How to Determine the Symmetry of a Rational Function

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The word "symmetry" refers to one half of something being a mirror image of the other half. For example, the capital letter E is symmetric about its horizontal axis and the capital letter A is symmetric about its vertical axis. The latter is called even symmetry. The letter S is not symmetric about either axis. However, it is rotationally symmetric in that it will look the same after rotating it 180 degrees. This is called odd symmetry.

Things You'll Need

  • Graphing calculator
  • Determine whether the function, f(x), exhibits even symmetry. Substitute -x for every occurrence of x in the equation. If the resulting function is equal to the original function, f(x) exhibits even symmetry. If the two functions are not equal, f(x) does not exhibit even symmetry. For example, if the function is f(x) = x^2 + x + 5, substituting -x would yield f(-x) = x^2 - x + 5. These two functions are not equivalent, and so f(x) does not exhibit even symmetry.

  • Determine whether the function, f(x), exhibits odd symmetry. Substitute -x for every occurrence of x in the equation. If the resulting function is equal to the original function multiplied by negative one, f(x) exhibits odd symmetry. Otherwise, f(x) does not exhibit odd symmetry. For example, if the function is f(x) = x^3 + x, substituting -x would yield f(-x) = -x^3 - x. This function is equivalent to -f(x), and so f(x) exhibits odd symmetry.

  • Use a graphing calculator to check your results. Graph f(x) and f(-x) and see if they overlap completely. If so, the function is even. Graph -f(x) and f(-x) and see if they overlap completely. If so, the function is odd.

References

  • "Precalculus"; Michael Sullivan, et al.; 1998
  • Photo Credit Hemera Technologies/AbleStock.com/Getty Images
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