The parabola is a conic section that is formed when a cone is cut by a plane parallel to one lateral side of the cone. The parabola has the main characteristic that all its points are located at the same distance from a point called the focus and a line called the directrix. Other important elements of a parabola are the vertex, the axis, the latus rectum, and the focal length.
Here, we will learn more about these elements and use diagrams to illustrate them. Also, we will learn about the different types of parabolas that we can have.
Definition of a parabola
A parabola is defined as a conic section. The parabolas are obtained by cutting a cone with a plane parallel to one of the lateral sides of the cone. Parabolas are formed by a set of points that are characterized by having the same distance from a fixed point, called the focus, and a straight line called the directrix.
Fundamental elements of a parabola
The following are the fundamental elements of a parabola:
- Focal length
- Latus rectum
The vertex of the parabola is its extreme point. If the parabola opens upwards, the vertex represents the lowest point in the parabola. If the parabola opens downwards, the vertex represents the highest point. In either case, the vertex is a point that changes the direction of the parabola. Frequently, the vertex is represented with the letter V.
The focus is a fixed point used to define the parabola. This point is not located on the parabola, but inside. The focus is denoted by F.
The focal length is the length between the vertex and the focus.
The latus rectum is a line perpendicular to the line joining the vertex and the focus and is four times the length of the focal length.
The directrix is a straight line in front of the parabola. We use d to represent the directrix. The distance between the directrix and the vertex is the same as the distance between the focus and the vertex.
The axis of the parabola is a line perpendicular to the directrix. The axis represents the line of symmetry of the parabola.
Types of parabolas
We can classify the parabolas depending on their orientation. We can have horizontally and vertically oriented parabolas. In addition, the parabolas can be opened to the right, to the left, up and down.
Horizontal parabola that opens to the right
This parabola is obtained when the directrix is vertical and the parameter p is positive.
Horizontal parabola that opens to the left
This parabola is obtained when the directrix is vertical and the parameter p is negative.
Vertical parabola that opens upwards
This parabola is obtained when the directrix is horizontal and the parameter p is positive.
Vertical parabola that opens downwards
This parabola is obtained when the directrix is horizontal and the parameter p is negative.
Interested in learning more about parabolas? Take a look at these pages: