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Transformations Online Practice

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Question
The function y=x2+2
undergoes a reflection. Which of the following could not represent f′(x)
, the image of f
after the reflection? (1 point)
Responses

y=−x2−2
y equals negative x squared minus 2

y=x2−4x+5
y equals x squared minus 4 x plus 5

y=x2+2
y equals x squared plus 2

y=−x2

1 answer

To determine which function could not represent \( f'(x) \), the image of \( f(x) = x^2 + 2 \) after a reflection, we need to understand what a reflection means in this context.

A reflection can happen across the x-axis or the y-axis:

  1. Reflection across the x-axis: This changes the sign of the entire function. For example, reflecting \( f(x) = x^2 + 2 \) across the x-axis would yield \( f'(x) = - (x^2 + 2) = -x^2 - 2 \).

  2. Reflection across the y-axis: This changes the sign of the x but leaves the value of the function unchanged. For instance, reflecting \( f(x) = x^2 + 2 \) across the y-axis would yield \( f'(-x) = (-x)^2 + 2 = x^2 + 2 \).

Let's analyze the given options:

  • Option 1: \( y = -x^2 - 2 \)

    • This is a reflection across the x-axis and is valid.

  • Option 2: \( y = x^2 - 4x + 5 \)

    • This is a different quadratic function and does not represent a direct reflection of the original function.

  • Option 3: \( y = x^2 + 2 \)

    • This is the original function itself and represents a reflection across the y-axis (which is not a typical reflection in this context).

  • Option 4: \( y = -x^2 \)

    • This is also a reflection across the x-axis but without the vertical shift.

Based on this analysis, the option that could not represent \( f'(x) \) after a reflection of \( y = x^2 + 2 \) is:

y = x^2 - 4x + 5.