To determine the order of transformations from \( f(x) = x^3 \) to \( f(-x + 2) + 4 \), we'll analyze the transformations step by step.
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Start with the original function: \( f(x) = x^3 \).
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Horizontal reflection: The transformation \( -x \) reflects the graph across the y-axis. Therefore, \( f(-x) = (-x)^3 = -x^3 \).
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Horizontal translation: The transformation \( -x + 2 \) can be rewritten as \( -(x - 2) \), which indicates a horizontal translation to the right by 2 units. Thus, \( f(-x + 2) \) can be computed as \( f(- (x - 2)) \) which becomes \( -(x-2)^3 \).
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Vertical translation: Finally, adding 4 to the function shifts the graph up by 4 units. Therefore, \( f(-x + 2) + 4 \) translates vertically, giving us \( -(x - 2)^3 + 4 \).
Now, summarizing the transformations:
- Horizontal reflection
- Horizontal translation (to the right by 2)
- Vertical translation (up by 4)
Given the order of transformations, the correct option is:
horizontal reflection, horizontal translation, vertical translation.