Quadratic Function

Quadratic Function

## Definition / Concept

A quadratic function is a function of the form:

$$f(x)=ax^2+bx+c,\qquad a\ne0$$

Its graph is a parabola.

## Formula

The axis of symmetry is:

$$x=-\frac{b}{2a}$$

The vertex is:

$$\left(-\frac{b}{2a},-\frac{D}{4a}\right)$$

where:

$$D=b^2-4ac$$

The vertex form is:

$$f(x)=a\left(x+\frac{b}{2a}\right)^2-\frac{D}{4a}$$

## Conditions / Special Cases

If:

$$a>0$$

the parabola opens upward.

If:

$$a<0$$

the parabola opens downward.

## Usage

- Used to determine the vertex, axis of symmetry and extremum of a quadratic function.