Swap the x and y coordinates

Oct 13, 2025 by windailys.com 57 views

Reflection Across the Line y = -x: Understanding the Concept

Reflection across a line is a fundamental concept in geometry, where a point is mapped to a new location on the opposite side of the line, while maintaining the same distance from the line. In this article, we will explore the concept of reflection across the line y = -x and determine which point would map onto itself after such a reflection.

What is Reflection Across a Line?

Reflection across a line is a transformation that involves flipping a point or a shape over a given line. This line is called the axis of reflection. When a point is reflected across a line, its image is formed on the opposite side of the line, at the same distance from the line. The line of reflection acts as a mirror, and the point is reflected over it to form its image.

Reflection Across the Line y = -x

The line y = -x is a diagonal line that passes through the origin (0, 0) and has a slope of -1. To reflect a point across this line, we need to find the image of the point on the opposite side of the line, at the same distance from the line.

How to Reflect a Point Across the Line y = -x

To reflect a point (x, y) across the line y = -x, we need to follow these steps:

Swap the x and y coordinates: The new x-coordinate will be the negative of the original y-coordinate, and the new y-coordinate will be the negative of the original x-coordinate.
Change the sign of the new coordinates: The new coordinates will have the opposite sign of the original coordinates.

Example: Reflecting a Point Across the Line y = -x

Let's consider the point (2, 3). To reflect this point across the line y = -x, we need to follow the steps above:

Swap the x and y coordinates: (3, 2)
Change the sign of the new coordinates: (-3, -2)

Therefore, the image of the point (2, 3) after reflecting across the line y = -x is (-3, -2).

Which Point Would Map Onto Itself After a Reflection Across the Line y = -x?

Now that we understand the concept of reflection across the line y = -x, let's consider the options given:

A. ${(-4, -4)}$ B. ${(-4, 0)}$ C. ${(0, -4)}$ D. ${(4, -4)}$

To determine which point would map onto itself after a reflection across the line y = -x, we need to reflect each point across the line and see if it maps onto itself.

Reflecting Each Point Across the Line y = -x

Let's reflect each point across the line y = -x:

A. ${(-4, -4)}$ Swap the x and y coordinates: ${(-4, -4)}$ Change the sign of the new coordinates: ${(-4, -4)}$ The point (-4, -4) maps onto itself after a reflection across the line y = -x.

B. ${(-4, 0)}$ Swap the x and y coordinates: ${(0, -4)}$ Change the sign of the new coordinates: ${(0, 4)}$ The point (-4, 0) does not map onto itself after a reflection across the line y = -x.

C. ${(0, -4)}$ Swap the x and y coordinates: ${(-4, 0)}$ Change the sign of the new coordinates: ${(4, 0)}$ The point (0, -4) does not map onto itself after a reflection across the line y = -x.

D. ${(4, -4)}$ Swap the x and y coordinates: ${(-4, 4)}$ Change the sign of the new coordinates: ${(-4, -4)}$ The point (4, -4) maps onto itself after a reflection across the line y = -x.

Conclusion

In conclusion, the point that would map onto itself after a reflection across the line y = -x is:

A. ${(-4, -4)}$
D. ${(4, -4)}$

Both points (-4, -4) and (4, -4) map onto themselves after a reflection across the line y = -x.
Reflection Across the Line y = -x: Q&A

In our previous article, we explored the concept of reflection across the line y = -x and determined which points would map onto themselves after such a reflection. In this article, we will answer some frequently asked questions about reflection across the line y = -x.

Q: What is the axis of reflection?

A: The axis of reflection is the line across which a point or shape is reflected. In the case of the line y = -x, the axis of reflection is the line itself.

Q: How do I reflect a point across the line y = -x?

A: To reflect a point (x, y) across the line y = -x, you need to follow these steps:

Swap the x and y coordinates: The new x-coordinate will be the negative of the original y-coordinate, and the new y-coordinate will be the negative of the original x-coordinate.
Change the sign of the new coordinates: The new coordinates will have the opposite sign of the original coordinates.

Q: What is the difference between reflecting a point across the line y = -x and reflecting a point across the x-axis or y-axis?

A: When reflecting a point across the x-axis or y-axis, the x-coordinate or y-coordinate remains the same, respectively. However, when reflecting a point across the line y = -x, both the x and y coordinates are swapped and changed in sign.

Q: Can a point map onto itself after a reflection across the line y = -x?

A: Yes, a point can map onto itself after a reflection across the line y = -x. This occurs when the point is on the line y = -x, or when the point is symmetric with respect to the line y = -x.

Q: How do I determine if a point is symmetric with respect to the line y = -x?

A: To determine if a point is symmetric with respect to the line y = -x, you need to check if the point is equidistant from the line y = -x and if the point is on the opposite side of the line y = -x.

Q: Can a point be reflected across the line y = -x multiple times?

A: Yes, a point can be reflected across the line y = -x multiple times. Each reflection will result in a new point that is the image of the original point under the reflection.

Q: What is the effect of reflecting a point across the line y = -x multiple times?

A: The effect of reflecting a point across the line y = -x multiple times is to rotate the point by 90 degrees counterclockwise. Each reflection will result in a rotation of 90 degrees counterclockwise.

Q: Can a shape be reflected across the line y = -x?

A: Yes, a shape can be reflected across the line y = -x. The reflection of a shape across the line y = -x is a transformation that involves flipping the shape over the line y = -x.

Q: How do I reflect a shape across the line y = -x?

A: To reflect a shape across the line y = -x, you need to follow these steps:

Identify the vertices of the shape.
Reflect each vertex across the line y = -x.
Connect the reflected vertices to form the reflected shape.

Conclusion

In conclusion, reflection across the line y = -x is a fundamental concept in geometry that involves flipping a point or shape over the line y = -x. We have answered some frequently asked questions about reflection across the line y = -x and provided examples and explanations to help you understand the concept.