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Introduction of Parabola

Parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. The other way to generate a parabola is to examine a point (the focus) and a line (directrix). The locus of points in that plane that are equidistant from both the line and point is a parabola. In algebra parabola is frequently encountered as graphs of quadratic functions, such as the line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point on the "axis of symmetry" that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. Parabolas can open up, left, down, right or in some other arbitrary direction. Any parabola can be rescaled and repositioned to fit exactly on any other parabola - that is, all parabolas are same.

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The parabola has many important applications from automobile headlight reflectors to the design of ballistic missiles. They are used in physics, engineering and so many other areas.

Equation of General Form of Conics

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

Check at the General Form for conics, an equation form will graph a Parabola when: 

  • Either A or C, the coefficients of x2 and y2, and it must equal zero
  • B, the coefficient of xy, must equal zero.
  • D, E and F must be equal any real values, D and E cannot both equal zero.

These equations all form parabolas on the coordinate x-y plane:

Some general forms of parabola in different coordinate plane

x2 - 2x + 36y + 28 = 0

4y2 - 50x - 75 = 0

9x2 + 4y - 36 = 0

To graph a parabola with mentioned equation in general form, we should first revise the equation into the Standard Form for Parabolas:

y - k = a(x - h)2     or      x - h = a(y- k)2

Where (h, k) is called the vertex of the parabola and  p =1/4a is known as the distance from the vertex to the focus & from the vertex to the directrix.  If the x-term is squared, the parabola opens either up or down and it y-term is squared the parabola opens to the right or left.

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