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