Quadratic Function
Functions & Advanced AlgebraA 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 when graphed. Its basic form is $f(x) = ax^2 + bx + c$.
Example
$f(x) = x^2$ gives the basic parabola opening upward. $f(x) = -x^2$ opens downward. The highest or lowest point is called the vertex.
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.
Definition
A quadratic function $f(x) = ax^2 + bx + c$ ($a \neq 0$) has degree $2$. Its graph is a parabola with vertex at $(-b/(2a), f(-b/(2a)))$. It opens upward if $a > 0$ and downward if $a < 0$.
Example
$f(x) = 2x^2 - 8x + 6$: vertex at $x = -(-8)/(2 \times 2) = 2$, $y = 2(4) - 16 + 6 = -2$. Vertex: $(2, -2)$. Parabola opens upward ($a = 2 > 0$). X-intercepts: factor or use quadratic formula.
Key Insight
Vertex form $f(x) = a(x - h)^2 + k$ directly reveals the vertex $(h, k)$ and transformations. Completing the square converts standard form to vertex form.
Definition
A quadratic function is a degree-$2$ polynomial $f(x) = ax^2 + bx + c$. Its discriminant $D = b^2 - 4ac$ determines roots: two real ($D > 0$), one repeated real ($D = 0$), or two complex conjugate ($D < 0$). Quadratics are the simplest nonlinear polynomial functions.
Example
Over $\mathbb{C}$, every quadratic has exactly two roots (counted with multiplicity). $f(x) = x^2 + 1$ has roots $i$ and $-i$ over $\mathbb{C}$ but no real roots, since $D = -4 < 0$.
Key Insight
Quadratic forms in $n$ variables, $f(x) = x^T A x$, generalize scalar quadratics to higher dimensions. Their sign-definiteness is central to optimization (second-order sufficient conditions) and differential geometry.