Parabola

Parabolic Paths and Reflectors

If air resistance is neglected and gravity is treated as constant, the flight of a basketball follows a parabolic path. Water leaving a fountain at an angle follows the same model while those assumptions are reasonable.

Its shape is also useful in technology. Satellite dishes collect signals, spotlights direct light, and parabolic arches appear in bridges and modern architecture.

The key feature is the reflection property. Rays arriving parallel to the axis are reflected toward the focus. Conversely, rays emitted from the focus leave a parabolic reflector parallel to the axis. This is why parabolic shapes can collect or direct energy effectively.

Parabola from a Fixed Point and a Fixed Line

Mathematically, a parabola is the locus of points that are equidistant from a fixed point and a fixed line. The fixed point is called the focus, while the fixed line is called the directrix.

Begin with one point, the focus, and one line, the directrix. Every point whose distance to the focus equals its perpendicular distance to the directrix lies on the parabola.

This definition works for every position and orientation. The defining condition is always the same: equal distance to the focus and the directrix.

Every point on a parabola is equally distant from the focus and the directrix.

Standard Parabola Equations

Start with a parabola whose vertex is at the origin $O(0,0)$. Its opening direction produces four standard forms.

The two aligned standard equations use a signed parameter $p$:

If $p > 0$, the parabola opens right or up. If $p < 0$, it opens left or down. The distance from the vertex to the focus is $|p|$, and a larger $|p|$ produces a wider parabola.

For example, for the parabola $y^2 = 4px$ (opens rightward):

Parabola with Arbitrary Vertex

A parabola does not have to lie at the origin. When its vertex is $(h, k)$, we use a shifted standard form.

General form of parabola equations with vertex at $(h, k)$:

For vertical parabola $(x-h)^2 = 4p(y-k)$:

For horizontal parabola $(y-k)^2 = 4p(x-h)$:

The sign of $p$ determines the opening direction. In the horizontal form, positive means right and negative means left. In the vertical form, positive means up and negative means down.

Determining Parabola Elements

The equation of a parabola determines its orientation, vertex, focus, directrix, and axis of symmetry. Start by identifying the orientation:

• If variable $x$ is squared → vertical parabola (opens up/down)
• If variable $y$ is squared → horizontal parabola (opens left/right)

In the following example, we complete the square to rewrite the equation in standard form. The standard form then gives the vertex, focus, directrix, and axis of symmetry.

Example: Given a parabola with equation $x^2 - 4x - 8y + 12 = 0$

The first step is to convert to standard form by completing the square:

From the form $(x-2)^2 = 8(y-1)$, we identify:

• $h = 2$, $k = 1$
• $4p = 8$, so $p = 2$

The parabola elements are:

From this analysis we obtain:

The parabola opens upward because $p=2>0$.

Finding Parabola Equations Vertices and Foci

Two worked problems now connect the standard form to the vertex and focus.

Problem 1: Determine the equation of a parabola that has vertex at $(1, -2)$ and focus at $(4, -2)$.

Solution:

Since the vertex and focus have the same $y$-coordinate, this parabola is horizontal.

From the focus condition, we get $h + p = 4$. Substituting $h = 1$:

The equation for a horizontal parabola is $(y-k)^2 = 4p(x-h)$:

Problem 2: For parabola $y^2 - 6y - 4x + 13 = 0$. Determine the coordinates of the vertex and focus.

Solution:

Complete the square for variable $y$:

From the form $(y-3)^2 = 4(x-1)$, we identify:

The calculation gives: