all real umbers that ca be writte i the frm 1 2 f x a x a x a x a x a, such that a 0. 1 2 1 0 The abve frm, with all its sub- ad super-scripts, is called the stadard frm f a plymial equati. Let s dissect the equati ad lear sme taxmy. i ai x ai represets the degree r rder f the plymial equati (the largest expet i the equati) W 0,1, 2,3, (THIS IS WHY THE DOMAIN IS ALL REAL NUMBERS!) represets a term f the plymial i is a idex, used t keep track f the terms represets all the cefficiets f the idividual terms a i a0 a represets the leadig cefficiet, a 0 represets the cstat x represets the idepedet variable f iterest f x represets the depedet variable f iterest (iterchageable with y) f represets the ame f the fucti Here s what a plymial lks like with umbers filled i a much mre cmfrtable lk. Page 1 f 9
Ntes 3.1: Ply Fucs Example 1: Idetify each f the fllwig as a plymial (P) r t a plymial (N). Be sure t uderstad why. 2 2 3 f () x 4 x l 3 x 1 (b) f () x 10 x 11 x 3 x (c) f () x 3 x x 9 (a) 2/3 5 2 (d) f x x2x 2 3x (d) 4 2 f () x 7 x x (e) 2 x 3 2 f () x x 7 x ex l1 x 2 (g) f x x 3 (f) f () x 3 x 7 x 3 2 2 x x We ca classify differet types f plymials by their degree. The graphs f plymials f degree 0 are cstats (hriztal lies), degree 1 graphs are liear (slated), ad degree 2 graphs are parablas. I geeral, the greater the degree, the mre cmplicated ad sphisticated the graph will be. Oe thig is fr sure, the graph f ay plymial fucti is always a smth, ctiuus curve, meaig it has breaks, jumps, gaps, chasms, VA s, etc r cusps ad sharp turs r ther dagerus, pity pits. Example 2: Idetify each f the fllwig as a graph f a plymial (P) r t a plymial (N). (a) (b) (c) (d) 0 The simplest frm f plymial fuctis f varius degrees are the sigle-termed plymials, r mmials, f the frm f x x. Yu ca thik f these as the paret fuctis fr all plymials f degree. Fr these mmials, there is a csistet patter i the shapes f the graphs. I geeral, as the degree icreases, the graphs becme flatter betwee x 1 ad steeper fr x 1. Why is that?? 2 y x y x y x y x y x Liear Quadratic Cubic Quartic Quitic 3 4 5 Page 2 f 9
Ntes 3.1: Ply Fucs Example 3: Sketch the fllwig trasfrmatis f the paret quartic fucti. Determie the ed behavir f each graph, as well as the umber f x-itercepts ad relative extrema. (a) f x x 3 4 2 (b) f x x 4 2 3 (d) f x 3 x 2 1 x 2 16 4 2 (c) f x 2x 8x There is a easy way t determie the ed behavir f a plymial equati i stadard frm, f x, simply by lkig at the leadig term, ax. The sig f the leadig cefficiet will give yu the right-ed behavir. D yu kw why this wrks? If 0 a, the lim f x If 0 x a, the lim f x x Oce the right-ed behavir is kw, kwig whether the degree is eve r dd will give yu the lefted behavir. If is eve, the lim f x lim f x x x If is dd, the lim f x lim f x x x Page 3 f 9
Ntes 3.1: Ply Fucs Here s a visual: is dd is dd is eve is eve a 0 a 0 a 0 a 0 Example 4: Determie the ed behavirs f the fllwig plymials: f x 2x P x 5x 3x 8 2x 4 3 4 (a) (b) 7 7 6 3 (c) g t 2t (d) N t 2t 4 6t 11t 8t I Example 3, we ticed that differet equatis f quartic plymials had differet umber f relative extrema as well as a differet umber f x-itercepts. Ifrmati abut bth relative extrema ad rts (zers/x-itercepts) ca be btaied frm the degree f a plymial equati. Therem A plymial fucti f degree has at mst zers ad at mst 1 relative extrema. This therem gives yu upper buds bth pssibilities. Ay plymial ca have fewer tha the maximum, ad we have t figure ut what thse ther pssibilities are. Page 4 f 9
