We are given the transformed function $f(x) = \frac{1}{8}h(x-6)-5$ and we want to determine the transformations that have been applied to the parent function $h(x)$. We need to identify any reflections, stretches/compressions, vertical shifts, and horizontal shifts.

1. Problem Description

We are given the transformed function $f(x) = \frac{1}{8}h(x-6)-5$ and we want to determine the transformations that have been applied to the parent function $h(x)$. We need to identify any reflections, stretches/compressions, vertical shifts, and horizontal shifts.

2. Solution Steps

* Reflection: The general form for a reflection across the x-axis is $-h(x)$ and a reflection across the y-axis is $h(-x)$. In our function, $f(x) = \frac{1}{8}h(x-6)-5$, there is no negative sign in front of $h(x-6)$ or a negative inside the $h(x)$ as in $h(-x)$, thus there is no reflection.

* Stretches/Compressions: The general form is $a \cdot h(x)$, where $a$ is a constant. If $|a| > 1$, then it's a vertical stretch. If $0 < |a| < 1$, then it's a vertical compression. In the function $f(x) = \frac{1}{8}h(x-6)-5$, the constant $a = \frac{1}{8}$. Since $0 < \frac{1}{8} < 1$, there is a vertical compression by a factor of $\frac{1}{8}$.

* Vertical Shift: The general form is $h(x) + c$, where $c$ is a constant. If $c > 0$, then the graph shifts upward. If $c < 0$, then the graph shifts downward. In the function $f(x) = \frac{1}{8}h(x-6)-5$, we have $-5$, so there is a vertical shift downward by 5 units.

* Horizontal Shift: The general form is $h(x - d)$, where $d$ is a constant. If $d > 0$, then the graph shifts to the right. If $d < 0$, then the graph shifts to the left. In the function $f(x) = \frac{1}{8}h(x-6)-5$, we have $x-6$. Here, $d = 6$, so the graph shifts to the right by 6 units.

3. Final Answer

a) No Reflection

b) Vertical Compression by a factor of 1/8

c) Vertical Shift Down 5 units

d) Horizontal Shift Right 6 units