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Symmetry of a Function: Testing For

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symmetry of a function

The polynomial function on the left is symmetrical to the y-axis; The function on the right is symmetric to the origin. The blue dashed line is the axis of symmetry.


A useful fact about polynomial functions is that they are symmetric with respect to the y-axis when every term is either a constant or has an even exponent. For other functions, you could just graph them to test for symmetry. However, it may not be easy to see symmetry on a graph. For example, the following graph is symmetric around the origin, but it’s challenging to see that because of the wild oscillations:
testing for symmetry

How to Test for Symmetry of a Function

A better way is to test for symmetry of a function using a little algebra. All you have to do is work your way down the list of three possibilities:

  • Replace x by -x. If you get the same function, then that function is symmetric over the y-axis.
  • Replace y by -y. If you get the same function, then that function is symmetric over the x-axis.
  • Replace x by -x and y by -y. If you get the same function, then that function is symmetric with respect to the origin.

Example question: Is y = 2x3 – x symmetric?

Solution:

  • Replace x by -x and then simplify:
    • y = 2x3 – x →
    • y = 2(-x)3 – (-x) →
    • y = 2x3 + x

    This gives a different function, so y = 2x3 – x is not symmetric to the y-axis.

  • Replace y by -y.
    • y = 2x3 – x →
    • -y = 2x3 – x

    This also gives a different function, so y = 2x3 – x is not symmetric to the x-axis either.

  • Replace x by -x and y by -y.
    • y = 2x3 – x →
    • Replacing x and y: -y = 2(-x)3 -(- x) →
    • Simplifying: -y = -2x3 + x→
    • Multiply by -1: y = 2x3

    This results in an equivalent function, so y = 2x3 – x is symmetric around the origin.

References

Larson, R. & Edwards, B. (2018). Calculus: Early Transcendental Functions. Cengage Learning.
Graphs drawn with Desmos.

CITE THIS AS:
Stephanie Glen. "Symmetry of a Function: Testing For" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/symmetry-of-a-function/
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