Understanding Parabolas with Focus and Directrix

When you first learn about parabolas, you usually think of them as the graph of a quadratic function like $y = ax^2 + bx + c$. However, in geometry, a parabola has a very specific definition based on distance.

The Geometric Definition

A parabola is the set of all points in a plane that are the exact same distance from a fixed point and a fixed line.

• Focus: The fixed point inside the curve of the parabola.
• Directrix: The fixed line outside the curve of the parabola.
• Vertex: The midpoint between the focus and the directrix. It is the turning point of the parabola.

The Standard Equation Using $4p$

To connect this geometric definition to algebra, we use a standard equation involving a value called $p$. The value $p$ is the directed distance from the vertex to the focus.

For a parabola with its vertex at the origin $(0,0)$:

• Vertical Parabola (opens up/down): $x^2 = 4py$
  • Focus: $(0, p)$
  • Directrix: $y = -p$
• Horizontal Parabola (opens left/right): $y^2 = 4px$
  • Focus: $(p, 0)$
  • Directrix: $x = -p$

Note: If $p > 0$, the parabola opens up or to the right. If $p < 0$, it opens down or to the left.

Example 1: Finding the Focus and Directrix

Problem: Find the focus and directrix of the parabola $y = \frac{1}{8}x^2$.

Solution:

1. Rearrange the equation to match the standard form $x^2 = 4py$. Multiply both sides by 8: $x^2 = 8y$
2. Set $4p$ equal to the coefficient of $y$: $4p = 8 \implies p = 2$
3. Because the $x$ term is squared, this is a vertical parabola. The vertex is at $(0,0)$.
4. The focus is at $(0, p)$, which gives $(0, 2)$.
5. The directrix is the line $y = -p$, which gives $y = -2$.

Example 2: Writing the Equation of a Parabola

Problem: Write the equation of a parabola with a focus at $(0, 3)$ and a directrix of $y = -3$.

Solution:

1. First, find the vertex. The vertex is exactly halfway between the focus $(0, 3)$ and the directrix $y = -3$. The midpoint is $(0, 0)$.
2. Next, find $p$. The distance from the vertex $(0,0)$ to the focus $(0, 3)$ is $3$. Since the focus is above the vertex, $p = 3$.
3. Since the directrix is horizontal and the focus is on the y-axis, the parabola opens vertically. We use the formula $x^2 = 4py$.
4. Substitute $p = 3$ into the formula: $x^2 = 4(3)y$ $x^2 = 12y$

You can also write this as $y = \frac{1}{12}x^2$.