Quadratic Function
A quadratic function is a polynomial function of degree 2 whose graph forms a U-shaped curve called a parabola.
Formula
f(x) = ax^2 + bx + c
Definition
A quadratic function has an $x^2$ term and forms a U-shaped curve (or upside-down U), called a parabola, with basic form $f(x) = ax^2 + bx + c$ ($a \neq 0$). Its graph has a vertex at $(-b/(2a), f(-b/(2a)))$, opening upward if $a > 0$ and downward if $a < 0$; the discriminant $D = b^2 - 4ac$ then determines the roots, two real ($D > 0$), one repeated real ($D = 0$), or two complex conjugates ($D < 0$).
Example
$f(x) = x^2$ gives the basic upward-opening parabola while $f(x) = -x^2$ opens downward, with the highest or lowest point called the vertex; for $f(x) = 2x^2 - 8x + 6$, the vertex is at $x = -(-8)/(2 \times 2) = 2$, $y = 2(4) - 16 + 6 = -2$, so the vertex is $(2, -2)$. Over $\mathbb{C}$, every quadratic has exactly two roots counted with multiplicity: $f(x) = x^2 + 1$ has roots $i$ and $-i$ but no real roots since $D = -4 < 0$. Quadratic forms in $n$ variables, $f(x) = x^T A x$, generalize scalar quadratics to higher dimensions, and their sign-definiteness is central to optimization and differential geometry.
Key Insight
Throw a ball in the air and its path traces a parabola: quadratics model anything with a maximum or minimum value, profit, projectile height, or the shape of a satellite dish. Vertex form $f(x) = a(x - h)^2 + k$ directly reveals the vertex $(h, k)$ and transformations, and completing the square converts standard form to vertex form; quadratics are the simplest nonlinear polynomial functions, which is why the discriminant and vertex are such powerful diagnostic tools.