Question Video: Finding the Average Rate of Change of Polynomial Functions between Two Points Mathematics • Second Year of Secondary School

Video Transcript

Determine the average rate of change for 𝑓 of 𝑥 is equal to six 𝑥 squared minus eight when 𝑥 changes from eight to 8.4.

We can find the average rate of change of a function 𝑓 of 𝑥 between 𝑥-values 𝑎 and 𝑏 using the formula the average rate of change is equal to 𝑓 at 𝑥 is equal to 𝑏 minus 𝑓 at 𝑥 is equal to 𝑎 over 𝑏 minus 𝑎. This is effectively a rewrite of the slope, which is 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. In terms of the average rate of change, this is just the slope of the line connecting the points 𝑎, 𝑓 of 𝑎 and 𝑏, 𝑓 of 𝑏.

In our question, we have 𝑓 of 𝑥 is equal to six 𝑥 squared minus eight. And we want to find the average rate of change for 𝑓 when 𝑥 changes from eight to 8.4. So with our function 𝑓 of 𝑥 is six 𝑥 squared minus eight, if we let 𝑎 equal to eight and 𝑏 equal to 8.4. In our formula for the average rate of change, we have 𝑓 of 8.4 minus 𝑓 of eight over 8.4 minus eight. That is 𝑓 of 8.4 minus 𝑓 of eight over 0.4.

To evaluate this, we need to work out 𝑓 of 8.4 and 𝑓 of eight. That is 𝑓 at 𝑥 is equal to eight and 𝑓 at 𝑥 is equal to 8.4. And substituting 𝑥 is equal to eight into our function 𝑓 of 𝑥 is six 𝑥 squared minus eight gives us six times eight squared minus eight. That is six times 64 minus eight, which is 376. Now, if we do the same for 𝑥 is equal to 8.4, we have six times 8.4 squared minus eight. That is six times 70.56 minus eight, which is 415.36.

So with 𝑓 of eight equal to 376 and 𝑓 of 8.4 equal to 415.36, we have the average rate of change equal to 415.36, that’s 𝑓 of 8.4, minus 376, which is 𝑓 of eight, over 0.4. Evaluating the numerator gives us 39.36 which we then divide by 0.4 to get 98.4. And we found that the average rate of change of our function 𝑓 of 𝑥 is six 𝑥 squared minus eight as 𝑥 changes from eight to 8.4 is 98.4.