Answer: The general form of a sine function is:
y = A sin (Bx + C) + D
Where:
A = amplitude
B = 2π/period
C = phase shift
D = vertical shift or midline
Given the values in the problem, we can substitute and simplify:
A = 3
midline = 2, so D = 2
period = 8π/7, so B = 2π/(8π/7) = 7/4
y = 3 sin (7/4 x + C) + 2
To find the phase shift, we need to use the fact that the sine function is at its maximum when the argument of the sine function is equal to π/2. That is:
Bx + C = π/2
We can solve for C:
C = π/2 - Bx
C = π/2 - (7/4) x
Substituting back the value of C in the equation, we get:
y = 3 sin (7/4 x + π/2 - 7/4 x) + 2
y = 3 sin (7/4 x - 7π/8) + 2
Therefore, the sine function with an amplitude of 3, a midline of y=2, and a period of 8π/7 is:
y = 3 sin (7/4 x - 7π/8) + 2