Final answer:
The function is f(x) = 4sin((8/7π)x) + 1, with an amplitude of 4, a period of 7π/2, no phase shift, and a vertical shift of 1.
Step-by-step explanation:
The graph of a sinusoidal function intersects its midline at (0, 1) and then has a maximum point at (7pi/4, 5). To find the formula of the function, we need to determine the amplitude, period, phase shift, and vertical shift.
The amplitude is the distance from the midline to the maximum point, which is 4 (5 - 1 = 4). The period can be calculated by noting that a full cycle (from midline to maximum and back to midline) is completed at x = 7pi/4, so the period, T, is 7pi/2 since the sine function repeats every half-cycle. There is no horizontal phase shift since the sine curve crosses the midline at x=0. The vertical shift is the midline value, which is 1.
The general form of a sine function is f(x) = A · sin(B(x - C)) + D, where A is the amplitude, B is the angular frequency (B = 2π/T), C is the phase shift, and D is the vertical shift.
Plugging in the values we get: f(x) = 4 · sin((2π/x)(x - 0)) + 1, or f(x) = 4sin((8/π7π)x) + 1.