Question
Describe the transformation of the graph of f(x)= sin x +2 to the graph labeled option 2 which is in the 4th quadrant and the f(x) =sin x +2 is in the 2nd quadrant
f(x-7)
f(-x)
-f(x)
-f(-x)
f(x-7)
f(-x)
-f(x)
-f(-x)
Answers
Answered by
GPT-4o mini
To understand the transformation from the graph of \( f(x) = \sin x + 2 \) to the graph labeled as option 2 in the fourth quadrant, we need to break down the transformations represented by the options provided:
1. **\( f(x - 7) \)**: This transformation translates the graph horizontally to the right by 7 units. This would not move the graph into the fourth quadrant.
2. **\( f(-x) \)**: This transformation reflects the graph across the y-axis. While this will change the orientation of the graph, it will not move the graph to the fourth quadrant, specifically if we are starting with \( f(x) = \sin x + 2 \).
3. **\( -f(x) \)**: This transformation reflects the graph across the x-axis. Since the original graph \( f(x) = \sin x + 2 \) is in the second quadrant, reflecting it across the x-axis would move parts of it into the fourth quadrant because the sine function oscillates between -1 and 1, leading to values between 1 and 3 before reflection which would create values between -3 and -1 after reflection.
4. **\( -f(-x) \)**: This transformation involves first reflecting the graph across the y-axis, and then reflecting it across the x-axis. This would indeed change the orientation and would also likely result in the graph being in the fourth quadrant.
### Conclusion:
The transformation that takes the graph of \( f(x) = \sin x + 2 \) from the second quadrant to the fourth quadrant is **\( -f(x) \)** or possibly **\( -f(-x) \)** depending on exactly how option 2 is defined.
If we specify that option 2 is in the fourth quadrant and reflects the original graph across the x-axis, then **the correct transformation is \( -f(x) \)**. But if it also involves a horizontal reflection as indicated by being labeled \( -f(-x) \), then that would also be applicable.
If you can provide more detail about the specific features of option 2, I can clarify further which transformation applies best!
1. **\( f(x - 7) \)**: This transformation translates the graph horizontally to the right by 7 units. This would not move the graph into the fourth quadrant.
2. **\( f(-x) \)**: This transformation reflects the graph across the y-axis. While this will change the orientation of the graph, it will not move the graph to the fourth quadrant, specifically if we are starting with \( f(x) = \sin x + 2 \).
3. **\( -f(x) \)**: This transformation reflects the graph across the x-axis. Since the original graph \( f(x) = \sin x + 2 \) is in the second quadrant, reflecting it across the x-axis would move parts of it into the fourth quadrant because the sine function oscillates between -1 and 1, leading to values between 1 and 3 before reflection which would create values between -3 and -1 after reflection.
4. **\( -f(-x) \)**: This transformation involves first reflecting the graph across the y-axis, and then reflecting it across the x-axis. This would indeed change the orientation and would also likely result in the graph being in the fourth quadrant.
### Conclusion:
The transformation that takes the graph of \( f(x) = \sin x + 2 \) from the second quadrant to the fourth quadrant is **\( -f(x) \)** or possibly **\( -f(-x) \)** depending on exactly how option 2 is defined.
If we specify that option 2 is in the fourth quadrant and reflects the original graph across the x-axis, then **the correct transformation is \( -f(x) \)**. But if it also involves a horizontal reflection as indicated by being labeled \( -f(-x) \), then that would also be applicable.
If you can provide more detail about the specific features of option 2, I can clarify further which transformation applies best!