Functions - Average Rate of Change - stuck on question for 2 hours please help

Question

For the following exercise, find the average rate of change of each function on the interval specified for real number h in simplest form.

a(t) = 1 / t + 4 on [9, 9 + h]

The textbook gives the answer of -1 / 13(13 + h) I can't seem to get it

Answer 1

The formula of the average rate of change is the same as the slope formula, which is (y2-y1)/(x2-x1).

First, lets find what is y2-y1. To do this, simply plug in 9+h first into a(t), then plug in 9 and finally subtract them. We get [1/(9+h+4)] - [1/(9+4)]. After taking the common denominator, we get [ (13 -13-h)/ 13(13+h) ]. Simplifying this gives us -h/13(13+h). This is our y2-y1. Now we need to get x2-x1. We do this by doing 9+h-9. We get h. Now finally, we need to divide y2-y1 with x2-x1, which is [-h/13(13+h) ] / h. The h cancels out, giving us -1 / 13(13 + h) . Although the math is a little difficult to explain over text, I have tried as best as I can. I hope I was helpful. Feel free to reach out with questions!