Overview:
A parabola is a conic section that forms an open curve. Graphs of quadratic functions have the shape of a parabola.
What Is A Parabola?
A parabola is the curve that is formed on either side of an axis of symmetry. If the parabola is vertical, a line to one side of the parabola (called the directrix) is vertical. If it is horizontal, the directrix will be horizontal. The axis of symmetry contains two named points, the focus of the parabola, which is usually inside the parabola, and the vertex, which is the point at the tip of the parabola, where it changes direction.
Why Is the Shape of a Quadratic Function Parabolic?
The equation of a parabolic function is a quadratic function. In general terms, the equation is ax2 + bx + c. If the vertex is at (h,k), the Distance Formula can be used can be derive a general equation, so that y = a(x-h)2 + k. Suppose the focus of a parabola is at (3, 5), and the directrix is y = 1. The axis of symmetry is at x =3. Using the equation for the vertex, the parabola will change direction at (3, 3). Its equation is (x-3)2+ 24 = 8y or x2 – 6x + 9 + 24 = 8y or (x2 – 6x + 33)/8 = y.
What Are Some Parabolic Shapes that Occur in Nature?
Some of the parabolic shapes that occur in nature include the trajectory of a moving body under a uniform gravitational field. For example, a baseball being thrown into the air at a low speed, or the center of gravity for a diver jumping off a diving board, without air resistance form a parabolic shape. The orbits of comets as they pass close to the Sun are also nearly parabolic.
How Is the Parabola Used in Calculation?
The parabola has many applications in calculation. For example, many bridges are constructed with parabolic arches, and it is a common architectural form. The focus points are also used to concentrate waves of sound in a parabolic antenna or in a parabolic mirror. Vertical curves in roads also form a parabolic design.
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