Figure 1

Traditional textbooks usually discuss the point symmetries of functions only in relation to the origin. For instance, a graph is considered point symmetrical in relation to the origin O when each point P of a graph as shown in Figure 1, has a corresponding point Q (also on the graph) under a reflection through O so that PO = OQ. Or equivalently in the plane, each point P of the graph can be mapped onto a corresponding point Q (also on the graph) by means of a half-turn (a rotation through 180 degrees) around O.1 Using the transformation formula for a half turn, it therefore follows that a graph is point symmetric in relation to the origin if y = f(x) ⇔ y =-f(-x); in other words if it remains invariant under a half-turn around the origin.