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Recognizing direct & inverse variation: table

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Determine whether the data in the table is an example of direct inverse or joint variation. Then identify the equation that represents the relationship. So let's just think about what direct inverse or joint variation even means. So if you have direct variation. Direct variation. So if y varied directly with x it literally means that y is equal to some constant multiple of x, or if you divide both sides of this by x it means that y over x is equal to k so the ration between y and x is a constant. And you could go the other way around. You could also say that x is equal to some constant not, not going to be the same constant times y. Or that x over y is going to be equal to some other constant. So these aren't necessarily the same k. All I'm just saying is that it's a constant relationship. These are all examples of direct variation. In dir, or I should say inverse variation is to some degree the opposite depending on how you view the opposite. And before I even talk about that, let's think about the telltale signs of direct variation. If x increases, y should increase. So if x increases. Let me do that in the same yellow. So the telltale signs of direct variation, if x increases then y will increase and vice versa. The other telltale sign is. Is if you increase x by some, by some factor. So, if you have x going to 3x then y should also increase by that same factor. And we could see that with some examples. So, I mean, you could pick a K, let's say that, let's say that K was one. So if y is equal to x, if you take, if x goes from one to three, then y is also going to go from one to three. So that's all we're talking about here. Let me actually, y should actually to three times y, that's what I'm talking about. If you triple x, you're also gonna end up tripling y. Inverse variation. You have y being equal to some constant times one over x. So instead of an x here you have a one over x or if you multiply both sides by x you get x times y is equal to some constant. And you could switch the x's and the y's around as well for inverse variation. Now what are the tale tale signs? Well if you increase x, if x goes up, then what happens to y? If x goes up then this becomes a smaller value cuz it's one over x so then y will go down. Then y will go down. And if you take X and if you're to say increase it by a factor of three then what's going to happen to Y? Well if you increase this by a factor of three, you're actually going to decrease this whole value by a factor of one-third, so Y is going to go, so then you're going to have one-third of y. So that's, these are the tell-tale signs for inverse variation. Now finally they talk about something called joint variation, and this one you won't necessarily see in introductory algebra course. But joint variation deals with more than one variable. So if I told you, if I told you that area of a rectangle is equal to the width of a rectangle times the length of rectangle, this is an example of joint variation. Area is proportionally to two. Is the proportional to two different quantities? So the main tell tale sign here for joint variation frankly is you're gonna be dealing with more than two variables. Joint, Joint, Joint variation. So when you look at this example, there only gi, giving us two variables. So you can rule out joint variation just right from the get go. Now let's look at the tell tale signs. So as x is increasing, as x goes from one to two, what is happening to y? Y went from 12 to six. So as x is going up by a factor of two, y is going, is, is going by a factor of one half. Or y is being multiplied by one half. So as x goes from one to three, it's being multiplied by three. Y is being multiplied or I guess could say is, is multiplied by one-third. So it's definitely not direct variation. As x increases, y is decreasing. So it's definitely not direct variation. And then really you can just rule out, since we rule out the other two, you can probably guess this is going to be inverse variation. But we can validate it. When X increases, Y is decreasing. When X increases by a certain factor, Y is increasing by one over that factor, which is actually decreasing. So, if you go from one to three, if X is being multiplied by three, then Y essentially becomes one third of its original value. When X is one Y is 12 when X is three Y is four so we have inverse variation in play. Now they ask us identify the equation that represents the relationship. Well we know with inverse variation the product of x and y need to be equal to some constant. So that if we take x times y over here so lets just multiply lets make another column here call this the x times y column. One times 12 is 12. Two times six is 12. Three times four is 12. Four times three is 12. So, clearly in every situation, x times y is, is a constant and it is 12. So, the equation that represents the relationship, it is, X, Y is equal to 12 and that is clearly an inverse