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Curve Sketching Calculator

Curve-sketching helper. Enter f(x) on [a,b] to scan extrema, inflection points, monotonicity and concavity into a report—handy for hand-drawing and analysis.

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f′ determines monotonicity; f″ determines concavity; a sign change at f″ = 0 marks an inflection point
Supports + - * / ^ and sin cos tan sqrt exp log pi e
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📖 Tutorial｜Curve Sketching

1. Definition
Sketch by analyzing domain, parity, period, asymptotes; f′ gives monotonicity and extrema, f″ gives concavity and inflection.
2. Symbols
AnalysisConclusion
f′>0/<0Increasing/decreasing
f′=0 & sign changeExtremum
f″>0/<0Concave up/down
f″=0 & sign changeInflection
3. How it works
  • We sample f′ and f″ on the interval;
  • Detect sign changes;
  • Report extrema and inflection points.
4. Steps
  1. Enter f(x);
  2. Enter [a,b];
  3. Click Calculate for sketch points.
5. Example
Example: f=x³−3x. f′=3x²−3 gives extrema at x=±1; f″=6x gives an inflection at x=0.
6. Pitfalls
This is numerical; pair it with domain and asymptotes;
f″=0 is not always an inflection (it must change sign);
Handle discontinuities separately.

❓ FAQ｜Curve Sketching

Steps to sketch a curve?
Domain, parity/period, asymptotes, f′ for extrema, f″ for concavity/inflection, then plot key points.
Extremum vs inflection?
Extrema come from f′ sign changes; inflections from f″ sign changes (concavity flips).
Is f″=0 always an inflection?
No. f=x⁴ at x=0 has f″=0 but stays concave up.
How to find asymptotes?
Vertical: undefined points blowing up; horizontal: limit as x→∞.
Does it draw the graph?
It lists sketch points (extrema, inflections) so you can draw accurately.