How to predict symmerty of parametric curve

Question

Suppose we are given a curve as $$x^{2/3} + y^{2/3} = 1 $$ In parametric form it can be written as $$x=\cos^{3}(\theta)$$ and $$y=\sin^{3}(\theta)$$ now how can we predict if curve will be symmetric about $x$ axis or $y$ axis? In Cartesian curve we just replace $$x=-x$$ and $$y=-y$$ to get an idea about symmetry. Someone told me to replace $$\theta=\pi-\theta$$ but why? How will that get me information on symmetry ?

Answer 1

Try and take a look at the parametric equation in this perspective:

The most $x^{2/3}+y^{2/3}$ will ever be equal to is 1. This may sound trivial but you can glean a bit of information out of this. If the $x^{2/3}$ term is 1, the $y$ term must be zero. Conversely, when the $y^{2/3}$ term is 1, the $x$ term must be zero. We can imagine these points on the Cartesian plane. Continuing to work from intuition alone, when the $x^{2/3}$ term increases from zero, the $y^{2/3}$ must decrease from one. Because $f(a)=a^{2/3}$ is strictly increasing when examined over the positive reals, and because the exponents for the parameters are the same, it may help to imagine $x+y=1$, and note that when you modify the equation to $x^{2/3}+y^{2/3}=1$, the curve is no longer linear - it will pucker towards the origin. (As for some intuition behind that, why is $({1/2},{1/2})$ not part of the solution set of your parametric equation?)

Answer 2

If there is no change in x by changing sign of x it has x axis symmetry.If there is no change in y by changing sign of y it has y axis symmetry. If either, symmetry about both axes. No change in sign .. this is brought about by even functions like cos, square of arguments etc.