To determine which of the given options could not represent \( f'(x) \), the image of the function \( f(x) = x^2 + 2 \) after a reflection, we need to understand how reflections affect the graph of a function.
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Reflections Across the X-axis: These transformations take the function \( f(x) \) and transform it to \( f'(x) = -f(x) = - (x^2 + 2) = -x^2 - 2 \).
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Reflections Across the Y-axis: These transformations result in \( f'(x) = f(-x) = (-x)^2 + 2 = x^2 + 2 \) which does not change the graph at all.
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Analyzing the provided options:
- Option 1: \( y = -x^2 \): This is a reflection across the x-axis but does not include the vertical shift of +2. This could represent a reflection but is different from the original function's shift.
- Option 2: \( y = x^2 + 2 \): This is exactly the same as the original function and does not represent a reflection.
- Option 3: \( y = x^2 - 4x + 5 \): This can be rewritten as \( y = (x - 2)^2 + 1 \), which is a parabola shifted to the right and down, but not a reflection of the original function.
- Option 4: \( y = -x^2 - 2 \): This is the correct reflection across the x-axis including the vertical shift of +2.
From the options considered, the function that could not represent the reflected image of \( f(x) = x^2 + 2 \) is:
y = x^2 + 2
This is simply the original function and does not show any reflection.