Hello and welcome to another common core algebra one lesson by E math instruction. My name is Kirk weiler, and today we're going to be doing unit three lesson number 6 on the average rate of change. This is a very, very important lesson that's going to show up a lot in future lessons. So, before we begin, just as a reminder, you can find the worksheet and a homework assignment that go along with this video by clicking on the video's description or by visiting our website at WWW dot E math instruction dot com. Also, don't forget on the upper right hand corner of every worksheet are our QR codes. Scan those with a smartphone or a tablet app and you can be brought right to this video. All right, let's begin. Remember, the whole idea of functions is that we are modeling outputs compared to inputs, right? Functions or rules that change inputs into outputs. Often what we want to do is measure how fast an output is changing as the input changes. Okay? That's known as a rate. So what we're going to be getting into today is called the average rate of change. On average, how fast is the output change in compared to the input. I love the first exercise. So let's jump right into it. Max and his younger sister Evie are having a race in the backyard. Max gives his sister a head start and they run for 20 seconds. The distance they are long in the race in feet is given below. With max's distance given by the function M of T and EV's distance given by the function E of T all right? Now one of the things that we're supposed to be able to do in common core math. Is interpret statements that kind of look like this. M of 12 equals 30. How do you interpret that? What would you write down? Pause the video now, and think about that a bit. All right. Well, let's take a look. Remember, this is the input, and this is the output. All right. So when the input is 12 and this X axis or T axis goes by two, then the output here on M of T is 30. So what that means is that after 12 seconds. Max, whoops. Probably capitalize his name. Max. Is 30 feet along in the race. He's 30 feet along in the race. Maybe he's 30 feet in front of the starting line. Let her be. If both runners start at T equals zero, which is right here. How much of a head start does max give as little sister? How can you tell? All right, think about this for a second, pause the video and see if you can see how generous was max was in this race. All right, well, let's take a look. Max starts at zero. On the other hand, where does EV start at? EV starts at right here, right? At 25 feet. So that's how much of a head start it gives her. Right? That's the head start. Now we had to know that max started at zero and EV started at 25, because if max started at ten and the EV started at 25, then he would have only given her a 15 foot head start. But he's starting, let's say, at the starting line, and she's starting 25 feet ahead. Does max catch up to his sister? How can you tell? Pause the video again and think about how you can interpret this graph and weather max catches up to his sister Evie. All right. Well, I'm hoping that you said no. Now you could give many different explanations, right? To me, maybe the best way of saying it would be at T equals 20 seconds, right? E of 20, where is EV, EV is at 60 feet, right? And M of 20 max is at 50 feet, right? Here's Eevee. At 60, and here's max, at 50. Right? So he's still ten feet behind. He's been catching up because he started 25 feet behind. But he hasn't caught up to his sister. Another way of putting it is no because the lines. Never cross. Right? They're going to be at the same place if those lines cross if they intersect with one another. But the lines never cross, and therefore max, at least during this 22nd race, never catches up to his sister. All right, now we're going to keep working with this problem, but I do have to clear out this text. All right? So copy down what you need to. All right, let's keep going. All right. Letter D, how far does max run during the 22nd race? How far does EV run? What calculation can you do to find EV's distance? Now, what I'd like you to do is I'd like you to pause the video because this is very important for letter E I don't know how that became letter D but for letter E, it's really important that we know how far each one of them ran. So pause the video now and see if you can figure that out. All right. Well, let's just keep track. Max starts at zero. Ends at 50, so he traveled 50 feet. EV on the other hand, well. She starts at 25. And she ends at 60. Now what that means is that we have to do a little bit of subtraction, right? 60 -25, she ran 35 feet. So max ran farther, right? He ran 50 feet. She ran 35 feet, that's why he gave her a head start, right? Because he knew that he was going to be the faster runner. But that's important, right? We found EV's distance by doing subtraction. And specifically, what we did was we found the change in her position, right? We found the change in her output. She started at 25, she ended at 60, and we subtracted the two to figure out that she ran 35 feet. So finally, in letter E, it asks us how fast do both EV and max travel. In other words, how many feet do each of them run per second, express your answers as decimals and attach units. Okay? So I'm going to ask you to pause the video right now and try to come up with their speeds. I think intuitively you're going to know how to do this. All right, so why don't you go ahead? All right. Well, let's go through it. Now it's just a real world example that's very simple. If I told you I had traveled a hundred miles in two hours. Then my speed would be a hundred divided by two. Or 50 mph. Max traveled 50 feet in 20 seconds, so he traveled, if you do that division, 2.5 feet per second. Every second that goes by, he travels 2.5 feet. EV, on the other hand, traveled 35 feet in 20 seconds. If we do that division so a little bit uglier, we get 1.75. Feet. Per second. Both of these are what are known as average rates of change. Average rate, I will put S on their rates of change. Because what they're doing is they are measuring how fast the output, in this case, the distance is changing. Compared to the input, which in this case is time. Right? So that's what average rate of change is. The change in the output divided by the change in the input. I'm going to clear this text down, write down what you need to. Okay. Now, exercise two. Finding the average rate of change, in other words, how fast the output or Y values are changing, compared to the input, the X values, is the same as finding the blank of a line. I'm hoping that this whole process feels kind of familiar because it's something you found with a line before. Pause the video right now. And think to yourself, finding the change in Y divided by the change in X is the same as finding what. All right. Well, from a lot of previous experience, especially last year, that is the same as finding the slope of the line. All right? So I'm going to introduce you to a formula. A formula for finding the average rate of change of any function. And it's going to look probably a little scary. But I don't want you to forget that all we're really doing is finding the slope between two points. Okay? So I'm going to clear this text out, then we're going to tackle this ugly looking formula and try to make sense of it. All right, the average rate of change. For the function, Y equals F