How to Analyze Rational Functions: Asymptotes, Holes, and Domains

Miacademy & MiaPrep Learning Channel

This comprehensive math lesson guides students through the algebraic analysis of rational functions, serving as a critical bridge between simplifying expressions and graphing. The video begins by defining rational functions as ratios of polynomials and explains how to identify domain restrictions where the denominator equals zero. It then categorizes these restrictions into two types of discontinuities: vertical asymptotes (infinite discontinuity) and holes (removable discontinuity), demonstrating how factoring determines which type occurs. The lesson progresses to analyzing end behavior through horizontal and oblique (slant) asymptotes. It provides clear, rule-based methods for determining asymptotes based on the comparative degrees of the numerator and denominator polynomials. By using specific examples, the video illustrates three scenarios: when the denominator's degree is larger, when degrees are equal, and when the numerator's degree is exactly one greater than the denominator's. Ideal for Algebra II and Precalculus classrooms, this resource helps students understand the "why" behind the shapes of rational graphs. It specifically addresses common misconceptions, such as the belief that functions can never cross an asymptote, clarifying that horizontal asymptotes only describe end behavior. Teachers can use this video to introduce the analytic steps required before students ever put pencil to graph paper.