 Imagine we have a scatterplot like we see in front of us. We probably want to find some type of mathematical function that can model the data presented. So we can use that function to make predictions, forecasts, interpolation, extrapolation, all that type of jazz, right? So how do we find such a function? Well, a lot of it comes down to what's the shape of the data? Oftentimes, people like to use a linear regression. But when you look at this one, a line doesn't quite work, right? Because if you kind of go here, it's not going to work. A line maybe here doesn't quite fit. Instead, this data kind of looks like a parabola. In which case we're tempted, can we find a parabola of best fit? Or in this case, we're going to draw it by hand and look for a parabola of good fit. And so if we were just to draw, intuitively we're seeing something that might look like the following. This isn't, of course, the best drawing in the world. Computers can do a much better job at this. But the idea is this visually looks like a parabola. Now, if we want to find the equation of a parabola, our function model would look something like the following. Y equals AX squared plus BX plus C. And therefore, we have to find the numbers A, B, and C. And that often comes down to picking three points on the parabola here. So we might have like, let's say right here, X1, Y1. We get X2, Y2. And over here, we'll get X3, Y3. So we get three specific points and then we can plug these into the equation because when we replace X with a specific X chord and Y with a specific Y chord and that gives you points on the parabola. So we get something like Y1 equals AX1 squared plus BX1 plus C. We'd also get Y2 is equal to AX2 squared plus BX2 plus a constant. And Y3 equals AX3 squared plus BX3 plus C. And so when we put this all together, we get this three by three linear system. That is a system of equations with three linear equations, three unknowns. Which just so we're clear, the unknowns now are gonna be the coefficients A, B, and C. We don't know what they are. The numbers X1 and Y1, X2, Y2, these are specific numbers we grabbed from the data. Like we could be like, oh, this point right here, it looks like, it's about maybe five and this right here maybe about 70. So we have the point five comma 70. And so this is what we put over here. We have a five squared times a five and a 70, right? We look at this point right here. This would look more like maybe eight comma, we'll say 35 and kind of just kind of guessing from the graph here. So we put an eight right here and eight right here, 35. And then with my last point, oh, what do we wanna say that is? We'll say it's 20 comma. And that looks about the same height, maybe like a 37 right there. And so we get 37, 20, and 20. So we have specific numbers for these X1s, the XIs and the YIs. And so the ABCs are the things we don't know. We could solve this three by three linear system. That's possible. And oftentimes that's how computers would try to model this data using a little bit more statistical and linear algebra, statistical analysis and linear algebra in the background we often don't see. But if you're trying to do this by hand, this will be a very difficult process, doable, but there's a lot easier way. Turns out linear systems, at least three by three linear systems is overkill. And that's because we're using the wrong form of the, we're using the wrong form of the quadratic equation. Instead of using the standard quadratic form, which you see on the screen, instead what I recommend is using the vertex form, Y equals A times X minus H squared plus K. The vertex of the parabola holds a lot more information than just a random point. The vertex we can see very quickly, it would be somewhere around here, which will again, we'll make a guess about 14 comma 15. Seems to be halfway up there. And so with that in mind, then our equation very quickly becomes A times X minus 14 squared plus 15. And so now we just have to figure out what is this coefficient A for which we can just go one step over to find it or just pick your favorite point on the graph. For me, I like to pick the Y intercept. So we get X equals zero and Y. We'll say that's about, maybe, because if we went halfway, I'm just kind of guessing right here, we'll say 110. Again, this might not be the most accurate, but I'm just trying to do this kind of by hand here. I like the Y intercept because the Y intercept is when the X coordinate is gonna be zero. I should have written that as a zero right there, like so. And so then we plug these in here, Y will be 110, X is gonna be zero, like so. So we get 110 is equal to 14 squared times A plus 15. And so this is the actual part of the process where we might actually need a calculator to help us out here. Again, not too difficult of calculations, but we're gonna do it now. So let's see the next thing to do here in solving this. I would subtract 15 from both sides. This will give us 95 equals 14 squared. What is that? Again, if you don't know off the top of your head, no big deal, use your calculator. 14 times 14 is 196 times A. And so to finish this thing up, your A coefficient is gonna be 95 over 196. That's 96 there. For which, again, use your calculator to help you out here. 95 divided by 196. That gives you, I mean, we could leave it as a fraction if we wanted to or that would give you about 0.48, whichever you prefer. I'm just gonna leave it as a fraction here. So we get Y equals 95 over 196 times X minus 14 squared plus 15. And so this right here gives me an equation for the parabola. And I did it actually quite effortlessly, right? Using the vertex and the y-nurcept, you can find this equation without much effort whatsoever. And this accuracy will be based on how close I got to the true vertex, how close I got to the y-nurcept. And as this is just a regression parabola, there is no like true vertex because the data is not actually a parabola. We're just using a parabola to model it. And so we can solve quadratic equation, quadratic regression, excuse me, just by eyeballing it. So to speak, we can look at specific points and go from there. The key observation here is the vertex is the golden standard here. That's the first point you wanna find when you try to build a quadratic function.