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.

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

Where $0 \leq t \leq 5$.

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

In table mode on the calculator:

Three calculator screens are shown:

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

Example: Write $x = 5t + 3$ and $y = 2t - 4$ where $0 \leq t \leq 5$ as a Cartesian equation.

1. Rearrange any of the two equations to say $t = \ldots$ From $x = 5t + 3$:

2. Substitute into the other equation: Substitute $t = \frac{x - 3}{5}$ into $y = 2t - 4$: