Find the formula for a function of the form y=Asin(Bx)+C with a maximum at (0.5,0), a minimum at (1.5,−4), and no critical points between these two points.

Answer 1

What do you call a critical point? No more maxima or minima between x = 0.5 and 1.5 ? I will assume so.

The wavelength would be two times the x distance between maximum and minimum, or 2 in this case. The amplitude is half the difference between max and min, or 2. The average value is -2. That is the C that gets added at the end

There will be a midpoint of the sine function at x = 0. That makes it easier.

Thus the function looks something like
y = 2 sin[pi*x] -2