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A parabola is a U-shaped mirror symmetric curve which has different characteristics and applications in **mathematics and physics**. A parabola is described through the **quadratic functions y = x ^{2}** in most of the

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To form a **parabola**, one needs to start with a line and a point off to one side. This line is called “**Directrix**” and the point is called the “**Focus**“. The parabola is a curve where the points on the curve are equidistant from the **directrix and focus**.

Any **geometrical structure** of parabola will reveal that it is a **U-shaped geometric image**. There is a sharp turning point in a parabola from which the graphical figure changes the direction abruptly. There may be two types of turning points –

- A minimum turning point which indicates that the turning point is at the lowest point of the graph.
- A maximum turning point which signifies that the turning point is at the highest point of the graph.

**The axis of symmetry**

If a parabola has a turning point (h,k), then the line of symmetry is given by x = h.

The turning point gives us information on the location of the line of symmetry along with the highest point or lowest point of the parabola.

**Vertex**

The point on the “**Axis of symmetry**” which is exactly the midway between the focus and Directrix is called the **vertex**.

**Quadratic forms of parabola**

Apart from y = x^{2}, parabolas can also be of the following forms –

y=ax^{2}+bx+c This is the general form of y interception.

y=a(x+d) (x+e), This is the x intercepted form with x as the primary factor.

y=a(x+k)^{2}+v This is known as the turning point form.

Each of these quadratic forms of parabola signifies some specific characteristics of the most popular and useful geometric figure.

It is evident from all these quadratic functions that a parabola has only one y intercept whereas the x intercepts may vary from zero to two.

Like **hyperbola, ellipse, and circle, parabola** is a special conic section with the above mentioned characteristics.

If a real life situation is considered, if any light or sound ray is inflowing the parabola, which is parallel to the axis of symmetry, and hitting the inner surface of the parabolic “**bowl**“, it will be reflected back to the focus.

Bionic ears and radio telescopes etc. are the real life examples of parabolic dishes. With the help of their parabolic property, a signal is received with a maximum concentration onto a receiver. In general, a parabola is differentiated from other conic sections, in the following aspects –

- The focus of a parabola is always inside the parabola;
- The vertex is always on the parabola;
- The directrix is always outside the parabola.

The assignments on parabola inevitably requires the following –

- Deriving the equation from the story given in the assignment.
- Simplifying the equation to give it a parabolic quadratic form.
- Using the information given in the assignment to solve the equation.
- Drawing the equation as per the solution.
- Coming to a conclusion.

In some assignments, real life problems may be given which may need some more formulas to solve successfully. Again, many problems on parabola may be solved in more than one ways. The examiner may ask the students to follow any particular process to solve those equations.

In any case, problems on parabola are not easy to solve. The students are expected to show the highest levels of acumen and knowledge on that subject to solve the equations and assignments.

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