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