Graphing Rational Functions

Pre-Calculus · Axiom Academy

LESSON Graphing Rational Functions One strategy turns any quotient of polynomials into a graph: find the asymptotes, holes, and intercepts, then let the sign of the function fill in the curve. 1. Factor First: Vertical Asymptotes and Holes Everything starts with factoring the top and bottom. The denominator's zeros are the only candidates for trouble. A zero that survives cancellation is a vertical asymptote ; a factor that cancels with the numerator leaves a single missing point — a hole . Factor top and bottom; the shared (x+2) cancels Surviving zero x=2 is the vertical asymptote 2. Compare Degrees: the End-Behavior Asymptote Far from the origin a rational function behaves like the ratio of its leading terms . Comparing the degree of the top, n , to the degree of the bottom, m , tells you the horizontal or slant asymptote without plotting a single point. The bottom grows faster, so . Horizontal asymptote y = 0 . Tops out at the ratio of leading coefficients: . No horizontal asymptote — a slant line. Divide to find it. End behavior follows a polynomial of degree n-m ; no linear asymptote. Both x^2 - x - 6 and x^2 - 4 are degree 2 with leading coefficient 1 , so the ends of the graph flatten onto . (A function may cross a horizontal asymptote in the middle, but it returns to it as .) 3. Intercepts, Signs, and the Finished Graph

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