 Now let's see what happens when we graph y equals 2 to the x. Incidentally, I am using the Desmos calculator, you find it at Desmos.com. I really like this application. It's wonderful. It's not like that clumsy TI-81, 82, 83. This is very nice and you can play with the scale. Very nice and easy. Especially if you have a touch screen computer. Let's look at that. We have that function. What happens when we change the base and we consider y equals instead of 2, say, 3 to the x. 3 to the x. Oh, we get a function that just increases at a faster rate, but it has the same general shape. And what if instead of 3 we go up a bit higher, like y equals 10 to the x. Oh, look at that. How sharply now it goes all the way up. Now what if we were to look, I'm going to keep y equals 2 to the x. And now I'm going to play a little bit with it. What if we were to, instead of a base greater than 1, we would choose a base between 0 and 1. Let's choose 1 half. y equals, say, 1 half to the x. Oh, notice, notice what happens. We get a function, we get a function that is a mirror reflection along the y axis. This is very neat. This is very neat. I mean, it all makes sense, right? Because when x is equal to, say, they intersect at x equals 0 because both are 1, but when x equals 1, the red one is 1, 2, and the purple one is 1 half. And as you move along, the red one is 2 to the fifth power, that's 32. So when x is 5, y is 32. And the purple one, when x is 5, y is 1 over 30. 1 over 32. Because the number just becomes smaller and smaller. But notice that if we have, instead of 5, negative 5, 1 half to the negative 5, then that becomes the reciprocal of that. So it will become 32 as well. And you see it on the left-hand side of the purple screen. Now, another nice thing that we may want to do using this calculator, this Desmos graphing calculator, is, and what happens if I, instead of this, I multiply by, say, negative 2. So what would happen? Well, say y equals, say, negative 2 to the x. Now, I am not saying that the base is negative, okay? Let me clarify that. All I am saying, I'm taking the opposite of that, okay? I'm taking the opposite of that, and that is why we get a reflection along the x-axis. As we discussed, we will see it in a separate video, why it is that we don't consider negative bases. But we can see that we get that. And looking at the red one, what happens when we say multiply it by, say, a factor of 5. Y equals, say, 5 times 2 to the x2 to the x. What do we get? Oh, isn't that nice? We get, that's the black graph. We get a graph that it just grows at a faster rate than the red one.