What math field is this called?

[kramer733]

Where you just study the graphs of equations but more abstractly. Don't you think that intuitively the cartesian coordinate system makes sense but at the same time, it's arbitrary? We could have made the left hand side the positive numbers and the right hand side the negative numbers.

Graphs depend on this don't they? So why don't we abstract these ideas more? We could have really inverted and more weird graphs. Don't the picture of the graphs depend on how we define how a function is drawn?

 

[CompuChip]

Yep, if you use a logarithmic axis then suddenly exponential functions are drawn as straight lines, and when you have a polar curve $r(\theta)$ you can plot it in polar coordinates ($x = r \cos\theta, y = r \sin\theta$) or you can plot r vs. $\theta$ ($x = r, y = r(\theta)$) and it looks completely different.
This is why mathematics usually tries to describe the functions in another way (e.g. as the solution of some equation, or by certain properties) rather than just giving it as a graph.

Note that some branches of mathematics - especially those like linear algebra and differential geometry, which are heavily influenced by physics - make quite a point out of writing things in a coordinate free way.

 

[mathwonk]

If f:S-->T is any function, the graph of f is a subset of the cartesian product SxT, consisting of all ordered pairs (s,t) such that t = f(s). You can picture this product space anyway you want or not at all.