The idea

Converting to Cartesian form

Eliminating the parameter to find the Cartesian equation of a parametric curve, by substitution or rearrangement, and carrying the domain restriction from the parameter through to the Cartesian equation.

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Pure · Coordinate geometry

Converting to Cartesian form

Why it works

Eliminating the parameter

A parametric pair

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

and a single Cartesian equation in

x

and

y

can describe the same curve. Converting from one to the other means getting rid of the parameter — eliminating $t$ — so that only

x

and

y

are left.

The standard method: isolate t, substitute

Rearrange the simpler of the two equations to get

t = \ldots

in terms of

x

(or

y

), and put that into the other equation. If

x = t + 1, \qquad y = t^2,

then the parameter falls in two moves:

x = t + 1, \quad y = t^2 \;\longrightarrow\; y = (x - 1)^2

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