What Are Quadratic Functions?

The parabola created 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. The graphs of quadratic functions are parabolas; they tend to look like a smile or a frown.

Points Within a Parabola

The points on a 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.

When to Use a Quadratic Function

Quadratic functions can be highly useful when trying to solve any number of problems involving measurements or quantities with unknown variables.

One such example would be if you were a rancher with a limited length of fencing and you wanted to fence in two equal-sized sections creating the largest square footage possible. You would use a 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.

Eight Characteristics of Quadratic Formulas

Regardless of what the quadratic function is expressing, whether it be a positive or negative parabolic curve, every quadratic formula shares eight 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.
  4. 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.
  5. The domain of a quadratic function consists entirely of real numbers.
  6. If the vertex is a minimum, the range is all real numbers greater than or equal to the y-value. If the vertex is a maximum, the range is all real numbers less than or equal to the y-value.
  7. 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.
  8. 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, you can use quadratic equations to solve a variety of real-life problems with missing variables and a range of possible solutions.

    Once you understand how to use these relatively simple equations to determine a range of results, you can easily solve problems involving unknown amounts and factors.