A function in mathematics is a very precise thing – one of the conditions is that a function can take a number in the domain to only one other number in the range at a time. For example $y=x^{2}$ will take 2 to 4 but no other number and $y=x^{3}$ will take $-3$ to $-27$ and no other number. This doesn’t mean that a number in the range can only have come from one number, for example $y=x^{2}$ will also take $-2$ to $4$.

We often see graphs of functions such as $y=x^{2}$ and $y=\mathrm{e}^{x}$ and it is a common mistake to think that every graph is the graph of a function – this is not true. For example take the graph of $y=\sqrt{x}$ as shown below

Graph drawn using SAGE Math

Graph of a curve that is not a function (?)

this graph is not the graph of a function because $x=4$ is taken to both $-2$ and $2$; this is not allowed by the definition of a function.

To resolve this we have to introduce a new independent variable (often $t$) called a parameter and make $x$ and $y$ functions of the parameter $t$. This is like adding a third axis to the usual $x$-$y$ plane – this new axis is the $t$-axis.

Here is a graph of the parametric equations $x=\mathrm{cos}t$ and $y=\mathrm{sin}^{3}t$

Graph of a parametric relation

2-dimensional graph of a parametric equation created in SAGE Math

This, however, appears not to satisfy the definition of a function since there are several values of $x$ that are taken to two different $y$-values. Lets look a little closer at what is going on. If we add a further axis – the $t$-axis – to the diagram then this is what we see,

SAGE Math 3D Parametric Plot

A 3-dimensional parametric plot created in SAGE_Math

As you can see from the plot each value of $t$ corresponds to exactly one point in three-dimensional space.

If we look at this plot from a certain angle – in particular, straight down the $t$-axis we see the following

SAGE Math 3D Parametric Plot

3-dimensional parametric plot created in SAGE Math

Now if we ignore the effects of perspective we see exactly the original graph that we started with – so even though we originally thought that this did not satisfy the conditions to be a function, it actually does. By using the trigonometric identity $\mathrm{sin}^{2}t+\mathrm{cos}^{2}t \equiv 1$ we can show that the Cartesian form of this curve is given by $y^{2}=(1-x^{2})^{3}$ which is much more difficult to deal with than the parametric form of $x=\mathrm{cos}t$ and $y=\mathrm{sin}^{3}t$

Similarly $y=\sqrt{x}$ can be represented by the parametric equations $x=t^{2}$ and $y=t$ and again looking at the three dimensional plot straight down the $t$-axis gives a graph like the one earlier in the post. So sometimes things aren’t always what they first seem. With the help of parametric equations we can produce graphs of some very interesting curves that we can now treat as functions – not only that but parameterisations of curves and paths are critical to being able to do higher level calculus.

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