Parametric equations

In a nutshell

The coordinates of a point on a curve can be written as functions of a third variable. This is called the parameter and is often represented by $t$ .

Parametric equations

Parametric equations may be used to define a curve, where $x=p(t)$ and $y=q(t)$ . Each point on the curve will have coordinates $(p(t),q(t))$ .

Note: A given $x-$ coordinate may correspond to more than one point on the curve.

Parametric and Cartesian equations

You can convert parametric equations into Cartesian equations using substitution.

Note: The range and domain of the cartesian function can be found using the range and domain of the parametric functions. If the parametric equations are given by $x=p(t)$ and $y=q(t)$ and the Cartesian equation has equation $y=f(x)$ , then:

The domain of $f(x)$ will be the range of $p(t)$ .
The range of $f(x)$ will be the range of $q(t)$ .

Example 1

A curve has parametric equations $x=t+1$ and $y=t^2-3, \ \ -1\lt t\lt 5$ .

i) Find a Cartesian equation of the curve in the form $y=f(x)$ .

First, eliminate $t$ :

$x=t+1$

So, $t=x-1$ .

$y=t^2-3$

This means $\underline{y=(x-1)^2-3}$ .

ii) Write down the range of $f(x)$ .

Find the range of $y(t)=t^2-3$ .

This is a parabola, so it will have a minimum point of $(0,-3)$ , which occurs when $t=0$ .

The maximum point in the given range is given by $f(5)=5^2-3=22$ .

The range is $\underline{-3 \lt f(x) \lt 22}$ .