Writing exponential functions to fit data and make predictions

0:00- [Instructor] We're told a group of biologists use a tagging system to track a deer population. They need an exponential function P to model the population after x years so that they can make predictions. Here is a table that contains some of their data. Okay, so after year two, this is the population, year four, this is the population.

0:21If the trend continues, and exponential trends don't always continue, maybe the deer run out of food or something, what is a reasonable function P of x to model this data? And what is a reasonable prediction for the population at year 12? So pause this video, have a go at this, and it'll probably be U.S.eful to have a scientific calculator to do this, or I guess you could do some of the math on your computer.

0:43All right, now let's do this together. So let's just think about the general form of an exponential function. So we could write it, the population is going to be equal to some constant a times some common ratio to the x power. So we need to figure out what are reasonable approximations for a and b based on this data right over here.

1:06So first let's think about what this common ratio is. And to do that, let's think about the ratio of the population every two years to the population two years before that. So what are we multiplying by when we go from 930 to 1,670? Let me get my calculator out here.

1:24So if I take 1,670 divided by 930, it is approximately, let's just round everything to the nearest 10th. This is a model, it's not going to be exact anyway. So this is approximately 1.8. So I'll say times approximately 1.8. Maybe I'll say, well, I'll just write times 1.8.

1:49Well, I'll keep saying approximately because it's not exactly 1.8. Now, let's see what happens when we go from four years to six years. Get the calculator out again. And so I'm going to take 2,940 deer divided by 1,670 deer. And so this is 1.76. If we round to the nearest 10th, once again, and this is a little bit different because when you're dealing with real data, you're not going to get exactly the same common ratio every time, but this is pretty close, once again, to times 1.8.

2:22Again, it's roughly 1.8 if we round to the nearest 10th. And then last but not least, it might be good form, let's just check that last one. Where'd my calculator go? We have 5,240 divided by 2,940. And so once again, if we round to the nearest 10th, it's times 1.8.

2:46It wasn't exactly the same thing every time, but once again, this is an approximation. Now, you might be tempted to say b is 1.8. b would be 1.8 if these were year one, year two, year three, year four, but two years goes by every time we increase our population by a factor of 1.8.

3:05So how do we deal with that? So let's think about it this way. We know, for example, that P of four is equal to a times b to the fourth power, and that is equal to 1,670. We also know, let's see two years before that, P of two is a times b squared, which is equal to, because that's when x is equal to two, that's equal to 930.

3:38Well, what happens if we divide P of four divided by P of two? So let's do that. If we divide this, and maybe one way we could say it, P of four over P of two is equal to, let me just make all of these just one equal sign, P of four over P of two. If we divide the population at year four, divided by the population at year two, that's the same thing as a times b to the fourth divided by a times b squared, which is the same thing as 1,670 divided by 930.

4:10So on the left here, this simplifies a divided by a, that's the same thing as one over one. b to the fourth divided by b squared is just b squared, and that's going to be equal to, we already know that this 1,670 divided by 930 is approximately 1.8, so let's just write 1.8 there.

4:28And so Now, we can figure out b by taking the square root of both sides. b is equal to the square root of 1.8, which is approximately equal to, and this is all a little bit of approximation because we are doing some modeling here. So if we were to take 1.8 and we were to just take the square root of that, we get, this is approximately, maybe I'll round to the nearest 100th here, this is roughly 1.34.

4:59So approximately 1.34. So now, we could say that P of x, maybe we might want to write is approximately equal to or if we want to write equal, but we have to keep in our mind that this is an estimate. It's not going to get every number exactly right. It's a mathematical model.

5:16The real world is seldom exactly that clean, but it's going to be equal to a times, and we could take 1.34 to the x power. Now, how do we solve for a? Well, we could take one of these points here and solve for a. For example, we know, let me do it over here, P of two is equal to a times 1.34 squared, which is equal to 930.

5:52930. Or another way to think about it, we already know what 1.34 squared is, this is 1.8. And so we can divide both sides by 1.8 and we are going to get a is equal to 930 divided by 1.8, which is, let me get the calculator back out, 930 divided by 1.8, which is equal to roughly 517.

6:26So let's call a is roughly, I'll just say equal to 517. And so an exponential that models this reasonably well could be P of x, the population, is going to be 517 times 1.34 to the x power. You might have gotten different things, you might have rounded in a different place.

6:46This problem wasn't very particular about what to round to. We just want to get a reasonable prediction of what might be happening here. But they say what could be a reasonable prediction for the population at year 12? Well, P of 12 would be 517 times 1.34 to the 12th power, which is equal to, get that calculator out again, so I'll start with 1.34 to the 12th power is equal to that and then times 517.

7:21So 17,328 deer. 17,328 deer is a reasonable prediction assuming that the exponential growth continues, the deer don't run out of food, predators don't start showing up in a larger way, whatever it might be. This is a reasonable prediction based on at least the data that we have and assuming that the exponential trend continues.