Join the growing community of curious minds on IDNLearn.com and get the answers you need. Discover prompt and accurate responses from our experts, ensuring you get the information you need quickly.
Sagot :
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
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
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.
