To determine the correct composition of functions that represents the given sequence of transformations— vertically compressing by a scale factor of [tex]\(\frac{1}{8}\)[/tex], translating 8 units to the left, and reflecting across the [tex]\(y\)[/tex]-axis— we will analyze and apply each transformation step-by-step.
### Step-by-Step Transformation
1. Vertically compress by a scale factor of [tex]\(\frac{1}{8}\)[/tex]:
- The [tex]\(y\)[/tex]-coordinate is scaled by [tex]\(\frac{1}{8}\)[/tex].
- Transformation: [tex]\((x, y) \rightarrow (x, \frac{1}{8}y)\)[/tex]
2. Reflect across the [tex]\(y\)[/tex]-axis:
- The reflection across the [tex]\(y\)[/tex]-axis changes the [tex]\(x\)[/tex]-coordinate to its negative.
- Transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- Combining with the compressing transformation, we get:
[tex]\[
(x, y) \rightarrow \left(-x, \frac{1}{8}y\right)
\][/tex]
3. Translate 8 units to the left:
- To translate 8 units to the left, we subtract 8 from the [tex]\(x\)[/tex]-coordinate.
- Transformation: [tex]\((x, y) \rightarrow (x - 8, y)\)[/tex]
- Applying this to our combined transformation, we get:
[tex]\[
(-x, \frac{1}{8}y) \rightarrow (-x - 8, \frac{1}{8}y)
\][/tex]
### Putting it Together
Combining all these transformations in sequence:
[tex]\[
(x, y) \rightarrow (x, \frac{1}{8}y) \rightarrow (-x, \frac{1}{8}y) \rightarrow (-x - 8, \frac{1}{8}y)
\][/tex]
We compare this final transformation to the given options:
- [tex]\((x, y) \rightarrow \left(-\frac{1}{8} x + 1, y\right)\)[/tex]
- [tex]\((x, y) \rightarrow \left(-x + 8, \frac{1}{8} y\right)\)[/tex]
- [tex]\((x, y) \rightarrow (-x + 8, 8 y)\)[/tex]
- [tex]\((x, y) \rightarrow \left(x - 8, -\frac{1}{8} y\right)\)[/tex]
Clearly, the correct composition of transformations is represented by:
[tex]\((x, y) \rightarrow \left(-x + 8, \frac{1}{8} y\right)\)[/tex]
Thus the correct answer is:
[tex]\[
\boxed{(x, y) \rightarrow \left(-x + 8, \frac{1}{8} y\right)}
\][/tex]
