Question
What is the correct description for the transformation from b(x)=3−12x−−−−−√
to m(x)=34x−−√
?(1 point)
Responses
horizontal stretch by a factor of 3 and a reflection over the x-axis
horizontal stretch by a factor of 3 and a reflection over the x -axis
horizontal compression by a factor of 13
and a reflection over the y-axis
horizontal compression by a factor of Start Fraction 1 over 3 End Fraction and a reflection over the y -axis
horizontal compression by a factor of 13
and a reflection over the x-axis
horizontal compression by a factor of Start Fraction 1 over 3 End Fraction and a reflection over the x -axis
horizontal stretch by a factor of 3 and a reflection over the y-axis
to m(x)=34x−−√
?(1 point)
Responses
horizontal stretch by a factor of 3 and a reflection over the x-axis
horizontal stretch by a factor of 3 and a reflection over the x -axis
horizontal compression by a factor of 13
and a reflection over the y-axis
horizontal compression by a factor of Start Fraction 1 over 3 End Fraction and a reflection over the y -axis
horizontal compression by a factor of 13
and a reflection over the x-axis
horizontal compression by a factor of Start Fraction 1 over 3 End Fraction and a reflection over the x -axis
horizontal stretch by a factor of 3 and a reflection over the y-axis
Answers
GPT-4o mini
To analyze the transformation from \( b(x) = 3 - \sqrt{12x} \) to \( m(x) = \sqrt{34x} \), we will decompose the transformations step by step:
1. **Identify the transformations involved**:
- The transformation involves moving from a function with a square root and subtraction to a different square root function.
2. **Rewrite the functions**:
- Start with \( b(x) = 3 - \sqrt{12x} \).
- To transform into \( m(x) = \sqrt{34x} \), you can consider the transformations applied to \( \sqrt{x} \).
3. **Breaking down each function**:
- In \( b(x) \), the expression involves \( \sqrt{12x} \). This can also be viewed as \( \sqrt{12} \cdot \sqrt{x} \).
- The factor inside the square root (12) means that it is a horizontal compression because \( \sqrt{kx} \) represents a horizontal compression by a factor of \( \frac{1}{\sqrt{k}} \) for \( k > 1 \).
Specifically:
- \( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \), which indicates a horizontal compression by a factor of \( \frac{1}{2\sqrt{3}} \).
4. **Compare to \( m(x) = \sqrt{34x} \)**:
- In \( m(x) \), we have a different factor inside the square root (34).
- This also signifies another horizontal compression since \( \sqrt{34} \) would provide a different factor for compression.
5. **Reflection**:
- The transformation also involves a change in vertical orientation (the negative sign in \( b(x) \)).
- The negative allows for a reflection across the horizontal axis, from \( 3 - \sqrt{...} \) to \( \sqrt{...} \) indicates a reflection across the x-axis.
Considering all these components, the transformations can be summarized as:
- A horizontal compression by a factor of \( \frac{1}{3} \) (as determined from the adjustments to \( x \)) combined with a reflection over the x-axis.
Based on this premise, the correct answer can be categorized as:
**horizontal compression by a factor of \( \frac{1}{3} \) and a reflection over the x-axis.**
1. **Identify the transformations involved**:
- The transformation involves moving from a function with a square root and subtraction to a different square root function.
2. **Rewrite the functions**:
- Start with \( b(x) = 3 - \sqrt{12x} \).
- To transform into \( m(x) = \sqrt{34x} \), you can consider the transformations applied to \( \sqrt{x} \).
3. **Breaking down each function**:
- In \( b(x) \), the expression involves \( \sqrt{12x} \). This can also be viewed as \( \sqrt{12} \cdot \sqrt{x} \).
- The factor inside the square root (12) means that it is a horizontal compression because \( \sqrt{kx} \) represents a horizontal compression by a factor of \( \frac{1}{\sqrt{k}} \) for \( k > 1 \).
Specifically:
- \( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \), which indicates a horizontal compression by a factor of \( \frac{1}{2\sqrt{3}} \).
4. **Compare to \( m(x) = \sqrt{34x} \)**:
- In \( m(x) \), we have a different factor inside the square root (34).
- This also signifies another horizontal compression since \( \sqrt{34} \) would provide a different factor for compression.
5. **Reflection**:
- The transformation also involves a change in vertical orientation (the negative sign in \( b(x) \)).
- The negative allows for a reflection across the horizontal axis, from \( 3 - \sqrt{...} \) to \( \sqrt{...} \) indicates a reflection across the x-axis.
Considering all these components, the transformations can be summarized as:
- A horizontal compression by a factor of \( \frac{1}{3} \) (as determined from the adjustments to \( x \)) combined with a reflection over the x-axis.
Based on this premise, the correct answer can be categorized as:
**horizontal compression by a factor of \( \frac{1}{3} \) and a reflection over the x-axis.**