Mathwords: Inverse Variation
proportion. With direct proportion, the two variables change at the same rate. a. d =? t = 2 From the formula, substitute the value of time and k. d = kt d = This has the mathematical formula of y = kx, where k is a constant. When there is a direct or inverse relationship, x and y are proportional to Pi is always the same, so if you double the value of D, the value of C doubles too. Direct and Inverse proportion revision along with proportionality resources. Choose Maths Then, expressing this as an equation we get. y = k x y = kx y=kx.
Let's try y is equal to negative 3x. So once again, let me do my x and my y.
Direct and inverse proportions
When x is equal to 1, y is equal to negative 3 times 1, which is negative 3. When x is equal to 2, so negative 3 times 2 is negative 6. So notice, we multiplied. So if we scaled-- let me do that in that same green color.
If we scale up x by it's a different green color, but it serves the purpose-- we're also scaling up y by 2. To go from 1 to 2, you multiply it by 2. To go from negative 3 to negative 6, you're also multiplying by 2. So we grew by the same scaling factor. To go from negative 3 to negative 1, we also divide by 3. 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. 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. And there's other ways we could do it.
We could divide both sides of this equation by negative 3.Direct and Inverse Proportion
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. 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.
Intro to direct & inverse variation
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.
Intro to direct & inverse variation (video) | Khan Academy
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?
Direct and inverse relationships - Math Central
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. In short, direct relationships increase or decrease together, but inverse relationships move in opposite directions.
A bigger diameter means a bigger circumference. In an inverse relationship, an increase in one quantity leads to a corresponding decrease in the other. Faster travel means a shorter journey time.
How Does y Vary with x? Scientists and mathematicians dealing with direct and inverse relationships are answering the general question, how does y vary with x? Here, x and y stand in for two variables that could be basically anything. By convention, x is the independent variable and y is the dependent variable. So the value of y depends on the value of x, not the other way around, and the mathematician has some control over x for example, she can choose the height from which to drop the ball.
When there is a direct or inverse relationship, x and y are proportional to each other in some way. Direct Relationships A direct relationship is proportional in the sense that when one variable increases, so does the other. Using the example from the last section, the higher from which you drop a ball, the higher it bounces back up.
A circle with a bigger diameter will have a bigger circumference. If you increase the independent variable x, such as the diameter of the circle or the height of the ball dropthe dependent variable increases too and vice-versa. Sciencing Video Vault A direct relationship is linear. Pi is always the same, so if you double the value of D, the value of C doubles too.