4.4 Rational Functions A Rational Function is a function whose rule is the quotient of two polynomials. i.e. f(x) = 1 𝑥 g(x) = 4𝑥−3 2𝑥+1 h(x) = 2 𝑥 3 +5𝑥+2 𝑥 2 −7𝑥+6 The graphs of rational functions have breaks.

Vertical Asymptotes A rational function has a vertical asymptote at the zeros of the denominator as long as that zero is NOT a zero of the numerator. Ex. 1 Identify the vertical asymptotes. a) f(x) = 𝑥 𝑥−5 b) g(x) = 𝑥−1 𝑥 2 −1

Holes Holes occur when a zero is found in both the numerator and denominator. Ex. 2 Identify any holes. a) f(x) = 𝑥 2 −4 𝑥 −2 b) g(x) = 𝑥+1 𝑥 2 −1

Horizontal Asymptotes H.A. occur if the degree of the numerator < degree of denominator. - H.A. at y = 0 (x-axis) H.A. occur if numerator & denominator have the same degree. - H.A. at y = 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑛𝑢𝑚. 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑑𝑒𝑛. Ex. 3 Identify the horizontal asymptotes. a) f(x) = 3 𝑥 2 −5 2 𝑥 2 +3𝑥+1 b) g(x) = 𝑥 −1 𝑥 2 −1

Ex. 4 Identify the oblique asymptotes. O.A. occur when the degree of the numerator is > than the degree of the denominator. O.A. is found by dividing. (long or synthetic) y = answer (w/o remainder) Ex. 4 Identify the oblique asymptotes. a) f(x) = 𝑥 2 −𝑥 −2 𝑥 − 5 b) g(x) = 𝑥 2 −𝑥 −6 𝑥+1

Ex. 5 Identify all asymptotes & holes. (V.A., Holes, H.A., O. A) f(x) = 3𝑥 −2 𝑥+3 g(x) = 𝑥 3 −1 𝑥 2 −4 h(x) = 4 𝑥 2 −4 2 𝑥 3 −3 𝑥 2 +𝑥