Parametric Equations

An alternative to defining curves with Cartesian equations (one equation per curve with only x and y variables), we can express curves as separate equations of x and y, each in terms of the third variable t.

A curve y = f(x) can also be described using x = p(t) and y = q(t).

The coordinates of each point on the curve are (p(t), q(t))

It is possible to convert between Cartesian and parametric forms with substitution to eliminate one variable:

To convert x = 4t, y = 2t ² + 4 into Cartesian:

• x = 4t

• t = ¼x Rearrange to make t the subject

• y = 2t ² + 4

• y = 2(¼x) ² + 4 Substitute into y = q(t)

• y = ⅛x² + 4 Simplify

Domain & Range

For parametric equations x = p(t) and y = q(t) where the Cartesian equation is y = f(x):

• The domain of f(x) is the range of p(t)

• The range of f(x) is the range of q(t)

Trigonometry

Often, parametric equations will be given as trigonometric functions. When this is the case, it is often possible to use trig identities (see notes sheet on trigonometry) to convert between parametric and Cartesian.

Angles should always be taken in radians.

To convert x = sin t + 2, y = cos t - 3 into Cartesian:

• x = sin t + 2

• sin t = x - 2 Rearrange to make sin t the subject

• y = cos t - 3

• cos t = y + 3 Rearrange to make cos t the subject

• (x - 2)² + (y + 3)² = 1 Use identity sin²t + cos²t ≡ 1

Note that this is the equation of a circle.

Sketching Parametric Curves

To sketch a curve from parametric equations, either convert into Cartesian form - for example the equation above turned out to be a simple circle, radius 1 and centre (2, -3) - or tabulate the coordinates for different values of t and then plot these.