Linear, quadratic and cubic moduli

\(|u|=u\) when \(u\ge0\), and \(|u|=-u\) when \(u<0\). To draw \(y=|\mathrm{f}(x)|\), keep every part on or above the horizontal axis and reflect only the parts below it. Zeros stay in place; negative turning values become positive. The domain is unchanged.

For \(y=|x-1|\), the vertex is \((1,0)\), with slopes \(-1\) and \(1\). For \(y=|x^2-4|\), reflect the portion between \(-2\) and \(2\); \((0,-4)\) becomes \((0,4)\).

For \(y=|(x+2)(x-1)(x-3)|\), mark roots \(-2,1,3\) and the vertical intercept \(6\). The original cubic changes sign at each simple root. A double root touches the axis; a triple root still crosses it. More generally, even multiplicity touches and odd multiplicity crosses.