M1410206 - 2.66 SECTION 2.6 RATIONAL FUNCTIONS PART A...

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2.66 SECTION 2.6: RATIONAL FUNCTIONS PART A: ASSUMPTIONS Assume f x ( ) is rational and written in the form f x ( ) = N x ( ) D x ( ) , where N x ( ) and D x ( ) are polynomials, and D x ( ) 0 (i.e., the zero polynomial). Assume for now that N x ( ) and D x ( ) have no real zeros in common. Note: The textbook essentially makes this last assumption when it assumes that N x ( ) and D x ( ) have no common factors (over R ) aside from ± 1 , though, in Part E , we will consider what happens when we relax this assumption. Warning: Even though x 2 + x x = x + 1 x 0 ( ) , we do not consider the rational function [rule] f x ( ) = x 2 + x x to be a polynomial function [rule]. PART B: VERTICAL ASYMPTOTES (VAs) An asymptote for a graph is a line that the graph approaches. Example Let f x ( ) = 1 x 2 . Find any VAs for the graph of f . Solution Observe that 1 and x 2 have no real zeros in common. x 2 = 0 x = 2 , so the only VA has equation x = 2 .
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2.67 The graph of y = 1 x (on the left) is translated 2 units to the right to obtain the graph of y = 1 x 2 (on the right). We typically use dashed lines to indicate asymptotes. x = 2 is a VA for the graph on the right, because: (Actually, just one of the two statements below would be sufficient.) f x ( ) →∞ as x 2 + (i.e., as x approaches 2 from the right, or from higher numbers), and f x ( ) → −∞ as x 2 (i.e., as x approaches 2 from the left, or from lesser numbers). Under our Assumptions in Part A , the graph of f x ( ) = N x ( ) D x ( ) has a VA at x = c c R ( ) c is a real zero of D x ( ) f x ( ) → ∞ or −∞ as x c + , and f x ( ) → ∞ or as x c . Note: When we study logarithmic functions in Chapter 3 , we will see graphs that have “one-sided” VAs. Graphs of rational functions “shoot off” on both sides of any VAs.
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2.68 PART C: HORIZONTAL ASYMPTOTES (HAs) The graph of f x ( ) = N x ( ) D x ( ) has a HA at y = L L R ( ) f x ( ) L as x →∞
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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M1410206 - 2.66 SECTION 2.6 RATIONAL FUNCTIONS PART A...

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