To find out the order of transformations that take the function \( f(x) = x^2 \) to \( f(-x + 4) + 3 \), we analyze the transformations step by step:
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Reflecting over the x-axis: The original function \( f(x) = x^2 \) transforms to \( f(-x) = (-x)^2 = x^2 \) when reflected. However, to reflect over the x-axis, we would write it as \( -f(x) = -x^2 \). In our function, there's no reflection taking place, thus this step is not applicable.
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Translating left 4 units: The expression \( f(-x + 4) \) can be analyzed to find the impact of the function. The inner part of the function gives us a hint: \[ -x + 4 \] This indicates a horizontal shift to the left. To get from \( x \) to \( -x + 4 \), we treat it as \[ f(x) \to f(x - 4) \] which means we first translate the function left by 4 units.
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Translating up 3 units: After performing the horizontal shift, we need to translate the entire function up by 3 units as indicated by the \( +3 \) at the end of the function.
Now we can summarize the transformations in the order they occurred:
- First Transformation: Translate left 4 units (Type 1)
- Second Transformation: Translate up 3 units (Type 2)
Since there is no reflection happening in this transformation, the reflection (Type 3) does not apply here based on the expression \( f(-x + 4) + 3 \).
Thus:
- First Transformation: 1
- Second Transformation: 2
- Third Transformation: (Not applicable; no reflection in this case)
Final answer:
- First Transformation: 1
- Second Transformation: 2
- Third Transformation: (No reflection, so leave blank or N/A)