Discover new information and get your questions answered with IDNLearn.com. Ask anything and receive well-informed answers from our community of experienced professionals.
Sagot :
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].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.
