| home
| contents
| previous
| next page
| send comment
| send link
| add bookmark |
9.4. Symmetry
A graph has the property of symmetry if there is an exact correspondence between points on the
graph that are on opposite sides of an axis, the origin, an arbitrary dividing line or, in three
dimensions, a plane.
Symmetry about the y-axis
The graph of an equation is symmetric with respect to the y-axis if the equation is unchanged
when x is replaced by -x.
For example: y = 2x2 + 3
y = 2(-x)2 + 3
y = 2x2 + 3
Symmetry about the x-axis
The graph of an equation is symmetric with respect to the x-axis if the equation is unchanged
when y is replaced by -y.
For example: y2 = 2x + 1
(-y)2 = 2x + 1
y2 = 2x + 1
Symmetry about the origin
The graph of an equation is symmetric with respect to the origin if the equation is unchanged
when x is replaced by -x and y replaced by -y.
For example: y = x3 - x
(-y) = (-x)3 - (-x)
-y = -x3 + x
y = x3 - x
Symmetry about the line y = x
The graph of an equation is symmetric with respect to the line y = x if the equation is
unchanged when x is replaced by y and y is replaced by x.
For example: y = 2/x
(x) = 2/(y)
x·y = 2
y = 2/x
Summary
The tests for the symmetry of graphs in two dimensions are summarized in the following table.
The graph is symmetric with
respect to the...
| If the equation is
unchanged when...
|
| y-axis
| x is replaced with -x
|
| x-axis
| y is replaced with -y
|
| origin
| x is replaced with -x and y is replaced with -y
|
| line y = x
| x is replaced with y and y is replaced with x
|
Table 9-1: Tests for symmetry
| home
| contents
| previous
| next page
| send comment
| send link
| add bookmark |
Copyright © 2005, Stephen R. Schmitt