For y=|q(x)|, positive quadratic portions stay fixed; negative portions reflect upwards. A crossing zero becomes a sharp join; a touching zero need not become a sharp join. Find the roots and vertex of q. Decide the sign on each interval, reflect the negative intervals, then check endpoints on a restricted domain. For q(x)=(x-1)2-1, roots are 0,2 and the vertex (1,-1) becomes (1,1); only 0<x<2 reflects. Check: |(x+1)2|=(x+1)2, and |(x+1)2+2| never reaches zero. Modulus alone does not guarantee a zero.

For \(y=|q(x)|\), positive quadratic portions stay fixed; negative portions reflect upwards. A crossing zero becomes a sharp join; a touching zero need not become a sharp join.

Find the roots and vertex of \(q\). Decide the sign on each interval, reflect the negative intervals, then check endpoints on a restricted domain.

For \(q(x)=(x-1)^2-1\), roots are \(0,2\) and the vertex \((1,-1)\) becomes \((1,1)\); only \(0<x<2\) reflects.

Check: \(|(x+1)^2|=(x+1)^2\), and \(|(x+1)^2+2|\) never reaches zero. Modulus alone does not guarantee a zero.

Modulus graphs of quadratic functions

Modulus graphs of quadratic functions