Rational Functions A function of the form where

Graphs of Rational Functions may have breaks in Continuity. Breaks in Continuity can appear

Vertical Asymptote If a rational expression is written in simplest form and the function

Point Discontinuity If the original function is undefined for x = a but the

Finding Horizontal Asymptotes for Rational Functions Given a rational function: f (x) = p(x)

• If m > n, then there are no horizontal asymptotes. • If

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Rational Functions A function of the form where p(x) and q(x) are polynomial functions and q(x) ≠ 0. Examples: (MCC 9 -12. F. IF. 7 d)

Graphs of Rational Functions may have breaks in Continuity. Breaks in Continuity can appear as: 1. Vertical Asympotes 2. Point Discontinuity (A hole in the graph) (MCC 9 -12. F. IF. 7 d)

Vertical Asymptote If a rational expression is written in simplest form and the function is undefined for x = a, then x = a is a vertical asymptote. Example: x = - 2 is vertical asymptote. (Note: Set the denominator equal to zero & solve for x. )

Point Discontinuity If the original function is undefined for x = a but the rational expression of the function in simplest form is defined for x = a, then there is a hole in the graph at x = a. Example: Point of Discontinuity as x = -2 (Note: If a factor cancels in the top & bottom, set it equal to zero & solve for x. )

Finding Horizontal Asymptotes for Rational Functions Given a rational function: f (x) = p(x) q(x) = am xm + lower degree terms bn xn + lower degree terms Let am be the leading coefficient of the numerator and m be the degree of the numerator. Let bn be the leading coefficient of the denominator and n be the degree of the denominator. (MCC 9 -12. F. IF. 7 d)

• If m > n, then there are no horizontal asymptotes. • If m < n, then y = 0 is a horizontal asymptote. • If m = n, then y = am is a horizontal asymptote. bn (MCC 9 -12. F. IF. 7 d)