There are actually 3 different things here: graphs of functions, "Cartesian Equations", and parametric equations.

The distinction between graphs of functions and "Cartesian Equations" (to use your terminology) is best seen by considering a circle.

The equation

is satisfied by a set of points (x, y) in the plane that form the unit circle at the origin. But it's not the graph of any function y = f(x).

(Would f(0) be 1, or -1? It can't be both for a function f. This is the origin of the so-called "vertical line rule", which the circle fails.)

So by "Cartesian Equations", I assuming you're including things like

, which is not the graph of any function. (It's the union of the graphs of 2 different functions,

and

.)

The following are the 3 types of sets in the plane it seems you want to consider:

1. Graphs of Functions:

(where

is the function's domain.)

2. "Cartesian Equations":

3. Parametric Equations:

, where

.

The differences? Well, every graph of a function y = f(x) could be represented by a "Cartesian Equation" or a parametric equation.

Ex:

when x is in f's domain, and define

however you like, other than zero, otherwise. And for parametric, set

, where the t-domain is f's domain.

However, not every "Cartesian Equation" can be

*continuously* parameterized.

, whose graph is the union of a parabola and the x-axis.

For practical purposes, the same is true in reverse (the formal claim would be a mess, and likely be false), meaning that not every parametric equation has the same image as a "Cartesian Equation"

Ex: Try to find F such that

where

.

Note that the set of points in the plane via "Cartesian Equation" is given by a function inverse at a point (it's the set

), while in the parametric case it's the image set of a function (the function Q).

Also, if you don't put some minimal restrictions - especially continuity - on the possible functions F and Q for "Cartesian Equations" and parametric equations respectively, then ANY subset of the plane comes from such an F or Q.

Ex: For F, you could use 1 minus the indicator function, meaning F of a point is 0 if the point is in the given set, and 1 otherwise. For Q, one could exploit a set bijection between

and

.

Even requiring continuity, there are very counter-intuitive examples possible under these definitions.

Ex: F(x,y) := 0 is the "Cartesian Equation" whose graph is the entire plane. Likewise, parametricly, there are

__continuous__ space-filling curves (like the Peano curve:

https://en.wikipedia.org/wiki/Peano_curve)... counter-intuitive indeed!

A set of points in a plane is not a function! A corresponding function is uniquely recoverable from that only in the first case, where you're told that the set of points is the graph of a function. In the other two cases, "Cartesian coordinates" and parametric equations, there are multiple functions F and Q, respectively, which could produce those sets.

Ex: For the "Cartesian Equation" case, if F defines the set in the plane, then kF defines the same set for any non-zero constant k. For parametric, consider the line y = x via

, OR

.

The functions F and Q contain more information than you get from just the set of points in the plane given by

and

respectively!

The "best" form in general is the parametric form, because it contains the most information, is the most flexible, is the most direct (it's just a function

) and is the most useful for modelling.

Parametric Equations common use of t is, of course, for "time". So you think of parametric equations as describing a point moving in the plane in time - which clearly contains vastly more information than just the set that moving point eventually covers. And even more valuable than all that additional information is that the description of a point moving through time is EXACTLY what's needed to model most dynamic real world scenarios.

At the level of generality of your question - asking for "the relationship" between them - that's about all I can think to say.