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parabola
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Parts of a parabola Two panels showing function graphing concepts: Panel A illustrates plotting an arbitrary function f by selecting values of t and marking corresponding f(t) points; Panel B graphs the specific parabola f(t) equals t squared, with plotted points showing how t values map to t squared. Both graphs use dashed lines to connect points to the axes for clarity. Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. For example, all circles have zero eccentricity, and all parabolas have unit eccentricity; hence, all circles (and all parabolas) have the same shape, only varying in size. (Under appropriate magnification they are indistinguishable.) In contrast, ellipses and hyperbolas vary greatly in shape. Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. A diagram of a double cone intersected by two green planes labeled PP (picture plane) and RP (reality plane), illustrating projected conic sections in projective geometry, with the apex and a point Omega marked. Illustration showing how a parabolic satellite dish antenna focuses incoming transmission signals onto a single point, with red lines representing signal paths reflecting off the dish surface toward the feedhorn. Insets provide a side view and a three-quarter view, highlighting the role of the paraboloid shape in signal reception. Archimedes' parabolic segment calculation Diagram showing the elements of a bullet's parabolic trajectory in ballistics, with a rifle at the left, labeled lines for line of sight, line of departure, and the curved bullet trajectory passing through mid-range height and hitting a target at 200 yards. Two graphs compare projectile motion. Panel A shows a parabolic path starting high on the Z axis and curving downward to the X axis. Panel B shows a parabolic path starting at the origin, rising upward, then curving back down to the X axis, illustrating an initial upward velocity. Figure 7: Parabola (see text).
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Parts of a parabola
Parts of a parabola Parts of a parabola. The curve is symmetric about the vertical axis (axis of symmetry). The vertex is the lowest point of the parabola, the focus is a fixed point inside the curve, and the directrix is a line below the vertex. Any point on the parabola is equidistant from the focus and the directrix. The latus rectum is the line segment through the focus that is perpendicular to the axis of symmetry.

parabola

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Top Questions

What is a parabola?

Who discovered the parabola as a conic section?

What contributions did Galileo make regarding parabolas?

parabola, an open plane curve that is the set of points that have the same distance from both a fixed line (the directrix) and a fixed point (the focus). It is also the conic section produced by the intersection of a right circular cone and a plane parallel to the cone’s surface. On the Cartesian plane, the equation of the parabola is a quadratic equationy = ax2 + bx+ c,where a, b, and c are real numbers. The same equation can also be expressed in vertex formy = a(x - h)2 + k,where h and k are real numbers and the vertex (the lowest or highest point of the parabola) is at (h, k). For a parabola whose axis is the x-axis and with vertex at the origin, the equation is y2 = 2px, in which p is the distance between the directrix and the focus.

The parabola’s vertex is also the point closest to the directrix; it is equidistant from the directrix and the focus. The line connecting the vertex and the focus is the axis of symmetry of the parabola. The line through the focus parallel to the directrix is the latus rectum (straight side). The parabola is symmetric about its axis, moving farther from the axis as the curve recedes in the direction away from its vertex.

History

The Greek mathematician Menaechmus (c. 380–320 bce) is credited with discovering the parabola as a conic section. Menaechmus used the parabola to solve geometric problems, most notably the problem of duplicating the cube (i.e., finding the cube root of two).

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The name “parabola” comes from the Greek word parabolē, meaning “comparison” or “application.” It is derived from para- (“alongside”) and bolē (“a throwing”). This term was coined by the Greek mathematician Apollonius of Perga (c. 240–190 bce). Apollonius studied conic sections and identified the parabola as a curve formed by the intersection of a plane and a cone.

Italian astronomer Galileo Galilei found through experiments that projectiles follow a parabolic path when air resistance is neglected. French philosopher René Descartes (1596–1650) invented the Cartesian coordinate system in his work La Géométrie (1637) and provided a way to represent the parabola algebraically.

The Editors of Encyclopaedia Britannica This article was most recently revised and updated by Erik Gregersen.