2-1 - 2-1 Rational Functions Asymptotes and asymptotic...

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Unformatted text preview: 2-1 Rational Functions Asymptotes and asymptotic behavior Think about the function 1 y= x the various routes x can take, and the consequent behavior of y: as x goes from 1 to +oo from 1 to 0 from -1 to -oo from -1 to 0 sign of y + + - magnitude of y gets smaller and smaller gets bigger and bigger gets smaller and smaller gets bigger and bigger In fact, the graph will look something like this: When a portion of a graph: • extends infinitely in such a way as to get closer and closer to a straight line without actually joining it • we say that the line is an asymptote of the graph, and • that the graph approaches the line asymptotically 2-1 p. 1 A rational function is a function of the form: P (x ) ax + ... f(x) = = Q( x ) bx +... where P and Q are polynomials. n m The key to understanding a rational function lies in finding its asymptotes VERTICAL ASYMPTOTES: • occur at zeros of denominator • graph cannot cross a vertical asymptote • it will go (off-scale) to either +oo or - oo as it approaches a vertical asymptote HORIZONTAL ASYMPTOTE: Three cases: Horizontal asymptote deg P > deg Q deg P = deg Q deg P < deg Q none horizontal line y = a/b x-axis • graph will approach the horizontal asymptote both at extreme left and extreme right of coordinate system 2-1 p. 2 Graphing a rational function 3n − 6 f ( n) = Example: n≥1 n −1 1. Draw in vertical asymptotes (at zeros of denominator). zero of denominator n = 1 2. Draw in horizontal asymptote (if any). degree of numerator = degree of denominator, so y = 3 is horizontal asymptote. 3. Plot a few representative points f(2) = 0, f(3) = 3/2, f(6) = 12/5, f(9) = 21/8 4. Draw. • graph must approach horizontal asymptote at extremities (in this case extreme right). • graph must go off-scale at vertical asymptotes f(n) 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 n 20 −1 −2 −3 −4 −5 2-1 p. 3 ...
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This note was uploaded on 12/11/2011 for the course MATH 1324 taught by Professor Staff during the Spring '11 term at Austin Community College.

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2-1 - 2-1 Rational Functions Asymptotes and asymptotic...

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