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Write a function of the form y= A sin (Bx-C)+D that has period 8, phase shift -2, and the range -12 ≤ y ≤-4.

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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

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