Parametric equations

The idea

Parametric equations give $x$ and $y$ in terms of a third variable $t$ . To get a Cartesian equation, make $t$ the subject of one equation and substitute it into the other. The range of $t$ limits which part of the curve is drawn.

Rearrange the simpler equation for $t$ .
Substitute into the other to remove $t$ .
Use the range of $t$ to give the allowed $x$ and $y$ values.

Watch out The Cartesian curve can be longer than the parametric one: check the range of $t$ .

See it worked

Worked example

$x = t + 1$ , $y = t^2 - 1$ . Find the Cartesian equation.

$t = x - 1$ .
$y = (x - 1)^2 - 1$ .
$y = x^2 - 2x$ .

Answer $y = x^2 - 2x$

Your turn

Question 1Non-calculator3 marks

$x = 2t$ , $y = t^2 + 3$ . Find $y$ in terms of $x$ .

Question 2Non-calculator3 marks

A curve has $x = t^3$ , $y = t^2$ . Find the exact gradient at $t = 1$ .

Question 3Calculator allowed3 marks

A ball's path is $x = 20t$ , $y = 15t - 4.9t^2$ (metres, $t \ge 0$ ). Find how far it travels horizontally before it lands ( $y = 0$ ). Give your answer to 3 significant figures.

In past papers

13 official questions in 13 papers since 2018. The latest:

All A-level Maths past papers

A-level Maths Pro, Pearson Edexcel 9MA0

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