Focus and Directrix of a Parabola

(Videos) Directrix and Focus

by PatrickJMT

Here's a video by patrickJMT showing you how to find the focus and directrix of a parabola. Below are some more examples by him if you need more practice.

###$\underline{\textbf{Focus and Directrix}}$

One important quality that defines a parabola is the fact that all points that line along it are the same distance away from a $\textit{point}$ called the $\underline{\textbf{focus}}$ and $\textit{line}$ called the $\underline{\textbf{directrix.}}$ Another way to think of it is that a parabola is the locus of points equidistant from both the focus and directrix.

###$\underline{\textbf{How to Find the Focus and Directrix}}$

The first think to note is that the focus lies on the axis of symmetry of the parabola a certain distance (a fixed point) away from its vertex (this distance is denoted $p$ and it will be defined below). The directrix, on the other hand, is a line that lies perpendicular to the line of symmetry, and is also a certain $p$ distance away from the vertex. So, the coordinates of the vertex will be an important tool we will use to help us determine the location of the focus and directrix. Additionally, it is also important that we factor in the direction the parabola opens. A parabola can open left or right(on the $x$-axis), or up or down (on the $y$-axis). Which one it is will change how we determine the focus and directrix.

The three things we must know are:

1. $\textit{Coordinate of the Vertex:}$ $(h, k)$, where $h$ is the $x$-coordinate of the vertex and $k$ in the $y$-coordinate of the vertex
2. $\textit{Direction Parabola Opens:}$ along $x$-axis or along $y$-axis
3. $\textit{Value of p}:$ which is the distance from the vertex

Knowing this information, we can now move on to locating the focus and directrix!

$\underline{\text{Case I: Opens on Y-axis}}$

You may recognize the above graph as that of a quadratic equation! From previous lessons, we know that the axis of symmetry for the above quadratic function is parallel to the $x$-axis. The vertex is on the origin.

When we have a parabola that opens along the direction of the $y$-axis, the equation for it can we written in the form, $$(x - k)^2 = 4p(y - k)$$ This is called vertex form. By writing the equation in this form, we can single out the value of both $p$ and the vertex and be able to use them to determine the focus and directrix. When the parabola has a $y$-axis line of symmetry, $p$ will be found on the same side of the vertex form equation as $y$.

In this case,

• $\textbf{focus:}$ $(h, k + p)$
• $\textbf{directrix:}$ $y = k - p$

$\underline{\text{Case II: Opens on X-axis}}$

When we have a parabola the opens along the direction of the $x$-axis, the equation for it can we written in the form, $$(y - k)^2 = 4p(x - h)$$ This is still vertex form, except in this case $p$ will be located on the same side of the equation as $x.$

In this case,

• $\textbf{focus:}$ $(h + p, k)$
• $\textbf{directrix:}$ $x = h - p$

Parabola: Focus and Directrix

A parabola is a curved shape that looks like a U. It can open up, down, left, or right. Every parabola has a point called the focus and a line called the directrix. These concepts build off the center and foci of conics. This concept will be helpful in solving word problems with parabolas.

Take a look at the graph above. We know that the vertex is the point in the middle of the curve, which lies on the axis of symmetry. The vertex is between the focus and directrix - a parabola is made up of all points that are the same distance from the focus and the directrix!

The focus is always inside the curve, and the directrix is outside the curve. The closer the focus is to the vertex, the narrower the parabola will be.