Worked example: average rate of change from table

What is the average rate of change of y of x over the interval negative 5 is less than x is less than negative 2? So this is x is equal to negative 5. When x is equal to negative 5, y of x is equal to 6. And when x is equal to negative 2, y of x is equal to 0. So to figure out the average rate of change, so the average rate of change, of y of x, with respect-- and we can assume it's with respect to x-- let me make that a little bit neater-- this is going to be the change in y of x over that interval over the change of x of that interval. And the shorthand for change is this triangle symbol, delta. Delta y-- I'll just write y. I could write delta y of x. It's delta y. Change in y over our change in x. That's going to be our average rate of change over this interval. So how much did y change over this interval? So y went from a 6 to a 0. So let's say that we can kind of view this as our endpoint right over here. So this is our end. This is our start. And we could have done it the other way around. We would get a consistent result. But since this is higher up on the list, let's call this the start. And the x is a lower value. We'll call that our start. This is our end. So we start at 6. We end at 0. So our change in y is going to be negative 6. We went down by 6 in the y direction. It's negative 6. You could say that's 0 minus 6. And our change in x, well, we are at negative 5, and we go up to negative 2. We increased by 3. So when we increased x by 3, we decreased y of x by 6. Or if we want to simplify this right over here, negative 6 over 3 is the same thing as negative 2. So our average rate of change of y of x over the interval from negative 5 to negative 2 is negative 2. Every time, on average, x increased 1, y went down by negative 2.