The key result on quadratic functions as compositions is very useful here for visualizing how to solve an inequality of the form $f(x)=Ax^2 + Bx +C > 0$ with $A \ne 0$ by considering the related inequality $A(x−h)^2+k > 0$.

The mapping diagram for $f$ considered as a composition helps visualize the inequality $A(x−h)^2+k > 0$. as well as the solution set:

when $A>0, k \le 0$ the mapping diagram has solution set $(-\infty, h - \sqrt{-\frac kA}) \cup (h + \sqrt{-\frac kA},\infty) $,

.

In the case, $A>0, k> 0$, the diagram visualizes why the
solution set is $(-\infty, ,\infty) $; when $A<0, k\le 0$ the
diagram visualizes why the solution set is empty.You can use this next dynamic example to solve quadratic equations like that in Example QF.QInEQ.0 visually with a mapping diagram of $f$.