AI Tutor
If you throw a ball into the air, its flight traces a curved path called a parabola. But what if you need to describe not just the flight path, but the entire zone beneath it where the ball could safely land?
To capture an entire two-dimensional territory on a coordinate plane, mathematicians turn ordinary equations into quadratic inequalities.
How do we decode a shaded graph and turn it back into an algebraic sentence?

The Anatomy of the Graph

Every inequality graph has two essential clues: the boundary curve that outlines the region, and the shaded area containing all coordinate pairs (x,y)(x, y) that make the statement true.
A dashed border tells you the boundary points are excluded, meaning the inequality must use << or >>. A solid border includes the boundary itself, demanding ≤\le or ≥\ge.
Before deciding which inequality symbol to write, how do we find the exact quadratic formula of the boundary curve itself?

Building the Boundary Equation

Every basic parabola y=x2y = x^2 can be shifted horizontally by hh units and vertically by kk units to position its turning point, called the vertex (h,k)(h, k), giving the vertex form y=a(x−h)2+ky = a(x - h)^2 + k.
The multiplier aa, known as the vertical stretch factor, dictates how wide or narrow the parabola opens, as well as whether it opens upward (a>0a > 0) or downward (a<0a < 0).
Once you have the vertex (h,k)(h, k) and any clear point (x,y)(x, y) on the curve, substitute them into y=a(x−h)2+ky = a(x - h)^2 + k and solve for aa.