 Imagine you have a scatter plot of data in front of you, like you do right now. It's useful to come up with a function that fits the data so we can use that function to interpolate or extrapolate or just make other predictions or forecasts relevant to this data set right here. But sometimes finding the right function to fit can be a little bit of a challenge. Like if you look at this thing right here, you might think, oh, I'll do a linear regression. And although you can, the linear regression doesn't quite seem to work, right? I mean, there's no one point that really captures it perfectly, right? I mean, you can. I mean, this one's pretty good right here. But just looking at it, it doesn't really feel like a line. It feels like the data's got some more curvature to it. And you might be like, okay, well maybe it's like a parabola or something like that. In which case you might get something like the following. But the problem is a parabola eventually will have to bend back down. This one kind of wants to go up like this. So if you really want to find the shape that kind of best fits it, we need a higher degree polynomial. Maybe something that kind of like levels out in the middle that spikes up again. This looks a lot like maybe y equals x cubed or something like that or some other odd monomial function. So maybe we try to make a fit there. Well, if we have zero right here and zero right here, the origin of our graph is gonna be about right here, which doesn't seem to be about the vertex of the parabola here. There's gonna be some type of shifting going on here. So we have to accept the fact that this monomial could be shifted in some kind, in some degree. And so the general formula we probably wanna use would be y equals a times x minus h cubed plus k. So we accept the fact that there could be a shift left or right, up or down shift. And then this scalar right here will compensate for any vertical or horizontal reflections or stretching or some kind. You will notice the in behavior it's pointing up on the right and pointing down on the left. So this already tells me that my coefficient a should be positive. I can see that. So we wanna find essentially the vertex of this graph right here. Where it was like the middle of this cubic, which in the untransformed version would correspond to the origin, that seems to be about right here, I would say. And so try to estimate that here. x-cornered seems to be about a one. I'll go with that. And then the y-coordinate, if we bring this over, it seems about to be zero, right? Okay. So I'm gonna go with just like eyeballing this thing here, one comma zero, I want that to be my vertex. And so with regard to my function then, I'm gonna get y equals a times x minus one cubed plus zero. That's great. You don't even need the plus zero there. So we still have to determine this coefficient a, right? So what we're gonna do is we're gonna look for a point on the graph, which seems like it matches up with the graph and we're gonna plug that in there. And so you could use an x-intercept or y-intercept if you wanted to. The y-intercept gets a little bit problematic because it looks like it's going off the screen and this is really steep. So we're not gonna go with that one as well. We could try the x, as I'm sorry, the y-intercept would be over here. What am I talking about? This isn't the y-axis. Shame on me. This is the y-axis right here. Notice this is x equals zero. So we could try to go with the y-axis. It's really close in height to this point. So I think for a better fit, we're gonna be better off looking for a point elsewhere. And so maybe if we grab something like this right here, this seems like a pretty good point. So we have x equals negative two and then the corresponding y-coordinate over here. So it's a little bit below negative 20. Here's a negative 40. So maybe we say something like negative 25. That seems feasible. And so then what we're gonna do is we're gonna plug in for y, negative 25. We're gonna plug in, well, we don't know a for x. We're gonna get a negative two and then we need to compute this thing right here. So we got a negative two minus one. So it's gonna give us a negative three. When you cube that, you're gonna get negative 27a have a negative 25. And so then solving for a, a would look like negative 25 over 27, negative 27. So it's just double negative makes it a positive. And so putting this all together, my function I'm estimating, this would be y equals 25 over 27 times x minus one cubed. This seems like a fairly good estimate of the cubic function that matches this data. And again, this isn't perfect. I just kind of eyeballed this one. Computer software and statistical analysis, linear algebra, some more advanced techniques than your typical college algebra could be used here to find a cubic curve of best fit. Now here, I'm just trying to talk about how one could find a cubic curve of good fit. And we're just gonna eyeball it here. If we want a better, as soon again, computer software is necessary, but I wanted to demonstrate how a human could just sort of use their intuition to find and some college algebra here. They could find a polynomial that fits the data fairly well.