So very commonly throughout all of physics, you'll see questions that ask you how a variable changes when another variable in the equation changes as well. And this is an area of math called proportional reasoning, which basically just analyzes how one thing increases or decreases with another. There are a couple of different situations that you'll commonly see, so I just want to review them really quickly. So first, let's say I had something like y = 2x. Basically, just go ahead and plug in a bunch of values. So if you have x = 2, you'll get y = 4; if x = 10, y = 20, and so on. As we can see here, there's a direct relationship between y and x. We say they're directly proportional. As the x values get bigger, so do the y values. As x gets smaller, so do the y values. These things are sort of directly correlated.
Let's take a look at another example. We have y = \frac{1}{x}. Let's just plug in a bunch of numbers. For x = 1, y = 1; for x = 2, y = \frac{1}{2}; for x = 3, y = \frac{1}{3}; and then y = \frac{1}{4}, y = \frac{1}{5} as x continues to increase. So these are actually fractions, and we can see here that the fractions actually get smaller with higher xs. These relationships are inversely proportional because as the x values increase, the y values decrease, and vice versa. As x values go down, the numbers get bigger for y. So these relationships are inversely proportional.
Now let's take a look at the last situation, which is actually going to be the more common one. Let's say you have something like f = ma, and these are actually two variables. We actually say that these are jointly proportional, and this is what happens here. So m and a could be any numbers, and we're just going to plug in a bunch of numbers here. But f = ma is one of the most important equations that you'll see in physics, and it's just a multiplication of two things. So for example, 5 \times 2 = 10, 4 \times 1 = 4, 3 \times 0 = 0, and so forth. There's a little bit of a pattern that emerges here because as m gets bigger and a gets bigger, we can see that f gets bigger. And then as m and a both get smaller, so does f. But there's also another kind of problem which you're going to see, which is that because it's two variables that are multiplied together, you can also have situations where you want a constant f. So for example, I want f to just be 20. There are actually just a lot of different combinations of m and a that will make that work. So for example, 1 \times 20 = 20, 2 \times 10 = 20, and 4 \times 5 = 20. And so what you're going to see here is that for constant f's, as the m gets bigger over here, the a actually has to get smaller. So as m gets larger, a has to be smaller in order to compensate so that they both make f = 20, and also the opposite happens. As m gets smaller over here, the a actually has to get bigger, so these relationships are also sort of, like, inversely proportional as well. So this is called jointly proportional. It's one of the more common things that you'll see in physics.