8.1PureCore

Parametric Equations

Instead of one equation linking x and y, parametric equations give both coordinates in terms of a third variable, the parameter t. This is natural for motion and for curves that fail the vertical line test.

26 min Video by Zeeshan Zamurred Parametric Equations

Edexcel A level Maths: 8.1 Parametric Equations (Domain and Range)Watch the full walkthrough before the notes below.

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What you'll be able to do

Understand x = f(t), y = g(t)
Convert parametric to Cartesian by eliminating t
Find the domain and range of a parametric curve
Plot points from parameter values

The idea

A parametric curve is given by $x = f (t)$ and $y = g (t)$ . Each value of $t$ produces a point $(x, y)$ . Eliminating $t$ recovers the Cartesian equation.

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

Both coordinates depend on the parameter t.

Eliminating the parameter

Make $t$ the subject of one equation and substitute into the other. The domain of $x$ becomes the domain of the Cartesian curve; the range of $y$ becomes its range.

1From the first,

t = x - 1

2Substitute:

y = (x - 1)^{2}

Answer

y = (x - 1)^{2}

Tip — The parameter’s range controls the domain and range of the final curve.

Formula recap

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

Parametric form.

t = f^{- 1} (x) \Rightarrow y = g (f^{- 1} (x))

Eliminate t.

Common mistakes to avoid

Ignoring the range of t.

The parameter range restricts the domain/range of the Cartesian curve.

Forgetting to substitute fully when eliminating t.

Replace every t so the result is purely in x and y.

Key takeaways

x = f(t), y = g(t) give a curve via a parameter.
Eliminate t to get the Cartesian equation.
The range of t sets the domain and range of the curve.

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