Reflection is a geometric transformation that produces a mirror image of a figure across a line, called the line of reflection. Each point and its image are equidistant from the line of reflection, and the line connecting them is perpendicular to the line of reflection.
When reflecting a point across the y-axis, the transformation rule is (x, y) becomes (-x, y). This means the x-coordinate changes sign while the y-coordinate stays the same. For example, the point (-2, 3) becomes (2, 3).
When reflecting a point across the x-axis, the transformation rule is (x, y) becomes (x, -y). This means the y-coordinate changes sign while the x-coordinate stays the same. For example, the point (-2, 3) becomes (-2, -3).
When reflecting a point across the line y equals x, the transformation rule is (x, y) becomes (y, x). This means the x and y coordinates swap positions. For example, the point (-2, 3) becomes (3, -2). The line connecting the point and its image is perpendicular to the line of reflection.
When reflecting a point across the line y equals negative x, the transformation rule is (x, y) becomes (-y, -x). This means both coordinates change sign and swap positions. For example, the point (-2, 3) becomes (-3, 2). The line connecting the point and its image is perpendicular to the line of reflection.
Reflections have several important properties. First, they are isometries, meaning they preserve distances, angles, and shapes. Second, the original figure and its image are equidistant from the line of reflection. Third, the line connecting any point and its image is perpendicular to the line of reflection. Finally, reflections reverse orientation - a clockwise figure becomes counterclockwise in its reflection.
Let's summarize what we've learned about reflections. When reflecting the point (-2, 3): Across the y-axis, it becomes (2, 3). Across the x-axis, it becomes (-2, -3). Across the line y equals x, it becomes (3, -2). And across the line y equals negative x, it becomes (-3, 2). These transformation rules apply to all points in the coordinate plane.