5.7 Graphs of Rational Functions

Rational Function Rational function: Ex 1) Sketch x y x y x-axis and y-axis are Asymptotes! Rational Function Rational function: Doesn’t touch y-axis Ex 1) Sketch x y x y As x ↑ , y gets closer to 0 1 1 -1 -1 2 ½ -2 -½ 3 1/3 -3 -1/3 As x ↓ , y ↑ 4 ¼ -4 -¼ ½ 2 -½ -2 Doesn’t touch x-axis ¼ 4 -¼ -4

Vertical & Horizontal Asymptotes Vertical Asymptotes: when denom = 0 Horizontal Asymptotes: If n < m, y = 0 If n = m, y = a/b If n > m, no H.A. num exp < denom exp exponents of leading coeff. num exp = denom exp num exp > denom exp

x0 Ex 2) Sketch the graph of V.A.: H.A.: Shift right 1 Stretch (x2) x Numer exp < denom exp → y = 0 Shift right 1 Stretch (x2) x y 2 2 3 1 5 ½ -2 -1 -1 -3 -½

Oblique Asymptotes Case 3) n > m: If expon on numer is exactly 1 more than expon on denom → oblique asymptote O.A.:

Ex 3) Find all asymptotes V.A.: H.A.: none O.A.: O.A.: ↓

T.O.O. Ex 3 cont…. (Find all asymptotes) V.A.: H.A.: V.A.: H.A.:

Ex 4) Sketch the graph of x y V.A.: H.A.: 1 -1 .5 -4/3 1.5 -4/3 3 1/3 .125 2.25 1.8 -1 1/3 -2 .125 -.5 .8

Homework #511 Pg. 330 1 – 19 odd