# Video: Finding the Average Rate of Change Expression of a Rational Function

Evaluate The average rate of change of π(π₯) = β1/(8π₯ β 5) when π₯ varies from π₯β to π₯β + β.

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### Video Transcript

Evaluate The average rate of change of π of π₯ equals negative one over eight π₯ minus five when π₯ varies from π₯ sub one to π₯ sub one plus β.

The average rate of change formula allows us to find the rate at which the functionβs output changes as compared to the functionβs input. For our function π¦ equals π of π₯ for values of π₯ from π to π, the average rate of change is given by π of π minus π of π over π minus π. Now weβre told that our function π of π₯ is negative one over eight π₯ minus five. But what do we define π and π to be? Well, our input, our value of π₯, varies from π₯ sub one to π₯ sub one plus β. So weβre going to let π be equal to π₯ sub one and π be equal to π₯ sub one plus β.

Then we evaluate π of π and π of π. We see π of π is π of π₯ sub one. So we substitute π₯ for π₯ sub one. Similarly, π of π₯ sub one plus β is found by substituting π₯ sub one plus β for π₯. And we get negative one over eight times π₯ sub one plus β minus five. And if we distribute the parentheses, our denominator becomes eight π₯ sub one plus eight β minus five.

Letβs substitute everything we know then into the average rate of change formula. When we do, we subtract our π of π₯ sub one from π of π₯ sub one plus β. And so our numerator is as shown. Our denominator is π minus π. So itβs π₯ sub one plus β minus π₯ sub one. And so we notice that on the denominator π₯ sub one minus π₯ sub one is zero. And weβre simply left with β. Weβre going to need to do something with the numerator, though.

And so before we do, we notice that weβre subtracting a negative. So this is the same as adding the fractions. And to add algebraic fractions, we need to find a common denominator. The common denominator here will simply be the product of the denominators we have. We multiply the numerator and denominator of our first fraction by eight π₯ sub one minus five and of our second by eight π₯ one plus eight β minus five. And that leaves us with the common denominator shown.

Since weβre subtracting eight π₯ sub one minus five, we get negative eight π₯ sub one plus five. And then we see that negative eight π₯ sub one plus eight π₯ sub one is zero and plus five minus five is zero. That leaves us with simply eight β on our numerator. And so we can divide our ββs. And so our average rate of change function simplifies really nicely. Eight β divided by β is eight. And so we get eight over eight π₯ sub one minus five times eight π₯ sub one plus eight β minus five.

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