Symmetry in Graphs of Functions and Relations

​
equation
y=
2
x
+3x
symmetry check
none
x axis
y axis
origin
list types of symmetry?
types of symmetry: none
This Demonstration shows the three types of symmetry commonly studied in graphs: symmetry with respect to the
x
axis, the
y
axis, or the origin.
A graph has symmetry with respect to the
x
axis if reflecting it across the
x
axis yields an identical graph. The graph of an equation has this symmetry if replacing
y
with
-y
in the equation yields the identical equation.
A graph has symmetry with respect to the
y
axis if reflecting it across the
y
axis yields an identical graph. The graph of an equation has this symmetry if replacing
x
with
-x
in the equation yields the identical equation.
A graph has symmetry with respect to the origin if reflecting the graph across both the
x
axis and
y
axis yields an identical graph. The graph of an equation has this symmetry if replacing
x
with
-x
and
y
with
-y
in the equation yields the identical equation.

Details

The graph of a function that is not identically zero is never symmetric with respect to the
x
axis as it necessarily fails the vertical line test (e.g.
2
y
=5x-
3
x
and
|y|=
3
x
+1
). A function whose graph is symmetric with respect to the
y
axis is called even (e.g.
y=-
8
x
+25
6
x
-30
4
x
+10
2
x
-3
and
y=-2
6
x
+21
4
x
+14
2
x
+1
). A function whose graph is symmetric with respect to the origin is called odd (e.g.
y=
1
x
).
Equations that are symmetric to both the
x
axis and
y
axis will necessarily also be symmetric with respect to the origin (e.g.
2
y
-
2
x
=1
). However, this is not a requirement for a graph to be symmetric with respect to the origin (e.g.
y=
1
x
).

External Links

Even and Odd Functions
Odd Function (Wolfram MathWorld)
Even Function (Wolfram MathWorld)

Permanent Citation

Laura R. Lynch
​
​"Symmetry in Graphs of Functions and Relations"​
​http://demonstrations.wolfram.com/SymmetryInGraphsOfFunctionsAndRelations/​
​Wolfram Demonstrations Project​
​Published: June 5, 2014