To find the equation of the graph of the function [tex]\( y = x^2 \)[/tex] reflected over the [tex]\( x \)[/tex]-axis, we need to understand what reflecting a graph over the [tex]\( x \)[/tex]-axis entails. Reflecting a graph over the [tex]\( x \)[/tex]-axis means that each point on the graph (that has the form [tex]\((x, y)\)[/tex]) will be transformed to a new point [tex]\((x, -y)\)[/tex].
Let's start with the original function [tex]\( y = x^2 \)[/tex]:
- For any point [tex]\((x, y)\)[/tex] on this graph, the [tex]\( y \)[/tex]-coordinate is given by [tex]\( y = x^2 \)[/tex].
When we reflect this graph over the [tex]\( x \)[/tex]-axis:
- The [tex]\( x \)[/tex]-coordinate remains the same.
- The new [tex]\( y \)[/tex]-coordinate becomes the negative of the original [tex]\( y \)[/tex]-coordinate.
Therefore, the new [tex]\( y \)[/tex]-coordinate will be:
[tex]\[ y = - (x^2) \][/tex]
or simply,
[tex]\[ y = -x^2 \][/tex]
Thus, the equation of the transformed graph, which is the reflection of [tex]\( y = x^2 \)[/tex] over the [tex]\( x \)[/tex]-axis, is:
[tex]\[ y = -x^2 \][/tex]
Let's examine the given options to identify which one matches our result:
1. [tex]\( y = -x^2 \)[/tex]
2. [tex]\( y = (-x)^2 \)[/tex]
3. [tex]\( y = \sqrt{-x} \)[/tex]
4. [tex]\( y = -\sqrt{x} \)[/tex]
The correct option is:
[tex]\[ \boxed{y = -x^2} \][/tex]
This matches the equation we determined for the graph after reflection over the [tex]\( x \)[/tex]-axis.
