Please help me on this question

Question

Examine the following graph of the quadratic function f(x)=−1/8(x+1)^2+8 and the linear function, g(x)=−6/7x−6.

For which of the following intervals is the average rate of change of f(x) faster than the average rate of change of g(x)

There is more than one correct answer. Select all correct answers.

A. −1≤x≤3

B. −9≤x≤2

C. −4≤x≤7

D. 2≤x≤6

E. 4≤x≤9

F. 1≤x≤8

Answer 1

We can visualize an average rate of change over an interval as the slope of the line connecting two points on the graph. We define it as follows: avg. rate of change of f(x) on the interval a ≤ x ≤ b = [f(b) - f(a)] / (b - a).

This should look to us like a very familiar definition of slope of a line: m = (y₂ - y₁) / (x₂ - x₁)

So, the average rate of change of f(x) given above on the interval -1 ≤ x ≤ 3 is calculated thus:

f(3) = -1/8(4)² + 8 = 6 and f(-1) = -1/8(-1)² + 8 = 63/8 so avg r.o.c. of f = (6 - 63/8) / (3 - (-1)) = - 15/32

The rates of change for f over the other intervals are calculated in the same way.

Because g(x) is linear, the average rate of change over any interval = m, in this case - 6/7.

So f(x) is NOT faster than g(x) on the interval -1 ≤ x ≤ 3.

By the way, we should assume that we are comparing the absolute values of the rates of change, since the question doesn't distinguish between a negative and positive average rate of change. So, you should calculate the average rate of change, let's call it r_f, for f(x) on the remaining intervals B - F and single out any of those for which |r_f| > 6/7