Consider the function [tex]f(x)=6 \sin \left(x-\frac{\pi}{8}\right)+8[/tex]. What transformation results in [tex]g(x)=6 \sin \left(x-\frac{7 \pi}{16}\right)+1[/tex]?

A. Translate [tex]\frac{5 \pi}{16}[/tex] units left and 7 units up.
B. Translate [tex]\frac{5 \pi}{16}[/tex] units right and 7 units up.
C. Translate [tex]\frac{5 \pi}{16}[/tex] units left and 7 units down.
D. Translate [tex]\frac{5 \pi}{16}[/tex] units right and 7 units down.



Answer :

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.