I have a point $A (-2k; 3)$ and a point $B$ that is symmetrical to the point A given the centre of symmetry $C (-1; 0)$.
I tried applying the following formula, where $x_o$ and $y_0$ are the coordinates of the centre of symmetry, $C$, and $x$ and $y$ are the coordinates of the point $A$:
$$\begin{cases} x_B = 2x_o - x = 2(-1) - (-2k ) = 2k - 2\\ y_B = 2y_0 - y\ = 2(0) - 3 = -3 \end{cases} $$
I find $B(2k-2; -3$) as coordinates of the point $B$. Though, putting that into a graph, it doesn't seem that the point $C$ is the centre of symmetry of $A$ and $B$.
I suppose I didn't calculate the coordinates of the point $B$ correctly. Could you please spot my mistake?
The calculations in OP are correct, but the graph is wrong. For $k=1$ the point $B$ is $(0,-3)$ and not $(0,0)$ as in the graph. If you correct than $C$ is the middle point between $A$ and $B$.