Quadratic sketches and graphical roots

A positive coefficient of \(x^2\) gives an upward-opening curve with a minimum; a negative coefficient gives a downward-opening curve with a maximum. In \(y=\pm(x-p)^2+q\), the turning point is \((p,q)\) and the symmetry line is \(x=p\). Set \(x=0\) for the \(y\)-intercept and \(y=0\) for roots.

Example

\(y=(x-1)^2-4=(x+1)(x-3)\) has minimum \((1,-4)\), roots \(-1,3\) and \(y\)-intercept \(-3\). The roots are the \(x\)-coordinates where the graph meets the horizontal axis.

For \(y=-(x-1)(x-3)=-(x-2)^2+1\), the roots are \(1,3\), the symmetry line is \(x=2\), and the maximum is \((2,1)\).

For a graph supplied on a grid, read roots only as accurately as the scale allows. A curve may meet the axis twice, touch it once, or not meet it; report only the real roots visible within the stated domain.