Parabolic Function

A parabola is a U-shaped curve. It is the graph of a quadratic function in the form of y = ax2 + bx + c, where a, b, and c are constants, and a ≠ 0. A parabola is a curve formed by all the points that are the same distance from a specific point, called the focus, and a straight line, called the directrix. 
 

The parabolic function is in the form of f(x) = ax2 + bx + c. The graph of a parabolic function forms a U-shaped curve called a parabola. This function involves squared terms like x2, which makes it a second-degree or a quadratic function. A parabolic function can have the same range for two different domain values, so it is called a ‘many-to-one’ function.
 

The graph of a parabolic function looks like a U-shaped curve called a parabola. A parabola has a special property that every point on it is the same distance from a fixed point and a fixed line. The graph of a parabola is symmetric, which means the left and right sides are mirror images of each other. The line that divides the parabola into two equal parts is called the axis of symmetry. Based on the orientation of a parabola, its axis can be vertical or horizontal. In some advanced cases, the axis of a parabola can also be tilted, and it is not strictly vertical or horizontal. 

Properties of Parabolic Function

Learning and understanding the properties of a parabolic function make it easier to analyze its behavior. Some key properties of parabolic functions are: 

• For two different domains (inputs), the codomain (output) can be the same. 
• Any two points on the graph that lie on the horizontal line (same ordinate) will have different x-values (abscissas).
• The domain of the parabolic function can have both positive and negative values, but the range of the parabolic function consists of only positive values or only negative values.
• The parabolic function is also known as a many-to-one function, because multiple x-values can produce the same y-value.
• The graph of a parabolic function is symmetric with respect to the vertical line known as the axis of symmetry.
• The equation of the parabolic function meets all the conditions that define a geometric parabola. 
   

Parabolic functions are used in everyday life, from the way the ball moves in the air to the shape of satellite dishes; parabolas are in many places. Here are some of the real-life examples of parabolic functions.

• Projectile Motion: When you throw a ball into the air, it does not go in a straight line; it travels along a curved path. This curved path is shaped like a parabola. Parabolic function helps athletes, engineers, and scientists to predict how far or high something will go.
• Satellite dishes and Radio telescopes: Satellite dishes are shaped like parabolas because they can focus signals from a wide area onto a single point at the focus. This parabolic design helps to collect weak signals from space or TV stations and send them to a receiver at the focus.
• Architecture and Bridges: The arches, bridges, and roller coasters use parabolic shapes not just for their beauty but also for structural strength. The parabolic curves help to distribute weight evenly along the structure by allowing it to support heavy loads more efficiently.