Parabolas

Parabolas are common in algebra as the graphs of quadratic functions, and they have many important real world applications.

Key Points

• Parabolas are frequently encountered as graphs of quadratic functions, including the very common equation y=x^2.

• All parabolas contain a focus, a directrix, and an axis of symmetry that vary in exact location depending on the equation used to define the parabola.

• Parabolas are frequently used in physics and engineering in places such as automobile headlight reflectors and in the design of ballistic missiles.

Terms

• A line used to define a curve or surface; especially a line, the distance from which a point on a conic has a constant ratio to that from the focus

• A point on the curve with a local minimum or maximum of curvature.

• Or relating to projectiles moving under their own momentum, air drag, gravity and sometimes rocket power

Figures

1. Interactive Graph: Parabola with Focus and Directrix

Graph of the parabola where . Parabolic curve showing directrix (L) and focus (F). The distance from any point on the parabola to the focus (PnF) equals the perpendicular distance from the same point on the parabola to the directrix (PnQn).

In mathematics, a parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point (the focus) and a line (the directrix), as can be visualized in Figure 1. The locus of points in that plane that are equidistant from both the line and point is a parabola. In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as: $y=x^2$

The line perpendicular to the directrix and passing through the focus, that is, the line that splits the parabola through the middle, is called the axis of symmetry. The point on the axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, all parabolas are similar.

To locate the x-coordinate of the vertex, cast the equation for y in terms of $ax^2+bx+c$. The vertex will be at the point $x= \frac{-b}{2a}$. For example, in the parabola $y=x^2$, a=1 b=0, c=0 and the vertex is at x=0.

Parabolas have the property that, if they are made of material that reflects light, then light which enters a parabola traveling parallel to its axis of symmetry is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected, or collimated, into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas.

The parabola has many important applications, from automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.

Key Term Glossary

ballistic
Or relating to projectiles moving under their own momentum, air drag, gravity and sometimes rocket power
Appears in these related concepts:
ballistics
The science of objects that predominately fly under the effects of gravity; observing their momentum and atmospheric drag; and dealing with details of their behaviour at the origin and destination of their flight, as of bullets, missiles or rockets.
Appears in this related concept:
conical
Shaped like a cone; of or relating to a cone or cones.
Appears in these related concepts:
conic section
Any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola and hyperbola.
Appears in these related concepts:
directrix
A line used to define a curve or surface; especially a line, the distance from which a point on a conic has a constant ratio to that from the focus
Appears in these related concepts:
distance
The amount of space between two points, measured along a straight line
Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign. E.g. x=5.
Appears in these related concepts:
function
A relation in which each element of the input is associated with exactly one element of the output.
Appears in these related concepts:
graph
A diagram displaying data; in particular one showing the relationship between two or more quantities, measurements or numbers.
Appears in these related concepts:
parabola
The conic section formed by the intersection of a cone with a plane parallel to a plane tangent to the cone; the locus of points equidistant from a fixed point (the focus) and line (the directrix).
Appears in these related concepts:
point
An entity that has a location in space or on a plane, but has no extent
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A quadratic polynomial, function or equation.
Appears in these related concepts:
any function whose value is the solution of a quadratic polynomial
reflection
a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points
Appears in these related concepts:
term
any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an overall expression or table.
Appears in these related concepts:
vertex
A point on the curve with a local minimum or maximum of curvature.