quadratic function

A quadratic function is a rule that turns each input x into an output by squaring, scaling, and shifting it. Think of a thrown ball: its height over time follows a quadratic — it rises, slows, peaks, and falls in a smooth arc. That arc is the signature of a degree-two rule.

Its formula is f(x) = ax^2 + bx + c with a ≠ 0. Unlike a quadratic equation, which asks where the output is zero, a quadratic function gives an output for every input x. Its graph is a parabola, and the sign of a tells you which way it opens: upward (like a smile) when a > 0, downward (like a frown) when a < 0.

The places where the function equals zero — its roots or x-intercepts — are exactly the solutions of the matching quadratic equation ax^2 + bx + c = 0. So the function and the equation are two views of the same algebra: one is the whole curve, the other asks only where that curve crosses the x-axis.