Use parametric equations in modelling in a variety of contexts.

The idea

Parametric equations describe a curve by giving x and y separately in terms of a third variable, the parameter, usually t or θ. In modelling, the parameter often represents time, so x(t) and y(t) track a moving point: a ball's horizontal and vertical positions, or a robot arm's coordinates. To use them, substitute values of t to generate points, eliminate the parameter to find a Cartesian equation, and differentiate to find gradients. For example, x = 3t, y = 2t² gives y = 2x²/9 after eliminating t. Contexts include projectile paths, circular motion, and curves that are awkward as y = f(x). Always state the range of the parameter and check whether the model's domain is restricted by the physical situation.

What earns the marks

• Correctly substitutes a given parameter value into both x(t) and y(t) to obtain a coordinate pair on the modelled curve.
• Eliminates the parameter correctly, for example rearranging x = 3t to t = x/3 and substituting into y = 2t² to obtain y = 2x²/9.
• Differentiates parametric equations using dy/dx = (dy/dt) ÷ (dx/dt), with dx/dt not equal to zero, to find a gradient at a stated point.
• Interprets the parameter in context, such as identifying t as time and restricting t to t ≥ 0 where the model requires it.
• States a suitable domain or range for the parameter and explains how the physical context limits it.

Common mistakes and how to put them right

• Eliminating the parameter by dividing one equation by the other without checking that the divisor is non-zero.

  Put it right: Rearrange one equation for t and substitute into the other.

• Forgetting to square the whole expression when substituting, for example writing y = 2x/3 instead of y = 2(x/3)².

  Put it right: Apply the exponent to the entire bracket.

• Ignoring the context's restriction on the parameter, such as allowing negative time.

  Put it right: State the valid interval for the parameter before interpreting the model.

Exam tips

• Write the parameter value and both coordinates clearly before substituting, so arithmetic slips are easy to spot.
• When eliminating the parameter, show the rearrangement step explicitly; this earns method credit even if later algebra slips.
• Check the final Cartesian equation against one known point from the parametric form to confirm the two forms agree.