Answer: (a) \(f(x)=\pm4(x+2)(x-1)(x-3)\); (b)(ii) \(x=\frac12\) or \(x=\frac52\).

Answer: (a) \(f(x)=\pm4(x+2)(x-1)(x-3)\); (b)(ii) \(x=\frac12\) or \(x=\frac52\).

(a) The graph of \(|f(x)|\) touches the \(x\)-axis at \(x=-2\), \(x=1\) and \(x=3\). Hence

\(f(x)=k(x+2)(x-1)(x-3).\)

The graph has \(|f(0)|=24\). Now

\(f(0)=k(2)(-1)(-3)=6k.\)

Thus \(|6k|=24\), so \(k=4\) or \(k=-4\). Therefore

\(f(x)=4(x+2)(x-1)(x-3)\)

or

\(f(x)=-4(x+2)(x-1)(x-3).\)

(b)(i) For \(y=|2x+1|\), the vertex and \(x\)-intercept are \(\left(-\frac12,0\right)\), and the \(y\)-intercept is \((0,1)\).

For \(y=|4(x-1)|\), the vertex and \(x\)-intercept are \((1,0)\), and the \(y\)-intercept is \((0,4)\).

(b)(ii) Solve the two cases:

\(2x+1=4(x-1)\quad\text{or}\quad2x+1=-4(x-1).\)

The first gives \(x=\frac52\), and the second gives \(x=\frac12\).