Rational Function. To Find the Domain. ( x) ( ) q( x) ( ) ( ) ( ) , 0. where p x and are polynomial functions. The degree of q x

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Brian Stone
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1 Graphig Ratioal Fuctios R Ratioal Fuctio p a + + a+ a 0 q q b + + b + b0 q, 0 where p a are polyomial fuctios p a + + a+ a0 q b + + b + b0 The egree of p The egree of q is is If > the f is a improper ratioal fuctio To Fi the Domai Give R p q q Solve q 0 Domai The omai of a ratioal fuctio is all real values ecept where the eomiator, q() 0

2 Domai: all reals ecept ( + )( ) ( + )( + ) + + Domai: all reals ecept a Poits Not i The Domai Poits ot i the omais of a ratioal fuctio are either holes or vertical asymptotes A poit is a vertical asymptote if it caot ot be compeletly factore out p p If R after factorig, the q q the zeroes of q 0 are vertical asymptotes (VA) the other zeroes of q 0 are the holes

3 Holes a Vertical Asymptotes { } D f, is a VA is a hole s Fi holes a vertical asymptotes H ( + )( ) is a hole a is a VA Fi holes a vertical asymptotes F ( + )( + 4) 4 a are VA

4 Practice f + + is a VA ( + )( + ) ( )( + ) is a hole f y-itercept If R p p q q after factorig, the the ( ) If 0 D R there is ot y-it. Otherwise ( 0) ( 0) ( 0) ( 0) p the y-it is y R( 0 ), q 0 0 q p or y R( 0 ), q 0 0 q Fi y-itercept f ( + )( ) ( + )( + 4) f ( 0) ( 0+ )( 0 ) ( 0+ )( 0+ 4 ) 4 4

5 If R -itercepts p p q q after factorig, the the zeroes of p 0 are -itercepts We o ot fi zeros of p because some of these zeros are ot i the omai, a hece are ot -itercepts Fi -itercept(s) f ( + )( ) ( + )( + 4) + 4 -itercept is Horizotal a Oblique Asymptotes horizotal asymptotes (HA) oblique (slat) asymptotes (OA) 5

6 E Behavior p a a a0 R q b b b 0 + There is a oblique (slat) asymptote a y b is the horizotal asymptote < y 0 is the horizotal asymptote A fuctio will ot have both a oblique a a horizotal asymptote Horizotal Asymptote A horizotal lie is a asymptote oly to the far left a the far right of the graph. "Far" left or "far" right is efie as aythig past the vertical asymptotes or -itercepts. Horizotal asymptotes are ot asymptotic i the mile. It is okay to cross a horizotal asymptote i the mile < y 0 is HA 6

7 + + + y is HA + + There is a OA > + There is o OA or HA 7

Practice Fi HA f y + + Oblique Asymptotes Whe the egree of the umerator is eactly

a oblique asymptote. Aother ame for a oblique asymptote is a slat asymptote.

will work) by iviig the eomiator ito the umerator a iscarig the remaier Fiig

) + q ( ) is a proper ratioal fuctio Divie by a obtai where f r or ivie p* by q*

8 Practice Fi HA f y + + Oblique Asymptotes Whe the egree of the umerator is eactly oe more tha the egree of the eomiator, the graph of the ratioal fuctio will have a oblique asymptote. Aother ame for a oblique asymptote is a slat asymptote. To fi the equatio of the oblique asymptote, perform log ivisio (sythetic if it will work) by iviig the eomiator ito the umerator a iscarig the remaier Fiig Oblique Asymptotes a + + a + a p p 0 If R b + + b + b0 q q a + p ( ) q ( r ) f ( ) + q ( ) is a proper ratioal fuctio Divie by a obtai where f r or ivie p* by q* a obtai f + q* where is a proper ratioal fuctio f y f is the oblique asymptote after factorig, 8

+ 4 + 0 + + 4 4 6 9 + 9 + + 4 + 4 9 + + 4 + Discar the remaier a the OA is y 4 + ( + )

9 Discar the remaier a the OA is y 4 + ( + ) Discar the remaier a the OA is y 4 ( + 4)

Fi OA + + 0+ + + + + + oblique asymptote y Fi OA + Usig sythetic

10 Fi OA oblique asymptote y Fi OA + Usig sythetic ivisio 0 5 R oblique asymptote y Our ratioal fuctio Our ratioal fuctio a OA 0

Fi OA + + + 5 + + 5 + 5 0 OA y + 5 Fi OA Usig sythetic ivisio 0

11 Fi OA OA y + 5 Fi OA Usig sythetic ivisio R OA y + 5 Our ratioal fuctio Our ratioal fuctio a OA

12 4 ( + )( ) ( + )( ) 0 D( f ) {,} sice + a o ot factor out. a are VA a there are o holes < y 0 is HA 0 is a -it 0 y is the y -it We see the VA We see the where D(f) ot efie We see the VA is y 0

Rational Functions. Rational Function. Example. The degree of q x. If n d then f x is an improper rational function. x x. We have three forms

Rational Functions. Rational Function. Example. The degree of q x. If n d then f x is an improper rational function. x x. We have three forms Ratioal Fuctios We have three forms R Ratioal Fuctio a a1 a p p 0 b b1 b0 q q p a a1 a0 q b b1 b0 The egree of p The egree of q is is If the f is a improper ratioal fuctio Compare forms Epae a a a b b1