Guidelines for Sketching a Curve

Curve sketching begins by analyzing a function that defines a curve on the coordinate plane with horizontal and vertical axes.

The domain shows where the function’s output is defined, setting the horizontal limits of the curve.

Intercepts provide key reference points for developing sketches, and identifying symmetries can simplify the process. Limiting behavior helps locate vertical and horizontal asymptotes that guide the overall shape of the curve.

First derivative tests give information about the slope of tangent lines and identify intervals where the curve increases or decreases. Critical points mark locations where this behavior may change, including possible local maxima or minima.

Analysis of the second derivative shows curvature by marking intervals where the graph is concave up or down and locating inflection points where the curvature changes. Together, these features provide essential information for accurate graph sketching.

One example is the logistic growth model, where population growth slows and levels off over time. Its curve includes a horizontal asymptote and an inflection point that shapes the S-curve.