Rational Functions Summary

Algebra 2 · Axiom Academy

Let's review how rational functions model rates and inverse relationships, and how asymptotes, holes, and algebra reveal their behavior. A rational function is a quotient of two polynomials ; the domain always excludes every x where Q(x)=0 — find those exclusions before anything else. Not every zero of Q(x) survives as a vertical asymptote: factor first and cancel common factors — a factor that cancels leaves a removable hole, not an asymptote. Horizontal asymptotes come from comparing degrees: gives y=0 , gives , and gives none — unless the degrees differ by exactly 1 , which gives an oblique asymptote instead. A graph can never cross its own vertical asymptote, but it's free to cross a horizontal or oblique one — an asymptote describes end behavior, not a forbidden line. Clearing denominators with the LCD or cross-multiplication can manufacture solutions that don't actually work — always check every solution against the original excluded values. Rational inequalities need a full sign chart, not just algebra: test every interval formed by the zeros and the vertical asymptotes together, since the sign can flip at either kind of critical value. Core Concept Rational Functions & Domain

This is the written version of the interactive lesson above. See the full Algebra 2 course.