Parts of a Parabola with Diagrams

The parabola has the main characteristic that each point on its curve is located at the same distance from a point, called the focus, and a line, called the directrix. Parabolas are conic sections formed when a cone is cut by a plane parallel to one of the sides of the cone. The most important parts of the parabolas are the focus, the directrix, the vertex, the axis, the latus rectum, and the focal length.

Here, we will learn about these parts in more detail.

PRECALCULUS
elements of a parabola

Relevant for

Learning about the important parts of a parabola.

See parts

PRECALCULUS
elements of a parabola

Relevant for

Learning about the important parts of a parabola.

See parts

What is a parabola?

A parabola is a curve that is formed at the intersection of a plane with a cone when the plane is parallel to one of the lateral sides of the cone. Parabolas are also formed by starting with a line, called the directrix, and a point called the focus and drawing the set of all points equidistant from the directrix and the focus.

focus, vertex and directrix of parabola

Important parts of a parabola

The following are the most important parts of a parabola:

  • Vertex
  • Focus
  • Focal length
  • Latus rectum
  • Directrix
  • Axis
elements of a parabola

Vertex

The vertices are the points that are located at the end of the parabola. For example, when the parabola opens upwards, the vertex is the lowest point of the parabola and when the parabola opens downwards, the vertex is the highest point of the parabola.

The vertex is a point where the direction of the parabola changes. We use V to represent the vertex.

Focus

The focus is a point that is not located on the curve, but inside. The focus is denoted by F and is used to define the parabola.

Focal length

The focal length of a parabola is the length between the vertex and the focus.

Latus rectum

The latus rectum is a line that is perpendicular to the line joining the vertex and the focus. The length of the latus rectum is equal to four times the length of the focal length.

Directrix

The directrix is a straight line outside the parabola. The directrix is represented by d and is also used to define the parabola. The directrix is located at the same distance from the vertex as the distance between the focus and the vertex.

Axis

The axis is a line perpendicular to the directrix that represents the line of symmetry of the parabola.


Types of parabolas

We can distinguish four types of parabolas based on their orientation. A parabola can be oriented horizontally and vertically and can open downwards, upwards, to the right, or to the left.

Vertical parabola that opens downwards

The parabola opens downward when the directrix is horizontal and the parameter p is negative.

equation of vertical parabola that opens downwards

Vertical parabola that opens upwards

The parabola opens upward when the directrix is horizontal and the parameter p is positive.

equation of vertical parabola that opens upwards

Horizontal parabola that opens to the right

The parabola opens to the right when the directrix is vertical and the parameter p is positive.

Horizontal parabola that opens to the right

Horizontal parabola that opens to the left

The parabola opens to the left when the directrix is vertical and the parameter p is negative.

equation of Horizontal parabola that opens to the left

See also

Interested in learning more about parabolas? Take a look at these pages:

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