Curve sketching from first and second derivatives
Build a graph from domain, intercepts, asymptotes, monotonicity, extrema, concavity, and end behavior—in that order.
LaTeX article Updated July 11, 2026
Start before the derivatives
Record the domain, symmetry, intercepts, and asymptotes first. These facts create the canvas on which derivative information must fit.
A sign chart cannot repair a graph that ignores a domain break or vertical asymptote.
First derivative: direction
Critical numbers occur where f′ is zero or undefined while f remains defined. Test intervals between them. A positive-to-negative sign change gives a local maximum; negative-to-positive gives a local minimum.
A zero derivative without a sign change is a stationary point, not an extremum.
Second derivative: shape
Positive f″ means slopes are increasing and the graph is concave up. Negative f″ means slopes are decreasing and the graph is concave down.
An inflection point requires a change in concavity. Solving f″ = 0 only produces candidates.
Worked example
Common mistakes
- Calling every f′ = 0 point an extremum.
- Calling every f″ = 0 point an inflection point.
- Skipping domain and asymptote analysis.
Keep these ideas
- Domain and end behavior frame the graph.
- Sign changes matter more than zeros alone.
- Combine first- and second-derivative information in one coherent sketch.