What makes a function quadratic?

Build an intuitive understanding through a clear explanation, visual model, and guided examples.

f(x) = ax² + bx + c

vertex

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Why this matters

Quadratic functions model motion, area, optimization, and many patterns where change itself changes. Learning to read their shape gives you a visual shortcut before you calculate anything.

The central idea

A quadratic function contains a squared term. Its graph is a parabola—a symmetric curve with a lowest or highest point called the vertex.

●heightdistance

Real-world connection

A ball’s path is a parabola

Ignoring air resistance, an object rises, slows at its highest point, and returns along a curved path. A quadratic equation describes that changing height.

What to notice

The sign of a tells you whether the graph opens upward or downward.
The vertex is the graph’s turning point.
The axis of symmetry divides the graph into two matching halves.