8.3PureCore

Sketching Parametric Curves

To sketch a parametric curve, build a table of values of t, compute the corresponding points, and trace the path in the direction of increasing t. Converting to Cartesian can confirm the shape.

24 min Video by Zeeshan Zamurred Parametric Equations

Edexcel A level Maths: 8.3 Sketching Parametric CurvesWatch the full walkthrough before the notes below.

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What you'll be able to do

Build a table of (t, x, y) values
Plot points and join them in t-order
Identify the direction of travel
Confirm shape via the Cartesian equation

Table of values

Pick several values of $t$ across its range, compute $x$ and $y$ for each, and plot. Joining points in increasing $t$ shows the direction the curve is traced.

t = - 2 : (4, - 4)

t = - 1 : (1, - 2)

t = 0 : (0, 0)

t = 1 : (1, 2)

t = 2 : (4, 4)

2These lie on

x = \frac{y ^{2}}{4}

— a sideways parabola.

AnswerA parabola opening in the positive x-direction.

Tip — Arrows on the sketch show increasing t — examiners often ask for direction.

Confirming the shape

Eliminate $t$ to get the Cartesian equation, which reveals whether the curve is a line, parabola, circle, ellipse, etc. Remember the parameter range may give only part of the full Cartesian curve.

Formula recap

(x, y) = (f (t), g (t))

Compute points from t.

x = t^{2}, y = 2 t \Rightarrow x = \frac{y ^{2}}{4}

Sideways parabola.

Common mistakes to avoid

Joining points in the wrong order.

Join in increasing t to show direction.

Assuming the whole Cartesian curve is traced.

The t-range may produce only part of it.

Key takeaways

Tabulate t, x, y and plot the points.
Join in increasing t and mark the direction.
Eliminate t to confirm the shape (mind the t-range).

Test yourself

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