Final answer:
The average rate of change for the function g(x) is negative in the intervals that are to the left of the vertex. Since the vertex here is at x=0, both intervals [-4,-2] and [-8,-4] show a consistent negative rate of change, but the interval [-4,-2] is the best choice as it is explicitly requested to pick one interval.The correct answer is option (a).
Step-by-step explanation:
The student is asking about the average rate of change of a quadratic function, which is determined by the function's slope over a specific interval. When asked about the interval where the function g(x) = - x²/4 +7 has a negative average rate of change, we are looking for where the graph of g(x) is sloping downwards. To find this, we can calculate the average rate of change between two points (x1, g(x1)) and (x2, g(x2)) on the interval [a, b]. The average rate of change is calculated as (g(x2) - g(x1)) / (x2 - x1).
To find intervals where g(x) has a negative average rate of change, we can look at the coefficients of the quadratic function. Since the coefficient of x² is negative (-1/4), the parabola opens downwards. This means that the function will have a negative slope to the left and the right of the vertex. The vertex of this function is at x=0, so any interval that does not include x=0 will have a consistent slope sign.
Considering the intervals provided:
- Interval a: [-4,-2] is to the left of the vertex, so the average rate of change here is negative.
- Interval b: [-8,-4] is also to the left of the vertex, meaning the average rate of change is again negative.
- Interval c: [-2,0] includes the vertex, and the rate of change switches from negative to positive.
- Interval d: [0,4] is to the right of the vertex; here the rate of change is also negative.
However, we are to choose only one correct interval. Because the interval [-2,0] includes the vertex where the average rate of change is zero (the slope switches from negative to positive), it cannot be consistently negative over this entire interval.
Therefore, the correct answer is intervals a and b, where the average rate of change is definitely negative. However, since the question requires selecting only one interval, the best answer would be a. [-4,-2] as it is entirely to the left of the vertex and will show a consistent negative average rate of change.