To find the image of the point [tex]\((0,0)\)[/tex] after two reflections, first across the [tex]\(x\)[/tex]-axis and then across the [tex]\(y\)[/tex]-axis, we can proceed as follows:
1. Reflection across the [tex]\(x\)[/tex]-axis:
The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis changes its coordinates to [tex]\((x, -y)\)[/tex]. This means we only change the sign of the [tex]\(y\)[/tex]-coordinate while keeping the [tex]\(x\)[/tex]-coordinate the same.
Let's consider the given point [tex]\((0,0)\)[/tex]:
- Reflecting [tex]\((0,0)\)[/tex] across the [tex]\(x\)[/tex]-axis:
[tex]\[
(0, 0) \rightarrow (0, -0) = (0, 0)
\][/tex]
So, after the reflection across the [tex]\(x\)[/tex]-axis, the coordinates remain [tex]\((0,0)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis changes its coordinates to [tex]\((-x, y)\)[/tex]. This means we only change the sign of the [tex]\(x\)[/tex]-coordinate while keeping the [tex]\(y\)[/tex]-coordinate the same.
Now, consider the point obtained from the previous step, which is [tex]\((0,0)\)[/tex]:
- Reflecting [tex]\((0,0)\)[/tex] across the [tex]\(y\)[/tex]-axis:
[tex]\[
(0, 0) \rightarrow (-0, 0) = (0, 0)
\][/tex]
Again, the coordinates remain [tex]\((0,0)\)[/tex].
Therefore, the image of the point [tex]\((0,0)\)[/tex] after reflecting first across the [tex]\(x\)[/tex]-axis and then across the [tex]\(y\)[/tex]-axis is [tex]\((0,0)\)[/tex].
