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 Parametric Equations

Edexcel A level Maths: 8.3 Sketching Parametric CurvesCurated by Kepler, not produced by us. Watch it 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.

Example 1For

x = t^2

y = 2t

, plot the points at

t = -2, -1, 0, 1, 2

t=-2:(4,-4)

t=-1:(1,-2)

t=0:(0,0)

t=1:(1,2)

t=2:(4,4)

2These lie on

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

— a sideways parabola.

Answer

A 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 = 2t \Rightarrow x = \tfrac{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

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

To sketch a parametric curve, first build a table of:
- t, x, y values
- x only
- gradients
- asymptotes
Show answer and explanation
Answer: t, x, y values
Compute points.
Points should be joined in order of:
- increasing t
- increasing x
- increasing y
- size
Show answer and explanation
Answer: increasing t
Shows direction.
$x = t^2$ , $y = 2t$ is a:
- sideways parabola
- circle
- straight line
- ellipse
Show answer and explanation
Answer: sideways parabola
x = y²/4.
The t-range may mean the curve is:
- only part of the Cartesian curve
- always the full curve
- a single point
- undefined
Show answer and explanation
Answer: only part of the Cartesian curve
Restricted domain.
Arrows on a parametric sketch indicate:
- direction of increasing t
- asymptotes
- maxima
- roots
Show answer and explanation
Answer: direction of increasing t
Path direction.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.