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Video: Computing the Average Rate of Change of Polynomial Functions between Two Points

Alex Cutbill

Evaluate the average rate of change for the function 𝑓(𝑥) = −7𝑥² − 3𝑥 + 3 when 𝑥 changes from 1 to 1.5.

03:22

Video Transcript

Evaluate the average rate of change for the function 𝑓 of 𝑥 equals negative seven 𝑥 squared minus three 𝑥 plus three when 𝑥 changes from one to 1.5.

To solve this, we’re going to use a formula for the average rate of change. So for any function 𝑓 of 𝑥, the average rate of change of 𝑓 of 𝑥 as 𝑥 changes from 𝑎 to 𝑏 is this thing here: 𝑓 of 𝑏 minus 𝑓 of 𝑎 all divided by 𝑏 minus 𝑎.

If we compare this definition to our question, we’ll see that the value of 𝑎 is one and the value of 𝑏 is 1.5. We also need to find the values of 𝑓 of 𝑎 and 𝑓 of 𝑏 which we do by substituting into this expression here. So substituting in one, we get negative seven times one squared minus three times one plus three. And evaluating that we get negative seven.

We also find 𝑓 of 𝑏 which is 𝑓 of 1.5, because as we’ve seen before 𝑏 is 1.5. And again, this is just a case of substituting in. This time we’re substituting 1.5 instead of one. And this time we get negative 17.25.

So now we have these values. Let’s substitute them into the formula we have, the average rate of change here. And of course we remember that 𝑓 of one is 𝑓 of 𝑎, because 𝑎 is one. And similarly, 𝑓 of 1.5 is 𝑓 of 𝑏. We can clearly see this when we substitute the values of 𝑎 and 𝑏 into our formula, we get 𝑓 of 1.5 minus 𝑓 of over 1.5 minus one.

Substituting the values of 𝑓 of 1.5 and 𝑓 of 1, we get negative 17.25 minus negative seven all over 1.5 minus one. Evaluating the numerator and denominator, we get negative 10.25 over 0.5. So finally, we get the answer of negative 20.5. This is the average rate of change of the function 𝑓 of 𝑥 equals negative seven 𝑥 squared minus three 𝑥 plus three when 𝑥 changes from one to 1.5. You can think about this in terms of the graph of 𝑓 of 𝑥 as 𝑥 changes from one to 1.5.

We’ve seen that the value of 𝑓 of 𝑥 changes from negative seven to negative 17.25. And the average rate of change of the function between those two values of 𝑥 turns out to be the gradient of the line segment between the two endpoints of the curve. So the gradient, also known as the slope, is negative 20.5. If this were a displacement time graph where 𝑓 represented a displacement of the time 𝑥, then this average rate of change of 𝑓 would be the average speed in the interval from one to 1.5.