Question about Weighted Vertex Cover algorithm


I need to generalize the algorithm given in CLRS book for vertex cover
problem and then show that for every C>0 exists a graph with weights
(each vertex has a weight) for which this expantion of the algorithm
gives a solution C times worst than the optimal one.

I think I found a way to generalize it but it's a 2 Approximation
algorithm, so it can't have a grapsh for which the solution is C times
worst since C might be bigger than 2.

Any ideas?