Symmetries of a Graph - Maple 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 $\left(x,y\right)$ is on the graph the point $\left(x,-y\right)$ is also on the graph.   The following graph is symmetric with respect to the $\mathrm{x}$-axis. The mirror image of the blue part of the graph in the $\mathrm{x}$-axis is just the red part, and vice versa.     This graph is that of the curve . If you replace $y$ with $-y$, the result is , 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 $\left(x,y\right)$ is on the graph the point $\left(-x,y\right)$ 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 . If you replace $x$ with $-x$ the result is , 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 $\left(x,y\right)$ is on the graph the point $\left(-x,-y\right)$ is also on the graph.   This graph is symmetric with respect to the origin.     This is the graph of the curve . If you replace $x$ with$-x$ and $y$ with $-y$ the result is , which on multiplication of both sides by $-1$ gives , the original equation. This mathematically shows that this graph is symmetric with respect to the origin.

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