Question

Test the equation for symmetry.

$x=y^4-y^2$

Short answer

The equation is symmetry about thex-axis.

Step-by-step solution

Step 1. Concept of symmetry. 

Symmetry with respect to the x-axis can be identified by replacing $y\mathrm{by}-y$. The resulting equation is equivalent to the original one.

Symmetry with respect to the y-axis can be identified by replacing $x\mathrm{by}-x$. The resulting equation is equivalent to the original one.

Symmetry with respect to the origin can be identified by replacing $x\mathrm{by}-x$and role="math" localid="1643720550901" $y\mathrm{by}-y$. The resulting equation is equivalent to the original one.

Step 2. Given equation. 

$x=y^4-y^2$

Step 3. Determining the symmetry. 

If y is replaced by$-y$in the equation$x=y^4-y^2$

role="math" localid="1643720713674" $$\begin{gathered} x={\left(-y\right)}^4-{\left(-y\right)}^2(\mathrm{Replacey}\mathrm{by}-y) \\ x=y^4-y^2\left(\mathrm{Simplify}\right) \end{gathered}$$

And so the equation is equivalent to the original one.

The graph is symmetry about x-axis.