 In this video, we're going to show how we can use data to create a power function as we want a power function to fit specific data. So it'll just be a generic power function f of x equals c times x to the a. We have to determine the parameters a and c such that the function passes through f of 1 equals 5 and f of 2 equals 80. And so what happens when we plug these values into the function here? So this first thing right here is f of 1 equals 5. This tells us that when x equals 1, you're going to get that y equals 5. This is a really good point to have when you have a power function that is x equals 1 because you'll see that f of 1 here equals c times 1 raised to the a. This is equal to 5. This is a great value to have because even though I don't know what a is, 1 to the a is always just going to equal 1. So this would become c times 1, which is equal to 5. You see very quickly here that c equals 5. And so when x equals 1, this value right here is none other than the c value. How much are we going to stretch our power function here? Once we have the c value, we then can look at this other value as well. f of 2 equals, well c is 5 times x. Well, we know what x is. It's 2. x to the a. This is equal to 80. And so at this moment, we want to solve for the a value, which we don't know what it is. To begin with, I'm going to divide both sides by 5. That's just going to be a little bit of arithmetic right there. And dividing both sides by 5, we end up with 16 on the right-hand side. So we get 2 to the a is equal to 16. Now normally, this would be a situation where we might have to introduce something called a logarithm. But the numbers were chosen in such a way here that we actually don't need that. Because we're looking for a power of 2 that's equal to 16. And we notice that 16 is the same thing as 2 to the fourth. And so that's what we need right here. A value is going to equal 4. A value equals 4. And so therefore, the function we were looking for was f of x equals 5x to the fourth. And so this is a power function, which will contain the points 1, 5, which is what we saw right here. And it'll also contain the point 2, 80. And like I said, this one worked out so nicely because the number and consideration was a perfect power of the base 2. In general, that might not be so easy. And like I said, one would need a logarithm to help us out with such a thing. And that's a topic we'll talk about in a much later point in this lecture series.