Parametric Equations

9.1.1 Parametric Equations - Basics

Parametric equations represent a curve by expressing the coordinates $x$ and $y$ as functions of a third variable, typically t (the parameter). Instead of a direct relationship between $x$ and $y$ , you have:

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

Key Concepts:

Parameter: t controls both x and y, defining the curve as t varies.
Elimination of the parameter: You can sometimes eliminate t to find a direct y = h(x) relationship.
Differentiation: To find the gradient, use the chain rule: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} .$
Area under a curve can be found using $\int y \frac{dx}{dt} dt .$ Parametric equations are useful for modelling motion or curves that are difficult to describe with standard $y = f(x)$ equations.

A parametric is a third, non-coordinate value that can be used to define coordinate variables in functions.

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Example: The following function is defined in terms of a third non-coordinate variable $t$ :

x = 5t + 3

y = 2t - 4

Where $0 \leq t \leq 5$ .

To sketch this, we can create a table of $x$ and $y$ values:

t	0	1	2	3	4	5
x	3	8	13	18	23	28
y	-4	-2	0	2	4	6

In table mode on the calculator:

Three calculator screens are shown:

The equation for $x(t) = 5t + 3$ .
The equation for $y(t) = 2t - 4$ .
The table range settings with Start: $0$ , End: $5$ , Step: $1$ .