Final answer:
The correct formula for the rate of change between the points (a, f(a)) and (6, f(6)) is (f(6) - f(a)) / (6 - a), which is option A. This measures the average rate of change of the function over that interval.
Step-by-step explanation:
The student's question refers to the calculation of the rate of change of a function at two distinct points given by the coordinates (a, f(a)) and (6, f(6)). The rate of change in this context is the slope of the line connecting these two points on the graph of the function, which is also known as the average rate of change over the interval from a to 6.
The correct formula for the rate of change is option A, which is (f(6) - f(a)) / (6 - a). This formula represents the difference in the function values divided by the difference in the x-values, which gives the average rate of change of the function between the two points.
Options B, C, and D provide formulas that are either incorrect applications of calculating the slope or relate to different scenarios not described in the student's question.