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Functions AII.7 e 2009 Objectives: • Find the Vertical Asymptotes • Find the Horizontal Asymptotes Rational Functions A rational function f(x) is a function that can be written as p ( x) f ( x) q ( x) where p(x) and q(x) are polynomial functions and q(x) 0 . A rational function can have more than one vertical asymptote, but it can have at most one horizontal asymptote. Vertical Asymptotes If p(x) and q(x) have no common factors, then f(x) has vertical asymptote(s) when q(x) = 0. Thus the graph has vertical asymptotes at the zeros of the denominator. Vertical Asymptotes V.A. is x = a, where a represents real zeros of q(x). Example: Find the vertical asymptote of f ( x) 2x 2 x 1 2 q ( x ) x 1 ( x 1)( x 1) the zeros Since are 1 and -1. Thus the vertical asymptotes are x = 1 and x = -1. . Horizontal Asymptotes A rational function f(x) is a function that can be written as p ( x) f ( x) q ( x) where p(x) and q(x) are polynomial functions and q(x) 0 . The horizontal asymptote is determined by looking at the degrees of p(x) and q(x). Horizontal Asymptotes f ( x) p ( x) q ( x) a. If the degree of p(x) is less than the degree of q(x), then the horizontal asymptote is y = 0. b. If the degree of p(x) is equal to the degree of q(x), then the horizontal asymptote is leading coefficien t of p( x) y . leading coefficien t of q( x) c. If the degree of p(x) is greater than the degree of q(x), then there is no horizontal asymptote. Horizontal Asymptotes • deg of p(x) < deg of q(x), then H.A. is y = 0 f ( x) deg of p(x) = deg of q(x), then H.A. is p ( x) • q ( x) Example: • leading coefficient of p( x) y . leading coefficient of q( x) deg of p(x) > deg of q(x), then no H.A. Find the horizontal asymptote: f ( x) Degree of numerator = 1 Degree of denominator = 2 3x x2 1 Since the degree of the numerator is less than the degree of the denominator, horizontal asymptote is y = 0. . Horizontal Asymptotes • deg of p(x) < deg of q(x), then H.A. is y = 0 f ( x) p ( x) q ( x) • deg of p(x) = deg of q(x), then H.A. is leading coefficient of p( x) leading coefficient of q( x) • deg of p(x) > deg of q(x), then no H.A. y Example: 3x 1 . Find the horizontal asymptote: f ( x) 2x 1 Degree of numerator = 1 Degree of denominator = 1 Since the degree of the numerator is equal to the degree of the denominator, horizontal asymptote is y 3 . 2 Horizontal Asymptotes • deg of p(x) < deg of q(x), then H.A. is y = 0 f ( x) p ( x) q ( x) • deg of p(x) = deg of q(x), then H.A. is y leading coefficient of p( x) leading coefficient of q( x) • deg of p(x) > deg of q(x), then no H.A. Example: 3 x2 1 . Find the horizontal asymptote: f ( x ) 2x 1 Degree of numerator = 2 Degree of denominator = 1 Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Vertical & Horizontal Asymptotes V.A. : x = a, where a represents real zeros of q(x). Practice: H.A. : • deg of p(x) < deg of q(x), then H.A. is y = 0 • deg of p(x) = deg of q(x), then H.A. is leading coefficient of p( x) y leading coefficient of q( x) • deg of p(x) > deg of q(x), then no H.A. Find the vertical and horizontal asymptotes: 3x2 1 f ( x) 2x 1 Answer Now Vertical & Horizontal Asymptotes V.A. : x = a, where a represents real zeros of q(x). H.A. : • deg of p(x) < deg of q(x), then H.A. is y = 0 • deg of p(x) = deg of q(x), then H.A. is y Practice: leading coefficient of p( x) leading coefficient of q( x) • deg of p(x) > deg of q(x), then no H.A. Find the vertical and horizontal asymptotes: 3 x2 1 f ( x) . 2x 1 1 V.A. : x = 2 H.A.: none Vertical & Horizontal Asymptotes V.A. : x = a, where a represents real zeros of q(x). H.A. : • deg of p(x) < deg of q(x), then H.A. is y = 0 • deg of p(x) = deg of q(x), then H.A. is y Practice: leading coefficient of p( x) leading coefficient of q( x) • deg of p(x) > deg of q(x), then no H.A. Find the vertical and horizontal asymptotes: f ( x) 2x x 1 2 Answer Now Vertical & Horizontal Asymptotes V.A. : x = a, where a represents real zeros of q(x). H.A. : • deg of p(x) < deg of q(x), then H.A. is y = 0 • deg of p(x) = deg of q(x), then H.A. is y leading coefficient of p( x) leading coefficient of q( x) • deg of p(x) > deg of q(x), then no H.A. Practice: Find the vertical and horizontal asymptotes: f ( x) x 1 2 2x x2 1 V.A. : none H.A.: y = 0 is not factorable and thus has no real roots. Vertical & Horizontal Asymptotes V.A. : x = a, where a represents real zeros of q(x). H.A. : • deg of p(x) < deg of q(x), then H.A. is y = 0 • deg of p(x) = deg of q(x), then H.A. is y Practice: leading coefficient of p( x) leading coefficient of q( x) • deg of p(x) > deg of q(x), then no H.A. Find the vertical and horizontal asymptotes: 2x 1 f ( x) x 1 Answer Now Vertical & Horizontal Asymptotes V.A. : x = a, where a represents real zeros of q(x). H.A. : • deg of p(x) < deg of q(x), then H.A. is y = 0 • deg of p(x) = deg of q(x), then H.A. is y Practice: leading coefficient of p( x) leading coefficient of q( x) • deg of p(x) > deg of q(x), then no H.A. Find the vertical and horizontal asymptotes: 2x 1 f ( x) x 1 V.A. : x = -1 H.A.: y = 2