Symmetry (Meta) Math Example 1

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Example 1

easy
Show that the equation x2+y2=25x^2 + y^2 = 25 is symmetric about both coordinate axes and the origin. Verify by substituting (x,y)=(3,4)(x,y) = (3,4) and its reflections.

Solution

  1. 1
    Check symmetry about the yy-axis: replace xx with −x-x: (−x)2+y2=x2+y2=25(-x)^2+y^2 = x^2+y^2=25. Unchanged — symmetric about yy-axis.
  2. 2
    Check symmetry about the xx-axis: replace yy with −y-y: x2+(−y)2=x2+y2=25x^2+(-y)^2=x^2+y^2=25. Unchanged — symmetric about xx-axis.
  3. 3
    Check symmetry about the origin: replace (x,y)(x,y) with (−x,−y)(-x,-y): (−x)2+(−y)2=25(-x)^2+(-y)^2=25. Unchanged.
  4. 4
    Verify: (3,4)(3,4): 9+16=259+16=25. (−3,4)(−3,4), (3,−4)(3,−4), (−3,−4)(−3,−4) all also satisfy the equation.

Answer

x2+y2=25 is symmetric about both axes and the originx^2+y^2=25 \text{ is symmetric about both axes and the origin}
An equation has a symmetry if replacing variables by their reflections leaves the equation unchanged. Squaring terms always produces this symmetry because (−x)2=x2(-x)^2=x^2.

About Symmetry (Meta)

A property of a mathematical object that remains unchanged under a specified transformation — reflection, rotation, translation, or algebraic substitution.

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More Symmetry (Meta) Examples

Example 2 medium

Use symmetry to evaluate [formula] for even [formula].

Example 3 easy

Determine whether [formula] is odd, even, or neither, by testing the symmetry condition.

Example 4 medium

Use the symmetry of [formula] (odd function) and [formula] (even function) to simplify: [formula].

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