Intro to direct & inverse variation (video) | Khan Academy
Two values are said to be in direct proportion when increase in one results in an increase in the other. Similarly, they are said to be in indirect proportion when. Work more hours, get more pay; in direct proportion. This could be Inversely Proportional: when one value decreases at the same rate that the other increases . In mathematics, two variables are proportional if there is always a constant ratio between them. . The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates.
We could have y is equal to negative pi times x. I don't want to beat a dead horse now. I think you get the point.
Any constant times x-- we are varying directly. And to understand this maybe a little bit more tangibly, let's think about what happens.
Directly Proportional and Inversely Proportional
And let's pick one of these scenarios. Well, I'll take a positive version and a negative version, just because it might not be completely intuitive.
So let's take the version of y is equal to 2x, and let's explore why we say they vary directly with each other. So let's pick a couple of values for x and see what the resulting y value would have to be. So if x is equal to 1, then y is 2 times 1, or is 2. If x is equal to 2, then y is 2 times 2, which is going to be equal to 4. So when we doubled x, when we went from 1 to so we doubled x-- the same thing happened to y.
So that's what it means when something varies directly. If we scale x up by a certain amount, we're going to scale up y by the same amount.
If we scale down x by some amount, we would scale down y by the same amount. And just to show you it works with all of these, let's try the situation with y is equal to negative 2x.
I'll do it in magenta. Let's try y is equal to negative 3x. So once again, let me do my x and my y. 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. 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. It's going to be essentially the inverse of that constant, but they're still directly varying.
Intro to direct & inverse variation
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. 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.
Direct and inverse proportions | Class 8 (India) | Math | Khan Academy
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. The gradient of the graph tells you the value of the constant. Inverse Relationships Inverse relationships work differently.