Unknown coefficients of two quadratic functions whose vertices lie on a given line

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined for \(x\in\mathbb{R}\) by \(\mathrm{f}(x)=x^2-6x+p\) and \(\mathrm{g}(x)=2x^2+qx+7\), where \(p\) and \(q\) are constants and \(q>0\).

The vertex of the graph of \(y=\mathrm{f}(x)\) is \(V\) and the vertex of the graph of \(y=\mathrm{g}(x)\) is \(W\). The line \(y=3-2x\) passes through both \(V\) and \(W\).

1. Show that \(p=6\).[3]
2. Find \(q\).[5]