# Video: Finding the Value of a Function given Its Average Rate of Change between Two 𝑥-Values

The average rate of change of 𝑓 as 𝑥 varies from 2 to 2.6 is −1.67. If 𝑓(2) = −13, what is 𝑓(2.6)?

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### Video Transcript

The average rate of change of 𝑓 as 𝑥 varies from two to 2.6 is negative 1.67. If 𝑓 of two is equal to negative 13, what is 𝑓 of 2.6?

Remember, the average rate of change of a function 𝑓 as 𝑥 varies from 𝑎 to 𝑎 plus ℎ is given by 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎 over ℎ. Now, in this question, we don’t actually know what 𝑓 of 𝑥 is. But we do see that 𝑥 varies from two to 2.6. So, we let 𝑎 be equal to two. And then, ℎ is the amount 𝑥 varies by. It’s 2.6 minus two, which is 0.6. We want to find the average rate of change function, so that’s 𝐴 of ℎ, which is 𝐴 of 0.6. And so, according to our formula, that’s 𝑓 of two plus 0.6 minus 𝑓 of two over 0.6. This simplifies to 𝑓 of 2.6 minus 𝑓 of two over 0.6.

We’re told, however, that this is equal to negative 1.67, and also that 𝑓 of two is equal to negative 13. So, we find that negative 1.67 must be equal to 𝑓 of 2.6 minus negative 13 over 0.6. To find 𝑓 of 2.6 as this question is asking us, we need to solve this equation for 𝑓 of 2.6. We’ll begin by multiplying through by 0.6. And that gives us negative 1.002 on the left. And then on the right, we’re left with 𝑓 of 2.6 minus negative 13, which is, of course, 𝑓 of 2.6 plus 13. Next, we subtract 13 from both sides, and we find that 𝑓 of 2.6 is negative 14.002. Correct to the nearest whole number, 𝑓 of 2.6 is negative 14.