What is the range of the quadratic function f(x,y) = (xy-x^2, xy-y^2)?

[JohnSimpson]

Consider the map $$f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$$
defined by
$$(x,y) \mapsto (xy-x^2, xy-y^2)$$

I'm interested in figuring out the range of this function, but I keep thinking myself in circles. What would be a systematic method for approaching something like this?

 

[haruspex]

Writing (u, v) for the new co-ordinates, you could look at a linear combination of these, u+m.v say, and find the extremal points as a function of m. This will give you a parametric equation describing the boundary.
In the present case, u+v is interesting.