Graphs of proportional relationships
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- [Voiceover] So I have three different relationships here between X and Y. And what I wanna think about which of these, if any, are proportional relationships. And then I wanna graph them to see if we can see anything visually that makes them obviously proportional. And just as a reminder, a proportional relationship is one where the ratio between the two variables, and let's say we took the ratio between Y and X, you could also go the other way around, the ratio between X and Y. But the ratio between Y and X is always going to be some number, some constant number. Or you could rewrite it another way, If you were to multiply both sides of this equation times X, you could say in a proportional relationship, Y is always going to be equal to some constant times X. So with that out of the way, let's look at these three relationships. So this one over here, let me draw another column here. Another column. This is, let me call this the Y over X column. I'm just gonna keep figuring out what this ratio is for each of these pairs. So for this first pair, when X is one, Y is one half, so this ratio is one half over one. Well one half over one is just the same thing as one half. When X is four, Y is two, this ratio is gonna be two over four, which is the same thing as one half. When X is negative two and Y is negative one, this ratio is negative one over negative two, which is the same thing as one half. So for at least these three points that we've sampled from this relationship, it looks like the ratio between Y and X is always one half. In this case K would be one half, we could write Y over X is always equal to one half. Or at least for these three points that we've sampled, and we'll say, well, maybe it's always the case, for this relationship between X and Y, or if you wanted to write it another way, you could write that Y is equal to one half X. Now let's graph this thing. Well, when X is one, Y is one half. When X is four, Y is two. When X is negative two, Y is negative one. I didn't put the marker for negative one, it would be right about there. And so if we say these three points are sampled on the entire relationship, and the entire relationship is Y is equal to one half X, well the line that represents, or the set of all points that would represent the possible X-Y pairs, it would be a line. It would be a line that goes through the origin. Because look, if X is zero, one half times zero is going to be equal to Y. And so let's think about some of the key characteristics. One, it is a line. This is a line here. It is a linear relationship. And it also goes through the origin. And it makes sense that it goes through an origin. Because in a proportional relationship, actually when you look over here, zero over zero, that's indeterminate form, and then that gets a little bit strange, but when you look at this right over here, well if X is zero and you multiply it by some constant, Y is going to need to be zero as well. So for any proportional relationship, if you're including when X equals zero, then Y would need to be equal to zero as well. And so if you were to plot its graph, it would be a line that goes through the origin. It would be a line that goes through the origin. And so this is a proportional relationship and its graph is represented by a line that goes through the origin. Now let's look at this one over here, this one in blue. So let's think about whether it is proportional. And we could do the same test, by calculating the ratio between Y and X. Y and X. So it's going to be, let's see, for this first one it's going to be three over one, which is just three. Then it's gonna be five over two. Five over two, well five over two is not the same thing as three. So already we know that this is not proportional. Not proportional. We don't even have to look at this third point right over here, where if we took the ratio between Y and X, it's negative one over negative one, which would just be one. Let's see, let's graph this just for fun, to see what it looks like. When X is one, Y is three. When X is one, Y is three. When X is two, Y is five. X is two, Y is five. And when X is negative one, Y is negative one. When X is negative one, Y is negative one. And I forgot to put the hash mark right there, it was right around there. And so if we said, okay, let's just give the benefit of the doubt that maybe these are three points from a line, because it looks like I can actually connect them with a line. Then the line would look something like this. The line would look something like this. So notice, this is linear. This is a line right over here. But it does not go through the origin. So if you're just looking at a relationship visually, linear is good, but it needs to go through the origin as well for it to be proportional relationship. And you see that right here. This is a linear relationship, or at least these three pairs could be sampled from a linear relationship, but the graph does not go through the origin. And we see here, when we look at the ratio, that it was indeed not proportional. So this is not proportional. Now let's look at this one over here. Let's look at what we have here. So I'll look at the ratios. Y over X. So for this first pair, one over one, then we have four over two, well we immediately see that we are not proportional. And then nine over three, it would be three. So clearly this is not a constant number here. We don't always have the same value here, and so this is also not proportional. Not proportional. But let's graph it just for fun. When X is one, Y is one. When X is two, Y is four. This actually looks like the graph of Y is equal to X squared. When X is three, Y is nine. At least these three points are consistent with it. So one, three, four, five, six, seven, eight, nine. So it's gonna look something... And so, if this really is, if these points are sampled from Y equals X squared, then when X is zero, Y would be zero. So this one actually would go through the origin, but notice, it's not a line. It's not a linear relationship. This right over here is the graph of Y equals X squared. So this one also is not proportional. So once again, these three points could be sampled from Y equals one half X, these three points could be sampled from, let's see, Y is equal to, let's see, it looks like a line when... this looks like it could be Y = 2x + 1. So it's a linear relationship, but it does not go through the origin, so it's not proportional. And these three points look like they could be sampled from Y equals X squared, which goes through the origin. When X is zero, Y is zero, but it's not a linear relationship. Any way you look at it, if you look at it visually, it has to be a line that goes through the origin, or if you look at a table of values, look at the ratios, and the ratios always have to be the same value. And that was only the case with this magenta one, right over here.