Consider the quadratic function $f(x) = Ax^2 + Bx + C$.
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) $, e.g., $2(x−1)^2-2 > 0$, the solution set is $(-\infty, 0)) \cup (2,\infty)$.
while when $A<0, k>0$
the mapping diagram helps visualize the solution set, $(h - \sqrt{-\frac kA}) , h +
\sqrt{-\frac kA}) $ , e.g., $-2(x−1)^2+2 > 0$, the solution set is $(0 , 2)$.
.
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.
. 