To determine the transformation that changes the function \( f(x) = 6 \sin \left(x - \frac{\pi}{8}\right) + 8 \) to \( g(x) = 6 \sin \left(x - \frac{7\pi}{16}\right) + 1 \), we need to analyze both the horizontal and vertical shifts.
### Horizontal shift
First, let's consider how the argument of the sine function changes:
1. For \( f(x) \), the argument is \( x - \frac{\pi}{8} \).
2. For \( g(x) \), the argument is \( x - \frac{7\pi}{16} \).
We need to determine the difference between these two arguments:
[tex]\[ x - \frac{7\pi}{16} - \left( x - \frac{\pi}{8} \right) = - \left( \frac{7\pi}{16} - \frac{\pi}{8} \right) \][/tex]
We need a common denominator to subtract these fractions:
[tex]\[ \frac{7\pi}{16} - \frac{2\pi}{16} = \frac{5\pi}{16} \][/tex]
Since the difference is \( \frac{5\pi}{16} \), the \( f(x) \) to \( g(x) \) sees a shift of \( \frac{5\pi}{16} \). Specifically, \( x - \frac{\pi}{8} \) must be adjusted to \( x - \frac{7\pi}{16} \). This adjustment means a rightward shift, which adds to the argument.
So, horizontally:
[tex]\[ \frac{\pi}{8} \rightarrow \frac{7\pi}{16} \][/tex]
This corresponds to translating \( \frac{5\pi}{16} \) units to the right.
### Vertical shift
Now, let's handle the vertical shift:
1. For \( f(x) \), the vertical position is \( +8 \).
2. For \( g(x) \), the vertical position is \( +1 \).
The difference here is:
[tex]\[ 1 - 8 = -7 \][/tex]
This indicates that to transform from \( f(x) \) to \( g(x) \), we need to shift the function 7 units down.
### Conclusion
Combining these two transformations:
- There is a horizontal shift to the right by \( \frac{5\pi}{16} \) units.
- There is a vertical shift down by 7 units.
Thus, the correct transformation that changes \( f(x) = 6 \sin \left(x - \frac{\pi}{8}\right) + 8 \) to \( g(x) = 6 \sin \left(x - \frac{7\pi}{16}\right) + 1 \) is:
Translate [tex]\( \frac{5\pi}{16} \)[/tex] units right and 7 units down.
