Complete Curve Sketch

Calculus 1 · Axiom Academy

Domain, intercepts, asymptotes, monotonicity, and concavity — the full analysis of a rational function. Work through a complete curve sketch of the rational function — find its domain, intercepts, asymptotes, where it rises and falls, and its concavity, then assemble the picture. Nice work — you ran a full curve sketch end to end. The systematic procedure: Domain: exclude where the denominator is zero — here . Intercepts: the y -intercept is f(0) ; the x -intercepts are where the numerator is zero — both land at the origin. Asymptotes: vertical where the denominator vanishes ( ); horizontal from the end behavior ( y = 0 , since the numerator has lower degree). First derivative: everywhere it is defined — f is decreasing on its whole domain, with no local extrema. Second derivative: flips sign at x = 0 , giving an inflection point at (0,0) . Synthesis: combine every piece — branches, asymptotes, slope, and bend — into one accurate sketch. This same checklist works for any function — practice it until the steps become automatic.

This is the written version of the interactive lesson above. See the full Calculus 1 course.