Curve Sketching using Calculus
Learn curve sketching using calculus for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-06). A careful sketch brings together everything the derivatives reveal, so you can draw an accurate graph of an unfamiliar function without a graphing app.
You will learn to combine intercepts, stationary points, points of inflection, domain, symmetry, asymptotes and end behaviour into one clear sketch, for polynomials, exponentials and other functions — a skill that pulls the calculus course together.
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.