Parabola Equation

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., parabola forms the most basic curve of the conic section. In physics, middle school students are introduced with the concept of projectile motion that follows a parabolic path. Due to the uniqueness of parabolic figure parabolic reflectors find variety of applications in daily life that can be better understood by defining it. So, prior to knowing about the mathematical expression for parabola equation, let us know the definition of parabola.

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 (called focus) and a straight fixed line (called directrix). A parabola can be oriented either vertically in upward or downward direction and horizontally, either in left or right position. Parabolas are shaped like the letter U and their arms extend into the infinity.

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

Parabola Equation in Standard Form
While considering the standard form of parabola equation, one has to consider the axis of symmetry of the parabola. The standard equation of parabola 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.

Parabola with 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 vertex and focus.

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

For values of constant P < 0, the orientation of the parabola is downwards

Parabola with horizontal axis of symmetry:

(y - k)² = 4p(x - h)

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

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

Definition of Terms in Parabola Equations
To understand the concept of parabola in a better way, we must be aware of various definitions that play a crucial role in understanding the concept of parabola.

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 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 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 always equals to 4p, where p = distance between vertex and focus.

How to Find the Vertex of Parabola
To find the vertex of the parabola, 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 = (h= -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 parabola 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.