Math A Reflection in a Line

For an intuitive lesson on reflections, see Intuitive Notion of Line Reflection.  Now, let's examine some reflections related to coordinate geometry.

When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite. 

The reflection of the point (x,y) across the x-axis is the point (x,-y).

Hint:  If you forget this "rule", simply fold your graph paper along the x-axis to see where your new figure will be located.  You can also measure how far your points are away from the axis as indicated in the picture above.

When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite. 

Hint:  If you forget this "rule", simply fold your graph paper along the y-axis to see where your new figure will be located.  You can also measure how far your points are away from the axis as indicated in the picture above.

When you reflect a point across the line y = x, the x-coordinate and the y-coordinate change places.  When you reflect a point across the line y = -x, the x-coordinate and the y-coordinate change places and are negated (the signs are changed). 

The reflection of the point (x,y) across the line y = x is the point (y,x). The reflection of the point (x,y) across the line y = -x is the point (-y,-x).

Each point of a reflected image is the same distance from the line of reflection as the corresponding point of the original figure.  In other words, the line of reflection lies directly in the middle between the figure and its image. Keep this idea in mind when working with lines of reflections that are neither the x-axis nor the y-axis.