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RATIONAL FUNCTIONS A rational function is a function of the form: px Rx where p and q qx are polynomials px Rx What would the domain of a rational qx function be? We’d need to make sure the denominator 0 2 Rx 5x 3 x x : x 3 Find the domain. x 3 H x x 2x 2 x : x 2, x 2 If you can’t see it in your x 1 head, set the denominator = 0 F x 2 and factor to find “illegal” x 5x 4 values. x 4x 1 0 x : x 4, x 1 The graph of f x 2 looks like this: 1 x If you choose x values close to 0, the graph gets close to the asymptote, but never touches it. Since x 0, the graph approaches 0 but never crosses or touches 0. A vertical line drawn at x = 0 is called a vertical asymptote. It is a sketching aid to figure out the graph of a rational function. There will be a vertical asymptote at x values that make the denominator = 0 Let’s consider the graph f x 1 x We recognize this function as the reciprocal function from our “library” of functions. Can you see the vertical asymptote? Let’s see why the graph looks like it does near 0 by putting in some numbers close to 0. 1 1 The closer to 0 you get f 10 for x (from positive 10 1 10 direction), the larger the function value will be Try some negatives 1 1 1 1 f 100 1 f 100 10 100 1 100 1 f 1 100 10 1 100 10 Does the function f x 1 have an x intercept? 1 0 x x There is NOT a value that you can plug in for x that would make the function = 0. The graph approaches but never crosses the horizontal line y = 0. This is called a horizontal asymptote. A graph will NEVER cross a vertical asymptote because the x value is “illegal” (would make the denominator 0) A graph may cross a horizontal asymptote near the middle of the graph but will approach it when you move to the far right or left Graph Q x 3 1 1 3 vertical translation, x x moved up 3 This is just the reciprocal function transformed. We can trade the terms places to make it easier to see this. The vertical asymptote Q x 3 1 remains the same because in x either function, x ≠ 0 The horizontal asymptote f x 1 will move up 3 like the graph x does. Finding Asymptotes VERTICAL ASYMPTOTES There will be a vertical asymptote at any “illegal” x value, so anywhere that would make the denominator = 0 So there are vertical x 2x 5 2 Rx 2 asymptotes at x = 4 x 1 x x 4 3x 4 0 and x = -1. Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be. HORIZONTAL ASYMPTOTES We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes. 1<2 degree of top = 1 If the degree of the numerator is less than the degree of the 2x 5 1 R x 2 the x axis is a denominator, (remember degree x 3x 4 is the highest power on any is horizontal asymptote. This x term) the x axis = 0. horizontal along the line y is a asymptote. degree of bottom = 2 HORIZONTAL ASYMPTOTES The leading coefficient is the number in front of If the degree of the numerator is the highest powered x equal to the degree of the term. denominator, then there is a degree of top = 2 horizontal asymptote at: y = leading coefficient of top 2x 4x 5 2 R x 2 leading coefficient of bottom 1 x 3x 4 degree of bottom = 2 horizontal asymptote at: 2 y 2 1 OBLIQUE ASYMPTOTES If the degree of the numerator is greater than the degree of the denominator, then there is not a degree of top = 3 horizontal asymptote, but an oblique one. The equation is x 2 x 3x 5 3 2 found by doing long division and Rx the quotient is the equation of x 3x 4 2 the oblique asymptote ignoring the remainder. degree of bottom = 2 x 5 a remainder x 2 3 x 4 x 3 2 x 2 3x 5 Oblique asymptote at y = x + 5 SUMMARY OF HOW TO FIND ASYMPTOTES Vertical Asymptotes are the values that are NOT in the domain. To find them, set the denominator = 0 and solve. To determine horizontal or oblique asymptotes, compare the degrees of the numerator and denominator. 1. If the degree of the top < the bottom, horizontal asymptote along the x axis (y = 0) 2. If the degree of the top = bottom, horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom 3. If the degree of the top > the bottom, oblique asymptote found by long division. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au