Question
Describe the transformations that bring f(x)=4x−−√−6
to g(x)=−24x−−√−6
.(1 point)
Responses
reflection over the line y=−6
and a vertical stretch by a factor of 2
reflection over the line y equals negative 6 and a vertical stretch by a factor of 2
reflection over the line y=−6
and a vertical compression by a factor of 2
reflection over the line y equals negative 6 and a vertical compression by a factor of 2
reflection over the line x=−6
and a vertical compression by a factor of 2
reflection over the line x equals negative 6 and a vertical compression by a factor of 2
reflection over the line x=−6
and a vertical stretch by a factor of 2
to g(x)=−24x−−√−6
.(1 point)
Responses
reflection over the line y=−6
and a vertical stretch by a factor of 2
reflection over the line y equals negative 6 and a vertical stretch by a factor of 2
reflection over the line y=−6
and a vertical compression by a factor of 2
reflection over the line y equals negative 6 and a vertical compression by a factor of 2
reflection over the line x=−6
and a vertical compression by a factor of 2
reflection over the line x equals negative 6 and a vertical compression by a factor of 2
reflection over the line x=−6
and a vertical stretch by a factor of 2
Answers
Answered by
GPT-4o mini
To understand the transformations that convert the function \( f(x) = 4\sqrt{x} - 6 \) into \( g(x) = -24\sqrt{x} - 6 \), let's analyze the changes step by step:
1. **Vertical Reflection**: The transformation from \( 4\sqrt{x} \) to \( -24\sqrt{x} \) includes a reflection over the horizontal axis (the x-axis) because \( g(x) \) has the negative coefficient of the square root function compared to \( f(x) \).
2. **Vertical Stretch/Compression**: We also notice that the coefficient of \( g(x) \) is \(-24\), while the coefficient of \( f(x) \) is \(4\). The factor of 24 compared to 4 represents a vertical stretch. Specifically, the vertical stretch factor can be calculated as:
\[
\text{Stretch factor} = \frac{|a|}{|b|} = \frac{24}{4} = 6.
\]
However, since it is a reflection over the x-axis, we consider it as a vertical stretch by a factor of 6 while reflecting.
3. **Vertical Shift**: The term \(-6\) in both functions indicates that there is no vertical shift occurring since both functions are shifted down by 6 units unchanged.
Based on this analysis, the transformations that bring \( f(x) \) to \( g(x) \) involve a reflection over the x-axis and a vertical stretch by a factor (in terms of absolute value, since we are reflecting). The correct response should clearly indicate the reflection and the type of stretch as analyzed.
Upon reviewing the provided responses, we see that none of them correctly describe both elements (reflection and the correct vertical stretch factor). Here is a summary:
- **Reflection over x-axis** (note: most options mention reflection over the line y = -6, which isn’t accurate).
- **Vertical stretch by a factor of 6** (not explicitly stated in any provided options).
For the purpose of answering, the closest option that aligns with the reflection characteristic, though not fully accurate, is as follows:
- **Reflection over the line \( y = -6 \) and a vertical stretch by a factor of 2** (incorrect stretch factor, should be 6).
Therefore, while these details clarify the transformations correctly, you may wish to select the response that is the closest to capturing the reflection while noting the inaccuracies in the vertical stretching factor. Subsequent corrections might be required if needed.
1. **Vertical Reflection**: The transformation from \( 4\sqrt{x} \) to \( -24\sqrt{x} \) includes a reflection over the horizontal axis (the x-axis) because \( g(x) \) has the negative coefficient of the square root function compared to \( f(x) \).
2. **Vertical Stretch/Compression**: We also notice that the coefficient of \( g(x) \) is \(-24\), while the coefficient of \( f(x) \) is \(4\). The factor of 24 compared to 4 represents a vertical stretch. Specifically, the vertical stretch factor can be calculated as:
\[
\text{Stretch factor} = \frac{|a|}{|b|} = \frac{24}{4} = 6.
\]
However, since it is a reflection over the x-axis, we consider it as a vertical stretch by a factor of 6 while reflecting.
3. **Vertical Shift**: The term \(-6\) in both functions indicates that there is no vertical shift occurring since both functions are shifted down by 6 units unchanged.
Based on this analysis, the transformations that bring \( f(x) \) to \( g(x) \) involve a reflection over the x-axis and a vertical stretch by a factor (in terms of absolute value, since we are reflecting). The correct response should clearly indicate the reflection and the type of stretch as analyzed.
Upon reviewing the provided responses, we see that none of them correctly describe both elements (reflection and the correct vertical stretch factor). Here is a summary:
- **Reflection over x-axis** (note: most options mention reflection over the line y = -6, which isn’t accurate).
- **Vertical stretch by a factor of 6** (not explicitly stated in any provided options).
For the purpose of answering, the closest option that aligns with the reflection characteristic, though not fully accurate, is as follows:
- **Reflection over the line \( y = -6 \) and a vertical stretch by a factor of 2** (incorrect stretch factor, should be 6).
Therefore, while these details clarify the transformations correctly, you may wish to select the response that is the closest to capturing the reflection while noting the inaccuracies in the vertical stretching factor. Subsequent corrections might be required if needed.