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
+ 6+ 9 + 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
Holes a Vertical Asymptotes { } + + 5+ 6 + + + 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
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
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
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. + 5 + < y 0 is HA 6
+ + + y is HA + + There is a OA 4 + + > + There is o OA or HA 7
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 + ( + ) + 4 + 0+ 4 + 4 6 9 Discar the remaier a the OA is y 4 ( + 4) + 9 + + 4 + 4 9 + + 4 + 9
Fi OA + + 0+ + + + + + 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 5 + + + 5 R OA y + 5 Our ratioal fuctio Our ratioal fuctio a OA
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 0 4 4 We see the VA We see the where D(f) ot efie We see the VA is y 0