Curve sketching from first and second derivatives
Build a graph from domain, intercepts, asymptotes, monotonicity, extrema, concavity, and end behavior—in that order.
How to recognize the method, run it, and know when it is the wrong choice.
Reviewed July 11, 2026Use the sign of f′ to determine increasing and decreasing intervals; use the sign of f″ to determine concavity. Critical numbers and inflection candidates organize the sign chart.
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.
Analyze f(x) = x³ − 3x.
Critical numbers are −1 and 1.
Concavity changes at zero.
Anchor extrema and the inflection point.
Common mistakes
- Calling every f′ = 0 point an extremum.
- Calling every f″ = 0 point an inflection point.
- Skipping domain and asymptote analysis.
Three takeaways
- Domain and end behavior frame the graph.
- Sign changes matter more than zeros alone.
- Combine first- and second-derivative information in one coherent sketch.