What Is a Quadratic Function?

Quadratic Function

A function of the form $f (x) = a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants and $a \neq = 0$ . The highest power of $x$ is 2, which gives the function its distinctive curved shape.

Quadratic functions are everywhere in the real world. When you throw a ball into the air, the path it follows is a parabola — the graph of a quadratic function. The same shape appears in the design of satellite dishes, suspension bridges, and even the trajectory of water from a fountain.

The word quadratic comes from the Latin quadratus, meaning square — referring to the $x^{2}$ term that defines these functions.

Analogy

Think of a quadratic function like a trampoline or a skateboard half-pipe. Whether it curves upward (like a bowl) or downward (like a hill) depends on the sign of the leading coefficient $a$ . Jump in the middle of a trampoline — that's the vertex!

The Parabola: Shape and Orientation

Parabola

The U-shaped (or inverted U-shaped) curve that is the graph of every quadratic function.

The shape and orientation of the parabola depend on the value of $a$ in $f (x) = a x^{2} + b x + c$ :

If $a > 0$ : the parabola opens upward (like a valley or a smile ☺)
If $a < 0$ : the parabola opens downward (like a hill or a frown ☹)
The larger $∣ a ∣$ is, the narrower the parabola
The smaller $∣ a ∣$ is (closer to zero), the wider the parabola

Example

Compare these three quadratics:

$f (x) = 3 x^{2}$ → opens upward, narrow
$g (x) = x^{2}$ → opens upward, standard width
$h (x) = 0.2 x^{2}$ → opens upward, wide
$k (x) = - x^{2}$ → opens downward, standard width

The coefficient $a$ controls both direction and width.

The Parabola: Shape and Orientation

Free preview

12 more sections in this topic

Pick this up in your Library: it holds the whole topic, notes, cheatsheet and questions.