which two points of a trignometric function will always give an average rate of change of 0?

To find the two points of a trigonometric function that will always give an average rate of change of 0, we need to first understand what average rate of change means.

The average rate of change of a function between two points is the difference between the function values at those points divided by the difference between the corresponding input values. In other words, it represents the average rate at which the function changes over a given interval.

For a trigonometric function, such as sine or cosine, the average rate of change is determined by the frequency, period, and amplitude of the function.

The general form of a trigonometric function is given by:

f(x) = A * sin(B * x + C) + D

Where:
- A represents the amplitude of the function,
- B represents the frequency, which is equal to 2π divided by the period of the function,
- C represents the phase shift, which determines the horizontal translation of the function,
- D represents the vertical shift, which determines the vertical translation of the function.

To find two points that will give an average rate of change of 0, we need to consider the points where the function reaches its maximum and minimum values over one period.

For example, if we consider the sine function:

f(x) = sin(x)

The maximum value of the sine function is 1, which occurs at x = π/2 + 2πn, where n is an integer. The minimum value of the sine function is -1, which occurs at x = 3π/2 + 2πn, where n is an integer.

Therefore, the two points that will always give an average rate of change of 0 for the sine function are:

(π/2 + 2πn, 1) and (3π/2 + 2πn, -1)

Similarly, for other trigonometric functions like cosine, you can find the maximum and minimum points and use them to determine the two points that will always give an average rate of change of 0.

Note: It's important to understand that these points only give an average rate of change of 0 within one complete period of the function. Beyond that interval, the average rate of change can be nonzero.