To determine which expression could NOT represent \( f'(x) \), we need to analyze the original function \( f(x) = x^2 + 2 \) and understand what happens during a reflection.
The original function \( f(x) = x^2 + 2 \) is a parabola that opens upwards. Reflecting it across the x-axis would change its sign. Thus, the reflected function would be:
\[ f'(x) = -f(x) = -(x^2 + 2) = -x^2 - 2 \]
Now, let's examine each of the given options to see if they could represent the image of \( f \) after the reflection.
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\( y = -x^2 \): This reflects \( f \) across the x-axis but does not include the constant \( -2 \). So, it's incorrect for the image of \( f(x) \) specifically, as it does not match the transformation we'd expect for a reflection of \( f(x) = x^2 + 2 \).
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\( y = x^2 - 4x + 5 \): This is a quadratic function but not necessarily the reflection of \( f(x) \). We can't definitively say it's incorrect without more context, but it's not the standard form of the reflected function derived from \( f(x) \).
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\( y = -x^2 - 2 \): This expression matches our derived reflection \( f'(x) = -x^2 - 2 \) precisely.
Based on this analysis:
The option that could NOT represent the reflection of \( f(x) = x^2 + 2 \) is:
\( y = -x^2 \)