Parametric Equations

Parametric equations define x and y separately in terms of a third variable (parameter), usually t or θ. This allows us to describe curves that are difficult or impossible to express as y = f(x).

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Introduction

Instead of y = f(x), a curve may be defined by:

• x = f(t)
• y = g(t)

As t varies, the point (x, y) traces out a curve.

Worked Example 1

A curve is defined by x = 2t, y = t² − 1.

t	x	y
−2	−4	3
−1	−2	0
0	0	−1
1	2	0
2	4	3

This traces out a parabola. To find the Cartesian equation: t = x/2, so y = (x/2)² − 1 = x²/4 − 1.

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Converting from Parametric to Cartesian

Method

1. Express t in terms of x (or y).
2. Substitute into the other equation.
3. Simplify.

Worked Example 2

x = 3 cos t, y = 3 sin t.