The 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.

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.

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.

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.

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

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.

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.

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).

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.

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.