Tracking a turning point through a combined transformation

The point \((p,q)\) lies on the curve \(y=\mathrm{f}(x)\). The constants \(a\), \(b\), \(c\) and \(d\) satisfy \(a\ne0\) and \(b\ne0\).

1. Show that the point \(\left(\dfrac{p-c}{b},\ aq+d\right)\) lies on the curve \(y=a\,\mathrm{f}(bx+c)+d\).[3]
2. The curve \(y=2\mathrm{f}(3x-6)+1\) has a minimum point at \((4,5)\). Find the minimum point of \(y=\mathrm{f}(x)\).[2]
3. Explain why the corresponding point on \(y=-\mathrm{f}(2x)+4\) is a maximum point, and find its coordinates.[3]