INSTRUCTOR: In today's lesson, we will learn how to perform exponential regression in the online graphing calculator, Desmos. So here, we have an example. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after 1985. And we want to know how many cell phone subscribers were in Centerville in 1994. Now, based on this information here, you could probably build a function model, substitute, evaluate, and figure it out, which is what I did down here. 

But what if you only had the data from each year and you wanted to go backwards? You wanted to build the model. That's where we can use regression to help us. So what I've done is I've taken this data and I've put it into an Excel spreadsheet. The reason I put it into an Excel spreadsheet was to make it easier to copy and paste it and put it into Desmos. But I will also show you how to build a table in Desmos if you don't want to first put this in Excel. 

So what I'm going to do is take this information, and I'm going to copy it. And then I'm going to go to Desmos. Once I'm here in Desmos, I merely have to click on a cell and paste my data. And notice it makes a table, giving my x values x1 and my y values a y1. But if I wanted to put this in manually, I can always go here to the Plus button and insert a table and then type in my values. I can put any value I want here in for x, and I can put any additional value I want in here for y. 

But let's go ahead and paste since we've already got the data. Notice over here on the right, I'm looking at my data. And you can see that this looks like an exponential curve. If I want to see my dots a little more clearly, I can go to the wrench. I can select Projector Mode. 

Notice I can also change my x-axis, the minimum and maximum values I'm displaying. For example, on here, it looks like I really only go to about 8. So I'm going to change this from 17 to 10 in order to help my data spread out a little more. I'm going to put this in projector mode, and then you can see that the dots are bigger. 

Now, what's a good equation to model this? Well, again, all I have to do is click in a box and type in my exponential model. I will start by typing in y1. Instead of equals, remember I need to use the tilde to approximate so that I will tell Desmos to perform regression. Then I'll type in my model. Remember that it's a, which stands for the initial value, b, which stands for the growth rate or decay rate. And then use the up caret in order to raise it to the power of x sub 1. 

Notice that it put in a line that fits my data very nicely. And if I look down here in Desmos, it tells me what my a value is, and it tells me what my growth rate is. Notice that my growth rate is 1.75. If we go back to the problem, remember that the words said that the number of subscribers increased by 75% So it's no coincidence that my growth rate was 1.75. Remember that I keep 100% of what I had the previous year, and then add an additional 75% to that. 

Notice that we started with 285 cell phones back in 1985. And when I go here to Desmos, my a value or my starting value is approximately 285. Remember that this is a line of regression. It's using an approximate line of best fit to model my data. So while it's very, very close, it might be off a small amount due to rounding. 

And this concludes our brief tutorial on how to use Desmos in order to perform regression. You can perform regression for linear models as well as exponential models or other models that we'll be studying in the future.