Algebra 2A. Algebra 2A- Unit 5
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1 Algeba 2A Algeba 2A- Ui 5
2 ALGEBRA 2A Less: 5.1 Name: Dae: Plymial fis O b j e i! I a evalae plymial fis! I a ideify geeal shapes f gaphs f plymial fis Plymial Fi: ly e vaiable (x) V a b l a y a :, ze a 0: : All he expes ae mbes. qai Algeba 2A- Ui 5 2
3 Vale f a fi: f ( k ) = Oe way evalae plymial fis is se die sbsii. Ahe way evalae a plymial is se syhei sbsii. V a b l a y Real Zes f a plymial fi: Maximm mbe f eal zes is eqal he degee. Real zes: whee he gaph sses he x-axis. f(x) = x + 2 g(x) = x 2 4 Hw may zes d he fllwig fis have? Zes may be eal mplex... The Fdameal Theem f Algeba: h degee plymial eqai has exaly slis. Ed behavi f a plymial fi: A B C D a x x + A) psiive eve B) psiive dd C) egaive eve D) egaive dd f (x) f (x) Algeba 2A- Ui 5 3
4 I s Example 1: Deide whehe he fi is a plymial fi. If i is, wie he fi i sadad fm ad sae is degee, ype ad leadig effiie. 1. f (x) = ½ x 2 3x f (x) = x x 2. f (x) = 6x x 1 + x 4. f (x) = 0.5 x + π x 2 2 Y Y T 1: Wha ae he degee ad leadig effiie f a) 3x 2-2x x 3 ) 4x 2 3xy + 16y 2 b) 100-5x x 7 d) 4x 6 + 6x 4 + 8x 8 10x I s Example 2: Vale f a fi sig Die Sbsii f (x) = 2 x 4-8 x x - 7 whe x = 3. Sli: Example 3: Vale f a fi sig Syhei Sbsii f (x) = 2 x 4-8 x x - 7 whe x = 3. Sli: Y T Y T 2: Use die sbsii If f(x)= 2x 2-3x + 1 a) f(-4) Y T 4: Usig Syhei Sbsii. Fid f(2) a) 3x 2 2x x 3 Y T 3: Use die sbsii If f(x)= x 2-4x 5 b) f(a 2-1) Y T 5: Usig Syhei Sbsii. Fid f(-5) b) 100 5x x 4 Algeba 2A- Ui 5 4
5 F eah gaph belw, desibe he ed behavi, deemie whehe i epeses a dd-degee a eve-degee fi. sae he mbe f eal zes. f(x)= f(x)= f(x)= f(x)= f(x)= f(x)= f(x)= f(x)= Algeba 2A- Ui 5 5
6 Y T 6: Y T 7: Y T Degee: # eal zes: Degee: # eal zes: Ed behavi: Ed behavi: Y T 8: Y T 9: Y w: Degee: # eal zes: Degee: # eal zes: Ed behavi: Ed behavi: Algeba 2A- Ui 5 6
7 Clse 5.1 Wam-p 5.1 p(3) = p(-5) = p(3) = p(-5) = Algeba 2A- Ui 5 7
8 ALGEBRA 2A Less: 5.2 Name: Dae: Gaphig Plymial Fis O b j e! I a gaph plymial fis ad lae hei eal zes! I a fid he maxima ad miima f plymial fis We have leaed hw gaph fis wih he fllwig degees: V a b l a y 0 Example: f(x) = 2 hizal lie 1 Example: f(x) = 2x 3 lie 2 Example: f(x) = x 2 + 2x 3 paabla Hw d y gaph plymial fis wih degees highe ha 2? We ll make a able f vales, he gaph... Gaphs f Plymial Fis: ae is (hee ae beaks) have smh s wih degee, have a ms 1 s Fllws ed behavi adig (eve dd) ad a (psiive egaive). Algeba 2A- Ui 5 8
9 I I s s i i Example 1: Gaph by makig a able f vales ad fid he eal zes, x-diae f elaive Example maxima 2 : Gaph & by miima. makig Desibe a able he f vales ed behavi ad fid f he he zes, gaph. x-diae f elaive maxima & miima. Desibe he ed behavi f he gaph. f (x) = x 3 + x 2 4 x 1 f (x) = x 4 2x 3 + 2x 2 + 4x. : a : # s: a ms # al zes: : a : # s: a ms # al zes: # eal zes: a ms # eal zes: a ms Ed Behavi: Ed Behavi: # Real zes: (x-diae) # Real zes: (x-diaes) Zes: bewee ad Zes: bewee ad bewee ad bewee ad bewee ad bewee ad Relaive maxima & miima Relaive maxima & miima Maxima a Maxima a Miima a Miima a Fid he appximae zes ( he eaes hdedhs) Algeba 2A- Ui 5 9
10 Y T 1: f(x) = 3x 3 9x + 1 x f(x) Relaive maxima & miima: Maxima a Miima a # Real zes: Zes: bewee ad bewee ad bewee ad Y T 2: f(x) = x 3 + 4x x f(x) Relaive maxima & miima: Maxima Real a zes: Miima a # Real zes: Zes: bewee Maxima: ad bewee Miima: ad bewee ad Algeba 2A- Ui 5 10
