Curve Sketching Fundamentals

Being able to sketch curves from their equations is an essential skill at A-Level. This lesson brings together the techniques needed to sketch polynomial, rational, parametric and other curves, as required by the Edexcel 9MA0 specification.

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The Curve Sketching Checklist

When asked to sketch a curve, systematically consider:

1. Intercepts — where the curve crosses the axes.
2. Symmetry — is the curve symmetric about the y-axis (even function) or the origin (odd function)?
3. Asymptotes — are there vertical, horizontal or oblique asymptotes?
4. Stationary points — where is dy/dx = 0? What is their nature?
5. End behaviour — what happens as x → ±∞?
6. Special points — any points where the function is undefined?

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Intercepts

y-intercept

Set x = 0 and find y.

x-intercepts (roots)

Set y = 0 and solve for x. Use factorisation, the quadratic formula, or the Factor Theorem as appropriate.

Example: y = x³ − 4x = x(x² − 4) = x(x − 2)(x + 2).

x-intercepts: x = 0, x = 2, x = −2. y-intercept: y = 0.

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Asymptotes

Vertical Asymptotes