240 CHAPTER 4 Polynomial and Rational Functions 2 Find the Vertical Asymptotes of a Rational Function The vertical asymptotes of a rational function ( ) ( ) ( ) = R x p x q x , in lowest terms, are located at the real zeros of the denominator ( ) q x . Suppose that r is a real zero of q, so −x r is a factor of q. As →x r, the values − → x r 0, causing the ratio to become unbounded; that is, ( ) →∞ R x . Based on the definition, we conclude that the line = x r is a vertical asymptote. Finding Vertical Asymptotes Find the vertical asymptotes, if any, of the graph of each rational function. (a) ( ) = + − F x x x 3 1 (b) ( ) = − R x x x 4 2 (c) ( ) = + H x x x 1 2 2 (d) ( ) = − + − G x x x x 9 4 21 2 2 Solution EXAMPLE 4 (a) ( ) = + − F x x x 3 1 is in lowest terms, and the only real zero of the denominator is 1. The line = x 1 is the vertical asymptote of the graph of F. (b) ( ) = − R x x x 4 2 is in lowest terms, and the real zeros of the denominator − x 4 2 are −2 and 2. The lines = − x 2 and = x 2 are the vertical asymptotes of the graph of R. (c) ( ) = + H x x x 1 2 2 is in lowest terms, and the denominator has no real zeros. The graph of H has no vertical asymptotes. (d) Factor the numerator and denominator of ( ) = − + − G x x x x 9 4 21 2 2 to determine whether G is in lowest terms. ( ) ( )( ) ( )( ) = − + − = + − + − = + + ≠ G x x x x x x x x x x x 9 4 21 3 3 7 3 3 7 3 2 2 The only real zero of the denominator of G in lowest terms is −7. The line = − x 7 is the only vertical asymptote of the graph of G. As Example 4 illustrates, rational functions can have no vertical asymptotes, one vertical asymptote, or more than one vertical asymptote. Multiplicity and Vertical Asymptotes Recall from Figure 12 in Section 4.1 on page 199 that the end behavior of a polynomial function is always one of four types. For polynomials of odd degree, the ends of the graph go in opposite directions (one up and one down), whereas for polynomials of even degree, the ends go in the same direction (both up or both down). THEOREM Locating Vertical Asymptotes The graph of a rational function ( ) ( ) ( ) = R x p x q x , in lowest terms , has a vertical asymptote = x r if r is a real zero of the denominator q. That is, if −x r is a factor of the denominator q of the rational function R, in lowest terms, the graph of R has a vertical asymptote = x r. CAUTION If a rational function is not in lowest terms, this theorem may result in an incorrect listing of vertical asymptotes. j CAUTION When identifying a vertical asymptote, as in the solution to Example 4(a), write the equation of the vertical asymptote as x 1. = Do not say that the vertical asymptote is 1. j
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