Curve Sketching using Calculus
Curve sketching using calculus combines intercepts, stationary points and concavity from \(f'\) and \(f''\) to draw an accurate graph.
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Applications of calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
Calculus lets you sketch a curve from its key features — intercepts, stationary points and concavity. This Year 12 Mathematics Advanced topic (MAV-12-06) combines them into a graph.
Find the intercepts (\(x=0\) and \(y=0\)), the stationary points (\(f'=0\), classified), and the concavity / inflection (\(f''\)). Then plot and join with the right shape.
Method
- Intercepts: set \(y=0\) and \(x=0\).
- Stationary points: \(f'=0\), then classify.
- Concavity: use \(f''\); then plot and join.
| \(x(x^{2}-3)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(0,\ \pm\sqrt3\) |
| \(3x^{2}-3\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(\pm1\) |
| \(y''(-1)\) | \(<\) | \(0\ (\text{max})\) |
| \(y''(1)\) | \(>\) | \(0\ (\text{min})\) |
| \(6x\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(0,\ y=0\) |
Common pitfalls
Frequently asked questions
How do you sketch a curve using calculus?
Find the intercepts, the stationary points from f prime and classify them, and the concavity from f double prime, then plot the points and join them smoothly.
How do you find intercepts?
Set y equal to zero for x-intercepts and set x equal to zero for the y-intercept.
Why use the second derivative when sketching?
It tells you the concavity and locates points of inflection, so the curve is drawn with the correct bend.
What features should a good sketch show?
Intercepts, stationary points (maxima and minima), and points of inflection, all clearly labelled.