How do you determine the equation of a parabola given its vertex and focus?

To find the equation of a parabola when you know its vertex and focus, you need to start by understanding the basic properties of parabolas.

A parabola is defined as the set of all points that are equidistant from a fixed point called the focus and a fixed line known as the directrix.

The standard form of a parabola’s equation can vary depending on its orientation:

• Vertical Parabola: If the parabola opens upwards or downwards, the standard form is:

(x – h)² = 4p(y – k)

• Horizontal Parabola: If the parabola opens to the right or left, the standard form is:

(y – k)² = 4p(x – h)

Here, (h, k) represents the vertex of the parabola and p is the distance from the vertex to the focus. The sign of p determines the direction in which the parabola opens:

• If p is positive, the parabola opens upwards (for vertical) or rightwards (for horizontal).
• If p is negative, the parabola opens downwards (for vertical) or leftwards (for horizontal).

### Steps to Find the Equation:

1. Identify the Vertex and Focus: Assume the vertex is at point (h, k) and the focus at point (h, k + p) for a vertical parabola or (h + p, k) for a horizontal parabola.
2. Calculate p: Determine the distance p between the vertex and the focus. This is simply the y-coordinate of the focus minus the y-coordinate of the vertex for vertical parabolas, or the x-coordinate of the focus minus the x-coordinate of the vertex for horizontal parabolas.
3. Plug Values into the Standard Form: Substitute the values of (h, k) and p into the respective standard form equation of the parabola.

### Example:

Let’s assume the vertex is at (2, 3) and the focus is at (2, 5).

• The vertex (h, k) = (2, 3).
• The focus (h, k + p) = (2, 5), which gives us p = 5 – 3 = 2.

In this case, since the focus is above the vertex, the equation of the parabola is:

(x – 2)² = 8(y – 3)

Thus, the equation of the parabola with the given vertex and focus has been successfully identified!