Final answer:
The rate of change of the function f(x) over the interval [1,4] is -9. This is calculated by dividing the difference in the function values f(4) - f(1), which is -27, by the length of the interval, which is 3 (4 - 1).
Step-by-step explanation:
The rate of change of the function f(x) over the interval [1,4] can be calculated using the formula for the slope of the line segment connecting the points (1, f(1)) and (4, f(4)) on the graph of f(x). The rate of change is the difference in the y-values divided by the difference in the x-values, which translates to a direct answer of -9 when we compute (f(4) - f(1)) / (4 - 1).
Using the given information that f(4) - f(1) = -27, the calculation would be:
- Subtract f(1) from f(4), which gives us -27 based on the information provided.
- Subtract the x-value corresponding to f(1) from the x-value corresponding to f(4), giving us 4 - 1 = 3.
- Divide the change in y by the change in x, giving us -27 / 3 = -9.
Thus, the average rate of change of the function over the interval [1, 4] is -9 units per x-unit.