Certainly! Let's break down each transformation applied to the parent sine function \( y = \sin(x) \) to get the function \( y = \frac{1}{4} \sin \left(4\left(x + \frac{\pi}{6}\right)\right) \).
### 1. Vertical Compression
The coefficient in front of the sine function \( \frac{1}{4} \) affects the amplitude:
- The amplitude of \( y = \sin(x) \) is 1.
- Multiplying by \( \frac{1}{4} \) compresses the amplitude to \( \frac{1}{4} \).
Therefore, this is a vertical compression by a factor of \( \frac{1}{4} \).
### 2. Horizontal Compression
The coefficient inside the sine function before \( x \) indicates a horizontal stretch/compression:
- The general form \( y = \sin(bx) \) has a period of \( \frac{2\pi}{b} \).
- Here, \( b = 4 \), so the period is \( \frac{2\pi}{4} = \frac{\pi}{2} \).
This represents a horizontal compression, reducing the period from \( 2\pi \) to \( \frac{\pi}{2} \).
### 3. Phase Shift
The term inside the sine function then affects the horizontal shift:
- The general form \( y = \sin(b(x - c)) \) shifts the graph horizontally by \( c \) units.
- Here, we have \( x + \frac{\pi}{6} \), written as \( 4 \left( x + \frac{\pi}{6} \right) \), resulting in a phase shift.
- Rewriting, it implies \( y = \sin \left( 4 \left(x + \frac{\pi}{6}\right) \right) \), a phase shift of \( \frac{\pi}{6} \) units to the left.
### Conclusion
Given those transformations:
- Vertical compression by a factor of \( \frac{1}{4} \)
- Horizontal compression to a period of \( \frac{\pi}{2} \)
- Phase shift of \( \frac{\pi}{6} \) units to the left
The choice that matches these transformations is:
#### Vertical compression of \( \frac{1}{4} \), horizontal compression to a period of \( \frac{\pi}{2} \), phase shift of \( \frac{\pi}{6} \) units to the left.
Thus, the correct answer is:
2. Vertical compression of [tex]\( \frac{1}{4} \)[/tex], horizontal compression to a period of [tex]\( \frac{\pi}{2} \)[/tex], phase shift of [tex]\( \frac{\pi}{6} \)[/tex] units to the left.
