A quadratic function f can be visualized by drawing its graph (parabola) in a (two-dimensional) coordinate system. The function graph of a quadratic function f can be defined mathematically as the set of all pairs of elements ( x | y ) for which y = f (x). The graph of the continuous quadratic function on a continuous interval forms a continuous curve.

f(x) = a⋅x^{2} + b⋅x + c

A parabola is a special kind of function described by a quadratic equation. A parabola has the shape of a symmetrical U-shape, which is called the basic shape of a parabola. The highest or lowest point of the parabola is called the vertex, and the x-axis where the parabola intersects the x-axis is called the axis of symmetry. The parabola also has two special points: the vertex S (h, k) and the focal point F (h, k+p). A parabola has many applications in various fields such as physics, engineering, architecture and astronomy. Examples are the flight of projectiles, parabolic mirrors in telescopes and microwave antennas.

The function plotter draws the function graphs of the real quadratic function. The derivative can be drawn with (d/dx) as dotted line in the graph. The integral can be started with select ∫. The integration range can adjusted with variation of the points at the function graph.

Equation

Derivative

Zeros

Vertex

The parabola is defined by the set of all points for which the distance from the focal point (marked F in the diagram) to the parabola is equal to the perpendicular distance from the guide line (green line in the diagram) to the parabola.

The black slider illustrates the course of a parallel beam, which is reflected at the tangent (dashed) to the focal point.

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Focal point

Equation

Distance focal point to parabola

Distance parabola to guide line

The focal point representation of a parabola is a special representation of a parabola that focuses on the focal point (F) and the distance of the focal point from the axis of symmetry (p). This representation makes it possible to describe and analyze a parabola more quickly and easily. The focal point representation of a parabola has the form: (x-h)^2 = 4p (y-k) where (h, k) is the focal point of the parabola and p is the distance of the focal point from the symmetry axis of the parabola. One can calculate the focal point representation back to the canonical form y = a(x-h)^2 +k by transforming the equation and determining the constants.

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