The general form of the function is:
y = A sin(Bx - C) + D
To determine the specific values of A, B, C, and D that satisfy the given conditions, we can use the following steps:
Period: The period of a sinusoidal function is given by 2π/B, where B is the coefficient of x in the argument of the sine function. In this case, the period is 8, so we have:
2π/B = 8
Solving for B, we get:
B = π/4
Phase shift: The phase shift of a sinusoidal function is given by C/B, where C is the constant inside the argument of the sine function. In this case, the phase shift is -2, so we have:
C/B = -2
Substituting B from step 1, we get:
C/(π/4) = -2
Solving for C, we get:
C = -π/2
Amplitude and vertical shift: The range of the function is given as -12 ≤ y ≤ -4. The amplitude of a sinusoidal function is half the distance between its maximum and minimum values. In this case, the amplitude is:
(amplitude) = (maximum - minimum)/2 = (-4 - (-12))/2 = 4
The vertical shift of the function is given by the constant term D. Since the minimum value of the function is -12, we have:
D + (amplitude) = -12
Substituting the value of the amplitude from above, we get:
D + 4 = -12
Solving for D, we get:
D = -16
Therefore, the function of the form y = A sin(Bx - C) + D that has period 8, phase shift -2, and the range -12 ≤ y ≤ -4 is:
y = 4 sin(π/4 x + π/2) - 16