Quadratic graphs

Every time a ball leaves your hand, its height follows a curved path , up, peak, back down. That curve is a parabola, and the equation behind it is a quadratic function. The same shape appears in suspension bridges, satellite dishes, and the trajectory of a penalty kick.

Quadratic Function: Any function that can be written as $y=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants and $a\ne 0$. The condition $a\ne 0$ is what makes it quadratic , without it, you just get a straight line.

The three constants each do a different job. $a$ decides the shape and direction of the curve. $b$ shifts where the curve sits horizontally. $c$ is simply where the graph cuts the vertical axis. Understanding what each one does means you can sketch a parabola from an equation without plotting dozens of points.

Here's the counterintuitive part: a quadratic with a smaller value of $|a|$ actually produces a wider curve, not a narrower one. If $|a|>1$, the parabola is steep and narrow. If $0<|a|<1$, it spreads out wide. Think of it like zooming a camera , a large $|a|$ zooms in and squishes the curve, a small $|a|$ zooms out and stretches it.