find out the symmetry of

Question

Find out the symmetry of equations.

                                                 y = x² - 6x⁴ + 2

Solution

First we'll check for symmetry around the x-axis. It means that we have to replace all the y's with -y. That's simple enough to do in this case since there is only one y.

- y = x² - 6x⁴ + 2

Now, it is not an equal equation as the terms onto the right are alike to the original equation & the term on the left is the opposite sign.  So, this equation doesn't have symmetry around the x-axis.

Next, let's verify symmetry around the y-axis.  we'll replace all x's with -x here.

                                  y = ( - x )² - 6 ( - x )⁴  + 2

                                   y = x² - 6x⁴ + 2

Later than simplifying we got precisely the similar equation back out that means that the two are equivalent. Thus, this equation does have symmetry around the y-axis.

At last, we have to check for symmetry around the origin.  Here we replace both variables.

- y = ( - x )²  - 6 ( - x )⁴  + 2

- y = x² - 6x⁴ + 2

Thus, as with the first test, the left side is distinct from the original equation & the right side is the same to the original equation. Thus, it isn't equivalent to the original equation & we don't contain symmetry regarding the origin.