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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.

Symmetry with Respect to the x-axis

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.

Symmetry with Respect to the y-axis

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.

Axis of Symmetry

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 180o 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.

Symmetry with Respect to the Origin

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