The Parabola Equation

Kundan Pandey Feb 28, 2019
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Parabola equations form one of the most basic equations of the wide branch of conic sections in mathematics. Understanding the parabola equation can give you an edge in understanding various other shapes and figures in conics.
In the world of mathematics, geometry is one of the most complex and yet interesting branches. The pictorial representation of two dimensional and three dimensional figures generates keen interest in this branch of mathematics.
Amongst various conic sections in mathematics like hyperbola, pyramid, ellipse, etc., the parabola forms the most basic curve of the conic section.
In physics, middle school students are introduced to the concept of projectile motion that follows a parabolic path. Due to the uniqueness of this figure, parabolic reflectors find a variety of applications in daily life that can be better understood by defining it. So, let us first get to know its definition.

What is a Parabola?

In geometry, a parabola is defined as the locus of all those moving points that are always at a fixed distance from a point (focus) and a straight fixed line (directrix). It can be oriented either vertically, in upward or downward direction, and horizontally, either in left or right position. It is shaped like the letter U and its arms extend up to infinity.

Equation

Generally, there is no fixed equation for the parabola curve, as there can be a variety of changes and modifications of this equation. For instance, consider, ax2 + 2bx + c = 0, where a ≠ 0. This is nothing but a parabolic equation that is of degree two and reduced to three variables. What we must know is the most standard form of a parabola equation.

Equation in the Standard Form

While considering the standard form of a parabola equation, one has to consider its axis of symmetry. The standard equation for vertical and horizontal axis of symmetries will be different. In the standard form, a parabola with the vertex at point (h,k) has the following equations.
Vertical axis of symmetry:

(x - h)2 = 4p(y - k)

where, (h,k) is the vertex of the parabola, and p is the distance between the vertex and the focus.

For values of constant P > 0, the orientation of the curve is upwards

For values of constant P < 0, the orientation of the curve is downward
Horizontal axis of symmetry:

(y - k)2 = 4p(x - h)

For values of constant P > 0, the orientation of the curve is towards the right.

For values of constant P < 0, the orientation of the curve is towards the left

Definition of Terms

Axis of Symmetry

It is a line that cuts the parabola in two half mirror images. The axis of symmetry passes through the vertex and focus, and it is always perpendicular to the directrix. For the horizontal axis, the axis is given by x = h and for vertical axis, the equation is given by y = k.

Directrix

The straight line perpendicular to the parabola axis is called directrix. For vertical axis, the directrix is represented by the equation, y = k - p and for horizontal axis, the directrix is given by x = h - p.

Focus

Along with the directrix, latus rectum, and axis of symmetry, the focus also determines the orientation of the parabola. Focus is the intersection of the axis of symmetry and the latus rectum. For vertical axis of symmetry, focus is at (h, k+p). For horizontal axis of symmetry, the focus is at (h+p, k).

Latus Rectum

It is the line segment that passes from one end of the parabola to the other and it's numerical value is always equal to 4p, where p = distance between vertex and focus.

How to Find the Vertex of a Parabola

To find the vertex, one can take the following steps. Here, we will consider the example of a quadratic equation.
  • Note down the values of a, b, and c from the quadratic equations ax2 + bx + c.
  • In the equation, h= -b/2a, put the values of b and a.
  • h forms the x - coordinates of the vertex
  • Put h in the vertex form of equation, i.e., y = a(x - h)2 + c.
  • Solve for y and it will be equal to y = ah2 + bh + c.
  • The values x = -b/2a and the obtained value of y = ah2 + bh + c are the vertex of the quadratic equation. Therefore, (x , y) = (-b/2a, ah2 + bh + c )
The equations can be modified and represented in alternate form like ax2 + bx + cy + d = 0 or Ay2 + by + cx + d = 0, depending on the axis of symmetry. a, b, c and d are constants, where a and c ≠ 0.
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