Symmetries of a Graph - Maple Programming Help

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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.

x-axis symmetry

A graph is symmetric with respect to the x-axis if whenever a point x,y is on the graph the point x,y is also on the graph.

 

The following graph is symmetric with respect to the x-axis. The mirror image of the blue part of the graph in the x-axis is just the red part, and vice versa.

 

 

This graph is that of the curve x = y21. If you replace y with y, the result is x = y21 = y21, which mathematically shows that this graph is symmetric about the x-axis.

y-axis Symmetry

A graph is symmetric with respect to the y-axis if whenever a point x,y is on the graph the point x,y is also on the graph.

 

This graph is symmetric with respect to the y-axis. The mirror image of the blue part of the graph in the y-axis is just the red part, and vice versa.

 

 

This graph is that of the curve y=x43 x2+1. If you replace x with x the result is y=x43x2+1=x43 x2+1, which mathematically shows that this graph is symmetric about the y-axis.

 

 

Symmetry with respect to a point

A graph is symmetric with respect to a point if rotating the graph 180° about that point leaves the graph unchanged.

Symmetry About the Origin

A graph is symmetric with respect to the origin if whenever a point x,y is on the graph the point x,y is also on the graph.

 

This graph is symmetric with respect to the origin.

 

 

This is the graph of the curve y=x32 x. If you replace x withx and y with y the result is y=x32 x=x32 x, which on multiplication of both sides by 1 gives y=x32 x, the original equation. This mathematically shows that this graph is symmetric with respect to the origin.

 

 

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