Analyzing data to model functions

0:00- [Instructor] What we have in these three different tables are three different functions, or I guess we could say, points input and output pairs from three different functions. And our goal is based purely on these pairs, on these inputs and outputs. Try to classify each of these as either being linear, quadratic, or exponential.

0:21So the key to doing that is to recognize linear function. If you have consecutive values of x, the difference in consecutive values of y should be the same. For exponential functions, I mean, you could think of an exponential function, let's say you had y is equal to, I don't know, 3 to the x.

0:42Let me make a little table here. So when x and y, so if I had 0, 1, 2, 3 to the 0 is 1, 3 to the 1 is 3, 3 to the 2 power is 9. For exponential functions like this, the ratio between consecutive terms, as long as this distance is going to be the same, the ratio will be the same.

1:04So notice, the ratio between 3 and 1 is 3. Or another way, you can multiply by 3 every time. For quadratic, let me give you an example. A very simple quadratic is y is equal to x squared. And so let me make a couple of columns here. If I have x, y, and then let me do the df and then let me call this the second difference or the difference of the differences, second difference.

1:35So x squared, so let me do 0, 1, 2, 3. So 0 squared to 0. 1 squared is 1, 2 squared is 4, 3 squared is 9. The differences aren't going to be all the same like in a linear function. So the difference between 0 and 1 is 1. Difference between 1 and 4 is 3. Difference between 4 and 9 is 5.

1:59So these are all different. But for a quadratic, the second differences will be the same. So to go from 1 to 3, this is 2. And I encourage you to think about why that is the case. But notice, this is 2 again. So if our first differences are the same, it's linear.

2:15If our second differences are the same, it's quadratic. And if neither of those are true, but our ratio between each consecutive term is the same, then we're exponential. So let's look at these cases. And pause the video like always and try to classify them yourself before I do it.

2:32So this one over here, let me actually extend all of these so we have those multiple columns here. And so first, I will do the difference and then I will do the second difference if necessary, and then I'll do the ratio if necessary. So I'll make three columns for each of these characters.

3:00Okay, three. Three columns. All right, so our x values, these are all just going up by one. And actually we see that in all of these cases. So we can make that type of comparison. You have to be careful. This went from like 0 to 5 and then 5 to 7, and then 7 to 18.

3:18You wouldn't be able to do the type of analysis that I just said. But here, we have the same change in x every time. So going from 1 to 5, this is, our difference is 4. Our difference going from 5 to 9 is 4. Our difference from 9 to 13 is 4. Our difference from 13 to 17 is 4.

3:36So based on what I see here, I can confidently say that this is linear. Now let's do this example. Going from 1 to 4 is 3, 4 to 13 is 9. So I can already tell you this is not going to be linear. 13 to 18 is 15. Sorry, 13 to 28, I should say, is 15. 28 to 49 is 21.

4:0649 to 76 is, let's see, 26, it is 27. So the difference is not the same, but the difference of the differences. So maybe let's call that the second differences. Lemme call this second differences. Well, between 3 and 9, it's 6. Between 9 and 15, it's 6. Between 15 and 21. Whoops!

4:33Between 15 and 21, it's 6, and 21 and 27, it's 6. So our second differences are all the same. So we can say confidently, based on the data we're seeing here, that this is quadratic. And then last but not least, let's look over here. If we look at our first, maybe I'll call that, I'll use Greek letter, delta for difference here.

4:53So 2 to 6 is 4, 6 to 18 is 12. I can already tell you it is not going to be linear. 18 to 54, let's see, 20 to 54 would be 34, so this is 36. So this is 36. So this is definitely not linear. And if I took the second difference right over here, from 4 to 12 is 8 and then from 12 to 36 is, well, it is 24, so this is not quadratic either.

5:28Let's look at the ratio between consecutive terms. So if we go from 2 to 6, we're multiplying by 3. So it's 3. From 6 to 18, we're multiplying by 3. From 18 to 54, multiplying by 3. 54 to 162, multiplying by 3. 162 to 486, it looks like that is also multiplying by 3.

5:49So this one checks out. So this one is for sure, based on the data we see, at least not expotential, exponential, exponential. And we're done.