Common Core Algebra II.Unit 3.Lesson 2.Average Rate of Change

So we're going to get into it today. We're going to review how to calculate the average rate of change and how to think about it. So let's jump right into the first exercise. All right. In this particular exercise, we've got the function F of X shown graph to the right. It's a nice function F of X sitting over here. Letter a is simple enough. It says evaluate each of the following based on the graph. So take a moment, pause the video and figure out F of zero F of four, et cetera. All right. Well, this is simple enough. We're just reading things off of a graph. F of zero is equal to one. F of four is one, two, three, four, 5, 6, 7. That's equal to 7. F of 712-345-6789, ten. That's equal to ten. And F of 13, wow, one, two, three, four, 5, 6, 7, 8, 9, ten, 11, 12, 13, 14, 15. It's equal to 15. Now, let her be says, find the change in the function. Remember this notation delta right change in over each of the following intervals. Find this both by subtraction and shoga on the graph. So anytime we're trying to figure out the change in Y that's always sort of Y two minus Y one. 

So from zero to four, what I would do is I'd say the change in the function's value is 7 minus one or 6 units. And again, what that means is that this has gone up by 6 units as we kind of measure those two heights. Here, right, we're going to say that the change in the function is equal to ten -7, or three, right? That's then showing us this, the change being three. Between these two values. Finally, here we have delta F is equal to 15 minus ten. Or 5. That's going to. Be that. Now letter C is an important important question. It says, why can't you simply compare the changes in F from part B to determine over which interval the function is changing the fastest? Why can't we do that? Why can't we just say, well, clearly it must be changing the fastest over the first interval because it increased by 6. And then the last interval because it increased by 5, and then the middle, the middle interval is only increasing by three. 

Why can't we use that to talk about how fast the function is changing? Well, we can't do it because. The domain intervals. X is the domain intervals. Aren't the same size. In other words, that first interval, where X is from zero to four, well there, delta X is four. On the other hand, as X goes from four to 7, there, delta X is three, and then as X goes from 7 to 13, delta X is 6. These things aren't the same. So, you know, it's silly. It would be silly to compare two people's weight gains if one person you're testing over the span of like a couple weeks and another person you're testing over the span of a couple of years. It wouldn't make any sense to compare their weight gains. Not if you really want to see who's gaining weight the fastest, right? So now there's got to be a way to take that into account. And we're going to in the next part of this problem. So pause the video now and write down anything you need to. All right. 

Let's clear it out. So now let's talk about average rate of change. Average rate of change is always the change in the function's value divided by the change in the inputs values. And what that does is it gives us a ratio that compares how fast the two are changing. So for instance, in letter I, delta F to delta X, if you remember on that last screen, what we had was that the change in F was 6, the change in X is four, and therefore delta F to delta X is three halves. All right, likewise, number two, well, in that case, delta F to delta X the change in F if you recall was three units over three units. So it's got an average rate of change of one. In this last interval, delta F to delta X, my change in F was 5, my change in X was 6, and that's as far as we can reduce it. So let's talk about where it's going the fastest. Well, that's where it's going the fastest, right? And that's actually the second fastest, and that's the third fastest. Now, in each case, what we've really found are slopes. Right? That is slope, right? Isn't slope. The change in Y divided by the change in X but this isn't a line, right? But there are lines whose slopes we just calculated. For instance, the first one is the slope of that line. Okay? That's the one. Whoops. I didn't want that. That's the one that has a slope. Of three halves. 

Let me do the next line in a different color. Let's go with blue. The next line is that one. Let me kind of extend it a little bit this way. And that one is the one that has a slope of one. One more. Let's go in green. The last line is the one that goes from this, that didn't work out very well. Let's try green line. There we go. And then that last line. Is the one with the slope of 5 sixths. All right? Now, what does this measure? It really measures steepness, right? So that's what slope always measures is steepness. So what we're really trying to do is kind of measure on average. How steep the function is. How quickly the function's values are growing. Compared to how quickly the inputs are growing. Sorry about the messy picture. All right. I'm going to clear this out. So pause the video now and write down anything you need to. Okay. Let's keep going. 

The average rate of change for a function over the interval a to B, the function's average rate of change is calculated by doing the change in the function, divided by the change in the input. All right, F of B minus F of a divided by B minus a this is the classic slope formula. Y two minus Y one divided by X two minus X one. It's truly the slope formula. All right. Truly the slope formula. Which is kind of cool. All right? It's a very kind of mechanical formula. And let's play around with it some. All right? Exercise two, consider the two functions F of X equals 5 X plus 7 and G of X equals two X squared plus one. Calculate the average rate of change for both these functions over the following intervals. Do your work carefully and show the calculations that lead to your answers. So real quick what I'm going to do is I'm going to just put a little divide or in here. 

