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Special Graphs

Symmetry

Terms

Problems

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 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.

Symmetry with Respect to the Origin

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

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