Opera House and Harbour Bridge
The Harbour Bridge in Sydney, Australia. The bridge is in the shape of a parabola.

# Basic Description

Quadratic functions are a family of functions, where as one variable changes, the other increases or decreases proportional to the square of the first variable. Quadratic functions follow the standard form $ax^2 + bx + c$. Quadratic equations can exist in factored forms, which differ from this standard form. Quadratic functions are identified by the $x^2$ term. The inverse of a quadratic function is a root function, meaning that the inverse function is not in the quadratic family. The end behavior of quadratics depend on the orientation of the function; as x gets closer to positive and negative infinity, $f(x)$ also increases either positively or negatively (but never both at once).

Below is a table for the equation $2x^2 + 4x + 7$:

x f(x)
-3 13
-2 7
-1 5
0 7
1 13
2 23
3 37

From the the table, several discoveries can be made. First of all, it is clear that the x-component of the parabola's midpoint is somewhere between -2 and 0. Furthermore, an examination of the first differences proves that the equation is not linear. An examination of the second differences between each data entry results in the identification of the data table as a quadratic function.

The shape of a quadratic function is called a parabola. Parabolas are distinguished by their unique shape, which can be described as a "U" or an arch. Parabolas are symmetrical along their vertex, which depending on the orientation of the parabola, is the minimum or maximum point of the function. Parabolas can occur in all four quadrants, though they do not need to. While parabolas do not display asymptotes, no points can exist beyond the minimum or maximum points.

# Analysis of Equations

As mentioned above, the standard equation for quadratic functions is $ax^2 + bx +c$. The term $ax^2$ serves multiple purposes; it defines the function as quadratic, determines the steepness of a parabola, and dictates the orientation of the parabola. A negative $a$ value makes the parabola "open downwards", and creates a maximum point, while a positive $a$ value means the parabola has a minimum point. The term $c$ is directly responsible for shifting the y-intercept of the function up and down.