Identifying constant of proportionality graphically

We're asked what is the constant of proportionality between y and x in the graph. Just as a reminder, when we're talking about the constant of proportionality, it sounds like a very fancy thing, but it's not too bad.

If we're thinking about any xy pair on this line, let's say right over here we have some (x, y). If y is proportionate or is proportional to x, then that means we can say that y is equal to some constant, y is equal to some constant times x. And that constant, that is our constant of proportionality right over there.

Sometimes you will see this expressed if you divide both sides by x; sometimes you'll see this as ( \frac{y}{x} ) is equal to the constant of proportionality. It shows, for any xy pair, if you take your y divided by x, what do you get? That's the same thing.

So with that out of the way, see if you can answer their question: What is the constant of proportionality between y and x in the graph? Well, they very clearly give us a point right over here. This point is the point (3, 2). So we could set it up a few ways.

We could say, look, when y is equal to 2, x is equal to 3, and so 2 would need to be equal to some constant of proportionality times 3. If you wanted to solve for this, you just divide both sides by 3. So divide both sides by 3, and you would get that your constant of proportionality is ( \frac{2}{3} ).

Another way to do it right over here: well, here we've kind of already solved for a constant of proportionality. When x is 3, when x is 3, y is equal to 2. In either case, our constant of proportionality is ( \frac{2}{3} ).

Let's do another example. So here we have, which line has a constant of proportionality between y and x of ( \frac{5}{4} )? So pause the video and see if you can figure that out.

The key realization is we should test points on these lines; we should test xy pairs and say, well, look, if we take our y divided by x, do we get ( \frac{5}{4} )? Because that would be our constant of proportionality.

So let's first try line A right over here. So line A, let me find a point that sits on it. So that looks like a point that sits on it, and so if I take this, is the point (2, 5). If I took y divided by x, I would get a constant of proportionality as ( \frac{5}{2} ). So A is not going to be our answer. We want to get to a constant of proportionality of ( \frac{5}{4} ).

All right, let's try B. Okay, B, let me find a point on B. Looks like this is a point on B that is the point (4, 5). And so in this situation, k would be our y, which is 5, divided by our x, which is 4. So it looks like B is our choice. For kicks, you could also look at the constant of proportionality right over here.

Now there is one interesting example that I just want to touch on before we finish these examples. What about a situation where y is equal to x? What is the constant of proportionality then, and what would it look like as a line? Pause this video and think about it.

Well, there's really nothing new here; you just might not really see the constant of proportionality when you see it expressed this way. But y is equal to x is the same thing as y is equal to 1 times x.

And so then it might jump out at you that the constant of proportionality is 1 in this scenario right over here. Or if you took y divided by x, or if you took ( \frac{y}{x} ), you divide both sides by x, you would be left with the constant of proportionality, which would be equal to 1.

And if you wanted to graph it, well, it would just look like this: y would be equal to x for all x's. So that's what, when your constant of proportionality is 1, those would represent points on this orange line that I just constructed.