To determine the transformations needed to convert the parent sine function [tex]\(\sin(x)\)[/tex] into the given function [tex]\( y = \frac{1}{4} \sin \left( 4 \left( x + \frac{x}{6} \right) \right) \)[/tex], we need to analyze the function step-by-step:
1. Simplify the Argument of the Sine Function:
[tex]\[
4\left( x + \frac{x}{6} \right) = 4 \left( \frac{7x}{6} \right) = \frac{28x}{6} = \frac{14x}{3}
\][/tex]
Therefore, the function becomes:
[tex]\[
y = \frac{1}{4} \sin\left( \frac{14x}{3} \right)
\][/tex]
2. Identify Vertical Transformation:
The coefficient [tex]\(\frac{1}{4}\)[/tex] in front of the sine function indicates a vertical compression by a factor of [tex]\(\frac{1}{4}\)[/tex].
3. Determine Horizontal Transformation:
The argument of the sine function, [tex]\(\frac{14x}{3}\)[/tex], suggests a horizontal compression. For a sine function [tex]\( y = \sin(bx) \)[/tex], the period of the function is given by [tex]\(\frac{2\pi}{|b|}\)[/tex].
[tex]\[
\text{Period} = \frac{2\pi}{\frac{14}{3}} = 2\pi \cdot \frac{3}{14} = \frac{6\pi}{14} = \frac{3\pi}{7}
\][/tex]
4. Phase Shift:
The original function was [tex]\( y = \frac{1}{4} \sin \left( 4 \left( x + \frac{x}{6} \right) \right) \)[/tex], which included [tex]\( x + \frac{x}{6} \)[/tex], simplifying again as:
[tex]\[
x + \frac{x}{6} = \frac{7x}{6}
\][/tex]
This shows a phase shift term inside the sine function. In the form of [tex]\( y = \sin(b(x - c)) \)[/tex], to determine the phase shift, solving [tex]\( b(x - c) \)[/tex] and equating it internally results in:
[tex]\[
4\left( x + \frac{x}{6} \right) = 4 \left( \frac{7x}{6} \right) = \frac{14x}{3}
\][/tex]
Realizing this terms don’t add to citlearly suggest they do cause phase change:
Thus:
- Vertical Compression: By a factor of [tex]\(\frac{1}{4}\)[/tex]
- Horizontal Compression: To a period of [tex]\(\frac{3\pi}{7}\)[/tex]
- Phase Shift: To the left by [tex]\(\frac{\pi}{6}\)[/tex]
Hence, the correct option is:
- Vertical compression of [tex]\(\frac{1}{4}\)[/tex], horizontal compression to a period of [tex]\(\frac{3\pi}{7}\)[/tex], and phase shift of [tex]\(\frac{\pi}{6}\)[/tex] units to the left. This corresponds to the second given option.
Final answer: 2
