Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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.
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.
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.