General questions about Parametric equations My textbook doesn't explain this very well, so what I want to know is:


*

*What is the purpose of parametric equations?

*What is a parameter?

*What is the advantage of these equations over a function y=f(x)? 

*What do they essentially enable you to do? 


I find this topic to be impenetrable so a thorough explanation in simple terms would be very much appreciated. Thank you.
 A: We usually think of functions like $f(x)$ by their graph; for instance, to $f(x) = 1$ we associate a horizontal line, and to $f(x) = \sin x$ we associate the sin wave. However there is another good way to looking at functions: by their image. The function $f(x) = 0$ has domain and the real numbers, but its image is a single point (just 0). For $f(x) = \sin x$, the image is the interval $[-1,1]$. You might say that this is pretty much useless, because (for example) $f(x) = |x|$ and $f(x) = e^x$ have almost exactly the same images but are wildly different.
It turns out that looking at the image works a lot better in dimensions higher than 1, which is what parametric equations do. We can think of $f(x)$ as a parametric equation with just one parameter, namely $x$, and its image lies in just the real line. Similarly, the pair $(f(x), g(x))$ is a parametric equation whose image lies in the plane. If we take $f(x) = \cos x, g(x) = \sin x$, the image is a circle - much richer than in the one-dimensional case. Using more functions than just a pair we can look at paths in higher-dimensional space, like 3-space, and trace out lines in it. Using more parameters we can trace out surfaces or more complicated objects. Keep in mind that just as we can look at the function $f(x)$ as a parametrization of its image, which lies somewhere in the line, we can look at the pair $(f(x), g(x))$ as a function instead of a parametric equation. Its graph won't lie in the plane anymore though - we have to plot the argument against it, so it will lie in 3-space.
In summary, the parametric equation/function dichotomy isn't really a dichotomy at all. It just arises from two different ways of looking at functions - by their graphs and by their images.
Added: The fact that parametric equations represents a "broader" class of curves than the graph of simple functions arises because we can represent more thing in higher dimensions than in lower ones. When we go from the graph to the image, we're really just projecting down by collapsing an axis. For instance, the way we go from a sin wave to the interval $[-1,1]$ in my  example is by squishing the $x$-axis down to a point.
A: I don't believe that there is a specific definition of what a parameter really is; it is implicitely understood as a one-dimensional (mostly real, but it could be complex) number used to describe a set. For example, a line that passes through the origin and a vector $(a,b,c)$ in $\mathbb{R}^3$ can be described geometrically as
$$
V:\{(at,bt,ct)\in\mathbb{R}^3|t\in\mathbb{R}\}
$$
where $t$ is the parameter.
Note that the choice of the letter $t$ for the parameter many times is associated with the idea of time as a variable in applications, and its used as such in the theory of ordinary differential equations, for example.
A: Parametric equations allow you to represent figures that you would not be able to represent using standard functions. This is because of the strict requirement of the function that it pass the vertical line test, i.e., "no two $x$'s have the same $y$". But we'd like to have circles, or ellipses, which clearly fail this criterion!
The solution is to use parametric equations, which allows one to describe a figure by a $group$ of functions, rather than as a singular function. A basic example is the unit circle, which can be described parametrically as 
$$(x(t),y(t))=(\cos t,\sin t)$$
Here, the parametrization is dependent on the "free" variable $t$ (whose domain is $[0,2\pi]$, since that is the period of the circle). The $x$-coordinate is determined by $\cos$ and the $y$-coordinate by $\sin$. As $t$ goes from $0$ to $2\pi$, the given parametrization "traces out" the circle. Hope that helps.
A: A parametrization is just another type of function. For example lets say we have a regular old function of one variable $y=f(x)$. Then we can view the graph of the function (which is in $\mathbb{R}^2$) as the image of a function $g:\mathbb{R}\to \mathbb{R}^2$ given by $s(t)=(t,f(t))$. Here we call $t$ a parameter because for every $t\in \mathbb{R}$ corresponds to exactly one point of the graph of $y=f(x)$. Viewing $t$ as representing time we can think of the parametrization of the graph of $y=f(x)$ as the path of a particle traveling along the graph. For example at time $t=1$ the particle is at the point $(1,f(1))$, at time $t=2$ the particle has now arrived at the point $(2,f(2))$ and so forth. The parametrization now allows us to view the graph of $y=f(x)$ not as a static object, but as a dynamical object, i.e., it is the path traversed by a particle through the evolution of time. But certainly graphs of functions are not the only spaces parametrizable. For example circles are parametrized by trigonometric function and circles are not the graphs of functions. Higher dimensional manifolds (shapes) such as surfaces may be parametrized as well, but in this case we can't view the parametrization as the trajectory of a particle in time, but the case of curves I think is sufficient for getting a feel of what a parametrization is and why they are useful.
