How to Find the Focus and Directrix of a Parabola

TL;DR

To find the focus and directrix of a parabola, identify the value of p, which is the distance from the vertex to the focus or directrix. The equation of the directrix will be x = h - p (for horizontal parabolas) or y = k - p (for vertical parabolas). The length of the lattice rectum is always 4p, providing essential points for accurately graphing the parabola.

Transcript

in this video we're going to focus on parabolas so let's talk about some equations that you need to know so hopefully you have a sheet of paper with you and a pen to write down some notes so for the parabola on the left this corresponds to the equation y squared is equal to 4px and you would use this if the vertex of the parabola is the origin for ... Read More

Key Insights

• 📌 Parabolas can be described by various equations depending on the location of the vertex, focus, and directrix.
• 🗨️ Positive p values indicate an opening to the right, while negative p values indicate an opening to the left.
• ❓ The length of the lattice rectum is 4p, and its endpoints can be used to sketch the parabola accurately.

Questions & Answers

Q: What is the equation for a parabola if the vertex is at the origin and the focus is to the right?

The equation is y^2 = 4px, where p is the distance between the vertex and the focus. The directrix is given by x = -p.

Q: How does the equation change if the parabola opens downward?

If the parabola opens downward, the equation becomes y^2 = -4px, with a negative value of p indicating the direction of the opening.

Q: How do you calculate the length of the lattice rectum?

The lattice rectum has a length of 4p, where p is the focal distance. It is the segment connecting the parabola's curve and passing through the focal point.

Q: What is the standard form of the equation for a parabola with a given focus and directrix?

The equation is written as (x - h)^2 = 4p(y - k), where (h, k) represents the vertex coordinates, p is the focal distance, and the focus is at (h + p, k).

Summary & Key Takeaways

• The video introduces equations for parabolas with the vertex at the origin and the focus to the right or left.

• The focal distance (p) determines the location of the focus and directrix relative to the vertex.

• The lattice rectum is the segment passing through the focal point and perpendicular to the axis of symmetry.