Proportionality: direct and indirect
Direct variation describes a simple relationship between two variables. We say y varies directly with x (or as x, in some textbooks) if. Here is a table for the types of variation we'll be discussing: Involves a combination of direct variation or joint variation, and indirect variation. is an extra fixed constant, so we'll have an equation like, which is our typical linear equation.). Definition of indirect relationship: The relationship between two variables which 1 and 3 both interface with group 2, but neither of them interfaces directly with.
We also scale down by a factor of 3. So whatever direction you scale x in, you're going to have the same scaling direction as y. That's what it means to vary directly. Now, it's not always so clear. Sometimes it will be obfuscated. So let's take this example right over here. And I'm saving this real estate for inverse variation in a second. You could write it like this, or you could algebraically manipulate it.
Or maybe you divide both sides by x, and then you divide both sides by y. These three statements, these three equations, are all saying the same thing. So sometimes the direct variation isn't quite in your face.
Direct, Inverse, Joint and Combined Variation
But if you do this, what I did right here with any of these, you will get the exact same result. Or you could just try to manipulate it back to this form over here.
- Recognizing direct & inverse variation: table
- indirect relationship
And there's other ways we could do it. We could divide both sides of this equation by negative 3. And now, this is kind of an interesting case here because here, this is x varies directly with y. Or we could say x is equal to some k times y. And in general, that's true.
Intro to direct & inverse variation
If y varies directly with x, then we can also say that x varies directly with y. It's not going to be the same constant. It's going to be essentially the inverse of that constant, but they're still directly varying.
Now with that said, so much said, about direct variation, let's explore inverse variation a little bit. Inverse variation-- the general form, if we use the same variables.
And it always doesn't have to be y and x. It could be an a and a b. It could be a m and an n. If I said m varies directly with n, we would say m is equal to some constant times n. Now let's do inverse variation.
So let me draw you a bunch of examples. And let's explore this, the inverse variation, the same way that we explored the direct variation. And let me do that same table over here. So I have my table.
I have my x values and my y values.
If x is 2, then 2 divided by 2 is 1. So if you multiply x by 2, if you scale it up by a factor of 2, what happens to y? You're dividing by 2 now.
Here, however we scaled x, we scaled up y by the same amount. Now, if we scale up x by a factor, when we have inverse variation, we're scaling down y by that same. So that's where the inverse is coming from. And we could go the other way. So if we were to scale down x, we're going to see that it's going to scale up y.
So here we are scaling up y. So they're going to do the opposite things.
What is indirect relationship? definition and meaning - omarcafini.info
And you could try it with the negative version of it, as well. So here we're multiplying by 2. And once again, it's not always neatly written for you like this.
It can be rearranged in a bunch of different ways. But it will still be inverse variation as long as they're algebraically equivalent. So you can multiply both sides of this equation right here by x. And you would get xy is equal to 2. This is also inverse variation.
You would get this exact same table over here. You could divide both sides of this equation by y. So notice, y varies inversely with x. 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. 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.
Recognizing direct & inverse variation: table (video) | Khan Academy
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. 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.