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

Find the change in 𝑓(π‘₯) = βˆ’3π‘₯Β² + 8π‘₯ βˆ’ 2 as π‘₯ varies from 8 to 8.4.

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Video Transcript

Find the change in 𝑓 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 𝑓 of π‘₯ equals negative three π‘₯ squared plus eight π‘₯ minus two. And we’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 𝑓 of eight and the value of the function 𝑓 when π‘₯ is 8.4 is written as 𝑓 of 8.4. How do we find these two values? Well, let’s start with 𝑓 of eight. We have an expression for 𝑓 of π‘₯. To find 𝑓 of eight, we take the expression for 𝑓 of π‘₯ and we replace π‘₯ by eight. Doing this, we get that 𝑓 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 𝑓 of 8.4 in the same way. Substituting 8.4 for π‘₯ in the expression we have for 𝑓 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 𝑓 of eight and 𝑓 of 8.4.

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

In general, the change in a function 𝑓 as π‘₯ varies from π‘Ž to 𝑏 is 𝑓 of 𝑏 minus 𝑓 of π‘Ž. It’s 𝑓 of the number you’re varying to minus 𝑓 of the number you’re varying from. Okay, so let’s find this change. Luckily, we have the values of both 𝑓 of 8.4 and 𝑓 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 that’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 you’d like to get the answer negative 16.48.

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