Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To determine which reflection will produce the image of the point [tex]\((0, k)\)[/tex] at the same coordinates, let's analyze the effect of each reflection option on the point:
1. Reflection across the [tex]\( x \)[/tex]-axis:
The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\( x \)[/tex]-axis changes the sign of the [tex]\( y \)[/tex]-coordinate. Thus, the reflection of [tex]\((0, k)\)[/tex] across the [tex]\( x \)[/tex]-axis is:
[tex]\[ (0, k) \rightarrow (0, -k) \][/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 the sign of the [tex]\( x \)[/tex]-coordinate. Thus, the reflection of [tex]\((0, k)\)[/tex] across the [tex]\( y \)[/tex]-axis is:
[tex]\[ (0, k) \rightarrow (0, k) \][/tex]
3. Reflection across the line [tex]\( y = x \)[/tex]:
The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = x \)[/tex] swaps the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates. Thus, the reflection of [tex]\((0, k)\)[/tex] across the line [tex]\( y = x \)[/tex] is:
[tex]\[ (0, k) \rightarrow (k, 0) \][/tex]
4. Reflection across the line [tex]\( y = -x \)[/tex]:
The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = -x \)[/tex] swaps the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates and changes their signs. Thus, the reflection of [tex]\((0, k)\)[/tex] across the line [tex]\( y = -x \)[/tex] is:
[tex]\[ (0, k) \rightarrow (-k, 0) \][/tex]
By analyzing each reflection, we see that only the reflection across the [tex]\( y \)[/tex]-axis leaves the point [tex]\((0, k)\)[/tex] unchanged at its original coordinates. Therefore, the correct transformation is the reflection of the point across the [tex]\( y \)[/tex]-axis.
The answer is:
[tex]\[ \boxed{2} \][/tex]
1. Reflection across the [tex]\( x \)[/tex]-axis:
The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\( x \)[/tex]-axis changes the sign of the [tex]\( y \)[/tex]-coordinate. Thus, the reflection of [tex]\((0, k)\)[/tex] across the [tex]\( x \)[/tex]-axis is:
[tex]\[ (0, k) \rightarrow (0, -k) \][/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 the sign of the [tex]\( x \)[/tex]-coordinate. Thus, the reflection of [tex]\((0, k)\)[/tex] across the [tex]\( y \)[/tex]-axis is:
[tex]\[ (0, k) \rightarrow (0, k) \][/tex]
3. Reflection across the line [tex]\( y = x \)[/tex]:
The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = x \)[/tex] swaps the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates. Thus, the reflection of [tex]\((0, k)\)[/tex] across the line [tex]\( y = x \)[/tex] is:
[tex]\[ (0, k) \rightarrow (k, 0) \][/tex]
4. Reflection across the line [tex]\( y = -x \)[/tex]:
The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = -x \)[/tex] swaps the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates and changes their signs. Thus, the reflection of [tex]\((0, k)\)[/tex] across the line [tex]\( y = -x \)[/tex] is:
[tex]\[ (0, k) \rightarrow (-k, 0) \][/tex]
By analyzing each reflection, we see that only the reflection across the [tex]\( y \)[/tex]-axis leaves the point [tex]\((0, k)\)[/tex] unchanged at its original coordinates. Therefore, the correct transformation is the reflection of the point across the [tex]\( y \)[/tex]-axis.
The answer is:
[tex]\[ \boxed{2} \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.