Linear vs. exponential growth: from data | High School Math | Khan Academy

The number of branches of an oak tree and a birch tree since 1950 are represented by the following tables.

So for the oak tree, we see when time equals 0 it has 34 branches. After three years, it has 46 branches, so on and so forth.

Then for the birch tree, they give us similar data. At the beginning, it has eight branches, in 10 years it has 33 branches, and they give us all of that.

What I want to think about in this video is how we should model these. If we want to model these with functions, the choices we'll give ourselves—there are other options, but the choices we'll give ourselves in this video—are linear and exponential functions.

Which of these are going to be better for modeling the data?

So let's first look at the oak tree. The key realization is whenever I have a fixed increase in time—each of these steps is plus three years—what happens to my number of branches? Is it going to be a fixed change, in which case a linear model might be good, or is it going to be a change that's dependent on where we were?

What am I talking about? So, 34 to 46 that is +12. 46 to 59 is +13. 59 to 70 is +11. 70 to 82 is +12.

At first, you might think, "Well, this isn’t an exactly fixed change." These numbers seem to average right around 12, but when you're looking at real-world data, you're never going to get something exact. The models are just going to give us a good fit, a good approximation of the behavior of the number of branches over time.

For me, this is pretty close to a constant 12 branches a year, so I would construct a linear model here. I would say branches as a function of time. Let me be clear: this isn't 12 branches per year; this is 12 branches every 3 years.

This was 13 branches over three years; this was 11 branches every 3 years. But we're going to average 12 branches over three years.

So, the number of branches we have—we're going to start at 34 branches, and then plus 4 branches every year. Here, you could test this out. B of 0 is going to give us 34 branches. B of 12, let's just really test out the extreme part of the model, B of 12, is going to be 34 + 48, which is equal to 82.

So this model works quite well. It's going to have a couple of places where it's not exactly fitting the data, but it fits it quite well.

So this is a linear model. Now let's look at the birch tree. Time equals zero, so fixed change in time—let me—not layer all right, so we have a fixed change in time. Every time we are moving into the future a decade, let's see our change in branches.

We go from 8 to 33, so what is that? That is +25 branches. Then we go from 33 to 128. Well, that’s way more than 25 branches.

That’s going to be what—less than five less than 100, so that's plus 95 branches. So this clearly is not a linear model.

Let's think in terms of an exponential model. How much do we have to multiply to go from—did I do that right? 128 minus—yeah, if it was 133, then it would be 100, and then it's five less than that, yep.

Okay, so now let's think about it in terms of an exponential model. In an exponential model, we care about what we have to multiply for each step.

So, if we have a constant step in time, what do we multiply for how much we increase our branches? To go from 8 to 33, that’ll be approximately—it's going to be approximately four. It’s going to be a little bit more than four.

33 to 128, well that’s going to be a little bit less than four, but it’s approximately four. 33 * 4 would be 132, so we're close. 128 to 512, that’s exactly four, right?

That’s exactly—120 * 4 is 480 + 32, yep, that is exactly four. So times four.

It looks like we keep multiplying by four every 10 years that go by.

One way to think about it is we could say here B of T, the branches of T, our initial condition, our initial state is eight. Now, we could say our common factor is four.

But, if we want T to be in years, well, every 10 years we multiply by a factor of four. T has to go to 10 before we increase the exponent to one or has to go to 20 until this exponent becomes two.

So, 8 * 4 to the T over 10 power seems like a pretty good model. You could even verify this for yourself if you like.

Try out what B of 30 is going to be. B of 30 would be 8 * 4 to the 30 / 10 to the 3 power.

What is that going to be? That’s going to be 4. The 3 is 64. 64 * 8 is—it's 480 + 32. It is 512.

So, once again, this exponential model—this exponential model for the data does a pretty good job.