Quadratic functions

quadratic functionLet f: such that f (x) = ax2 + 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.

concavity

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

Maximus and minimous

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 symmetry

Axis intersection points

When a> 0

cut points - quadratic functions

When a <0

cut points - quadratic functions

The parabola always intersects the y axis at point c which is the independent term of the quadratic function y = ax2 + bx + c = 0 and can cut the x axis at two x points1 and x2, at a point x1 = x2 or at no point, as these represent the roots of the equation.

Example: Graph the following quadratic function: 2x2 + 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:

x1 = -1; x2 = - 3/2; c = 3

quadratic function

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 ± √ (b2 - 4ac)] / 2a

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

We solve

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

x1 = (- 5 + √1) / 4 = -1

x2 = (- 5 - √1) / 4 = - 3/2