To determine which option could NOT represent \( f'(x) \), the image of \( f(x) = x^2 + 2 \) after a reflection, we need to consider the properties of reflection and the form of the original function.
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Reflection across the x-axis: This changes \( f(x) \) to \( -f(x) \). For the given function, this would be: \[ -f(x) = -(x^2 + 2) = -x^2 - 2. \]
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Reflection across the y-axis: This changes \( f(x) \) to \( f(-x) \). For the original function: \[ f(-x) = (-x)^2 + 2 = x^2 + 2, \] which is the same as the original function.
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Other types of reflections: These involve combinations of transformations, but we'll primarily focus on the following potential transformations that are related to the given options.
Now, let's analyze each option:
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Option a: \( y = x^2 - 4x + 5 \) can be rewritten in vertex form. This is a translation and does not represent a simple reflection of \( f(x) \).
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Option b: \( y = x^2 \) represents a reflection across the horizontal axis followed by shifting down, and would still reflect the general parabola structure of the original function.
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Option c: \( 160 \) is a horizontal line. A simple constant function does not represent the reflection of a quadratic function such as \( f(x) \).
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Option d: \( y = -x^2 \) would represent a reflection of \( f(x) \) across the x-axis for the whole function, yielding a function that opens downward.
Conclusion: The option that could NOT represent \( f'(x) \), the image of \( f \) after reflection, is:
c. 160 (a constant function does not reflect a quadratic form).
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