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 b0 1 0 R 48 71 Factore p R q p Simplifie R q 4 4 4 4 1
To Fi the Domai Give R p q Solve q 0 Domai The omai of a ratioal fuctio is all real values ecept where the eomiator, q() = 0 69 1 Domai: all reals ecept 1 1 1 Domai: all reals ecept 1 a 1 1 Domai: all reals ecept 1 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 If R the zeroes of q p p q q after factorig, the 0 are vertical asymptotes (VA) the other zeroes of q 0 are the holes
Holes a Vertical Asymptotes 56 D f, 1 4 1 1 1 is a VA is a hole s R 5 H Fi holes a vertical asymptotes is a VA is a hole a is a VA F 1 1 4 4 a 1 are VA Horizotal a Oblique Asymptotes horizotal asymptotes (HA) oblique (slat) asymptotes (OA)
E Behavior p a a1 a0 R q b b b 1 0 1 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 4 1 y 0 is HA y is HA 4 1 There is a OA 4 1 1 There is o OA or HA The ability to aalyze these is importat i Math 15 4
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 p a a a a 1 1 0 R q b b1 b0 Divie p by q a obtai R f where f y f is a proper ratioal fuctio is the oblique asymptote r q 4 0 4 4 4 4 16 19 19 4 4 4 4 Discar the remaier a the OA is y 4 5
4 1 0 1 Fi OA 1 1 1 1 oblique asymptote y Fi OA 1 Usig sythetic ivisio 1 0 5 R1 oblique asymptote y 6
Our ratioal fuctio Our ratioal fuctio a OA Fi OA 5 5 5 10 1 1 10 1 5 1 OA y5 Our ratioal fuctio Our ratioal fuctio a OA 7
R Review a a1 a p p 0 b b1 b0 q q To fi the omai solve q 0 a To fi horizotal asymptotes eamie b To fi oblique asymptotes ivie a iscar remaier p q To fi the vertical asymptotes solve q 0 Remarks I busiess we frequetly ecouter problems ivolvig averages. For eample Average Cost is Cost ivie by AC C C Whe C() is a polyomial AC is a ratioal fuctio (a) Fi average cost (b) Fi the umber of uits to miimize average cost 1000 0 0.1 C 1000 0 0.1 C AC 1000 0 0.1 Note: I Math 15 we ofte fi it easier to use 1 AC 1000 0 0.1 8
1000 00.1 AC (a) Fi average cost (b) Fi the umber of uits to miimize average cost 1000 0 0.1 C 1000 00.1 AC (a) Fi average cost (b) Fi the umber of uits to miimize average cost 1000 0 0.1 C 1000 00.1 AC (a) Fi average cost (b) Fi the umber of uits to miimize average cost 1000 0 0.1 C = 100 uits 9
Solvig Ratioal Equatios To solve aalytically, multiply each term by the Least Commo Deomiator to obtai a polyomial equatio Remember, we ca solve ay equatio by graphig usig either the Zero Metho or the Itercept Metho (a) combie terms to form ratioal fuctio (b) fi umber of hours that will give a average sales of $60,000 The commo eomiator is 4 The average mothly sales (i thousas of ollars) epes o the umber of hours of traiig 0 S 40,4 150 4 0 4 4 S 40 10 160 4 4 4 4 10 160 4 10 160 40 The average mothly sales (i thousas of ollars) epes o the umber of hours of traiig 0 S 40,4 150 60 4 8010 0 80 80 410 80 590 Discar 1.5, because less tha 4 1.5, 78.47 78.47 hours of traiig 10
The average mothly sales (i thousas of ollars) epes o 0 the umber of hours of traiig S 40 4 Solve by graphig Itercept Metho 0 S 40 4 Choose wiow The average mothly sales (i thousas of ollars) epes o the umber of hours of traiig 0 S 40 4 Itercept Metho The average mothly sales (i thousas of ollars) epes o the umber of hours of traiig appro 78.5 hours of traiig 11