Answer: \(\mathrm f(x)=3(x+4)(2x-1)(x-2)\) or \(\mathrm f(x)=-3(x+4)(2x-1)(x-2)\).

Start by taking the key information from the graph or diagram, such as intercepts, turning points, amplitudes, periods, or intersection points. Then use the relevant algebra to justify the result.

The graph of \(y=|\mathrm f(x)|\) touches the \(x\)-axis where \(\mathrm f(x)=0\).

So the roots of \(\mathrm f(x)\) are \(x=-4\), \(x=\frac12\), and \(x=2\).

Using integer linear factors,

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

From the graph, \(|\mathrm f(0)|=24\). But

\((0+4)(2\cdot0-1)(0-2)=8\).

Therefore \(|8k|=24\), so \(|k|=3\).

Hence the two possible expressions are

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

This gives the required answer: \(\mathrm f(x)=3(x+4)(2x-1)(x-2)\) or \(\mathrm f(x)=-3(x+4)(2x-1)(x-2)\).