8 2 Rational Functions and Their Graphs Objectives

8. 2 Rational Functions and Their Graphs 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. Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

8. 2 Rational Functions and Their Graphs Glossary Terms excluded values hole in the graph horizontal asymptote rational expression rational function vertical asymptote Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

8. 2 Rational Functions and Their Graphs Rules and Properties Vertical Asymptote 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. Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

8. 2 Rational Functions and Their Graphs Rules and Properties Horizontal Asymptote P R(x) = is a rational function; Q P and Q are polynomials • If degree of P < degree of Q, then the horizontal asymptote of R is y = 0. • If degree of P = degree of Q and a and b are the leading coefficients of P and Q, then a the horizontal asymptote of R is y =. b Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

8. 2 Rational Functions and Their Graphs Rules and Properties Horizontal Asymptote P R(x) = is a rational function; Q P and Q are polynomials • If degree of P > degree of Q, then there is no horizontal asymptote Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

8. 2 Rational Functions and Their Graphs Rules and Properties Hole in the Graph 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). Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

8. 2 Rational Functions and Their Graphs Key Skills Identify all excluded values, asymptotes, and holes in the graph of a rational function. 2 x 2 + 2 x f(x) = x 2 – 1 factor: 2 x(x + 1) f(x) = (x + 1)(x – 1) Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

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