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To determine the coordinates of the reflected vertex (2, -3) across different axes and lines, we'll reflect the vertex through each specified line and axis step-by-step. Here's the process for each reflection:
1. Reflection across the x-axis:
- When reflecting a point [tex]\((x, y)\)[/tex] across the x-axis, the y-coordinate changes sign. So, [tex]\((x, y) \rightarrow (x, -y)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the x-axis: [tex]\((2, -3) \rightarrow (2, 3)\)[/tex].
2. Reflection across the y-axis:
- When reflecting a point [tex]\((x, y)\)[/tex] across the y-axis, the x-coordinate changes sign. So, [tex]\((x, y) \rightarrow (-x, y)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the y-axis: [tex]\((2, -3) \rightarrow (-2, -3)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex], we swap the x and y coordinates. So, [tex]\((x, y) \rightarrow (y, x)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the line [tex]\(y = x\)[/tex]: [tex]\((2, -3) \rightarrow (-3, 2)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex], we swap the x and y coordinates and then change the signs of both coordinates. So, [tex]\((x, y) \rightarrow (-y, -x)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the line [tex]\(y = -x\)[/tex]: [tex]\((2, -3) \rightarrow (3, -2)\)[/tex].
Therefore, the coordinates of the reflected vertex (2, -3) for each reflection are:
- Reflection across the x-axis: [tex]\((2, 3)\)[/tex]
- Reflection across the y-axis: [tex]\((-2, -3)\)[/tex]
- Reflection across the line [tex]\(y = x\)[/tex]: [tex]\((-3, 2)\)[/tex]
- Reflection across the line [tex]\(y = -x\)[/tex]: [tex]\((3, -2)\)[/tex]
So, an image of the vertex at [tex]\((2, -3)\)[/tex] will be produced at [tex]\((2, 3)\)[/tex], [tex]\((-2, -3)\)[/tex], [tex]\((-3, 2)\)[/tex], or [tex]\((3, -2)\)[/tex] depending on which reflection is applied:
- Reflection across the x-axis produces [tex]\((2, 3)\)[/tex].
- Reflection across the y-axis produces [tex]\((-2, -3)\)[/tex].
- Reflection across the line [tex]\(y = x\)[/tex] produces [tex]\((-3, 2)\)[/tex].
- Reflection across the line [tex]\(y = -x\)[/tex] produces [tex]\((3, -2)\)[/tex].
1. Reflection across the x-axis:
- When reflecting a point [tex]\((x, y)\)[/tex] across the x-axis, the y-coordinate changes sign. So, [tex]\((x, y) \rightarrow (x, -y)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the x-axis: [tex]\((2, -3) \rightarrow (2, 3)\)[/tex].
2. Reflection across the y-axis:
- When reflecting a point [tex]\((x, y)\)[/tex] across the y-axis, the x-coordinate changes sign. So, [tex]\((x, y) \rightarrow (-x, y)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the y-axis: [tex]\((2, -3) \rightarrow (-2, -3)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex], we swap the x and y coordinates. So, [tex]\((x, y) \rightarrow (y, x)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the line [tex]\(y = x\)[/tex]: [tex]\((2, -3) \rightarrow (-3, 2)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex], we swap the x and y coordinates and then change the signs of both coordinates. So, [tex]\((x, y) \rightarrow (-y, -x)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the line [tex]\(y = -x\)[/tex]: [tex]\((2, -3) \rightarrow (3, -2)\)[/tex].
Therefore, the coordinates of the reflected vertex (2, -3) for each reflection are:
- Reflection across the x-axis: [tex]\((2, 3)\)[/tex]
- Reflection across the y-axis: [tex]\((-2, -3)\)[/tex]
- Reflection across the line [tex]\(y = x\)[/tex]: [tex]\((-3, 2)\)[/tex]
- Reflection across the line [tex]\(y = -x\)[/tex]: [tex]\((3, -2)\)[/tex]
So, an image of the vertex at [tex]\((2, -3)\)[/tex] will be produced at [tex]\((2, 3)\)[/tex], [tex]\((-2, -3)\)[/tex], [tex]\((-3, 2)\)[/tex], or [tex]\((3, -2)\)[/tex] depending on which reflection is applied:
- Reflection across the x-axis produces [tex]\((2, 3)\)[/tex].
- Reflection across the y-axis produces [tex]\((-2, -3)\)[/tex].
- Reflection across the line [tex]\(y = x\)[/tex] produces [tex]\((-3, 2)\)[/tex].
- Reflection across the line [tex]\(y = -x\)[/tex] produces [tex]\((3, -2)\)[/tex].
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