Special Graphs: Symmetry

If a graph does not change when reflected over a line or rotated around a point, the graph is symmetric with respect to that line or point. The following graph is symmetric with respect to the x-axis (y = 0). Note that if (x, y) is a point on the graph, then (x, - y) is also a point on the graph.

If a function is symmetric with respect to the x-axis, then f (x) = - f (x).

The following graph is symmetric with respect to the y-axis (x = 0). Note that if (x, y) is a point on the graph, then (- x, y) is also a point on the graph.

If a function is symmetric with respect to the y-axis, then f (x) = f (- x).

If a graph can be reflected over a line without altering the graph, then that line is called the axis of symmetry. In the following graph, x = 2 is the axis of symmetry. Note that if (2 + x, y) is a point on the graph, then (2 - x, y) is also a point on the graph.

If a function has an axis of symmetry x = a, then f (x) = f (- x + 2a).

The following graph is symmetric with respect to the origin. In other words, it can be rotated 180^o around the origin without altering the graph. Note that if (x, y) is a point on the graph, then (- x, - y) is also a point on the graph.