I've been told by people that a graph with a straight line on it can be proportional but only if it passes through the origin. I fail to see why that's true. If a translation was applied and it was moved 1 unit to the right then, all of a sudden, x is not proportional to y anymore? That doesn't make sense to me.

In math, two quantities are proportional, by definition, if their ratio is constant.
i.e. if y is proportional to x, then y/x=k - a constant.
The graph would be y=kx - which is a special case of a straight line.

If you translated the graph, changing the reference point for measuring x for instance, then the equation of the line is:

y=k(x+a) and the graph of x vs y no longer passes through the origin.
The quantities x and y are no longer proportional (y/x=k+ka/x - not a constant) because it is a different x - instead it is x+a that is proportional to y ... which is fair, because x+a was the original quantity.

However, we can still say that

y1 = k(x1+a)
y2 = k(x2+a)

y2-y1 = k(x2-x1)

so changes in y are proportional to changes in x.

If two quantities x and y are related by some line y=mx+c, then the relationship is just called "linear".

So if the graph is translated left or right, I can still say that "changes in y are proportional to changes in x" but I can't say "y is proportional to x" - is that correct?