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Geometrically, a function is called odd if it is symmetric in the origin. The picture below is an example of an odd function because every point \$$A\$$ on the curve can be reflected through the origin to produce another point on the curve \$$B\$$.

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Here the line from \$$A\$$ to the vertical axis is shown as a solid line, and the reflection through the origin to \$$B\$$ is shown as a dashed line.

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{graph(a)}

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Algebraicly, a function has this kind of symmetry whenever

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\$$f(-x) \\equiv -f(x).\$$

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A graphical introduction to the concept of even functions a symmery

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If the \$$x\$$ coordinate of the point \$$A\$$ is \$$\\var{x1}\$$ then what is the \$$x\$$-coordinate of the point \$$B\$$?

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If the \$$y\$$ coordinate of the point \$$A\$$ is \$$\\var{a*x1^3}\$$ then what is the \$$y\$$-coordinate of the point \$$B\$$?

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Suppose that the point \$$A\$$ has coordinates \$$(\\var{x1}, \\var{a*x1^3})\$$. Then by the symmetry of the graph the point \$$B\$$ has coordinates \$$(\\var{-1*x1}, \\var{-a*x1^3})\$$.

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