Quadratic function

A function defined by a polynomial of degree two, usually written f(x) = ax² + bx + c with a ≠ 0. Its graph is a parabola.

Vertex form: A quadratic expression written as a(x − h)² + k, exposing its vertex and vertical scale. The parameters h and k give the graph’s vertex directly.
Polynomial: An expression formed from variables and coefficients using nonnegative integer powers and arithmetic operations. A quadratic function is the degree-two member of this broader family.
Projectile motion: The motion of an object moving under gravity after launch, when air resistance is neglected. Height as a function of time is quadratic in the idealized vertical-motion model.
Linear function: A function of the form f(x) = mx + b, whose graph is a straight line. A linear function has constant slope, unlike a quadratic’s changing slope.
Completing the square: An algebraic method that rewrites a quadratic expression as a squared binomial plus a constant. It converts standard form into vertex form and reveals the turning point.
Function: A rule that assigns each input in its domain exactly one output. The polynomial defines the rule; the function also specifies its domain and outputs.
Optimization: The process of finding the greatest or least value of a quantity under given conditions. A quadratic’s vertex gives its maximum or minimum value.
Cubic function: A function defined by a polynomial of degree three. Its graph can have an inflection point and up to three real roots.
Discriminant: For ax² + bx + c, the quantity b² − 4ac, which determines the number and type of roots. Its sign tells whether the parabola crosses the x-axis twice, touches it, or misses it.
Parabola: A plane curve whose points are equidistant from a fixed point and a fixed line. It is the characteristic graph of a one-variable quadratic function.

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