consider the function ( f(x)=6sinleft(x - \frac{pi}{8}\right)+8 ). what transformation results in (…

Answer:

B. Translate (\frac{5\pi}{16}) units right and 7 units down.

Explanation:

Step1: Analyze horizontal translation

For a function (y = A\sin(B(x - h))+k), the horizontal translation is determined by the change in (h). Original function (f(x)=6\sin\left(x-\frac{\pi}{8}\right)+8), new function (g(x)=6\sin\left(x - \frac{7\pi}{16}\right)+1). We find the difference in the (x) - terms: (\frac{7\pi}{16}-\frac{\pi}{8}=\frac{7\pi - 2\pi}{16}=\frac{5\pi}{16}). Since the value of (h) increased (from (\frac{\pi}{8}=\frac{2\pi}{16}) to (\frac{7\pi}{16})), the graph is translated (\frac{5\pi}{16}) units to the right (for (y = f(x - c)) where (c>0) is a right - hand translation).

Step2: Analyze vertical translation

The vertical translation is determined by the change in (k). Original (k = 8), new (k = 1). The difference is (1 - 8=-7). So the graph is translated 7 units down (for (y=f(x)+d) where (d=-7)).