What is the rule for a reflection across the y axis?
The correct answer and explanation is:
A reflection across the y-axis changes the x-coordinates of the points in a given figure while leaving the y-coordinates unchanged. The rule for reflecting a point across the y-axis is: (x,y)→(−x,y)(x, y) \rightarrow (-x, y)
Explanation:
In a reflection across the y-axis, every point on the figure moves horizontally to the opposite side of the y-axis, while its vertical position remains the same. This transformation is a type of rigid motion, meaning that the shape and size of the figure do not change; only its position does.
Consider a point with coordinates (x,y)(x, y). When reflected across the y-axis, the x-coordinate becomes its opposite (negative of the original), while the y-coordinate stays the same. Therefore, the new coordinates after the reflection are (−x,y)(-x, y).
Example:
- Take the point (3,4)(3, 4). After reflecting across the y-axis, the new coordinates will be (−3,4)(-3, 4). Notice that the point moves from the right of the y-axis to the left, but its vertical position (4) stays the same.
- Similarly, for the point (−2,5)(-2, 5), after reflection, the coordinates become (2,5)(2, 5). The point moves from the left side of the y-axis to the right, but the vertical position (5) remains unchanged.
General Behavior of Reflections:
Reflections across the y-axis result in a mirror image of the original figure. If the original figure lies to the right of the y-axis, its reflection will appear to the left, and vice versa. Importantly, all distances between points on the figure are preserved, meaning the shape stays exactly the same, just flipped horizontally.
This rule can be applied to geometric shapes like triangles, squares, or any polygon, and to functions or graphs. When reflecting a graph across the y-axis, every point on the graph undergoes the same transformation: its x-coordinate changes sign.