1. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Objectives Identify and evaluate rational functions. Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph. 8.2 Rational Functions and Their Graphs

2. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Glossary Terms excluded values hole in the graph horizontal asymptote rational expression rational function vertical asymptote 8.2 Rational Functions and Their Graphs

3. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Vertical Asymptote 8.2 Rational Functions and Their Graphs In a rational function R, if x – a is a factor of the denominator but not a factor of the numerator, x = a is vertical asymptote of the graph of R.

4. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Horizontal Asymptote 8.2 Rational Functions and Their Graphs If degree of P < degree of Q, then the horizontal asymptote of R is y = 0. R(x) = is a rational function; P and Q are polynomials P Q If degree of P = degree of Q and a and b are the leading coefficients of P and Q, then the horizontal asymptote of R is y =. a b

5. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Horizontal Asymptote 8.2 Rational Functions and Their Graphs If degree of P > degree of Q, then there is no horizontal asymptote R(x) = is a rational function; P and Q are polynomials P Q

6. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Rules and Properties Hole in the Graph 8.2 Rational Functions and Their Graphs In a rational function R, if x – b is a factor of the numerator and the denominator, there is a hole in the graph of R when x = b (unless x = b is a vertical asymptote).

7. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Key Skills Identify all excluded values, asymptotes, and holes in the graph of a rational function. 8.2 Rational Functions and Their Graphs f(x) = 2x 2 + 2x x 2 – 1 factor: f(x) = 2x(x + 1) (x + 1)(x – 1)

8. Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Key Skills Identify all excluded values, asymptotes, and holes in the graph of a rational function. 8.2 Rational Functions and Their Graphs excluded values: x = –1 and x = 1 hole in the graph: x = –1 vertical asymptote: x = 1 horizontal asymptote: y = 2 TOC