in this lesson we're going to focus on calculating the average rate of change of a function over an interval so here's the formula the average rate of change is f of b minus f of a over b minus a on the interval a to b so a and b are x values f of a and f b you could think of them as the y values so first let's calculate f of a and f b in this example a is one b is three so let's calculate f of one we just need to plug it into this equation so it's going to be one squared plus four times one minus five so that's going to be one plus four minus five one plus four is five five minus five is zero so f of one is zero now what about f of three that's going to be three squared plus four times three minus five three squared three times three is nine four times three is twelve and 12 minus 5 is 7 so we have 9 plus 7 which is equal to 16. so that's f of 3. so now let's use this expression so that's going to be f of three minus f of one divided by three minus one f of three is sixteen f of one is zero and three minus one is two sixteen divided by two is eight so the average rate of change of the function over the interval one comma three is equal to eight and that really represents the slope of the secant line so let's plot a generic graph one that's not related to the last example and let's say we have a graph that looks like this and we want to find the average rate of change between points a and b so that's on the interval a comma b the average rate of change it gives us the slope of the secant line and a secant line touches two points on a graph a tangent line only touches one point on a graph so that would be a tangent line so keep that in mind the average rate of change represents the slope of the secant line number two calculate the average rate of change of f of x equals x cubed minus four from x equals two to x equals five so it's going to be f of b well first we need to know what a and b are a is 2 b is 5. so it's f of b minus f of a over b minus a that's the average rate of change so this is going to be f of 5 minus f of 2 divided by 5 minus 2. so to calculate f of 5 we just got to plug it in to that formula so that's going to be 5 to the third minus four and to calculate f of two it's going to be two to the third minus four and five minus two is three now what is five to the third power five to the third power that's five times five times five five times five is twenty-five and twenty-five times five is 125. now two to the third that's two times two times 2 which is 8. 125 minus 4 is 121. 8 minus 4 is 4 and 121 minus 4 is 117. so it's 117 divided by 3 and 117 divided by 3 is 39 so 39 is the average rate of change from 2 to 5 or any interval 2 comma 5. so now you know how to calculate the average rate of change of a function so if you want to find more videos on pre-calculus algebra or even calculus check out my channel you can find a lot of other videos that can be helpful i have some videos on chemistry and physics as well if you need help in that area too you