1. A rational function is a quotient of two polynomials: where and are polynomials of degree n and m respectively. Most questions about a rational function can be answered by first factoring the numerator and denominator Example:

2. It’s all about finding the zeros of the numerator and the denominator. x (numerator zeros!) & y intercepts (x = 0) domain/vertical asymptotes (denominator zeros!) end behavior/horizontal asymptotes/ limits at ∞

3. More on Asymptotes For vertical asymptotes check the sign of to the left and right of the asymptote (LHL and RHL limits are useful here!) For end behavior/horizontal asymptotes, to evaluate the limit first factor out the largest power of x in the denominator and use the result Asymptotes “shape” the graph of a rational function!

4. End Behavior Examples (3 cases) 1.(n < m) 2. (n = m) 3.(n > m)

5. Putting it all together- Graphing Rational Functions 1.Factor numerator & denominator. 2.Find & plot x (numerator zeros) and y intercepts 3.Find domain/vertical asymptotes (denominator zeros) – sketch lines (check signs to left and right of vertical asymptotes – use sign lines) 4.End behavior/horizontal asymptotes: (divide by largest power of x in denominator) - sketch line 5.If needed plot additional points to fill in details

6. More Examples 1. 2. 3. Think! How can you factor these cubic equations?