Mth141s42 - Section 4.2 More on Graphs of Rational...

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Section 4.2 - More on Graphs of Rational Functions There are three situations for horizontal asymptotes (H.A.): (1) deg. of num. = deg. of denom. H.A. : y = ratio of leading coefficients (2) deg. of num. < deg. of denom. H.A. : y = 0 , the x-axis (3) deg. of num. > deg. of denom. no H.A. ! Ex1. 2 2 2 3 2 ( ) 2 x x f x x x + - = + - Ex2. 2 4 ( ) 12 x f x x x + = - - Ex3. 2 12 ( ) 4 x x f x x - - = + What happens in these as x gets very large (- & +)?? Remember our reciprocal function ( 1 ( ) f x x = ). It has both vertical and horizontal asymptotes. Ex. 2 2 1 ( ) 4 x f x x - = - ( 1)( 1) ( 2)( 2) x x x x + - = + - The denominator becomes 0 for x = -2, or 2. So this function will have two vertical asymptotes at x = -2 and x = 2. See graph. x=-2 x=2 Note: The H.A. & V.A. are not part of the graph. Y-INTERCEPT: In our linear functions, quadratic functions, and higher degree polynomial functions, the y-intercept was the constant on the end of the function. But in our rational functions we have two constants on the end (one in top, one in bottom). For a rational function the y- intercept is the ratio of the two constant terms. Notice in the above function, the y- intercept is + 1 4 . Note if the numerator does not have a constant term you can think of it as 0. So ( ) 3 x f x x = + will have a y-intercept of 0 3 = 0. Remember that really to find
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the y-intercept we plug in 0 for x into the function to find the point (0,?). But in so doing, all the terms with x in them become 0, we are just left with the constant terms!
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