To determine the transformation of the graph of \( f(x) \) to the graph labeled Option 3, we need to assess the changes in the characteristics of the waves. The transformations you listed correspond to specific modifications to the original function \( f(x) \). Here's a brief overview of those transformations:
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Vertical Shift: \( f(x) + k \) shifts the graph vertically by \( k \) units. Positive \( k \) moves it up, while negative \( k \) moves it down.
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Reflection Across the x-axis: \( -f(x) \) reflects the graph across the x-axis.
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Reflection Across the y-axis: \( f(-x) \) reflects the graph across the y-axis.
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Horizontal Reflection: \( -f(-x) \) reflects the graph across both axes, effectively flipping it upside down and reversing it.
Now, without the visual representation but based on the provided information, if Option 3 significantly alters the amplitude, direction, or phase of the wave, we can correlate it to the transformations mentioned.
To summarize the options:
- If the graph in Option 3 appears shifted vertically without affecting the overall shape, it is \( f(x) + 1 \).
- If it appears to be a mirror image horizontally, it is \( f(-x) \).
- If it is flipped upside down, it is \( -f(x) \).
- If it is flipped both ways (upside down and side), it is \( -f(-x) \).
Based on the transformations, review your image and see how the graph of Option 3 relates to the original function \( f(x) \). This will help identify the correct answer from the provided choices.
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