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Parametric Equations - A Level Maths Definition

Reviewed by: Mark Curtis

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What are parametric equations?

In A Level Maths, parametric equations are a pair of equations (one for x and one for y) that define a curve in the x y-plane, but in terms of a third variable, t (or theta), called a parameter. They are written in the form

table row x equals cell straight f open parentheses t close parentheses end cell row y equals cell straight g open parentheses t close parentheses end cell end table

Example 1: the curve y equals open parentheses x plus 1 close parentheses squared can be written as y equals t squared where t equals x plus 1, which rearranges to give

table row x equals cell t minus 1 end cell row y equals cell t squared end cell end table

Example 2: the circle x squared plus y squared equals r squared can be written in terms of the parameter theta, where 0 less or equal than theta less than 2 pi, as

table row x equals cell r space cos space theta end cell row y equals cell r space sin space theta end cell end table

The two examples above are curves with traditional shapes, however parametric equations can represent far more complicated shapes, such as a loops and spirals, that would be hard (or even impossible) to write in terms of x and y only.

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Mark Curtis

Reviewer: Mark Curtis

Expertise: Maths Content Creator

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

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