I have a point (point $a$). The graph, which is centered at point $o$ (not necessarily $(0, 0)$) is broken into four sections diagonally. For example:
The graph is split into 4 sections by the two orange dotted lines. The point $o$ is at the intersection of the two lines, and point $a$ is at a random point on the graph (this point could move). By looking at the picture, you can tell it is in the top section. However, is there any mathematical way of determining which section the point is in?
The two dotted orange lines will always intersect perpendicularly in an 'X'.
A straight line is described by an equation like $$ ax + by = c . $$
The regions on either side of that line are described by the inequalities $$ ax + by < c \text{ and } ax + by < c . $$
In your example, the two lines are $$ x + y = h \text{ and } x - y = k $$ for some constants $h$ and $k$ that depend on point $o$, so you know which inequalities to check.