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 x squared plus 2

y=−x2
y equals negative x squared

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

y=−x2−2

Answer 1

To determine which function could not represent the image of \( f(x) = x^2 + 2 \) after a reflection, we need to consider different types of reflections.

1. **Reflection across the x-axis**: This negates the function's output, turning \( f(x) \) into \( f'(x) = -f(x) \). For the function \( f(x) = x^2 + 2 \), this would yield:

\[
f'(x) = -(x^2 + 2) = -x^2 - 2
\]

2. **Reflection across the y-axis**: This maintains the same output as the original function since it only affects how we input the x-values; thus, it would not change the function.

3. **Reflection across the line y = x**: This would require finding the inverse of the function, which typically is not a simple transformation of the original function.

From the options provided:

- **Option 1**: \( y = x^2 + 2 \) is the original function and reflects across the y-axis.

- **Option 2**: \( y = -x^2 \) is a reflection across the x-axis but it is a different form from the original, as it does not include the "+2" component.

- **Option 3**: \( y = x^2 - 4x + 5 \) can be rewritten by completing the square. It does not represent a direct reflection about any axis or line as it shifts the entire graph and doesn't have to do with flipping over axes directly related to the original parabola.

- **Option 4**: \( y = -x^2 - 2 \) is also a valid reflection across the x-axis, translating downwards by 2 units.

Looking at all options, option \( y = x^2 - 4x + 5 \) could not represent a reflection of \( f(x) \) as it changes the shape and position of the graph, rather than simply reflecting it.

Thus, the answer is:

**y = x^2 - 4x + 5**