Ntes 3.1: Ply Fucs Example 5: Draw several pssibilities fr the shape f a plymial f dd degree. Use a degree 5 plymial as e such fucti. Aalyze the pssibilities fr zers ad relative extrema. Example 6: Draw several pssibilities fr the shape f a plymial f eve degree. Use a degree 4 plymial as e such fucti. Aalyze the pssibilities fr zers ad relative extrema. Page 5 f 9
Ntes 3.1: Ply Fucs I all the examples abve, d yu tice the relatiship betwee the degree, the umber f relative extrema, ad the umber f wiggles (iflecti pits)? Befre we summarize what we ve leared, we eed t talk abut repeated rts f plymial fuctis. Whe a zer repeats itself d times, we say that zer has a multiplicity f d, r (md). The sum f the multiplicities f all the rts f a plymial will equal the degree f the plymial! I this class, yu are respsible fr idetifyig graphically, ad frm factred frms f plymials, sigle rts (m1), duble rts (m2), ad triple rts (m3). Multiplicities i graphical ad factred frm A plymial f x i factred frm f x A x a x b x c 2 3, will have rts f x a (m1), x b (m2), ad x c (m3). The degree f this plymial will be 1 2 3 6. A pssible graph wuld lk like this (f curse, here, A wuld be a egative umber.) m1 crss x a x b m2 buce m3 iflecti pit x c f x A direct csequece f this is a very, very, very imprtat therem called the Factr Therem. The Factr Therem x a is a factr f a plymial fucti if ad ly if x a is a rt f the same plymial fucti. Example 7: Sketch a graph f the fllwig fucti: f x 5x x 3 2 x 2 3 x 5 2 x 6 term ad the degree f the plymial.. State the leadig Page 6 f 9
Ntes 3.1: Ply Fucs Example 8: 4 3 2 Let P x 2x x 3x. Fid the zers f P, the sketch the graph f P. Example 9: 3 2 Let hm m 2m 4m 8. Fid the rts f h usig factr-by-grupig, the sketch the graph f h. Example 10: Write the (a) geeral equati f the plymial fucti whse ly rts are x 5(m1), x 1(m2), x 2 (m3). Write the particular equatis f the plymial fucti frm part (a) that passes thrugh each the fllwig pits: (i) 0,10 (ii) 1, 4 HINT: It might help t sketch each first! Page 7 f 9
Ntes 3.1: Ply Fucs Example 11: Write the particular equati f a degree 2 plymial, f, with the fllwig prperties: f f f 1 5. 3 4 0, Example 12: Write a geeral equati i factred frm f the fllwig graph: Example 13: Write a (a) geeral ad (b) particular equati i factred frm fr the fllwig graph: Page 8 f 9
Ntes 3.1: Ply Fucs Nw I thik we ca summarize all we ve leared abut plymial fuctis. First, let s start with what ALL plymial fuctis have i cmm: Imprtat Chart I: What ALL Plymials f degree have i cmm Fu, h s much fu Dmai f all real umbers Smth ad ctiuus graphs Have a sigle y-itercept Have maximum x-itercepts/rts/zers Have up t 1 relative extrema, ccurrig i pairs after that Have expets that are Whle Numbers Ifiite ed behavirs N Asympttes T determie the degree f a plymial with all real rts, add the multiplicities f all the real rts. Imprtat Chart II: Hw Plymials f degree differ by whether is eve r dd If is EVEN If is ODD Same ed behavir Oppsite ed behavir a 0, tp i, tp ut a 0, bttm i, tp ut a 0, bttm i, bttm ut a 0, tp i, bttm ut Odd umber f relative extrema, up t 1 Eve umber f relative extrema up t 1 N guaratee f a zer Guaratee f at least e real rt Number f pssible rts: 0, 1, 2,..., Number f pssible rts: 1, 2, 3,..., Buded y-values (restricted rage) Ubuded y-values (Rage f all real umbers) Page 9 f 9