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⋅x2 + 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.

## Parabola graph

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.

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## Parabola definition using focal point and guide line

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
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## Focal point representation of a parabola

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.

### Screenshot of the quadratic function plot

Print or save the image via right mouse click.

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