 Consider the scatter plots you see on the screen right now. When you look at these different data sets, what type of shape does the data seem to form? If we were trying to connect the dots, like in this first example, A, and we try to do it in some continuous manner, what we see here is a graph that kind of looks like it's some type of exponential function of some kind, right? It does have that same kind of basic shape. And when you look at these other pictures, you kind of see the same idea as well that they appear to be these exponential graphs. All right, so this is something like that. And maybe you get something like this. These have these various different exponential growth and decay type graphs right here. And so I wanna kind of mention, how could you come up with a function to model these type of data sets if you believed your data looked like an exponential function? Well, when it comes to exponential functions, one of the most distinguishing characteristics is going to be its horizontal asymptote. So if this truly is an exponential function, that is the data model something exponential, there has to be some limiting value. So like either a maximum or a minimum value on the graph that the function doesn't seem to surpass. Now, if it's a growth model, right? I mean, we actually see two different types of growth models here. There's this one where basically emerges from nothing, or in this case, around two, and then it increases without bound really rapidly. That'd be exponential growth there. It could also be that you have this growth that's exponential, but even still, there's this limiting value. It doesn't get past something like this. So this one right here would be a good example of like population growth. That there's some baseline population, which in this case, you know, it probably would be zero, but even still, there's something like baseline that it grows off of going up and up and up and up without bound. But like this one right here, this might be something like if you take a very cold object and insert it into a hot environment, like Newton's Law of Cooling, it would start getting hotter, hotter, hotter. At first when it's really cold, it would change its temperature rapidly, but then it would slow down as you get closer, getting closer towards the reservoir's temperature right there. And then for the last one right here, again, this would be some type of decay model as you're going off towards some limiting value. Again, like Newton's Law of Cooling or radioactive decay or something like that. So you have to look for these horizontal acetotes. If there's no horizontal acetote, then it really isn't an exponential graph. So what you wanna know about an exponential function is the most general form of an exponential function is gonna look like Y equals what we'll call C times A to the X plus K, where you have some type of vertical stretch, some type of base. The only thing we know about the base is the base has to be positive and not equal to one, like so, and then there's gonna be some type of shift that goes on there, shift up and down. Now, what's nice about these exponential functions is if you wanna determine the shift, all you have to do is find the horizontal acetote. Where is the horizontal acetote? So for this one, we put it at Y equals two. For this one right here, we got it at Y equals 10 and this one looks to be about Y equals one. So we can see the horizontal shift very easily just by looking at, of course, we can look at the horizontal, the location of the horizontal acetote is the horizontal acetote's location. It's Y coordinate gives you the vertical shift for there. So the next thing to look for, I would then say is look at this number C right here. So like we observed, K equals the location, right? The location of the horizontal acetote. This number C on the other hand, you can determine C here by taking the difference. This is gonna be the X intercepts, excuse me, the Y intercepts location. Take away the horizontal acetote's location basically to take away the K. So if you take the difference between the horizontal acetote and the Y intercept, that is you take from the Y intercept the horizontal acetote, that gives you this value C. So then you'd have to look at your pictures and kind of see what do these Y coordinates look like right here. We'd have to kind of make an estimate of some kind. Something like this. So for this picture, it looks like it's gonna be like 9.5 would be my point right there. And so you take that, we're gonna subtract those then. We see that C is gonna equal 9.5, take away one. So we get 8.5 as your coefficient, okay? And then you can do similar things for these other ones. Like this one right here, you're looking like, let's say 0.5, take away 10. So you get negative 9.5. And that's good because the negative sign does indicate that it's kind of reflected downward. It's concave down compared to concave up like most exponentials are. And like this one right here, what do we wanna say it is? It's pretty close, right? So we might say something like 2.1. And so we get 2.1 minus two, which give us 0.1, like so. And so then we get these C values. And so the next thing then, then the determinant is going to be your base, right? So we have to determine the base. Now some of you