Complete Rational Function Analysis

Pre-Calculus · Axiom Academy

EXAMPLE Complete Rational Function Analysis Work through every key feature of one rational function — domain, intercepts, and asymptotes — then read it off the graph. Fully analyze the rational function : find its domain, its x - and y -intercepts, its vertical asymptotes (and any holes), and its horizontal asymptote — then sketch it. Vertical asymptotes (dashed red) at x = −2 and x = 2; horizontal asymptote (dashed green) at y = 1; intercepts marked in orange. No holes — numerator and denominator share no common factor. Nice work — you analyzed feature by feature, then assembled the full graph. Domain first: exclude the denominator's zeros — here x = 2 and x = -2 , so the domain is . Check for holes before asymptotes: a denominator zero is a vertical asymptote only if it doesn't cancel. The numerator (x-1)(x+1) and denominator (x-2)(x+2) share no factor, so both zeros are asymptotes and there are no holes. Equal degrees → horizontal asymptote from leading coefficients: , so y = 1 . Intercepts: numerator = 0 gives the x -intercepts (-1,0) and (1,0) ; gives the y -intercept (0,0.25) . The same checklist — domain, holes vs. asymptotes, end behavior, intercepts — analyzes any rational function before you ever plot a point.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.