Question
How is the function f left parenthesis x right parenthesis equals negative one half sine 5 x related to the function g left parenthesis x right parenthesis equals 4 sine 5 x?
(1 point)
Responses
f left parenthesis x right parenthesis is g left parenthesis x right parenthesis stretched horizontally by a factor of 8 and reflected about the y-axis.
Image with alt text: f left parenthesis x right parenthesis is Image with alt text: g left parenthesis x right parenthesis stretched horizontally by a factor of 8 and reflected about the y -axis.
f left parenthesis x right parenthesis is g left parenthesis x right parenthesis compressed vertically by a factor of 8 and reflected about the y-axis.
Image with alt text: f left parenthesis x right parenthesis is Image with alt text: g left parenthesis x right parenthesis compressed vertically by a factor of 8 and reflected about the y -axis.
f left parenthesis x right parenthesis is g left parenthesis x right parenthesis stretched vertically by a factor of 8 and reflected about the x-axis.
Image with alt text: f left parenthesis x right parenthesis is Image with alt text: g left parenthesis x right parenthesis stretched vertically by a factor of 8 and reflected about the x -axis.
f left parenthesis x right parenthesis is g left parenthesis x right parenthesis compressed vertically by a factor of 8 and reflected about the x-axis.
Image with alt text: f left parenthesis x right parenthesis is Image with alt text: g left parenthesis x right parenthesis compressed vertically by a factor of 8 and reflected about the x -axis.
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(1 point)
Responses
f left parenthesis x right parenthesis is g left parenthesis x right parenthesis stretched horizontally by a factor of 8 and reflected about the y-axis.
Image with alt text: f left parenthesis x right parenthesis is Image with alt text: g left parenthesis x right parenthesis stretched horizontally by a factor of 8 and reflected about the y -axis.
f left parenthesis x right parenthesis is g left parenthesis x right parenthesis compressed vertically by a factor of 8 and reflected about the y-axis.
Image with alt text: f left parenthesis x right parenthesis is Image with alt text: g left parenthesis x right parenthesis compressed vertically by a factor of 8 and reflected about the y -axis.
f left parenthesis x right parenthesis is g left parenthesis x right parenthesis stretched vertically by a factor of 8 and reflected about the x-axis.
Image with alt text: f left parenthesis x right parenthesis is Image with alt text: g left parenthesis x right parenthesis stretched vertically by a factor of 8 and reflected about the x -axis.
f left parenthesis x right parenthesis is g left parenthesis x right parenthesis compressed vertically by a factor of 8 and reflected about the x-axis.
Image with alt text: f left parenthesis x right parenthesis is Image with alt text: g left parenthesis x right parenthesis compressed vertically by a factor of 8 and reflected about the x -axis.
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Answers
GPT-4o mini
To analyze the relationship between the functions \( f(x) = -\frac{1}{2} \sin(5x) \) and \( g(x) = 4 \sin(5x) \), we can break down the transformations applied to \( g(x) \) to get to \( f(x) \).
1. **Vertical Stretch/Compression**: The function \( g(x) = 4 \sin(5x) \) has an amplitude of 4. The function \( f(x) = -\frac{1}{2} \sin(5x) \) has an amplitude of \(\frac{1}{2}\). Since \(\frac{1}{2}\) is \( \frac{1}{8} \times 4 \), this means \( f(x) \) is vertically compressed by a factor of 8.
2. **Reflection**: The negative sign in front of \(\frac{1}{2} \sin(5x)\) indicates that \( f(x) \) is reflected across the x-axis.
Putting these transformations together, we can conclude that:
- \( f(x) \) is the result of compressing \( g(x) \) vertically by a factor of 8 and reflecting it about the x-axis.
The correct response is:
- **f(x) is g(x) compressed vertically by a factor of 8 and reflected about the x-axis.**
1. **Vertical Stretch/Compression**: The function \( g(x) = 4 \sin(5x) \) has an amplitude of 4. The function \( f(x) = -\frac{1}{2} \sin(5x) \) has an amplitude of \(\frac{1}{2}\). Since \(\frac{1}{2}\) is \( \frac{1}{8} \times 4 \), this means \( f(x) \) is vertically compressed by a factor of 8.
2. **Reflection**: The negative sign in front of \(\frac{1}{2} \sin(5x)\) indicates that \( f(x) \) is reflected across the x-axis.
Putting these transformations together, we can conclude that:
- \( f(x) \) is the result of compressing \( g(x) \) vertically by a factor of 8 and reflecting it about the x-axis.
The correct response is:
- **f(x) is g(x) compressed vertically by a factor of 8 and reflected about the x-axis.**