Sketching Graphs of Functions and Their Derivatives

A graph is built from a small set of facts, and each derivative hands you a different kind of fact. f itself gives intercepts and asymptotes, the anchor points and boundaries. f prime tells you where the curve climbs or falls and where it turns around, at the zeros of f prime where the sign flips. f double prime tells you where the curve is cupped up or cupped down, and the inflection points are where that bend switches sign. None of these facts alone draws a curve, but together they lock down its whole shape, since a continuous function can only change behavior where one of these signs changes.

To sketch, work in stages. Find intercepts and any asymptotes first, since they set the frame. Then find f prime, locate its zeros and undefined points, and build a sign chart to mark increasing and decreasing intervals and classify local max and min points. Then find f double prime, locate its zeros, and build a second sign chart to mark concave up and concave down intervals and locate inflection points. Plot the key points, connect them respecting the sign charts, and let the curve approach any asymptotes correctly.

This topic is the payoff for everything else in the unit: the extreme value ideas, the first and second derivative tests, and concavity all exist so you can finally draw the full picture instead of just answering isolated questions about a function. It also previews later work, since reading a shape from its derivatives runs the opposite direction of sketching a derivative's graph from a given function's graph.