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. In mathematics, two variables are proportional if there is always a constant ratio between them. The statement "y is inversely proportional to x" is written mathematically as "y = c/x." This is . Correlation · Eudoxus of Cnidus · Golden ratio · Proportional font · Ratio · Rule of three (mathematics) · Sample size · Similarity · Basic. If we are dealing with an inverse proportionality, we apply the rule of three we need to observe the relationship that exists between the two.
Direct Relationships A direct relationship is proportional in the sense that when one variable increases, so does the other.
- Direct and inverse proportion
- Inversely proportional function
- Inversely Proportional
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. The gradient of the graph tells you the value of the constant. Inverse Relationships Inverse relationships work differently. If you increase x, the value of y decreases. For example, if you move more quickly to your destination, your journey time will decrease. In this example, x is your speed and y is the journey time. Doubling your speed halves the journey time, and increasing the speed by ten times makes the journey time ten times shorter.
Mathematically, this type of relationship has the form: As you start to increase x, y decreases really quickly, but as you continue increasing x the rate of decrease of y gets slower. In this case, y is inversely related to x. At first an increase of 3 in x decreases y by 2, but then an increase of 6 in x only decreases y by 1. 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.
Intro to direct & inverse variation
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.
Inversely Proportional | Definition of Inversely Proportional by Merriam-Webster
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. 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.
Direct and inverse proportions
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. 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. And you could just manipulate this algebraically to show that x varies inversely with y.