“This is the most magnificent discarded living room set I've ever seen.” 3.3: Properties of Rational Functions Rational Functions A rational function is a function of the form p(x) R(x) q(x) where p and q are polynomial functions. The domain consists of all real numbers except those for which the denominator is zero. Asymptotes An asymptote is a line that a function approaches as x or y goes to ∞ or -∞. Horizontal asymptotes The graph of a function may cross a HA Vertical asymptotes The graph of a function will never cross a VA Oblique asymptotes The graph of a function may cross an OA Vertical Asymptotes Given a rational function R(x), a vertical asymptote x = r occurs for all values of r for which the denominator is zero but the numerator is not. x R(x) 2 x 4 Vertical Asymptotes Given a rational function R(x), a vertical asymptote x = r occurs for all values of r for which the denominator is zero but the numerator is not. x2 9 G(x) 2 x 4 x 21 Vertical Asymptotes Find the vertical asymptotes and holes, if any, of the graphs of the following functions. State the domain of each function x4 1 F(x) 3 x x x2 G(x) 2 x x 12 x 2 2x 24 H(x) x 2 16 Horizontal Asymptotes Given a rational function R(x) p(x) , a horizontal asymptote q(x) y = b occurs if the degrees of p and q are equal OR if the degree of q > p (in other words, if R(x) is bottom heavy) Equal: Degree Bottom heavy: Degree q > p q = p HA occurs at the quotient of the leading coefficients 3x 2 3x F ( x) 2 2 x 5x 3 6 Always have a HA at y = 0 G(x) 5 4 3 2x 1 2x 3 4 x 2 2 6 5 4 3 2 2 1 1 -6 -5 -4 -3 -2 -1 -1 -2 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 -1 -2 -3 -3 -4 -4 -5 -5 -6 -6 1 2 3 4 5 6 Oblique Asymptotes Given a rational function R(x) p(x) , an oblique asymptote q(x) y = ax + b occurs if the degrees of q < p (in other words, if R(x) is top heavy). Use long division to find the OA. 3 x H(x) 2 2x 8 8 6 4 2 -8 -7 -6 -5 -4 -3 -2 -1 -2 -4 -6 -8 1 2 3 4 5 6 7 8 Horizontal or Oblique Asymptote? State whether the following functions contain a horizontal or oblique asymptote. Then, find it! x4 1 F(x) 3 x x x2 G(x) 2 x x 12 x 2 2x 24 H(x) x 2 16 Recap Let’s recap.. how ‘bout it?! Type Vertical Asymptotes x=r Horizontal Asymptotes y=b Oblique Asymptotes y = ax + b Can function cross it? When does a rational function have one? How do you find it? 3.3: Properties of Rational Functions HW #5: p.196 #11-27 odd, 44, 41-51 odd