### Video Transcript

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

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

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

To evaluate this, we need to work out π of 8.4 and π of eight. That is π at π₯ is equal to eight and π at π₯ is equal to 8.4. And substituting π₯ is equal to eight into our function π of π₯ is six π₯ 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 π₯ 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 π of eight equal to 376 and π of 8.4 equal to 415.36, we have the average rate of change equal to 415.36, thatβs π of 8.4, minus 376, which is π 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 π of π₯ is six π₯ squared minus eight as π₯ changes from eight to 8.4 is 98.4.