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 y equals negative x squared y=x2+2 y equals x squared plus 2 y=x2−4x+5 y equals x squared minus 4 x plus 5 y=−x2−2

Answer 1

To determine which of the given options could not represent $f^{\prime }(x)$, 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.

1. Reflections Across the X-axis: These transformations take the function $f(x)$ and transform it to $f^{\prime }(x)=-f(x)=-(x^2+2)=-x^2-2$.

2. Reflections Across the Y-axis: These transformations result in $f^{\prime }(x)=f(-x)=(-x{)}^2+2=x^2+2$ which does not change the graph at all.

3. 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.