Verified answer

Extreme values of the sine occurs when the argument is equal to π/2 and 3π/2.

x = π/8: b(x-c) = b(π/8 - c) = π/2

x = 3π/8: b(x-c) = b(3π/8 - c) = 3π/2

We have two equations in two unknowns. Solving for b and c

π/8 - c = π/(2b)

3π/8 - c = 3π/(2b)

2π/8 = 2π/(2b)

b = 4

c = 0

Maximum value of the sine function = 1

Minimum value of the sine function = -1

At the maximum value, x = π/8, 

y = 5 = a+d

At the minimum value, x = 3π/8

y = -1 = -a+d

Adding the two equations, 

2d = 4

d = 2

5 = a+2

a = 3

Ans:

a = 3

b = 4

c = 0

d = 2

b * (pi/8 - c) = pi/2

b * (3pi/8 - c) = 3pi/2

(pi/8) * b - bc = pi/2

(3pi/8) * b - bc = 3pi/2

(3pi/8) * b - bc - (pi/8) * b + bc = 3pi/2 - pi/2

(2pi/8) * b = 2pi/2

(pi/4) * b = pi

b = 4

4 * (pi/8 - c) = pi/2

pi/8 - c = pi/8

c = 0

y = a * sin(4 * (x - 0)) + d

y = a * sin(4x) + d

a = (max - min) / 2

a = (5 - (-1)) / 2

a = 6/2

a = 3

d = (max + min) / 2

d = (5 + (-1)) / 2

d = 4/2

d = 2

y = 3 * sin(4x) + 2