We are asked to explain how we can determine the of a figure after a over either axis. Reflecting a figure is the same as reflecting every of the figure, so it is enough to consider how to determine the coordinates of a single point after a reflection. Let's consider the reflection over each axis separately.
Reflection Over the x-axis
Let's consider an example point, A.
The A' of a reflection of A over the x-axis is a point that lies the same distance from the x-axis as A. Note that the distance of a point (x, y) from the x-axis is given by its y-coordinate.
The image of the reflection should be the same distance away from the x-axis, but on the other side of it. Therefore, the reflection of a point with coordinates (x, y) has coordinates (x, - y). In our case, the image A' has coordinates (2, - 1).
To find the coordinates of the image of a point after a reflection over the x-axis, we change the of the y-coordinate. This is the same as multiplying the y-coordinate by - 1.
Reflection Over the y-axis
Let's consider the same example point A.
This time, the image A' of a reflection of A over the y-axis is a point that lies the same distance from the y-axis as A. The distance of a point (x, y) from the y-axis is given by its x-coordinate.
The image of the reflection should be the same distance away from the y-axis, but on the other side of it. Therefore, the reflection of a point with coordinates (x, y) has coordinates (- x, y). In our case, the image A' has coordinates (- 2, 1).
To find the coordinates of the image of a point after a reflection over the y-axis, we change the sign of the x-coordinate. This is the same as multiplying the x-coordinate by - 1.
Conclusion
To find the coordinates of a figure after a reflection over the x-axis, we multiply the y-coordinate by - 1. Similarly, to find the coordinates of a figure after a reflection over the y-axis, we multiply the x-coordinate by - 1.
