 This video is going to look at data sets and their patterns and see if we can find equations just from looking at data. Again, this video is going to assume that you've already learned about linear, quadratic, and exponential functions because we're going to look at all three of those. So our first example here, we want to determine whether it's linear, quadratic, or exponential and then see what patterns we see and then see if we can find the equation. So this looks like x is changing by one all the way through. And this looks like I might have maybe subtracted two or divided in half, but then here I didn't do and divided in half or to track two, I actually added zero. And then I have that I added to here or multiplied by two. And if I look at here, it looks like I might have multiplied by two. But because I see that I decreased and then I increased, that tells me that it's going to be quadratic. This looks like maybe doubling here at the bottom, but I decreased at the beginning and exponential is always increased or always decreased. So it's not exponential, this one's going to be quadratic. Now quadratic, I can tell by looking at the pattern that it increases and decreases or decreases and increases, but I really have a hard time finding the equation by just looking at that table. The easiest way to do that is to go into my stats and just put my data in there. I already put the x in there to save some time, but now I'm going to put it in my y values or my f of x values, four, two, enter, two, enter, four, enter, eight, enter. Now we've decided it's a quadratic. So when I go back to stat and over to calculate, I'm going to choose a quadratic regression, which is five, and then enter, and I find out that f of x is going to be equal to one x squared minus three x plus four, or x squared minus three x plus four, if you'd rather. Alright, let's look at this one. Again, x is changing by one, all the way down. So that means that nice things will happen on the other side for me. When I look at this one, it looks like maybe I added three, and if I go from five to eight, I added three again, and eight plus three is eleven, and eleven plus three is fourteen, so I've added the same thing. So when you add the same thing in consistent amount every time, that's going to be a linear function. And with linear functions, if x changed by one, that makes it really nice because my slope, even if it hadn't changed by one, as long as it's changing by a consistent thing as well as my y's changing by a consistent thing, remembers it's the difference of the y's, so that's a three, over the difference of the x's, which in this case is just one. So my slope is three, and then I just need to know my y-intercept, and this table told me my y-intercept. When x is zero, I know that my y-intercept, this two is my b. So then I can write g of x is equal to three, my slope, times x, plus my b, which is two. And then finally, we have this one, and you might be thinking, oh, it's got to be exponential because we haven't done one, but let's prove it to ourselves. Again, x is changing by one all the way down, and here it looks like I've gone maybe two times three if I want to think about it that way, or two plus four, but when I do six, I didn't add four, and I did multiply by three, and six times three would be 18, 18 times three is 54, and 54 times three is 162. So I can see that this one, since I'm multiplying a consistent amount every time, this one is exponential, and exponential functions, we need an a times b to the x. Remember, this one is our y-intercept. Do we have a y-intercept? We do, because we had the x equals zero, again our y-intercept, in this case it's a, is equal to two, and our b is what we were multiplying every time, so that's this times three, so h of x is equal to a, which is two times my base, which is three to the x. So again, quadratics have an increasing and decreasing thing going in a, in the table, and you can also see a symmetry to it. We've got these twos that are right next to each other, but on either side of that we have a four, and if we had gone this, could see negative one, it would be an eight, just like the eight is on the other side of the four below it. And linears, we add the same thing consistently from one point to the next. That tells us it's linear, and exponentials we multiply consistently from one to the next, so that we can have an exponential function.