Parabola

Understanding Parabola

Have you ever thrown a basketball to the hoop? Or seen a water fountain spraying water into the air? The trajectory formed by these movements creates a very special curve in mathematics, called a parabola.

A parabola is not just an ordinary curve. Its unique shape makes it very useful in real life. Satellite antennas are parabolic in shape to capture signals, spotlights use parabolic reflectors to focus light, and even bridges and modern architecture often use parabolic arches for their strength.

What's interesting about parabolas is their perfect reflection property. All rays that come parallel to the parabola's axis will be reflected toward one point called the focus. This is why parabolas are so effective for collecting or transmitting energy.

Mathematical Definition

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.

Imagine you have one point (focus) and one straight line (directrix). Now find all points whose distance to the focus equals their distance to the directrix. The collection of these points is what forms a parabola!

This definition gives us a systematic way to understand parabolas. No matter what the orientation or position, as long as it satisfies the condition of equal distance to the focus and directrix, then it is a parabola.

This unique property of parabolas makes every point on the curve have a perfect "balance" of distance between the focus and directrix.

Standard Parabola Equations

Let's start with the simplest parabola: a parabola with vertex at the origin $O(0,0)$. There are four basic standard parabola forms depending on the direction of the curve's opening.

The four standard parabola equation forms are:

The value $p$ in these equations represents the distance from vertex to focus. The larger the value of $p$, the more "open" the parabola becomes.

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

Remember that the sign of $p$ determines the opening direction: positive for right/up, negative for left/down.

Parabola with Arbitrary Vertex

In real applications, parabolas don't always center at the origin. A parabola can have its vertex at any point $(h, k)$. This is the general form of parabola.

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 parabola's opening direction. Positive for up/right, negative for down/left.

Determining Parabola Elements

When given a parabola equation, we can identify all its important elements systematically. The first step is to recognize the parabola's orientation:

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

Let's see how to do this with an example.

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 $p = 2$

Therefore, the parabola elements are:

From this analysis we obtain:

Example Problems

Let's work through some problems to deepen our understanding of parabolas.

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:

Therefore: