Consider the quadratic function [math]f(x)=Ax^2+Bx+C[/math] .[br][br]The key result on quadratic functions as compositions is very useful here for [br]visualizing how to solve an inequality of the form [math]f(x)=Ax^2+Bx+C>0[/math]with [math]A\ne0[/math] by considering the related inequality [math]A(x−h)^2+k>0.[/math][br][br]The mapping diagram for [math]f[/math] considered as a [br]composition helps visualize the inequality [math]A(x−h)^2+k>0[/math]. as well as the solution set: [br][br]When [math]A>0,k\le0[/math] the apping diagram has solution set [math]\left(−∞,h-\sqrt{-\frac{k}{A}} \right)\cup\left(h+\sqrt{-\frac{k}{A}},\infty\right)[/math].[br][b]For example,[/b] for the inequality [math]2(x−1)^2−2>0[/math], the solution set is ([math]\left\langle-\infty,0\right\rangle\cup(2,∞)[/math] .[br][br]When [math]A<0,k>0[/math] the mapping diagram helps visualize the solution set, [math]\left(h-\sqrt{-\frac{k}{A}} , h+\sqrt{-\frac{k}{A}}\right)[/math].[br][b]For example[/b][i], [/i]for the inequality[math]−2(x−1)^2+2>0,[/math] the solution set is ([math]0,2)[/math].[br][br]In the case, [math]A>0,k>0,[/math] the diagram visualizes why the solution set is [math](−∞,,∞);[/math] , and when[math]A<0,k\le0[/math] the [br]diagram visualizes why the solution set is empty.[br][br]