Question

Given the function h(x)=-x^2-4x+6h(x)=−x 2 −4x+6, determine the average rate of change of the function over the interval -5\le x \le -1−5≤x≤−1..

A) -9
B) 3
C) 6
D) -3

Answer 1

The average rate of change of the function h(x)=-x^2-4x+6 over the interval -5≤x≤-1 is calculated to be 2, by evaluating the function at both endpoints and then dividing the difference in function values by the difference in x-values. This answer is not among the provided options, indicating a possible mistake in the question or choices.

Step-by-step explanation:

The average rate of change of a function over an interval −a≤x≤b is calculated as the difference in the function values at the endpoints divided by the difference in the endpoints. For the function h(x)=-x2-4x+6, we want to find the average rate of change over the interval −5≤x≤−1.

First, we calculate the function values at the endpoints:

• h(−5) = -(−5)2 - 4(−5) + 6 = -25 + 20 + 6 = 1
• h(−1) = -(−1)2 - 4(−1) + 6 = -1 + 4 + 6 = 9

Next, we use these values to find the average rate of change:

∆h(x) / ∆x = (h(−1) - h(−5)) / (−1 - (−5)) = (9 - 1) / (-1 - (-5)) = 8 / 4 = 2

The average rate of change of the function over the interval −5≤x≤−1 is 2, which is not listed as an option in the multiple-choice answers provided by the student. Therefore, the student may need to double-check the initial function or the options presented.