### Video Transcript

The average rate of change of a
function π between π₯ and π₯ plus β is π of π₯ plus β minus π of π₯ over β. Compute this quantity for π of π₯
equals negative four π₯ squared minus eight at π₯ equals negative four, and for β
equals 0.3.

If weβd like to, we can ignore the
context and just focus on evaluating this expression here, πof π₯ plus β minus πof
π₯ all over β. Weβre told that β is 0.3. And so we can substitute this value
in the denominator. We are also told that π₯ is
negative four. And so π of π₯ is π of negative
four. Similarly, πof π₯ plus β is πof
negative four plus 0.3. And so we get π of negative
3.7. So now what do we do?

We need to use the definition of π
of π₯ to evaluate π of negative 3.7 and π of negative four. We find π of negative 3.7 by
substituting a 3.7 into our definition of π of π₯. And we get negative four times
negative 3.7 squared minus eight. And itβs a similar story for π of
negative four. Thatβs negative four times negative
four squared minus eight. Thereβs nothing we can do to the
denominator to make it any simpler. So we keep it as 0.3.

And now we have a numerical
expression which we can evaluate using a calculator. And doing so, we get an answer of
30.8 exactly. We managed to get this answer
without worrying what the average rate of change of our function actually is. But of course, in general, itβs a
good idea to know what youβre doing and why youβre doing it.

Hereβs a graph of our function. Weβre told in the question that the
average rate of change of a function π between π₯ and π₯ plus β is π of π₯ plus β
minus π of π₯ all over β. We are also told that π₯ is
negative four and β is 0.3. So we need to find the average rate
of change of our function between negative four and negative four plus 0.3, or
negative 3.7. We mark the two points on the
graph, which corresponds to the two inputs, negative four and negative 3.7. This point, therefore, has
coordinates π₯, π of π₯, where π₯ is negative four. And the other point has coordinates
π₯ plus β, πof π₯ plus β, where π₯ is negative four and β is 0.3, as defined in the
question.

To get from the first point to the
second, we need to go across by β units to get from an π₯-coordinate of π₯ to an
π₯-coordinate of π₯ plus β. And we need to go up by πof π₯
plus β minus π of π₯ units to get from a π¦-coordinate of π of π₯ to a
π¦-coordinate of π of π₯ plus β. Alternatively, we could travel in a
straight line path. Now we can notice that the average
rate of change of the function between these two points is just the slope of this
straight line segment. And the value of this slope gives a
measure of how much the output of the function π is changing as its input changes
from negative four to negative 3.7.

Itβs probably worth mentioning that
the line segment just doesnβt look like it has a slope of 30.8. But this is because of the scales
on the π₯- and π¦-axes. And also, despite appearances, the
graph of the function that we have, π of π₯ equals negative four π₯ squared minus
eight, is in fact a parabola, part of which we can see plotted, and not a straight
line. The reason that it looks like a
straight line is, again, because of the scale chosen.