Graphmatica's equation parser will first attempt to isolate the variable y wherever it is in the equation. If that fails, it will try to isolate x instead. It will graph some relations, like circles (x^2 + y^2 = 36) and ellipses (x^2/3 + y^2/4 = 20), as well as hyperbolas, sideways parabolas (x = y^2) and many other conic sections. (Consult a good Algebra II textbook for help on their formulas.) The only limitation for graphing an equation as a Cartesian function is that there must be a single occurrence of one of the variables y or x. The relation graphing module (for graphs which may have more than one y-value for a given x value) works like this: if in isolating the y in an equation Graphmatica finds an even power of it (i.e. y^2), it makes two equations for that graph, one with the positive and one with the negative root. This method by no means covers all possible relations, but it is adequate for the most common.
Implicit functions involving multiple instances of both x and y (e.g. x+cos x = y^2+3y) can be graphed as well, provided the partial derivatives with repect to x and y can both be computed. However, some features available for Cartesian functions (finding intersections, critical points, and derviatives/integrals) are not supported for implicit functions.