 Today I would like to just discuss the average rate of change. So if you think back to a previous math course, you probably have talked about how to find the slope of a line. And one way to find the slope is to find just two points on the line. And then the slope is our average change. So a lot of times it is described as the rise over the run. Well the rise can be found by finding the change in the y values. Oh sorry, that's supposed to be a 1 there. So we can do y2 minus y1 to find the rise. And the run is just our change in our x values. And so we can find that by doing x2 minus x1. And our slope formula is simply the rise y2 minus y1 over the run x2 minus x1. Now today we're essentially going to be doing the same thing, except instead of just finding the slope, we're going to look at how you can find the slope when it is a nonlinear function. So rather than having a line, you can find what's called the average rate of change of a function when it's not just a line. But you can use that same idea. So for instance, if you look at this graph shown on your paper, if we wanted to find the average rate of change, which I'm just going to abbreviate AROC, on the interval negative 3 to 1, what we would do is just find where that point, negative 3 is for our x value where it hits the function and then we would find 1 for the x value. So both of these describe our x values and we're trying to find the rate of change between those two points. So essentially we're trying to find the slope of this line and you can use the same idea that we just did. This is the point negative 3,3 and this is the point 1,3 and we can find the slope between those two points by doing 3 minus 3,1 minus a negative 3, which is 0 over 4 or just 0 and that makes sense because it's a horizontal line. Now the difference with average rate of change is the average rate of change will be different depending on what interval you're looking at. So if I just erase this a second, let's now find the average rate of change on the interval. We'll just go from negative 4 to 1. So I've just shifted 1 point over and I've kept my second interval the same but notice this is getting me a much different line. So this now, the average rate of change will be positive and that's just giving you the overall change just on average what's happening. It doesn't matter what happens in between. On the graph we're just looking at what the overall average changes. So here if we wanted to find that average rate of change we'd have our points negative 4,0 and 1,3 and you can use your slope formula again, the rise or the change in our y's over the run, the change in your x's which gives us an average rate of change of 3 fifths. So what that means is on average when our x's go from negative 4 to 1 the y's are changing on average 3 fifths for every unit we go to the right. And you can see that that works because here we go up 3 and over 5 giving us an average rate of change of 3 fifths. Now pause the video for a minute and find the average rate of change for the remaining two graphs that are shown on your paper. And after you're done I'll go through a few examples of how we can find the average rate of change when we just are given an equation. Now as I said before our formula for the slope is the change in y's over the change in x's and the formula for the average rate of change is very similar. We just are simply relabeling things. So let me draw a little diagram here. We'll see if I can fit it in. Say we have this function here we'll call this function f of x and we're going to find the average rate of change from this point we'll call it x1 till f of x1 that's just in function notation to this point up here x2 f of x2. Well if we wanted to find the average rate of change we need to find the slope of this line that connects the two. And remember our slope is found by doing the rise over the run. And if we use the same idea as we do here the rise is the change in your y values. Well here our y values I know it looks weird in function notation but it would just be f of x2 minus f of x1 and the run is just your change in x's so that stays the same x2 minus x1. So if we put that in a fraction we get our formula for the average rate of change f of x2 minus f of x1 all over x2 minus x1. So this is really the same formula as our slope it's just written in function notation and can be used for non-linear equations. Okay suppose you have the function f of x equals x2 and we are going to find the average rate of change on the interval 1 to 3 using our formula. So what this means is these are our two x values. x1 is 1 and x2 is 3. So when we are plugging it into the formula we really need two points. These are both the x values to find the y values or the output we just need to plug these x values into our function. So we would plug in 1 and we would get the point 1, 1 we would plug in 3 to get the point 3, 9 and once we have two points to find the average rate of change we simply use our average rate of change formula also known as our slope formula and we find the change in y over the change in x's. 9 minus 1 is 8 3 minus 1 is 2 8 over 2 is 4. So the average rate of change on that interval would simply be 4.