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* + 2*a*)
.

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*)
.