Parabolic Function

A parabola is a U-shaped 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. It is the graph of a quadratic equation in the form of y = ax² + bx + c, where a, b, and c are constants, and a ≠ 0.

Have you ever ridden on a roller coaster and wondered about the tracks? Yes, they are parabola.

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.  Some important point to know about graph of parabolic functions are:

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

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

1. For two different domains (inputs), the codomain (output) can be the same. 
    
2. Any two points on the graph that lie on the horizontal line (same ordinate) will have different x-values (abscissas).
    
3. 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.
    
4. The parabolic function is also known as a many-to-one function, because multiple x-values can produce the same y-value.
    
5. The graph of a parabolic function is symmetric with respect to the vertical line known as the axis of symmetry.
    
6. The equation of the parabolic function meets all the conditions that define a geometric parabola. 

For students finding parabolic functions difficult, here are some tips and tricks:

1. Remember to check the value of 'a'. It tells about the shape of the parabola.
    
2. Factorize the equation to find the zeros easily.
    
3. Use parabola calculator to verify your answers for vertex, focus, directrix, and axis of symmetry.
    
4. Memorize the formula of vertex (\(x = {-b \over 2a}\)).
    
5. You can use vertex calculator to verify your calculated value of vertex.

Parent Tips: 
 

• Explain parabolas with the help of real life examples like roller coasters, water coming out of fountains, etc.
• You can also illustrate the projectile motion to explain 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.

1. 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 to predict how far or high something will go.
   
    
2. Satellite dishes and Radio telescopes
   
   Satellite dishes are shaped like parabolas because they focus signals from a wide area onto a single point at the focus. This helps to collect weak signals from space or TV stations and send them to a receiver at the focus.
   
    
3. Architecture and Bridges
   
   The arches, bridges, and roller coasters use parabolic shapes not just for their beauty but also for structural strength. It distributes the weight evenly along the structure by allowing it to support heavy loads more efficiently. 
   
    
4. Car Headlights
   
   The reflector used behind the bulb acts as a paraboloid. The position of the bulb is at the focus, such that the light beams are reflected and travel long distances.
   
    
5. Growth and Decay
   
   Parabolic functions are used to model population growth, chemical reactions or any physical process that behaves in a non-linear way.