The decreasing average rate of change over consecutive intervals indicates a concave-down graph for the function f, leading to the conclusion that option D is consistent with the values in the table.
To determine the concavity of the function represented by the given table, we can analyze the behavior of the average rate of change over consecutive equal-length intervals. The average rate of change between two points is given by the formula "change in f over change in x," where the change in f is the difference in the function values, and the change in x is the difference in the corresponding x-values.
Observing the table values, we can calculate the average rate of change between each pair of consecutive x-values:
Between 5 and 6: (-27 - (-17))/(6 - 5) = -10
Between 6 and 7: (-39 - (-27))/(7 - 6) = -12
Between 7 and 8: (-53 - (-39))/(8 - 7) = -14
Between 8 and 9: (-69 - (-53))/(9 - 8) = -16
Between 9 and 10: (-87 - (-69))/(10 - 9) = -18
The average rate of change is decreasing as x increases, suggesting that the graph of f is concave down. Therefore, the correct conclusion is:
The graph of f is concave down because the average rate of change over consecutive equal-length intervals is decreasing.