Parabola
The shape of the graph of a quadratic function
A parabola is a type of curve that can be defined as a symmetrical U-shaped curve that appears in a variety of contexts, including mathematics, physics, engineering, and architecture. It is created by the intersection of a plane with a cone parallel to its side.
The parabola is characterized by the equation y = ax^2 + bx + c, where a, b, and c are constants. The value of a determines the shape of the parabola, with a positive value creating an upwards opening curve, and a negative value creating a downwards opening curve.
The vertex of the parabola is the point where it changes direction and is located at (-b/2a, c – b^2/4a). The axis of symmetry is the vertical line passing through the vertex, and the focus is a fixed point on the axis of symmetry that determines how the curve is formed.
The directrix is a fixed line perpendicular to the axis of symmetry and located at a distance of 1/4a units below the vertex. All points on the parabola are equidistant from the focus and the directrix.
Applications of parabolas include satellite dishes, headlights of cars, bridges, arches, and many others. One of the most famous parabolic structures is the Gateway Arch in St. Louis, Missouri, which is a 630-foot tall monument that uses parabolic curves to create its unique shape.
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