Symmetries of a Graph
Symmetry with respect to a line

A graph is symmetric with respect to a line if reflecting the graph over that line leaves the graph unchanged. This line is called an axis of symmetry of the graph.

xaxis symmetry


A graph is symmetric with respect to the xaxis if whenever a point is on the graph the point is also on the graph.
The following graph is symmetric with respect to the axis. The mirror image of the blue part of the graph in the axis is just the red part, and vice versa.
This graph is that of the curve . If you replace with , the result is , which mathematically shows that this graph is symmetric about the xaxis.


axis Symmetry


A graph is symmetric with respect to the yaxis if whenever a point is on the graph the point is also on the graph.
This graph is symmetric with respect to the axis. The mirror image of the blue part of the graph in the yaxis is just the red part, and vice versa.
This graph is that of the curve . If you replace with the result is , which mathematically shows that this graph is symmetric about the yaxis.




Symmetry with respect to a point

A graph is symmetric with respect to a point if rotating the graph about that point leaves the graph unchanged.

Symmetry About the Origin


A graph is symmetric with respect to the origin if whenever a point is on the graph the point is also on the graph.
This graph is symmetric with respect to the origin.
This is the graph of the curve . If you replace with and with the result is , which on multiplication of both sides by gives , the original equation. This mathematically shows that this graph is symmetric with respect to the origin.





Visualizing symmetry


Explore the concepts of symmetry with respect to the xaxis, yaxis, and origin using this interactive demonstration. Select a type of symmetry by using the radio buttons below the graph. By clicking and dragging in the graph window below you can create a graph which exhibits that symmetry. Clear the graph by clicking on it.


Download Help Document
Was this information helpful?