Curve Sketching Strategies

This lesson brings together all the tools of coordinate geometry, differentiation, and algebraic analysis to develop a systematic approach to sketching curves. Being able to produce clear, accurate sketches is a vital skill at A-Level — it helps you solve problems, check answers, and communicate mathematical ideas.

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The Systematic Approach

When asked to sketch a curve, work through the following steps:

1. Intercepts — where the curve crosses the axes.
2. Symmetry — is the function even, odd, or periodic?
3. Asymptotes — vertical, horizontal, and oblique.
4. Stationary points — maxima, minima, and points of inflection.
5. Behaviour for large |x| — what happens as x → ±∞?
6. Sign analysis — where is y positive or negative?
7. Put it all together — sketch the curve.

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Step 1: Intercepts

y-intercept

Set x = 0 and find y.

x-intercepts

Set y = 0 and solve for x (these are the roots).

Example 1: Sketch y = x³ − 4x.

y-intercept: x = 0 gives y = 0. Point: (0, 0).