Graphing Rational Functions

H4 Graphing Rational Functions ______________________________________________________________________________________ 9 Rational functions: Equations of the form f ( x) = P( x) Q( x) where P (x ) and Q (x ) are both polynomials. 9 y intercept: This is the point where the curve crosses the y axis. Set x = 0 and solve for y. 9 x intercept: This is the point where the curve crosses the x axis. Set the numerator equal to zero and solve for x. 9 Vertical Asymptotes: Find the values of x which make the denominator equal to zero. Warning: The function must be in lowest terms! Why? 9 Horizontal Asymptotes: 1. If the degree of numerator is less than degree of denominator, the graph has a horizontal asymptote y = 0. Why? 2. If the degree of numerator equals the degree of denominator then the horizontal asymptote is equal to the ratio of the leading coefficients. Example: y = 5x 2 + 6 x − 7 3x 2 Horizontal asymptote y = 5 3 9 Slant Asymptotes: 1. If the degree of the numerator is one more than the degree of the denominator, perform the long division P ( x) ÷ Q ( x) . The quotient is the slant asymptote. (We ignore the remainder). Warning: The rational function must be fully reduced before asymptote analysis takes place! H4 Example: Graph • x - intercepts – (4, 0) • y - intercepts – (0, 4/3) • vertical asymptotes – The lines x = 2 and x = 6 are the vertical asymptotes. • horizontal asymptotes – Degree of the numerator equals the degree of the denominator Î The horizontal asymptote is the ratio of the leading coefficients. The horizontal asymptote is y = 1 (from 1/1). Graph other points as needed….