Parametric Equations

In many situations it is convenient to express the x- and y-coordinates of a curve separately in terms of a third variable called a parameter (usually t or θ). This lesson introduces parametric equations and their graphical interpretation, as required by the Edexcel 9MA0 specification.

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What Are Parametric Equations?

Instead of writing y as a function of x directly (the Cartesian equation), we write:

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

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

Example: x = 2t, y = t² defines a parabola. When t = 0: (0, 0). When t = 1: (2, 1). When t = −1: (−2, 1). When t = 2: (4, 4).

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Why Use Parametric Equations?

• Some curves cannot be written as y = f(x) because they fail the vertical line test (e.g. circles).
• Parametric equations can describe the direction and speed of motion along a curve.
• Many physical applications (projectile motion, circular motion) naturally use a parameter for time.

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Common Parametric Forms

Circle

A circle with centre (a, b) and radius r: