Answer: we have a vertical shift of 7 units down, and a horizontal shift of 5*pi/16 units to the right.
Explanation:
First, let's define the transformations:
Vertical shift.
If we have a function f(x), a vertical shift of N units is written as:
g(x) = f(x) + N
This will move the graph of f(x) up or down a distance of N units.
if N is positive, then the shift is upwards
if N is negative, then the shift is downwards.
Horizontal shift.
If we have a function f(x), a horizontal shift of N units is written as:
g(x) = f(x + N)
This will move the graph of f(x) to the right or left a distance of N units.
if N is positive, then the shift is to the left
if N is negative, then the shift is to the right.
Here we have:
f(x) = 6*sin(x - pi/8) + 8
and the transformed function:
g(x) = 6*sin(x - (7/16)*pi) + 1
We can assume that the transformations are a vertical shift of A units, and an horizontal shift of B units, then we can write:
g(x) = f(x + B) + A
g(x) = 6*sin(x + B - pi/8) + (8 + A) = 6*sin(x - (7/16)*pi) + 1
Then we must have that:
(x + B - pi/8) = x - (7/16)*pi
B - pi/8 = -(7/16)*pi
B = -(7/16)*pi + pi/8 = -(7/16)*pi +2*pi/16 = (-7 + 2)*pi/16 = -5*pi/16
And we also must have that:
8 + A = 1
A = 1 - 8 = -7
Then the transformation is:
g(x) = f(x - 5*pi/16) - 7
This means that we have a vertical shift of 7 units down, and a horizontal shift of 5*pi/16 units to the right.