of X, the average rate that F of X changes from a from X equals a to X equals B is given by F of B minus F of a divided by B minus a all right? I want to illustrate this just for a moment on a graph. Okay? So I'm not going to even draw anything in. We got a, right? And then at some point, we've got some point, and we have its output F of a. Remember, this is a Y value. Well, okay, that didn't help. Let's try that again. This is a Y value. Specifically that one. Then you add some other point B and F of B almost looks like an F, but not really. Right? That's also a Y value. All right, so what I do F of B minus F of a, what I've found is this. Any time you subtract two numbers, you find the distance between them. Although that can be negative. So take that with a grain of salt. And then this distance is B minus a so this is the change in the Y how much the Y values have changed. And this is the change in X, how much the X values have changed. When you divide the two, you find out how fast the Y values you're changing compared to how fast the X values are changing. Simple ratio. In the last problem, specially good with EV, right? What we found is that the change in the Y values were given by 60 -25, that was the change in the Y the change in the X was really 20 minus zero, which gave us that 35 divided by 20 or that 1.75. Feet per second. All right? That's it. F of B minus F of a divided by B minus a better yet, how much did the Y values change? Divided by how much the X values changed. Now I'm going to clear this out. We're going to do an exercise still on this page so that formula's just sitting in front of us. So copy down anything you need to before I bring up the next exercise. All right, let's go on. So exercise three says consider the function given by F of X equals X squared X times X plus three. Find its average rate of change from X equals negative one to X equals three carefully show your work that leads to your final answer. All right. So let's kind of really lay this out. We start with X equals negative one, and let's figure out what Y value we start at. If I plug negative one into my function to figure out my Y value, remember the output is the Y value. Negative one times negative one is positive one. So we started a Y value of four. Okay. Then as we go to X equals three, again, let's see what we have. Three times three is 9. Plus three is 12. So we started an X value of negative one, and a Y value of four, and we end in an X value of three, and ended a Y value of 12. So the change in X that little triangle means the change in X is three minus negative one, which is four. That's just your B minus a now, we could. We could have known that anyway, right? X increased by four. Our change in Y is 12 minus four or 8. Our Y values went up by 8. So the average rate of change, which is the change in Y divided by the change in X is 8 divided by four, which is just two. Now what that means, what that means, and I know the problem didn't ask us to interpret our answer. But what that means is that for every one unit that X increases. Y, on average, goes up by two units. That's all it means. You know, it's kind of like when I say I'm driving at 60 mph, that means for every hour I drive, right? Then I go an additional 60 miles. Here, an average rate of change of two means for every one unit that X increases. Y increases by two. On average. All right. I'm going to clear out this text, so write down what you need to, and then we're going to do some more on these types of problems. All right, here it goes. Let's move on to the next one. All right, now we got to be able to deal with functions and all their different forms. So here we have a function given with a table. The function H of X is given in the table below, which of the following gives its average rate of change, average rate of change over the interval two to 6. Two to 6. Well, the formula is H of 6 minus H of two divided by 6 minus two. But really all that is is the change in Y divided by the change in X now watch out because Y goes down. It goes from 9 to three, so we always do the second one minus the first one. Second one minus the first one now. This is important because three -9 is negative 6. 6 minus two is positive four, and of course we can reduce this fraction by dividing both by two and getting negative three halves. All right? So the change in Y divided by the change in X Y went down by 6 X went up by four X always increases. Always increases. We're always going from left to right on the graph. Okay? But why could either increase or decrease? All right, I'm going to clear this out. So again, copy down anything you need to. All right, here we go. All right. Let's do one more problem. Exercise number 5. Francis is selling glasses of lemonade. The function G of T equals T squared plus four divided by two. Models, the number of glasses she is sold, G after T hours. So in other words, make sure you understand this. The input is the number of hours she's been selling lemonade. And the output G is how many glasses of lemonade she sold. What is the average rate at which she is selling lemonade between two and 6 hours, include proper units. Well, let's figure out how many glasses she sold after two hours. According to our formula, that would be two squared plus four divided by two, two squared is four, and two times two. So now I get 8 divided by two, which is four. Now watch me keep track of units though. She's sold four glasses after two hours. Maybe not the greatest sales. Anyway, on the other hand, after 6 hours. Right? We've got 6 squared, which is 36 plus four divided by two. 40, that's a four divided by two. Gives me 20 glasses. So the change in my output right is 20 minus four or 16 glasses. That's in the numerator. That's in the top of my division. Right in the bottom of my division, if you will, actually, let me get rid of that arrow. I like the arrow being in a little bit of a different place. If you will, the bottom is simply the change in time, which is 6 minus two, or four hours. So finally, I can combine the two when I can say the change and the output divided by the change in the input, 16 glasses divided by four hours, gives me four glasses per hour. So on average, on average, Francis is selling four glasses of lemonade per hour. Between the two hour and the 6 hour mark. I don't know. Depending on how expensive the glasses of lemonade are, she might be making quite a bit of money or not so much. All right. Anyhow, I'm going to clear this out, so write down what you need to. Here we go. All right, let's finish up the lesson. So functions are all about modeling outputs compared to inputs. All right? In this lesson, what we did is we took a first glance at how we measure how fast those outputs are changing compared to how fast the inputs are changing. And we do that by measuring how much the output has changed. And dividing it by how much the input is changed. That's called the average rate of change. And you're going to get a lot of practice on that, obviously, on the homework set. And it's going to pop up from time to time in the rest of our lessons. But for now, let me just thank you for joining me for another common core algebra one lesson by email. My name is Kirk weiler. And until next time, keep thinking and keep solving problems.
Average Rate Of Change Common Core Algebra 1 Homework
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