Difference between variable & parameter Just started calculus 3, and I'm doing vector functions. What I don't understand is how a parameter function is any different from a standard $y(x) = 5x$ function, $x$ is the parameter in this function right?
If we could graph in $4$ dimensions, then wouldn't we be able to have $t$ as an axis for the parametric equations?
Is it just so simple that we use "parameter" instead of "variable" just cause we can't graph in $4$ dimensions, in which saying that it's a parameter rather then a variable separates it from the $x,y,z$ terms?
 A: In general:

*

*We call "parameter(s)" the variable(s) of a given parameterization, where a "parameterization" of a given set $S$ means a function $f$ such that $Im(f)=S$. Thus, every parameter is a variable of a certain function.


*Every function is a parameterization (of its image). Therefore, every variable is also a parameter (of a certain parameterization).
The basic ideia is that when we talk about parameterization and parameter we are not interested in the function itself, but in the set that the function describes (i.e., the set parameterized by the function).
For example:

*

*$f:[-1,1]\to\mathbb R^2$ defined by $f(t)=(t,\sqrt{1-t^2})$ is a vector function of a single variable $t$.


*$g:[0,\pi]\to\mathbb R^2$ defined by $g(s)=(\cos(s),\sin(s))$ is a vector function of a single variable $s$.
We have $f\neq g$ (e.g., the domains are different). However, $Im(f)=Im(g)$, that is, $f$ and $g$ are different parameterizations of the same curve (semicircle, graph of the real valued function $y(x)=\sqrt{1-x^2}$). If you are interested in this particular curve, then you can choose the most convenient parameterization. Let us say that our parameterization is intended to describe the motion of a particle on this semicircle from the left to the right in function of time. Then:

*

*$f$ does not work because time should be positive.

*$g$ does not work because its motion goes from the right to the left.

We can fix it by changing the parameter $s$ of $g$ to $\pi-s$. The result is the different parameterization
$$h(s)=(\cos(\pi-s),\sin(\pi-s)),\quad s\in[0,\pi]$$
of the same curve, which goes from the left to the right with a positive parameter and thus suitable for our purposes.
