Final answer:
The average rate of change of the function h(x) = x² - 6x + 4 over the interval -2 ≤ x ≤ 5 is calculated using the endpoints of the interval. Substituting -2 and 5 into the function and then calculating the change in h(x) over the change in x, we find that the average rate of change is -3.
Step-by-step explanation:
To determine the average rate of change of the function h(x) = x² - 6x + 4 over the interval -2 ≤ x ≤ 5, we use the formula:
average rate of change = ∆h / ∆x = (h(x2) - h(x1)) / (x2 - x1)
where (x1, h(x1)) and (x2, h(x2)) are the points on the interval's endpoints. For our function:
- h(-2) = (-2)² - 6(-2) + 4 = 4 + 12 + 4 = 20
- h(5) = 5² - 6(5) + 4 = 25 - 30 + 4 = -1
Now, we calculate the average rate of change:
average rate of change = (h(5) - h(-2)) / (5 - (-2)) = (-1 - 20) / (5 + 2) = -21 / 7 = -3
The average rate of change of the function h(x) over the interval from -2 to 5 is -3.