Compare the estimated average rate of change for the graphed cubic function b (x) = ^3square root 3x + 9 to the estimated average rate of change of the square root function d (x) = square root of -3x + 9 over the interval [-12, -3]. Which comparison is true?
A. The estimated average rate of change of d (x) is greater than the estimated average rate of change of b (x), but both rates are negative.
B. The estimated average rate of change of b (x) is greater than the estimated average rate of change of d (x) because b (x) is increasing over the interval but d (x) is decreasing.
C.The estimated average rate of change of d (x) is greater than the estimated average rate of change of b (x) because d (x) is increasing over the interval but b (x) is negative.
D. The estimated average rate of change of b (x) is greater than the estimated average rate of change of d (x), but both rates are positive.
To compare the estimated average rate of change for the given cubic function b (x) = ^3square root 3x + 9 and the square root function d (x) = square root of -3x + 9 over the interval [-12, -3], we need to find the average rate of change for both functions.
For the cubic function b (x) = ^3square root 3x + 9, the average rate of change over the interval [-12, -3] can be found using the formula:
Average rate of change = (b(-3) - b(-12))/(-3 - (-12))
Substituting the values into the formula, we get:
Average rate of change for b (x) = (^3square root 3(-3) + 9 - ^3square root 3(-12) + 9)/(-3 - (-12))
Simplifying this, we have:
Average rate of change for b (x) = (^3square root 9 + 9 - ^3square root 36 + 9)/(-3 + 12)
Average rate of change for b (x) = (3 + 9 - 6 + 9)/(9)
Average rate of change for b (x) = (15)/(9)
Average rate of change for b (x) = 5/3
Next, let's find the average rate of change for the square root function d (x) = square root of -3x + 9 over the same interval [-12, -3].
Average rate of change for d (x) = (d(-3) - d(-12))/(-3 - (-12))
Substituting the values into the formula, we get:
Average rate of change for d (x) = (square root of -3(-3) + 9 - square root of -3(-12) + 9)/(-3 - (-12))
Average rate of change for d (x) = (square root of 9 + 9 - square root of 36 + 9)/(-3 + 12)
Average rate of change for d (x) = (3 + 9 - 6 + 9)/(9)
Average rate of change for d (x) = (15)/(9)
Average rate of change for d (x) = 5/3
Thus, the average rate of change for both functions b (x) and d (x) over the interval [-12, -3] is the same, which is 5/3. None of the options provided in the question correctly describe this result.