What Are Quadratic Functions?

The parabola created by using a quadratic function.

In Algebra, quadratic functions are any form of the equation y = ax+ bx + c, where a is not equal to 0, which can be used to solve complex math equations that attempt to evaluate missing factors in the equation by plotting them on a u-shaped figure called a parabola (pictured to left).

The points on this graph represent possible solutions to the equation based on high and low points on the parabola; the minimum and maximum points can be used in tandem with known numbers and variables to average the other points on the graph into one solution for each missing variable in the above formula.

Quadratic functions can be highly useful when trying to solve any number of problems that involve measurements or quantities with unknown variables. One such example would be a rancher with a limited length of fencing who wants to fence in two equal-sized sections creating the largest square footage possible.

This farmer would use the quadratic equation to plot the longest and shortest of the two different sizes of fence sections and use the median number from those points on a graph to determine the appropriate length for each of the missing variables.

Characteristics of Quadratic Formulas

No matter what the quadratic function is expressing, whether it be a positive or negative parabolic curve, every quadratic formula shares these 8 core characteristics:

  1. y = ax2 + bx + c, where a is not equal to 0
  2. The graph this creates is a parabola, a u-shaped figure.
  3. The parabola will open upward or downward.
  1. A parabola that opens upward contains a vertex that is a minimum point; a parabola that opens downward contains a vertex that is a maximum point.
  2. The domain of a quadratic function consists entirely of real numbers.
  3. To determine the range of a quadratic function, one know whether the vertex is a minimum or maximum and what the y-value is of the vertex because if the vertex is a minimum, the range is all real numbers greater than or equal to the y-value, while maximum vertexes' ranges are all real numbers less than or equal to the y-value.
  1. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form x = n, where n is a real number, and its axis of symmetry is the vertical line x =0.
  2. The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots, solutions, and solution sets. Each quadratic function will have two, one, or no x-intercepts.

By identifying and understanding these core concepts related to quadratic functions, students can use quadratic equations to solve a variety of real-life problems with missing variables and a range of possible solutions.

Although some may find these equations utterly useless after completing Algebra II in 10th or 11th grade, skilled workers who understand how to use these relatively simple equations to determine a range of results can much more easily solve problems that involve unknown amounts and factors.