Video: Computing the Average Rate of Change of Polynomial Functions at a Point

The average rate of change of a function 𝑓 between π‘₯ and π‘₯ + β„Ž is (𝑓(π‘₯ + β„Ž) βˆ’ 𝑓(π‘₯))/β„Ž. Compute this quantity for 𝑓(π‘₯) = βˆ’4π‘₯Β² βˆ’ 8 at π‘₯ = βˆ’4, and for β„Ž = 0.3.

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Video Transcript

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

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

We need to use the definition of 𝑓 of π‘₯ to evaluate 𝑓 of negative 3.7 and 𝑓 of negative four. We find 𝑓 of negative 3.7 by substituting a 3.7 into our definition of 𝑓 of π‘₯. And we get negative four times negative 3.7 squared minus eight. And it’s a similar story for 𝑓 of negative four. That’s negative four times negative four squared minus eight. There’s 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, it’s a good idea to know what you’re doing and why you’re doing it.

Here’s a graph of our function. We’re told in the question that the average rate of change of a function 𝑓 between π‘₯ and π‘₯ plus β„Ž is 𝑓 of π‘₯ plus β„Ž minus 𝑓 of π‘₯ all over β„Ž. We are also told that π‘₯ is negative four and β„Ž 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‘₯, 𝑓 of π‘₯, where π‘₯ is negative four. And the other point has coordinates π‘₯ plus β„Ž, 𝑓of π‘₯ plus β„Ž, where π‘₯ is negative four and β„Ž is 0.3, as defined in the question.

To get from the first point to the second, we need to go across by β„Ž units to get from an π‘₯-coordinate of π‘₯ to an π‘₯-coordinate of π‘₯ plus β„Ž. And we need to go up by 𝑓of π‘₯ plus β„Ž minus 𝑓 of π‘₯ units to get from a 𝑦-coordinate of 𝑓 of π‘₯ to a 𝑦-coordinate of 𝑓 of π‘₯ plus β„Ž. 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 𝑓 is changing as its input changes from negative four to negative 3.7.

It’s probably worth mentioning that the line segment just doesn’t look like it has a slope of 30.8. But this is because of the scales on the π‘₯- and 𝑦-axes. And also, despite appearances, the graph of the function that we have, 𝑓 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.

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