To determine the new coordinates of the vertex after the reflections, let's go through each type of reflection step-by-step:
1. Reflection across the [tex]\( x \)[/tex]-axis:
When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\( x \)[/tex]-axis, its new coordinates are [tex]\((x, -y)\)[/tex].
For the vertex [tex]\( (2, -3) \)[/tex]:
[tex]\[
(2, -(-3)) = (2, 3)
\][/tex]
2. Reflection across the [tex]\( y \)[/tex]-axis:
When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\( y \)[/tex]-axis, its new coordinates are [tex]\((-x, y)\)[/tex].
For the vertex [tex]\( (2, -3) \)[/tex]:
[tex]\[
(-2, -3)
\][/tex]
3. Reflection across the line [tex]\( y = x \)[/tex]:
When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\( y = x \)[/tex], its new coordinates are [tex]\((y, x)\)[/tex].
For the vertex [tex]\( (2, -3) \)[/tex]:
[tex]\[
(-3, 2)
\][/tex]
4. Reflection across the line [tex]\( y = -x \)[/tex]:
When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\( y = -x \)[/tex], its new coordinates are [tex]\((-y, -x)\)[/tex].
For the vertex [tex]\( (2, -3) \)[/tex]:
[tex]\[
(-(-3), -2) = (3, -2)
\][/tex]
Now, if we compare these results:
- The reflection across the [tex]\( x \)[/tex]-axis gives us [tex]\((2, 3)\)[/tex]
- The reflection across the [tex]\( y \)[/tex]-axis gives us [tex]\((-2, -3)\)[/tex]
- The reflection across the line [tex]\( y = x \)[/tex] gives us [tex]\((-3, 2)\)[/tex]
- The reflection across the line [tex]\( y = -x \)[/tex] gives us [tex]\((3, -2)\)[/tex]
Therefore, the new positions of the vertex [tex]\( (2, -3) \)[/tex] after the reflections are:
- Reflecting across the [tex]\( x \)[/tex]-axis will produce an image with the vertex at [tex]\( (2, 3) \)[/tex]
- Reflecting across the [tex]\( y \)[/tex]-axis will produce an image with the vertex at [tex]\( (-2, -3) \)[/tex]
- Reflecting across the line [tex]\( y = x \)[/tex] will produce an image with the vertex at [tex]\( (-3, 2) \)[/tex]
- Reflecting across the line [tex]\( y = -x \)[/tex] will produce an image with the vertex at [tex]\( (3, -2) \)[/tex]
These corresponding reflections provide the correct new positions for the vertex after each type of reflection.
