In geometry, a reflection is a transformation that creates a mirror image of a point or shape across a line called the line of reflection. Let's start with the point (-3, 2) on a coordinate plane.
When we reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate changes sign. So (-3, 2) becomes (-3, -2). The line y equals zero is the line of reflection.
When we reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate changes sign. So (-3, 2) becomes (3, 2). The line x equals zero is the line of reflection.
When we reflect a point across the origin, both coordinates change sign. So (-3, 2) becomes (3, -2). This transformation is equivalent to rotating the point 180 degrees around the origin.
When we reflect a point across the line y equals x, the coordinates are swapped. So (-3, 2) becomes (2, -3). The line y equals x is the line of reflection.
When we reflect a point across the line y equals negative x, the coordinates are swapped and both change sign. So (-3, 2) becomes (-2, 3).
An important property of reflections is that they preserve distances. The original point and its reflection are always the same distance from the line of reflection. In this example, both (-3, 2) and (-3, -2) are 2 units away from the x-axis.
Let's summarize all the reflections of the point (-3, 2). Across the x-axis we get (-3, -2). Across the y-axis we get (3, 2). Across the origin we get (3, -2). Across the line y equals x we get (2, -3). And across the line y equals negative x we get (-2, 3). All these transformations preserve distances and angles.