Final answer:
The maximum and minimum values of the function indicate an amplitude of 2 and a vertical translation of 4. The period calculation and the given points suggest the phase shift is π and the value of b should be 2. However, without the correct graph or more information, it's not possible to definitively select one of the given options.
Step-by-step explanation:
The trigonometric function in question is of the form f(x) = a*cos(bx + c) + d. To determine the correct function from the given choices, we need to look at the amplitude, period, phase shift and vertical translation of the given maximum and minimum points. The maximum value of the function is 6 and the minimum value is 2, so the amplitude a is half the distance between these two points, which is (6 - 2)/2 = 2. The vertical translation d is the average of the maximum and minimum values: (6 + 2)/2 = 4.
The period of the cosine function is determined by the value of b in the equation and is equal to 2π/b. Since the maximum at π leads to a minimum at -3π/4, the period is the difference, which is π - (-3π/4) = 7π/4. Therefore, b is 2π divided by the period: 2π / (7π/4) = 8/7. However, since the wave needs to pass through two maxima at a distance of 2π, we need to find a multiple of 8/7 that would give us a whole number to fit with the graph. The smallest value for b which is a multiple of 2 is suitable, b = 2.
The phase shift c can be determined by looking at the displacement of the graph from the standard cosine function, which peaks at x=0. Given that the maximum point in this graph is at x=π, we can calculate the phase shift to be -π. Finally, our function that combines amplitude, period, vertical shift, and phase shift would match the option A) f(x) = 2cos(2x + π) + 4. However, as this option is not listed, it seems there might be an error in the provided options or in the assumptions I made about the period being 7π/4, the vertical translation being 4, and the fact that no phase shift other than π could create a cosine wave peaking at x=π if b = 2. Without the correct graph or additional information, it's impossible to definitively choose between the provided options A, B, C, or D.