might be inclined to use the natural exponential for which you have to modify your function as y equals C times E to say like the Rx plus K. You can make that adjustment. It's really okay. You can either do A or E to the R. If you wanna do exponential, like natural exponential, you can do that. That all involves some logarithms to solve for the R value right there, the rate of growth. I'm just gonna stick with just my basis arbitrary right here. In order to do this, then you just have to pick a point, right? Pick a point that looks like it's on the graph and use that as a baseline to figure things out. So like if you take this one, for example, right here, it's like feels like that's a point on there. And so it looks like three comma, I'm looking at the picture here and guessing, but we probably would have a dataset that would tell it to me. This looks like 6.5, something like that. For which then we can now work with our equation. We have y equals 0.1 times A to the x plus two, for which then if we plug in the x and the y, we're gonna get 6.5 is equal to 0.1 times a cubed, right? Plus two. And so then you can just solve for A in this equation, right? You're gonna have to subtract two, divide by 0.1 and take the cube root. You can do all of that. A little bit of a tip, if you can find the point when x equals one, that is a great point to use. So if I come over here, oh, that looks pretty good. That looks like one comma three. I like it when the x coordinates one because then you'll see y equals 0.1 times A to the first plus two, excuse me. Oh, I need to put in the y coordinate. The y coordinate turned out to be three. And then I think about when you take the first power is that you just get back in A, right? So you don't have to take any roots or anything like that. So we'll subtract two from both sides. Three minus two is one. We get 0.1 times A. And so you can divide by 0.1. So A equals one over 0.1, like so. Which of course that just gives you back a 10. And so our function, our model, model first one would appear to look like from my observations, y equals 0.1 times 10 to the x plus a three. Like so. And so that gives us an equation for the first one. If we do the second one, right? We wanna pick a point. Preferably the x coordinate one is really good. Like I said, it just makes the calculation a little bit easier. It's not necessary. If we take x equals one, that'd be about this point right here. So we'll say one comma, that's about four and a half again. Again, I'm just kind of eyeballing it. We can use technology to improve this calculation dramatically. But really in this video, I just wanna emphasize how do you do it if you're just looking at it using a scientific calculator, right? Could we do this? So our C value turned out to be negative 9.5. A we don't know yet. And then our K value turned out to be 10, I believe. That's what we saw, yes. And so then we're gonna plug it out. We have 4.5, negative 9.5 divided by A to the first plus 10. So we're then gonna proceed from there, right? So track 10 from both sides. That's a 4.5 right there. So we take 4.5, take away 10. That gives you negative 5.5 equals negative 9.5 A. We're then gonna divide both sides by negative 9.5. So we get 5.5 divided by 9.5 for which that would then give us, put it in the calculator. You're gonna get basically 0.5789, you know, we're gonna wrap all around the five decimal places so we get something like that. Again, with a computer estimate, we can do a much better estimate here. And so we then can write this down. Y turned out to be negative 9.5 times 0.57895 to the X plus 10. That equation would model the data you see right there. And then for the last example, let's finish this one up. So our equation will look like, oops, Y equals our C value on this one turned out to be 8.5. We don't know what A is yet, so that's what we have to solve for. And then our K value is a plus one. So we wanna look, again, if we can, find the point when X equals one. That way we can just kind of simplify things a little bit. So that's about this point right here. Let's say that's one and five and a half, right? Again, I'm just kind of estimating using my eyeballs for which then we're gonna plug those in there. So Y turns out to be 5.5. We get A to the first plus one. So you subtract one from both sides, you're gonna get 4.5 is equal to 8.5 A, for which then A, A should then be 4.5 divided by 8.5, for which when you throw that in your calculator, you get 4.5 divided by 8.5. That, whoops, that then gives us 0.52941. And that would then give us the equation we want. Y equals 8.5 times 0.52941 to the X plus one. And so we can model exponential functions by using the data here. My first recommendation is look for the horizontal acetote, then look for the y intercept, and then look for any other point on the graph, preferably the point one. And if you do that, we can minimize the calculations, the arithmetic necessary to model this thing. And again, this gives us a good estimate of these exponential functions that'll match the data pretty accurately. Again, we can always improve this by using statistics and computer software. But just so you have an idea of what's going on here, we're looking for parameters to fit the data, and this is how we can do it for exponential functions.