I'd like to get a graphical approach to Analysis in higher dimension vector spaces, such as $\mathbb{R^n}$. To make this easier, my goal is to be able to visualize the graph of functions from $\mathbb{R^2}$ to $\mathbb{R}$. If you take one of the most basic functions in one dimension $ f(x) = x $, it's pretty straightforward to see that it's a line with slope 1. Now, consider $ f(x,y) = xy $. By considering $ y $ as a constant $y_0 $ and looking at the graph of $ f(x, y_0) $ , we see that the slope will get bigger and bigger the more we continue along the y-axis. Our functions is also symmetric for $x$ and $y$ but I don't see any other clues to get a better idea of its graph. Help me out if you've got some tricks plz.

Thanks in advance