# 1140 L17 - p(t 3000 0 t

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Lecture 17: Section 2.6 Rational Functions (cont.) Holes The graph of a rational function f ( x ) = p ( x ) q ( x ) has a hole at x = a if: 1. both p ( x ) and q ( x ) have the common factor ( x ± a ), and 2. the simpli±ed denominator DOES NOT have the factor ( x ± a ). NOTE: To ±nd the y -coordinate of the hole in the graph, plug x = a into the simpli±ed expression of f ( x ).

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ex. Find all vertical asymptotes and holes of the function: 1) g ( x ) = 2 x 3 ( x + 1)( x ± 3) x ( x + 1) 2 ( x ± 2)( x ± 3) Properties of Horizontal Asymptotes{HA Although a rational function can have many verti- cal asymptotes, it can have at most one horizontal asymptote. The graph of a rational function will never intersect a vertical asymptote but may intersect a horizontal asymptote. A rational funcion f ( x ) = p ( x ) =q ( x ) that is sim- pli±ed (all common factors have been canceled) will have a horizontal asymptote whenever the degree of q ( x ) is greater than or equal to the degree of p ( x ).
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1140 L17 - p(t 3000 0 t

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