To determine the order of transformations for the function \( f(x) = x^2 \) to become \( f(-x + 4) + 3 \), we can analyze the changes step by step.
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Start with the basic function: \( f(x) = x^2 \).
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First Transformation: The function is reflected over the y-axis (which can also be seen as reflecting over the x-axis, since for our standard function \( f(x) = x^2 \), reflection would affect how we view the input). This is represented by \( f(-x) \). Reflection does impact the shape of the graph without changing its vertex's vertical position. Thus this corresponds to reflecting over the y-axis (or x-axis in horizontal context).
- Transformation 3 (reflect): \( f(-x) \) is the first transformation.
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Second Transformation: Next, we modify the x-value of the function with the addition of \( +4 \). This shifts the graph to the right (as the input is \(-x + 4\) instead of \(-x\)). Remember that within the function due to the reflection, it shifts left due to how the argument inside function behaves with subtraction, but conceptually here we correct to relate moving the graph to the right.
- Transformation 1 (translate left 4 units): The transformation \( f(-x + 4) \) indicates shifting (or wrapping) the graph left by converting it to implement \( +4\) inside the argument means you take it from typical orientation.
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Third Transformation: Finally, we add +3 outside of the function, which moves the whole graph upwards.
- Transformation 2 (translate up 3 units): Adding 3 to get \( f(-x + 4) + 3 \) represents a vertical shift up.
Putting this all together, the transformations in order are:
- First Transformation: 3 (reflect over the x-axis)
- Second Transformation: 1 (translate left 4 units)
- Third Transformation: 2 (translate up 3 units)
So the final answer is:
- First Transformation: 3
- Second Transformation: 1
- Third Transformation: 2