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How can one test the symmetry of x axis, y axis and origin?

MIT 18.06 Linear Algebra, Spring 2010
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May I assume that you mean test some function for symmetry across the x and y axes and the origin ? :-) Generally it means applying some transformation and seeing if there is no change ie. swap x for -x and see if f(x) = f(-x) so with \[f(x) = x^{2}\]then \[f(-x)=(-x)^{2} = (-1)^{2}x^{2}= x^{2}=f(x)\]if we are plotting this function in two dimensions as \[y =x^{2}\] |dw:1359934489024:dw|then this implies symmetry of reflection across the y-axis ( pardon my drawing skills ). Symmetry across the x-axis is likewise. As regards the origin : then that is a reflection across the y-axis and then another reflection across the x-axis. Thus a circle |dw:1359935010242:dw|is symmetric across x and y axes and hence the origin too. You can deduce this from the equation for a circle, say : \[x^{2}+ y^{2} = 1\]is invariant if we swap x for -x and y for -y \[(-x)^{2} + (-y)^{2} = (-1)^{2}x^{2} + (-1)^2y^{2} = x^{2} + y^{2}\]To be complete I ought mention that reflections across the x then y axes ( or vice versa ) is equivalent to a rotation of 180 degrees around the origin ....
Err, the circle was in that second graphic earlier today.

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