How do lines of symmetry change as a graph scales?

Question

Suppose $y=f(x)$ is symmetric in the line $ax+by+c=0$.

Then what is the corresponding line of symmetry for the following ?

1. $y=df(x)$ and
2. $y=f(dx)$

I guessed that it'd be

1. $ax+by/d+c=0$ and
2. $adx+by+c=0$

But my guess is wrong: $y=1/x$ has lines of symmetry $y=x$ and $y=-x$. So does $y=2/x$. (In particular, contrary to my above guess, $y=2/x$ doesn't have lines of symmetry $y=2x$ and $y=-2x$.)