Lesson 8 2 Graphing Rational Functions Rational Functions

Lesson 8. 2: Graphing Rational Functions

Rational Functions A Rational function is a function of the form where P(x) and Q(x) are polynomials.

Graphing Rational Functions When graphing do the following first: 1. Factor both numerator and denominator 2. Find any vertical and horizontal (or slant) asymptotes 3. Find any holes 4. Find any zeros and the y-intercept if possible 5. Use these to help you sketch the graph 6. A Sign Chart may be useful as well, if needed.

Horizontal Asymptotes (Think End Behavior) Check degree of numerator and denominator to quickly identify if the rational function has a horizontal asymptote. • If deg. of numerator < deg. of denominator, then y = 0 is horizontal asymptote • If deg. of numerator = deg. of denominator, then divide leading coefficients from numerator and denominator to get HA. • If deg. of num. > deg. of den. , then no HA

Example Sketch

Example Sketch

Slant Asymptotes If the degree of the numerator is one degree greater than that of the denominator, then the rational function has a slant asymptote. This means that the curve becomes more and more linear as x increases or decreases without bound. To find the slant asymptote for which the curve approaches, use long division to find the Quotient: y = Quotient is slant asymptote.

Example Sketch

More Examples Sketch the graph of the following: a. b. c. d.