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Identifying constant of proportionality graphically

CCSS Math: 7.RP.A.2, 7.RP.A.2b

Video transcript

- We're asked what is the constant of proportionality between Y and X in the graph? And just as a reminder, when we're talking about constant of proportionality, it sounds like a very fancy thing, but it's not too bad. If we're thinking about any X-Y pair on this, on this line, let's say, right over here. We have some X comma 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'll see this expressed, if you divide both sides by X, sometimes you'll see this as Y over X, is equal to the constant of proportionality. It shows for any X-Y pair, if you take your Y divided by X, what do you get? That's the same, 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 three, three comma two. And so we could set it up a few ways. We could say, look when Y is equal to two, X is equal to three, and so two would need to be equal to some constant of proportionality times three. And if you wanted to solve for this, you just divide both sides by the three. So divide both sides by three, and you would get your constant of proportionality is two-thirds. Another way to do it, right over here. Well here, we've kind of already solved for our constant of proportionality. When Y, or we could say, when X is three, when X is three, Y is equal to two. In either case, our constant of proportionality is two-thirds. Let's do another example. So here we have which line has a constant of proportionality between Y and X of five over four. So pause the video and see if you can figure that out. So the key realization is we should test points on these lines, we should test X-Y pairs, and say, look if we take our Y divided by X, do we get five-fourths? Because that would be our constant of proportionality. So let's first try, let's 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 two comma five. And so if I took Y divided by X, I would get a constant of proportionality as five halves. So A is not going to be our answer. We wanna get to a constant of proportionality of five-fourths. Alright, 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 four comma five, four comma five. And so in this situation, K would be our Y, which is five, divided by our X, which is four. 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 wanna 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. It's just, you 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 one times X. And so then it might jump out at you that the constant of proportionality is one in this scenario right over here. Or if you took Y divided by X, Y over... Let me do it in that black color. Or if you took Y over X, you divided both sides by X, you would be left with the constant of proportionality, which would be equal to one. 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 one, you would... Those would represent points on this orange line that I just constructed.