Proportionality and Translations in Graphs

[italia458]

Quote from Wikipedia:

To determine experimentally whether two physical quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system. If the points lie on or close to a straight line that passes through the origin (0, 0), then the two variables are probably proportional, with the proportionality constant given by the line's slope.

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.

 

[Simon Bridge]

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".

 

[italia458]

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.

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?