11 ALGEBRA 2A Less: 5.3 Name: Dae: Slvig Eqais by sig Qadai Tehiqes O b j e i! I a wie expessis i qadai fm! I a se qadai ehiqes slve eqais Qadai Fm: Qadai Fmla: V a b l a I s i Example 1: Wie he give expessi i qadai fm, if pssible. Example 2: Wie he give expessi i qadai fm, if pssible. Algeba 2A- Ui 5 11
12 I s i Example 3: Wie he give expessi i qadai fm, if pssible. Example 4: Wie he give expessi i qadai fm, if pssible. I y w wds: Wha is eessay f a expessi be wie i qadai fm? Y T 1: Wie eah expessi i qadai fm, if pssible. Y T a) 2x 4 + x b) x ) x 6 + x d) x - 2x 1/2 + 3 Algeba 2A- Ui 5 12
13 Example 5: Slve: Y T 2: Slve Y x 4-10x = 0 T Algeba 2A- Ui 5 13
14 Example 7: Slve. Example 8: Slve. Y T 3: Slve Algeba 2A- Ui 5 14
15 ALGEBRA 2A Less: 5.4 Name: Dae: The Remaide ad Fa Theems O b j e i! I a deemie whehe a bimial is a fa f a plymial by sig syhei sbsii Use syhei divisi: Example 1: (2x 2 + 3x 4) (x 2) R e v i e w Example 2: (p 3 6) (p 1) Algeba 2A- Ui 5 15
16 Y T 1: (2x 3 7x 2 8x + 16) (x 4) The bimial (x a) is a fa f he plymial f(x) if ad ly if f(a) = 0. This meas ha he emaide f he syhei divisi lg divisi is. Example 3: Shw ha x+5 is a fa f plymial.. The fid he emaiig fas f he Algeba 2A- Ui 5 16
17 Example 4: Give ha (x+2) is a fa f f(x), fid he emaiig fas f he plymial Y T 2: Algeba 2A- Ui 5 17
18 ALGEBRA 2A Less: 5.5 Name: Dae: Rs ad Zes O b j e i! I a fid all he zes (eal & imagiay) f a plymial fi! I a fid exa zes by sig he gaphig alla, syhei sbsii, ad he Qadai Fmla I s i Cmplex Zes ae always i pais! A plymial fi may have ay # f mplex zes. Examples: & (is jgae) & (is jgae) & (is jgae) Algeba 2A- Ui 5 18
19 Example 1: Fid all he zes f. Sep 1: Ty sme pssible zes by sig syhei sbsii: y may hea wih Gaph.Cal.! Sep 2: Oe y ge a plymial wih degee 2 y a slve he qadai eqai! Sep 3: Give he Aswe: Zes ae Example 2: Fid all he zes f f(x)= x 4 21x Sep 1: Ty sme pssible zes by sig syhei sbsii: y may hea wih Gaph.Cal.! Ty ahe ze il y ge a depessed plymial wih degee 2. Sep 2: Oe y ge a plymial wih degee 2 y a slve he qadai eqai! Sep 3: Give he Aswe: Zes ae Algeba 2A- Ui 5 19
20 Y T 1: Fid all he zes f Sep 1: Sep 2: Sep 3: Aswe Example 3: Wie a plymial fi f leas degee wih iege effiies whse zes ilde & (is jgae) Remembe: " Imagiay s always me i pais!!! " If p & q ae s f a eqai, he (x-p) ad (x-q) ae fas!!! S, bease hee ae zes, he leas degee will be:. Ad we ge he plymial fi wih he leas degee by mliplyig: Use FOIL disibive ppey. Hi: Dawig he aws may help y avid misakes! Simplify by mbiig like ems. Remembe: i 2 = -1 Aswe: Algeba 2A- Ui 5 20
21 Y T 2: Wie a plymial fis f leas degee wih iege effiies whse zes ilde & 4i. Whih e is missig? S, bease hee ae zes, he leas degee will be:. Ad we ge he plymial fi wih he leas degee by mliplyig: Use FOIL disibive ppey. Simplify by mbiig like ems. Aswe: Algeba 2A- Ui 5 21
22 ALGEBRA 2A Less: 5.6 Name: Dae: Opeais Fis O b j e! I a fid he sm, diffeee, pd, ad qie f fis.! I a fid he mpsii f fis. V a b l a y Cmpsii f Fis Thee is a 40% ff sale a Old Navy ad as a emplyee y eeive a 10% dis, hw mh will y pay a $299 jake? Y d ge 50% ff...his is a example f a mpsie fi. Y will pay 90% f he s (10% dis) afe y pay 60% (40% dis). The w fis lk like his f(x) = 0.9x g(x) = 0.6x We a p hese gehe i a mpsie fi ha lks like his f(g(x)) f f g f x Algeba 2A- Ui 5 22
23 Example 1: I s i esii: g(x) = 0 bease: Y T 1: Y D fge he esii sie he demia a eve be eqal! T Algeba 2A- Ui 5 23
24 I s i Example 2: Example 3: If f(x) = x 2 5 ad g(x) = 3x fid f[g(2)] fid g[f(2)] Y T 2: Y T Y T 3: Algeba 2A- Ui 5 24
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