A Point Has The Coordinates { (m, 0)$}$ And { M \neq 0$}$. Which Reflection Of The Point Will Produce An Image Located At { (0, -m)$}$?A. A Reflection Of The Point Across The { X$}$-axis B. A Reflection Of The

Mar 8, 2025 by ADMIN 211 views

**Reflection of a Point Across the x-axis and y-axis**

Understanding Reflections in Mathematics

In mathematics, a reflection is a transformation that flips a point or a shape over a line or a plane. This concept is crucial in geometry and is used to solve various problems in mathematics and real-life situations. In this article, we will discuss the reflection of a point across the x-axis and y-axis, and how it can be used to find the image of a point.

Reflection Across the x-axis

A reflection across the x-axis is a transformation that flips a point over the x-axis. This means that the x-coordinate of the point remains the same, but the y-coordinate changes its sign. In other words, if a point has coordinates (x, y), its reflection across the x-axis will have coordinates (x, -y).

Example 1: Reflection of a Point Across the x-axis

Let's consider a point with coordinates (3, 4). To find its reflection across the x-axis, we need to change the sign of the y-coordinate. Therefore, the reflection of the point (3, 4) across the x-axis is (3, -4).

Example 2: Reflection of a Point Across the x-axis

Now, let's consider a point with coordinates (-2, 5). To find its reflection across the x-axis, we need to change the sign of the y-coordinate. Therefore, the reflection of the point (-2, 5) across the x-axis is (-2, -5).

Reflection Across the y-axis

A reflection across the y-axis is a transformation that flips a point over the y-axis. This means that the y-coordinate of the point remains the same, but the x-coordinate changes its sign. In other words, if a point has coordinates (x, y), its reflection across the y-axis will have coordinates (-x, y).

Example 1: Reflection of a Point Across the y-axis

Let's consider a point with coordinates (4, 3). To find its reflection across the y-axis, we need to change the sign of the x-coordinate. Therefore, the reflection of the point (4, 3) across the y-axis is (-4, 3).

Example 2: Reflection of a Point Across the y-axis

Now, let's consider a point with coordinates (-5, 2). To find its reflection across the y-axis, we need to change the sign of the x-coordinate. Therefore, the reflection of the point (-5, 2) across the y-axis is (5, 2).

Reflection of a Point Across the x-axis and y-axis

Now, let's consider a point with coordinates (m, 0) and m ≠ 0. We want to find the reflection of this point across the x-axis and y-axis.

Reflection Across the x-axis

To find the reflection of the point (m, 0) across the x-axis, we need to change the sign of the y-coordinate. Therefore, the reflection of the point (m, 0) across the x-axis is (m, 0).

Reflection Across the y-axis

To find the reflection of the point (m, 0) across the y-axis, we need to change the sign of the x-coordinate. Therefore, the reflection of the point (m, 0) across the y-axis is (-m, 0).

Which Reflection Produces an Image Located at (0, -m)?

Now, let's consider the point (m, 0) and its reflections across the x-axis and y-axis. We want to find the reflection that produces an image located at (0, -m).

Reflection Across the x-axis

The reflection of the point (m, 0) across the x-axis is (m, 0). This is not the image we are looking for, since the y-coordinate is 0, not -m.

Reflection Across the y-axis

The reflection of the point (m, 0) across the y-axis is (-m, 0). This is not the image we are looking for, since the y-coordinate is 0, not -m.

Reflection Across the x-axis and y-axis

Since the point (m, 0) is located on the x-axis, we need to consider a reflection that involves both the x-axis and y-axis. This is a rotation of 90 degrees counterclockwise, followed by a reflection across the x-axis.

Solution

To find the reflection of the point (m, 0) that produces an image located at (0, -m), we need to perform a rotation of 90 degrees counterclockwise, followed by a reflection across the x-axis.

Conclusion

In this article, we discussed the reflection of a point across the x-axis and y-axis. We also considered a point with coordinates (m, 0) and m ≠ 0, and found the reflection that produces an image located at (0, -m). The reflection of the point (m, 0) that produces an image located at (0, -m) is a rotation of 90 degrees counterclockwise, followed by a reflection across the x-axis.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "Geometry: A High School Course" by Harold R. Jacobs

Glossary

Reflection: A transformation that flips a point or a shape over a line or a plane.
x-axis: A horizontal line that passes through the origin (0, 0).
y-axis: A vertical line that passes through the origin (0, 0).
Rotation: A transformation that turns a point or a shape around a fixed point.
Reflection across the x-axis: A transformation that flips a point over the x-axis.
Reflection across the y-axis: A transformation that flips a point over the y-axis.

Understanding Reflections in Mathematics

Q&A: Reflection of a Point Across the x-axis and y-axis

Q: What is a reflection in mathematics?

A: A reflection is a transformation that flips a point or a shape over a line or a plane.

Q: What is the difference between a reflection across the x-axis and a reflection across the y-axis?

A: A reflection across the x-axis flips a point over the x-axis, while a reflection across the y-axis flips a point over the y-axis.

Q: How do you find the reflection of a point across the x-axis?

A: To find the reflection of a point across the x-axis, you need to change the sign of the y-coordinate.

Q: How do you find the reflection of a point across the y-axis?

A: To find the reflection of a point across the y-axis, you need to change the sign of the x-coordinate.

Q: What is the reflection of a point (m, 0) across the x-axis and y-axis?

A: The reflection of a point (m, 0) across the x-axis is (m, 0), and the reflection of a point (m, 0) across the y-axis is (-m, 0).

Q: Which reflection produces an image located at (0, -m)?

A: The reflection of a point (m, 0) that produces an image located at (0, -m) is a rotation of 90 degrees counterclockwise, followed by a reflection across the x-axis.

Q: What is the difference between a rotation and a reflection?

A: A rotation is a transformation that turns a point or a shape around a fixed point, while a reflection is a transformation that flips a point or a shape over a line or a plane.

Q: How do you perform a rotation of 90 degrees counterclockwise?

A: To perform a rotation of 90 degrees counterclockwise, you need to multiply the coordinates of the point by the rotation matrix:

x' = -y y' = x

Q: How do you perform a reflection across the x-axis?

A: To perform a reflection across the x-axis, you need to change the sign of the y-coordinate:

x' = x y' = -y

Q: How do you perform a reflection across the y-axis?

A: To perform a reflection across the y-axis, you need to change the sign of the x-coordinate:

x' = -x y' = y

Conclusion

In this article, we discussed the reflection of a point across the x-axis and y-axis, and how it can be used to find the image of a point. We also answered some common questions about reflections in mathematics.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "Geometry: A High School Course" by Harold R. Jacobs

Glossary

Reflection: A transformation that flips a point or a shape over a line or a plane.
x-axis: A horizontal line that passes through the origin (0, 0).
y-axis: A vertical line that passes through the origin (0, 0).
Rotation: A transformation that turns a point or a shape around a fixed point.
Reflection across the x-axis: A transformation that flips a point over the x-axis.
Reflection across the y-axis: A transformation that flips a point over the y-axis.