Determining whether real world model is linear or exponential

The table represents the cost of buying a small piece of land in a remote village since the year 1990.

Which kind of function best models this relationship? I'm using this as an example from the Khan Academy exercises, and we're really trying to pick between whether a linear model or a linear function models this relationship, or an exponential model or exponential function will model this relationship. So, like always, pause this video and see if you can figure it out on your own.

All right, so now let's think about this together. As the time goes by, or on this, the time variable right over here, we see that we keep incrementing it by two. We go from zero to two, two to four, four to six, and so on and so forth. It keeps going up by two.

So, if this is a linear relationship, given that our change in time is constant, our change in cost should increase by a constant amount. It doesn't have to be this constant, but it has to be a constant amount. If we were dealing with an exponential relationship, we would multiply by the same amount for a constant change in time.

Let's see what's going on here. Let's just first look at the difference between these numbers. To go from 30 to 36.9, you would have to add 6.9. Now, to go from 36.9 to 44.1, what do you have to add? You have to add 7.2. And now, to go from 44.1 to 51.1, you would have to add seven.

Now, to go to 51.1 to 57.9, you are adding 6.8. And then finally, going from 57.9 to 65.1, let's see, this is almost eight. Seven point one, this is what? Seven point two. We're adding plus seven point two.

So, you might say, "Hey, wait, we're not adding the exact same amount every time." But remember, this is intended to be data that you might actually get from a real-world situation, and the data you get from a real-world situation will never be exactly a linear model or exactly an exponential model.

But every time we add two years, it does look like we're getting pretty close to adding seven thousand dollars in cost. Six point nine is pretty close to seven. That's pretty close to seven. That is seven. It's pretty close to seven, so this is looking like a linear model to me.

You could test whether it's an exponential model. You see, well, what factor am I multiplying each time? But that doesn't seem to be, as this doesn't seem to be growing exponentially. It doesn't seem like we're multiplying by the same factor every time. It seems like we're multiplying by a slightly lower factor as we get to higher costs.

So, the linear model seems to be a pretty good thing. If I see every time I increase by two years, I'm increasing cost by 6.9 or 7.2 or seven, it's pretty close to seven. So, it's not exactly the cost, but the model predicts it pretty well.

If you were to plot these on a coordinate plane and try to connect the dots, you could. It would look pretty close to a line, or you could draw a line that gets pretty close to those dots.