Let f: such that** f (x) = ax ^{2 }+ bx + c = 0**, where "x" is the variable or unknown, and the letters a, b and c are the coefficients, which can have any value, except that a = 0, is called

**quadratic function**, whose graphical representation is a curve called

**parabola**.

In a quadratic function the variable appears raised to the exponent, this being its maximum exponent. This function has some interesting features, it can be a strictly increasing or strictly decreasing function and can reach a minimum value (or a maximum value), which is called a vertex.

**Graphical representation of the parabola**

If we relate the quadratic function to its graphical representation we have:

**Concavity**

When a> 0, the parabola opens upwards and we say that it is **concave up**; when at <0, the parabola opens downwards and we say that it is **concave down**.

**Vertex, maximum and minimum**

The parabola always has a value of f (x) = y which is **minimum** when a> 0 and **maximum **when a <0. The value of "x" and the value of "y" that is maximum or minimum is a point of the parabola that is called **vertex**, whose abscissa is calculated with the following formula:

x = - b / 2a

**Growth and decrease**

When a> 0 from -∞ to the vertex abscissa the parabola is **strictly decreasing** and from the abscissa to + ∞, the parabola is **strictly growing**.

When a <0 from -∞ to the vertex abscissa the parabola is **strictly growing** and from the abscissa to + ∞, the parabola is **strictly decreasing**.

**Axis of symmetry**

The line that passes through the vertex and is parallel to the y axis is the **axis of symmetry** of the parable.

**Axis intersection points**

When a> 0

When a <0

The parabola always intersects the y axis at point c which is the independent term of the quadratic function y = ax^{2 }+ bx + c = 0 and can cut the x axis at two x points_{1} and x_{2}, at a point x_{1} = x_{2 }or at no point, as these represent the roots of the equation.

**Example**: Graph the following quadratic function: 2x^{2 }+ 5x + 3 = 0

- Since a> 0 → a = 2, the parabola is concave upwards.
- Since a> 0 → a = 2, the parabola has a minimum point. The vertex abscissa is:

x = - b / 2a = - 5/2 (2) = - 5/4

- The parabola is strictly decreasing from -∞ to the vertex abscissa.
- The parabola is strictly increasing from the vertex abscissa to + ∞
- The intersection points are:

x_{1} = -1; x_{2} = - 3/2; c = 3

**Roots calculation (cut points):**

From the equation we see that the coefficients are:

a = 2; b = 5 and c = 3

We substitute the coefficients in the formula:

x = [- b ± √ (b^{2 }- 4ac)] / 2a

x = {- 5 ± √ [(5)^{2 }- 4 (2) (3)]} / 2 (2)

We solve

x = [- 5 ± √ (25 - 24)] / 4 = (- 5 ± √1) / 4

x_{1} = (- 5 + √1) / 4 = -1

x_{2} = (- 5 - √1) / 4 = - 3/2