Definition
Quadratic function
A quadratic function has degree 2 and, over the real numbers, a parabolic graph.
What it means
The x² term is the highest variable power. That squared term bends the graph into a parabola instead of one straight line. A positive a opens the parabola upward; a negative a opens it downward.
Remember it
Highest power 2; the graph is a U or an upside-down U.
Example
For q(x) = x² − 4x + 4 = (x − 2)², the x² coefficient is 1, so q is quadratic. Inputs 0, 1, 2, 3, 4 give outputs 4, 1, 0, 1, 4, forming a symmetric parabola with vertex (2, 0).
Watch for
A squared number alone does not make a function quadratic. For example, f(x) = 3x + 2² simplifies to 3x + 4, which is linear because the variable is not squared.
Common mix-up
Every quadratic function is nonlinear, but not every nonlinear function is quadratic. Functions such as x³ and |x| are nonlinear without having degree 2.
Exact meaning
A quadratic function is a real polynomial function that can be written as f(x) = ax² + bx + c, where a, b, and c are real constants and a ≠ 0. Its graph is a parabola.