Lets assume we have a very simple transformation in 1 Dimension from $(x, p_x)\rightarrow (y,p_y)$ given as $$\begin{aligned} y &= cx \\ p_y &= c^{-1} p_x \end{aligned}$$
Is this a strictly canonical transformation, an extended canonical transformation or a scaling transformation?
Let the Hamiltonian in $x,p_x$ be $$ H(x, p_x) = \frac{p_x^2}{2m} + V(x) $$
A quick test with Poisson brackets $$\{f,H \}_{(x,p_x)}\equiv \frac{\partial f}{\partial x}\frac{\partial H}{\partial p_x} - \frac{\partial f}{\partial p_x}\frac{\partial H}{\partial x} $$
yields $$\begin{aligned} \dot y &= \{y,H \}_{(x,p_x)} = \frac{c}{m}p_x = c \dot x \\ \dot p_y &= \{p_y,H \}_{(x,p_x)} = -c^{-1}V'(x) = c^{-1}\dot p_x \\ \end{aligned}$$
which looks like it satisfies the strict canonical transformation conditions.
Is this a special case of a scaling transformation that is at the same time a canonical transformation? And if it is a canonical transformation, what is a generating function for this transformation? I fail at matching it with the four common types and have a hard time applying the formalism for canonical transformations to it.