Introduction to exponential decay

What we're going to do in this video is quickly review exponential growth and then use that as our platform to introduce ourselves to exponential decay. So let's review exponential growth. Let's say we have something that... and I'll do this on a table here so we make that straight.

So let's say this is our x and this is our y. And let's say when x is 0, y is equal to 3, and every time we increase x by 1, we double y. So y is going to go from 3 to 6. If x increases by 1 again, so we go to 2, then we're going to double y again and so 6 times 2 is 12. This right over here is exponential growth.

And you could even go for negative x's. When x is negative one, well if we're going back one in x, we would divide by two, so this is going to be three halves. Notice if you go from negative 1 to 0, you once again keep multiplying by 2, and this will keep on happening.

You can describe this with an equation. You could say that y is equal to... and sometimes people might call this your y-intercept or your initial value, is equal to three. Essentially, what happens when x equals zero is equal to three times our common ratio.

And a common ratio is, well, what are we multiplying by every time we increase x by 1? So 3 times our common ratio, 2 to the x power. And you can verify that. Pick any of these when x is equal to 2, it's going to be 3 times 2 squared, which is 3 times 4, which is indeed equal to 12.

And we can see that on a graph, so let me draw a quick graph right over here. So I'm having trouble drawing a straight line. All right, there we go. And let's see, we could go... and they're going to be on a slightly different scale, my x and y axes.

So this is the x-axis, y-axis, and we go from negative 1 to 1 to 2 unless we're going all the way up to 12. So let's say that this is 3, 6, 9, and let's say this is 12. And we could just plot these points here. When x is negative 1, y is 3 halves, so it looks like that.

Then at y equals 0, when x is 0, y is 3. When x equals 1, y has doubled; it's now at 6. When x is equal to 2, y is 12. And you will see this telltale curve.

There are a couple of key features that we've talked about several of them. But if you go to increasingly negative x values, you will asymptote towards the x-axis. It'll never quite get to zero as you get to more and more negative values, but it'll definitely approach it.

And as you get to more and more positive values, it just kind of skyrockets up. We've talked about in previous videos how this will pass up any linear function or any linear graph eventually.

Now let's compare that to exponential decay. Exponential decay... and an easy way to think about it: instead of growing every time you're increasing x, you're going to shrink by a certain amount; you are going to decay.

So let's set up another table here with x and y values. That was really a very... I'm supposed to... this is supposed to... when I press shift, it should create a straight line, but my computer... I've been eating next to my computer; maybe there's crumbs in the keyboard or something. All right, so here we go.

We have x and we have y. And so let's start with... let's say we start in the same place, so when x is 0, y is 3. But instead of doubling every time we increase x by 1, let's go by half every time we increase x by 1.

So when x is equal to 1, we're going to multiply by one-half, and so we're going to get to 3 halves. And then when x is equal to 2, we'll multiply by one-half again, and so we're going to get to three-fourths, and so on and so forth.

And if we were to go to negative values, when x is equal to negative one, well, to go... if we're going backwards in x by one, we would divide by one half, and so we would get to six. Or going from negative 1 to 0, as we increase x by 1, once again we're multiplying by one half.

And so how would we write this as an equation? I encourage you to pause the video and see if you can write it in a similar way. Well, it's going to look something like this. It's going to be y is equal to... you could have your y-intercept here, the value of y when x is equal to 0, so it's 3 times.

What's our common ratio now? Well, every time we increase x by 1, we're multiplying by one half, so one half. And we're going to raise that to the x power.

And so notice these are both exponentials. We have some, you could say, y-intercept or initial value, and it's being multiplied by some common ratio to the power x, some common ratio to the power x.

But notice when you're growing our common ratio... and it actually turns out to be a general idea: when you're growing, the absolute value of your common ratio is going to be greater than one. Let me write it then.

So the absolute value of two, in this case, is greater than one. But when you're shrinking, the absolute value of it is less than one. And that makes sense because if you have something where the absolute value is less than one, like one-half or three-fourths or 0.9, every time you multiply it, you're going to get a lower and lower value.

And you can actually see that in a graph. Let's graph the same information right over here. And let me do it in a different color; I'll do it in a blue color.

So when x is equal to negative 1, y is equal to six. When x is equal to zero, y is equal to three. When x is equal to one, y is equal to three halves. When x is equal to two, y is equal to three-fourths, and so on and so forth.

And notice because our common ratios are the reciprocal of each other, these two graphs look like they've been flipped over. They look like they've been flipped horizontally or flipped over the y-axis; they're symmetric around that y-axis.

What you will see in exponential decay is that things will get smaller and smaller and smaller, but they'll never quite exactly get to zero. It'll approach zero; it'll asymptote towards the x-axis as x becomes more and more positive. Just as for exponential growth, if x becomes more and more and more negative, we asymptote towards the x-axis.

So that's the introduction. I'd use a very specific example, but in general, if you have an equation of the form y is equal to a times some common ratio to the x power, we could write it like that just to make it a little bit clearer.

Well, there's a bunch of different ways that we could write it. This is going to be exponential growth, so if the absolute value of r is greater than one, then we're dealing with growth because every time you multiply... every time you increase x, you're multiplying by more and more r's; is one way to think about it.

And if the absolute value of r is less than one, you're dealing with decay. You are shrinking as x increases. And I'll let you think about what happens when... what happens when r is equal to one? What are we dealing with in that situation?

And it's a bit of a trick question because it's actually quite straight... oh, I'll just tell you. If r is equal to 1, well then this thing right over here is always going to be equal to 1, and you boil down to just the constant equation y is equal to a, so this would just be a horizontal line.