Final answer:
The ARC of the function over the intervals [0, 2] is -4 and over the interval [-2, 0] is 0, indicating that the function does not increase in either interval given these calculations. None of the option choices match these calculated values indicating a possible error in the question or the provided options.
Step-by-step explanation:
The question asks us to calculate the Average Rate of Change (ARC) of the function f(x) = – x^2 – 2x + 3 over the intervals [0, 2] and [-2, 0]. To find the ARC, we'll use the formula:
ARC = (f(x2) - f(x1)) / (x2 - x1)
For the interval [0, 2]:
- f(0) = 3
- f(2) = -2^2 - 2*2 + 3 = -4 - 4 + 3 = -5
- ARC = (f(2) - f(0)) / (2 - 0) = (-5 - 3) / 2 = -8 / 2 = -4
For the interval [-2, 0]:
- f(-2) = -(-2)^2 - 2*(-2) + 3 = -4 + 4 + 3 = 3
- f(0) = 3
- ARC = (f(0) - f(-2)) / (0 - (-2)) = (3 - 3) / 2 = 0 / 2 = 0
Comparing the Average Rate of Changes, the function does not increase in either interval; however, for the given option choices, none actually match our calculated values. There may be an error with the options listed or the calculation of the ARC. It's expected to calculate the same value for the ARC when considering opposite intervals of a quadratic function, given they are equidistant from the vertex. Since the interval [0, 2] has an ARC of -4 and the interval [-2, 0] has an ARC of 0, according to our calculations, the function does not increase over either interval.