How to Find the Equation of a Parabola Given the Focus and a Horizontal Directrix

TL;DR

Given focus (3,4) and directrix y=-2, find parabola equation opening upwards: (x-3)^2 = 12(y-1).

Transcript

hi everyone in this problem we're being asked to find the equation of the parabola given the focus and the directrix so problems like this the way i like to do it is to draw a little picture and from the picture we can usually figure out you know the question let's go ahead and plot everything we have so here's the y-axis and here is the x-axis so ... Read More

Key Insights

• 🆘 The focus and directrix help determine the vertex and orientation of a parabola.
• 🟨 The equation for a parabola opening upwards uses x squared, while left/right openings utilize y squared.
• 😀 The value c in the equation influences the focal length and curvature of the parabola.
• 🤩 Understanding the relationship between focus, directrix, vertex, and opening direction is key in identifying parabolas.
• 🛟 The midpoint between focus and directrix serves as the vertex for a parabola.
• 😀 Solving for c involves measuring the distance between the vertex and focus.
• 👈 The equation (x-h)^2 = 4c(y-k) is crucial in determining the mathematical representation of a parabola.

Questions & Answers

Q: How do you determine the vertex of a parabola given the focus and directrix?

The vertex is the midpoint between the focus and directrix, found by counting the distance and directions appropriately.

Q: Why does the equation for the parabola have x squared, not y squared?

Since the parabola opens upwards, the equation uses x squared; parabolas opening left/right use y squared.

Q: How is the value of c determined in the parabola equation?

The absolute value of c is the distance between the vertex and focus, with the sign depending on the direction of opening.

Q: What does the value of c signify in the equation (x-h)^2 = 4c(y-k)?

The value of c affects the focal length and curvature of the parabola, crucial in determining its shape and orientation.

Summary & Key Takeaways

• Focus at (3,4) and directrix y = -2.

• Vertex at (3,1), parabola opens upwards.

• Equation: (x-3)^2 = 12(y-1).