Final answer:
The order of reflections across the x-axis and y-axis does not affect the final image, which will be the same regardless of whether the reflection is performed first on the x-axis or y-axis, due to the commutative property of reflections in geometry.
Step-by-step explanation:
The question from the student is asking about the commutative property of reflections in geometry. Specifically, it inquires whether the order of reflections across two perpendicular axes—namely, the x-axis and y-axis—affects the final position and orientation of an image. To address this, we must recognize that reflections over the x-axis and y-axis are isometric transformations, which preserve the size and shape of geometric figures while possibly changing the figure's position and orientation.
When we reflect an object first across the y-axis and then the x-axis or vice versa, the final image is found to be in the same position regardless of the order of reflections. This is because the combined transformations are commutative; that is, the resultant effect on the shape does not depend on the order of their application. We can demonstrate this through a step-by-step approach:
- Reflection over the y-axis: a point (x, y) would be moved to (-x, y).
- Reflection over the x-axis: a point (-x, y) would be moved to (-x, -y).
If we change the order of reflections:
- Reflection over the x-axis: a point (x, y) would be moved to (x, -y).
- Reflection over the y-axis: a point (x, -y) would be moved to (-x, -y).
In both cases, the coordinates of the final image are (-x, -y), showing that the final image is the same regardless of the order of reflections. Therefore, the order in this context does not matter, and the resulting image will be the same.