A Point Has The Coordinates $(0, K)$. Which Reflection Of The Point Will Produce An Image At The Same Coordinates, $(0, K)$?A. A Reflection Of The Point Across The $x$-axis B. A Reflection Of The Point Across The

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 involving points, lines, and shapes. In this article, we will explore the reflection of a point across the coordinate plane and determine which reflection will produce an image at the same coordinates.

The Coordinate Plane

The coordinate plane is a two-dimensional plane that consists of two axes: the x-axis and the y-axis. The x-axis is horizontal, and the y-axis is vertical. The point of intersection between the two axes is called the origin, denoted as (0, 0). Any point on the coordinate plane can be represented by an ordered pair (x, y), where x is the x-coordinate and y is the y-coordinate.

Reflection Across the X-Axis

A reflection across the x-axis is a transformation that flips a point or a shape over the x-axis. This means that the y-coordinate of the point is negated, while the x-coordinate remains the same. For example, if a point has coordinates (0, k), its reflection across the x-axis will have coordinates (0, -k).

Reflection Across the Y-Axis

A reflection across the y-axis is a transformation that flips a point or a shape over the y-axis. This means that the x-coordinate of the point is negated, while the y-coordinate remains the same. For example, if a point has coordinates (0, k), its reflection across the y-axis will have coordinates (-0, k), which simplifies to (0, k).

Reflection Across the Origin

A reflection across the origin is a transformation that flips a point or a shape over the origin. This means that both the x-coordinate and the y-coordinate of the point are negated. For example, if a point has coordinates (0, k), its reflection across the origin will have coordinates (0, -k).

Which Reflection Produces an Image at the Same Coordinates?

To determine which reflection produces an image at the same coordinates, we need to analyze the reflections across the x-axis, y-axis, and origin. As we have seen, a reflection across the x-axis will produce an image with coordinates (0, -k), while a reflection across the y-axis will produce an image with coordinates (0, k). A reflection across the origin will also produce an image with coordinates (0, -k).

Conclusion

Based on our analysis, we can conclude that a reflection of the point across the y-axis will produce an image at the same coordinates, (0, k). This is because the y-coordinate remains the same, while the x-coordinate is negated, resulting in the same coordinates.

Final Answer

The final answer is: B. A reflection of the point across the 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 involving points, lines, and shapes. In this article, we will explore the reflection of a point across the coordinate plane and answer some frequently asked questions.

Q&A: Reflection of a Point Across the Coordinate Plane

Q: What is a reflection in mathematics?

A: A reflection in mathematics is a transformation that flips a point or a shape over a line or a plane. This means that the point or shape is mirrored over the line or plane, resulting in a new image.

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 negates the y-coordinate of a point, while a reflection across the y-axis negates the x-coordinate of a point. This means that the x-coordinate remains the same in a reflection across the y-axis, while the y-coordinate remains the same in a reflection across the x-axis.

Q: What is a reflection across the origin?

A: A reflection across the origin is a transformation that flips a point or a shape over the origin. This means that both the x-coordinate and the y-coordinate of the point are negated.

Q: Which reflection produces an image at the same coordinates?

A: A reflection of the point across the y-axis will produce an image at the same coordinates, (0, k). This is because the y-coordinate remains the same, while the x-coordinate is negated, resulting in the same coordinates.

Q: What is the importance of reflections in mathematics?

A: Reflections are an essential concept in mathematics, particularly in geometry. They are used to solve various problems involving points, lines, and shapes. Reflections are also used in real-world applications, such as architecture, engineering, and computer graphics.

Q: How do I determine the type of reflection?

A: To determine the type of reflection, you need to analyze the coordinates of the point. If the x-coordinate is negated, it is a reflection across the y-axis. If the y-coordinate is negated, it is a reflection across the x-axis. If both coordinates are negated, it is a reflection across the origin.

Q: Can a point have multiple reflections?

A: Yes, a point can have multiple reflections. For example, a point can be reflected across the x-axis and then reflected across the y-axis, resulting in a new image.

Conclusion

In conclusion, reflections are an essential concept in mathematics, particularly in geometry. Understanding reflections is crucial in solving various problems involving points, lines, and shapes. By analyzing the coordinates of a point, you can determine the type of reflection and solve problems involving reflections.

Final Answer

The final answer is: B. A reflection of the point across the y-axis.