Final answer:
The formula for the student's trigonometric function is not fully determinable with the given details, but includes an amplitude (a) of 5, a vertical shift (d) of 6, and requires further information for precise values of the horizontal stretch (b) and phase shift (c).
Step-by-step explanation:
The student is seeking to find the trigonometric function for the given graph with the form g(x) = acos(bx+c) + d. The function's key points are given as a maximum at (2,11) and a minimum at (3.5,1). From these points, we can infer the amplitude a, the vertical shift d, and work on determining the phase shift c and the horizontal stretch b.
The amplitude is half the distance between the maximum and minimum values of the function. Therefore, a = (11 - 1) / 2 = 5. The function's vertical shift d can be determined by averaging the maximum and minimum values, yielding d = (11 + 1) / 2 = 6. As the cosine function starts at its maximum, c will be the amount we need to shift the function to the right to match the maximum point at x = 2. Without further information, we cannot determine c exactly, but we know it would be such that cos(b x 2 + c) equals 1.
Finally, the period T of the cosine function is the distance between two successive maxima or two successive minima. In this case, it's the distance from the maximum to the minimum, T = 3.5 - 2, which means half a period equals 1.5. Since T = 2π / b, we can find b by rearranging the formula to b = 2π / (2 * 1.5). However, without seeing the graph or subsequent maxima or minima points to fully confirm the period, b cannot be accurately determined. With the given information, the full determination of c and b isn't possible.