# Question Video: Finding the Average Rate of Change of a Polynomial Function between Two Points Mathematics

Find the change in π(π₯) = β3π₯Β² + 8π₯ β 2 as π₯ varies from 8 to 8.4.

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### Video Transcript

Find the change in π of π₯ equals negative three π₯ squared plus eight π₯ minus two as π₯ varies from eight to 8.4.

We have the quadratic function π of π₯ equals negative three π₯ squared plus eight π₯ minus two. And weβre interested in how the value of this function changes as π₯ changes β in particular, as π₯ changes from eight to 8.4. It makes sense then to find the value of the function when π₯ is eight and the value of the function when π₯ is 8.4.

The value of the function when π₯ is eight is written as π of eight and the value of the function π when π₯ is 8.4 is written as π of 8.4. How do we find these two values? Well, letβs start with π of eight. We have an expression for π of π₯. To find π of eight, we take the expression for π of π₯ and we replace π₯ by eight. Doing this, we get that π of eight is negative three times eight squared plus eight times eight minus two. This is something that you can put into a calculator to evaluate. Doing so, we get negative 130.

We find π of 8.4 in the same way. Substituting 8.4 for π₯ in the expression we have for π of π₯, we get negative three times 8.4 squared plus eight times 8.4 minus two. And putting this into our calculator, we get negative 146.48. Okay, so we found the values of π of eight and π of 8.4.

Now, what is the change in π of π₯ as π₯ varies from eight to 8.4? This is defined to be π of 8.4 minus π of eight. Notice that the order is important here. We have π of 8.4 first and from this we subtract π of eight. If we swapped π of eight and π of 8.4, weβd get the wrong answer.

In general, the change in a function π as π₯ varies from π to π is π of π minus π of π. Itβs π of the number youβre varying to minus π of the number youβre varying from. Okay, so letβs find this change. Luckily, we have the values of both π of 8.4 and π of eight. So we can substitute. We get negative 146.48 minus negative 130. This is something we could put into our calculators.

Alternatively, we can find this by hand. Remembering thatβs two minuses make a plus, subtracting negative 130 is the same as adding 130. And now, you can think of this as 130 minus 146.48 if youβd like to get the answer negative 16.48.