Confirming symmetry about a curve/line?

Question

I'm a bit confused, is symmetric about Origin same as symmetric about $y=x$ line?

Yes or no, how can I check for myself? I mean how can I do it on paper, let's take a random line/curve and

Answer 1

Symmetry around a point in the plane usually means rotational symmetry. Symmetry across a line usually means reflectional symmetry. These are very different symmetries.

If you take the graph of $y=\sin(x)$ on $[−π/2,π/2]$ and rotate it around the origin, you get the same graph (the function is odd). But if you reflect it across $y=x$, you get the graph of the arcsine function.

It's no coincidence that the graph of the inverse of a function $f$ is the reflection of the graph of $f$ in the line $y=x$. If you work out the algebra, the reflection of the point $(a,b)$ in $y=x$ is $(b,a)$. That is, this reflection switches the coordinates. When dealing with functions, the inverse to a function switches the roles of $x$ and $y$ as well.

Answer 2

I'm not sure to well understand your question, but I suppose that you intend a reflection symmetry with line $y=x$ and a point reflection (or central symmetry) with center $(0,0)$.

If this is the case the figure that you find can be as this:

where you see that from the same point $C$ we find two different points $D$ ( symmetric with respect the line) and $E$ ( symmetric with respect the origin).