Welcome to a lesson on average rate of change of a function. The goals of this video are to define the average rate of change and then determine the average rate of change of a function. Looking at this graph here, this black line is a secant line. A secant line is a linear approximation of a function between two points. So in this graph, this is our first point, and this is our second point. The slope of the secant line represents the average rate of change of the function between these two points. So when you're asked to determine the average rate of change of a function, you're really just finding the slope of a line. Remember the equation for the slope of a line given two points is slope equals y sub two minus y sub one divided by x sub two minus x sub one. So again, when you're finding the average rate of change of a function, you're really just determining the slope of a secant line. To get a better feel for this, let's take a look at an animation. This red secant line represents the average rate of change of this black function between these two points. So notice that the slope of this red line is negative. Notice how the slope of the secant line is approximately negative thirty-seven, which means the average rate of change of the black function from this first point to the second point is approximately negative thirty-seven. As we animate this secant line, you can see the slope of the secant line changes, meaning the average rate of change of the function is also changing between those two points. At this location here, the average rate of change of the function between those two points is almost negative three. But if we progress a little bit further along this function, notice how now the average rate of change is approximately positive two. Again, the slope of this secant line represents the function's average rate of change between those two points. To get a better feel for average rate of change, let's say you take a trip from Phoenix, Arizona, to Los Angeles, California, and let's say you travel the three hundred eighty-mile trip in five and a half hours. We know from algebra that distance is equal to rate times time. So if we solve this equation for rate, we would have rate equal to distance divided by time. This represents an average rate. If we take the distance and divide by the time, our average speed or average rate of change on this trip would be approximately sixty-nine miles per hour. If we compare the time and distance traveled on our trip, the graph might look something like this. Notice how at certain times the distance traveled is less than at other times. Meaning when the graph is steep, we're traveling faster, and when the graph flattens out or doesn't increase as fast, our speed would be slower. If we picked two points on this graph and sketch the secant line, the slope of that line would be our average rate of change on that given interval of time. For example, if you look at the secant line from where we started to where we end, the slope of this red secant line will be approximately sixty-nine miles per hour. But if we were to pick two different times on this trip, let's say at two hours and then at three hours, the slope of this secant line would give us our average speed during this time interval. Notice that our average speed from two to three hours would be less than the average speed of the entire trip, and you can see that it looks like we may have stopped to get gas or maybe at a rest stop somewhere around this time. Let's take a look at a few of our own examples. Let's determine the average rate of change of the given function from x equals zero to x equals three. Well, the first thing we need to do is determine the coordinates of the points when x is zero and when x is three. Once we have our two points, we can then just use the slope formula to determine our average rate of change. So we'll have x and f of x. When x is zero, f of zero would be zero squared plus one, so f of x would be one. Remember, f of x is the same as y. When x is three, we'd have three squared plus one. That would be ten. We can check that over here on our graph. Here's the point zero, one, and here's the point three, ten. So the average rate of change of this function will be the slope of the line containing this segment here. So the average rate of change again is y sub two minus y sub one over x sub two minus x sub one. Just to keep things organized, I'll call this x sub one, y sub one, and this will be x sub two, y sub two. So we're going to have ten minus one in the numerator, and then x sub two minus x sub one is going to be three minus zero. So we'll have nine divided by three, which is equal to three. So the average rate of change from x equals zero to x equals three is three. To determine the meaning of this, it may be helpful to write it as a fraction. Remember, slope tells us the change in y with respect to the change in x. So on the interval from zero to three, the average rate of change of this function is, as x increases by one, y will increase by three. Remember, x is the independent or control variable, and y is the dependent variable. Let's take a look at one more example where we have the same function, but now we want the average rate of change from x equals negative two to x equals one. So start by filling out our table for x equals negative two and x equals one, and then we'll find f of x, which again is the same as y. So when x equals negative two, we'll have negative two squared, that's four plus one. So y will be five. Then when x is one, we'll have one squared plus one. That'll give us two. Again, we can verify that over here on the graph. Here's the point negative two, five, and here's the point one, two. Notice how the slope of this segment that would be contained within the secant line is now going downhill and therefore negative. Therefore, the average rate of change of this function from negative two to one is negative. Now you need to determine the slope of the secant line passing through these two points. Again, to keep things organized, I'll call this x sub one, y sub one, and this point x sub two, y sub two. So y sub two minus y sub one is going to be two minus five, and the denominator is going to be x sub two minus x sub one. Well, one minus negative two. Here we're going to have negative three over three. So the average rate of change is equal to negative one. To add some meaning to this, we'll go ahead and write this as a fraction. Negative one over one tells us that from x equals negative two to x equals one, as x increases by one, y decreases by one. Even though in fraction form it's more meaningful, when giving your answer, you always want to have it in the most simplest form. Meaning your average rate of change would be negative one for this problem. The same thing on the previous example, even though it's more meaningful to write it as three over one, you should express the average rate of change as positive three. I hope you found this helpful. Thank you for watching.