# Introduction To A Parabola And To Find The Vertex, Focus And Directrix Of The Parabola

A plane curve where any point is equidistant from a fixed point called the focus and the fixed straight line called the directrix is defined as a parabola. It is a U-shaped curve.

A parabola is also defined as a conic section that is obtained from the intersection of a right circular surface of a cone and a plane that is parallel to the other plane which is being tangential to the surface of a cone.

**The Parts Of A Parabola Include:**

**Line of symmetry:**A line that passes through the focus and perpendicular to the directrix.**Vertex:**It is a point at which the parabola intersects the line of symmetry. The parabola is sharply curved at this point.**Focal length:**It is the distance measured along the axis of symmetry between the vertex and the focus.**Latus rectum:**The chord of the parabola that passes through the focus and parallel to the directrix.

The general equation of a * parabola* is

*where a is the distance between the origin and the focus. The vertex form of the parabola is*

**y2 = 4ax,**

**y = a (x – h)2 + k.**

#### To Find The Vertex, Focus And Directrix Of The Parabola

The standard equation of the parabola is of the form ax2 + bx + c = 0. If a > 0 in ax2 + bx + c = 0, then the parabola is opening upwards and if a < 0, then the parabola is opening downwards.

#### Vertex Of The Parabola

It is a point (h, k) on the parabola. It becomes either the base or the top depending upon the orientation of the parabola [opening upward or downward].

The point **h = (–b / 2a)**

The point **k = [4ac – b2] / 4a**

#### Focus Of The Parabola

The axis of symmetry of the parabola is a vertical line that cuts the parabola into half and is given * by x = h*.

It lies on the axis of symmetry of the parabola at

**F (h, k + p) where p = 1 / 4a.**

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#### Directrix Of The Parabola

The line which is opposite to the focus and on the side of the vertex with an equation* y = k – p* is the directrix of the parabola.

Example 1: **How to find the vertex, focus and directrix of the parabola** **y = x² – 6x + 15.**

**Solution:**

The x-coordinate of the vertex is x = h = (–b / 2a)

The standard equation of the parabola is of the form ax2 + bx + c = 0.

On comparing, a = 1, b = -6, and c = 15.

x = -b / (2a)

=’yoast-text-mark’>= – (-6) / (2 * 1)

=’yoast-text-mark’>= 6 / 2

=’yoast-text-mark’>= 3

The x-coordinate of the vertex is y = k = [4ac – b2] / 4a

= [4 * 1 * 15 – (-6)2] / [4 * 1]

= [24 / 4]

= 6

The vertex is (3, 6).

The focus is F (h, k + p) where p = 1 / 4a.

The vertex is (3, 6) and hence the axis of symmetry is x =’yoast-text-mark’>= 3.

The x-coordinate of the focus is 3.

The y-coordinate of the focus is k + [1 / (4a)]

= 6 + [1 / (4 * 1)]

= 6 + [1 / 4]

= 6 + 0.25

= 6.25

The focus is (3, 6.25).

The equation y = k – p is the directrix of the parabola.

y = k – p

= 6 – (1 / 4)

= 5.75

#### Applications Of A Parabola

Approximations of parabolas can be found in the shape of the main cables on a simple suspension bridge.

Paraboloids can be seen in the surface of a liquid confined to a container and when rotated around the central axis.

The vertical curves in roads are usually parabolic by design in the United States.