Describe the symmetries of the graph $$\{(x,y)\in \mathbb R^2\mid y^3-x \in\Bbb Z\}$$ in the plane.
A graph is symmetric about any axis if we reflect (or taking image) graph about that axis,then we get the same image of graph. Say for $x$-axis symmetry, a point $(x ,y)$ on the graph changes to point $(x, - y)$ on the reflecting graph.
So, need to first plot and then find the axis of symmetry. Algebraically it is difficult as there are two variables.
So, need plot points on $y^3 -x$, say for $x=8, y=2$, have value $0$, etc.
But, is there any shortcut, based on context.
I plotted the graph of $y^3=x+k$ for $k=-3,-2,\dots,2,3$ using desmos graphing calculator. The symmetries of the set will be the symmetries that preserve the family of the curves $y^3=x+k$ where $k$ is an integer.
From the graphs, it can be deduced that the symmetries are the identity, translations to the left or right by $k$ units and rotation $180^\circ$ about the center $(k,0)$ where $k$ is an integer.