A quadratic function or quadratic function with unknown \(x\) has a function definition that can be reduced to the standard form \[f(x)=ax^2+bx+c\] where \(a\), \(b\) and \(c\) are given numbers and \(a\neq 0\).

In the examples on the right-hand side, the function definition of the same quadratic function \(f(x)\) has been specified in different ways.

Examples

\[\begin{aligned}f(x) &= x^2+3x+2\\[0.25cm] f(x)&=(x+\tfrac{3}{2})^2-\tfrac{1}{4}\\[0.25cm]
f(x)&=(x+1)(x+2)\end{aligned}\]

The graph of a quadratic function is called a parabola. This consists of the points \((x,y)\) in the plane for which \(y=ax^2+bx+c\) holds.

If \(a>0\), the graph is a valley parabola, i.e. a U-shaped graph that opens upward

If \(a<0\), the graph is a mountain parabola, i.e. an n-shaped graph that opens downward

The graphs below illustrate this naming.

You can inspect other examples of parabolas by using the sliders in the figure on the right-hand side.

A valley parabola has a minimum and a mountain parabola has a maximum. In both cases, we call it the vertex of the parabola.
The \(x\)-coordinate of the vertex is equal to \(\displaystyle \frac{-b}{2a}\).

Each parabola also has an axis of symmetry, namely, the vertical line through the vertex which separates the parabola in two halves that are each other's mirror image in a certain sense.

In the above two examples, the parabolas have two points in common with the horizontal axis. These are called the zeros or roots of the quadratic function.

A parabola has 0, 1 or 2 zeros.