Trigonometry - Trigonometric Graphs and Parameters

Consider a function \(f\) such that \(f(x)=3\sin{(2x+\alpha)}\) where \(0 \leq x \leq 2\pi\) and \(0 \leq \alpha \leq \frac{\pi}{2}\).

Write down the amplitude of \(f\)[1]

The graph of \(y=f(x)\) passes through the point \(\left(\frac{\pi}{3},\frac{3}{2}\right)\).

The function \(g\) is given by \(g(x)=pf(x)+q\) where \(0\leq x \leq 2\pi\) and \(p,q \in \mathbb{R}\).

The graph of \(y=g(x)\) passes through the origin and intercepts the graph of \(y=f(x)\) at the points where \(f(x)\) is at its maximum value.

Find the values of \(p\) and \(q\).[5]

In the case where \(\alpha=0\) and \(p\) and \(q\) have the same values found in part (c), show that \(g\) can be written in the form \(g(x)=k(m\sin{x}\cos{x}+n)\), where \(k, m\) and \(n\) are integers to be found and where \(k>0, k\neq 1\).[3]