Eq. 1: \(y=ax^2+bx+c\)

A quadratic expression is a specific type of polynomial function where the largest exponent of an x containing term is 2, i.e. the expression contains the term x² (Eq. 1). This gives the quadratic expression its distinctive parabolic shape (Figure 1). Unlike a linear expression which continues on a straight line on a graph indefinitely, a parabolic expression follows a curved shape, increasing exponentially (relative to x²⁾ in each direction from the midpoint (or vertex) of the parabola.

Figure 1: The features of a parabola formed by a quadratic expression. The parabola contains a midline that is a line of reflection about which the parabola extends in each direction and forms a mirror image either side of this central line of symmetry. The point where this line meets the parabola is called the vertex. When the parabola opens towards the x axis there are also x intercepts, which are points where the parabola meets the x axis. There may be 2, 1 or none of these points, depending on the location of the vertex and the direction that the parabola opens relative to the x axis.

Quadratic expressions have a number of key points and features which can be generalized when describing all quadratic expressions. What these points relate to, and their significance depends on what the quadratic expression is describing, however these features are common to all quadratic expressions (except for x intercepts, as explained below). The shape of the quadratic expression is known as the parabola. This is a curve that forms due to the x² term in the expression. This is because the parabola describes an exponential increase in either direction on the x axis from the starting point. Exponential increases are not consistent, and grow as the base value of x increases (Table 1), which is what leads to the curved shape of the parabola. The direction that the parabola opens, and its degree of stretch can vary, but it has the same basic shape due to the exponential trend created by the expression, and the exact direction and stretch and modifications of this.

Table 1: Values of x and values of x² in the domain of 0 <= x <= 6. As indicated by the first differences, the x² values increase in a non-linear fashion.

A quadratic expression has a line of symmetry that passes vertically through the midpoint of the parabola produced by the expression. The line of symmetry is the line across which each side of the function is a mirror image of the other side. This symmetry is created because negative values of x when squared produce positive results, as a negative number multiplied by another negative number is a positive, and a negative number multiplied by itself is by definition a negative number multiplied by itself (Table 2)

Table 2: Values of x and values of x² in the domain of -3 <= x <= 3. As can be seen, the -3 and 3 x values have the same x² values, giving quadratics their distinctive parabolic shape.

The vertex is the point where this line of symmetry meets the line formed by the expression, and is also the point around which the exponential trend grows in either direction. In the expression, there will be a point where the expression leads to a calculation of 0². As the result of this is 0, this is the point of the parabola where the expression opens and grows each side positive or negative away from this. The domain of a quadratic function covers all numbers (since both positive and negative values of x have valid values as solutions), or at least, they have a domain that is not limited by the properties of the quadratic function. The range of the function is only those values that are either greater than or less than the vertex, depending on the direction of the parabola. Parabolas that open upwards will have ranges greater than the y value of the vertex, while parabolas which open downwards have a range which is below the y value of the vertex.