Final answer:
The graph of the sinusoidal function, given points on the midline and a minimum value, allows one to determine the amplitude, period, and midline shift, resulting in the function y(x) = -5sin(2x/3\pi) -2.
Step-by-step explanation:
The question involves determining the equation of a sinusoidal function given two points on its graph: an intersection with the midline and a minimum point. The sinusoidal function passes through the midline at the point (0, -2) and has a minimum point at (\frac{3\pi}{2}, -7).
Since the minimum value is -7 and it occurs below the midline of -2, we can infer that the amplitude (A) of the function is 5 units (because -2 minus 5 equals -7).
Knowing that it crosses the midline at x=0, we can see that the horizontal shift (D) is 0. The period (T) of the function is four times the distance from the midline to the minimum, as a full sinusoidal cycle consists of two amplitudes: one above and one below the midline.
Therefore, T equals (\frac{3\pi}{2} times 2), which simplifies to 3\pi. The equation of the sinusoidal function can now be expressed as y(x) = -5sin(\frac{2x}{3\pi})-2, where -5 represents the amplitude, 3\pi is the period, and -2 is the vertical shift (midline).
The negative sign before the sine function indicates that the graph starts with a downward curve towards the minimum point.