1. Rational Functions A function of the form where p(x) and q(x) are polynomial functions and q(x) ≠ 0. Examples: (MCC9-12.F.IF.7d)

2. 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) (MCC9-12.F.IF.7d)

3. 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.)

4. 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.)

5. Finding Horizontal Asymptotes for Rational Functions Given a rational function: f (x) = p(x) a m x m + lower degree terms q(x) b n x n + lower degree terms = Let a m be the leading coefficient of the numerator and m be the degree of the numerator. Let b n be the leading coefficient of the denominator and n be the degree of the denominator. (MCC9-12.F.IF.7d)

6. If m > n, then there are no horizontal asymptotes. If m < n, then y = 0 is a horizontal asymptote. If m = n, then y = a m is a horizontal asymptote. bnbn (MCC9-12.F.IF.7d)