So we've got a clear sense of space. Let's start with F of X all right. And let's do it. You can be very mechanical about this. I like to calculate F of negative two. That's going to be 5 times negative two plus 7. I'm going to do this in my head. That's negative three. And I'm going to calculate F three. 5 times three plus 7. That's 15 plus 7 is 22. So the change in F divided by the change in X will be 22 minus negative three all over three minus negative two. 25 divided by 5 is 5. All right. Let's do it for G of X these calculations can be a little bit laborious, but it's not a bad thing, you know, try to, okay, G of negative two is going to be two times negative two squared plus one. That's going to be four. Times two is 8 plus one is 9. G of three will be two times three squared plus one. Three squared is 9 times two is 18 plus one is the 19. So change in G divided by change in X will be 19 -9. Divided by three minus negative two. It's going to be ten divided by 5. 

For an average rate of change of two. All right, pause the video now, see if you can do the two average rates of change called for in, I guess, part two. All right, let's take a look. Again, for F of X, we'll do F of one, 5 times one plus 7 gives me 12 F of 5, 5 times 5 plus 7 25 plus 7 gives me 32. Change in F divided by change in X 32 -12 divided by 5 minus one gives me 20 divided by four gives me 5. All right, let's do G of X let's see whoops. How about G? Of one will be two times one squared plus one, one squared is one times two is two, plus one is three. And G of 5 will be two times 5 squared plus one. 5 squared is 25 times two is 50 plus one is 51. So now we're ready, delta G divided by delta X 51 minus three divided by. 5 minus one will give me 48 divided by four, which will give me 12. Let her be asks, the average rate of change for F was the same in both one and two. 

Take a look at that. Right? In both one and two, the average rate of change for F was the same. But it was different for G why is that? And as a hint, let me just say that I could give you any interval and the average rate of change for F would be 5. So why is that? Did you figure it out? Something came up a lot in common core algebra one. F. Is a linear function. Whose average rate of change, I wish there was a simpler way of doing that. Whose average rate of change is always a constant. That's what makes linear functions linear. All right? What makes a linear function linear is the fact that its average rate of change does not differ. It's always the same, no matter what interval you give it. Which is why we can talk about the slope of a line, but we can't talk about the slope of a parabola. Lines have only one average rate of change. The parabola that we were dealing with in function G of X obviously its average rate of change is changing. All right, pause the video. Write down anything you need to, and then we'll go on and we'll continue to work. Okay, here we go. Exercise three. 

The table below represents a linear function. Good for us. Fill in the missing entries. Oh, this is cool. So remember, this is a linear function. It's not, it's not directly proportional. It's a linear function. See if you can use what we just learned about to fill in the missing entries. It's a little bit challenging. So take your time. All right. Well, what do we know about linear functions? What we know is that the change in Y divided by the change in X is a constant, not Y divided by X that was with that direct variation. The change in Y divided by the change in X is a constant. So let's figure out what that is right here. All right? Let's do that for a second. Well, the change in Y would be one minus negative 5. One minus negative 5, all divided by 5 minus one. Which will be a positive 6 divided by four, which will be three halves. Now what does that tell you? What it says is that for every increase of two in X, we get an increase. Of three in Y for every increase of two and X, we get an increase of three and Y well, let's just take a look here, here, right X goes up by 6. 

All right, so what's Y going to go up by? Well, let's see if delta Y divided by delta X is equal to three halves, and I don't know my change in Y, but I know my change in X 6, well then my change in Y has to be three halves times 6. And that's 9. So why has got to go up by 9? Again, it makes sense. I mean, I went up an X three times by two, so I got to go up by 9, that means that's got to be a ten. Now watch this, I can do the same thing right here. Here I see that my Y goes up by 12. Right? I just don't know my delta X, but I do know that the change in Y to the change in X has to be three halves. So now I could do a little cross multiplying three times my change in X would have to be 24. So my change in X would have to be 8. If that goes up by 12, X has to go up by 8. Which means that's got to be 19. Oh, this is a big one. 45 -19 is 26. Wow. All right, and then I got to find room for it over here. 

So again, I don't know what my change in Y is. But I know my change in X is 26, and I know they're ratio. Has to be three halves, so my change in Y must be three halves times 26. That's got to be 39. And therefore I'm going to add that 39 to the 22. And I must be at 61. All right. What this really relied on, though, is understanding that in a linear function. It's not that Y divided by X is a constant. It's that the change in Y divided by the change in X is a constant. And that can be kind of tricky. Pause the video, write down anything you need to. All right, here we go. So today, we reviewed the basic formula for how to calculate the average rate of change. And that formula is the slope formula. It's the change in Y divided by the change in X we also reviewed the idea that if you have a linear function that change in Y divided by that change in X, is a constant. 

And we'll call that constant the slope of the line. All right? And we're going to work with slopes and lines a lot more in the next lesson. For now, though, let me just thank you for joining me for another common core algebra two lessons by E math instruction. My name is Kirk Weiler, and until next time, keep thinking and keep solving problems.