Determining the Equation of a Parabola with Given Focus and Vertex

Creating a parabola equation given its focus and vertex is a classic problem in mathematics, crucial for understanding the geometric properties of parabolic shapes. In this guide, we will explore the process step-by-step using a specific example where the focus is at (-6, -6) and the vertex is at (-2, -2).

Identifying the Orientation

The first step in determining the equation of a parabola based on its focus and vertex is to identify the orientation of the parabola. The orientation helps us understand how the parabola will open relative to the given points.

In our example, the focus is at (-6, -6) and the vertex is at (-2, -2). Since the x-coordinate of the focus (-6) is to the left of the x-coordinate of the vertex (-2), the parabola opens to the left. This step is crucial in choosing the correct form of the parabola's equation.

Calculating the Directrix

The directrix is a key component in understanding the geometric properties of a parabola. It can be found using the relationship between the vertex, the focus, and the directrix.

The distance from the vertex to the focus (denoted as p) is:

p sqrt{(-6 - (-2))^2 (-6 - (-2))^2}

Plugging in the coordinates:

p sqrt{(-6 2)^2 (-6 2)^2} sqrt{(-4)^2 (-4)^2} sqrt{16 16} sqrt{32} 4sqrt{2}

Since the parabola opens to the left, the directrix will be a vertical line located p units to the right of the vertex:

x -2 4sqrt{2}

Formulating the Equation of the Parabola

The standard form of a parabola that opens to the left is given by:

(y - k)^2 -4p(x - h)

Where (h, k) is the vertex of the parabola. For our example, the vertex is (-2, -2), and p 4sqrt{2}.

Substituting these values into the equation:

(y - (-2))^2 -4(4sqrt{2})(x - (-2))

Simplifying the equation:

y 2^2 -16sqrt{2}(x 2)

Therefore, the equation of the parabola is:

y 2^2 -16sqrt{2}x - 2^2

Summary and Additional Insights

By following these steps, we can successfully determine the equation of a parabola given its focus and vertex. This process is essential in various fields such as physics, engineering, and architecture, where understanding the properties of parabolic shapes is crucial.

For further exploration, you can apply this method to other example problems, such as finding the equation of a parabola when the focus is at (a, 0) and the vertex is at (0, 0), and then normalizing the lines passing through these points.

Understanding how to find the equation of a parabola is not just an academic exercise; it has practical applications in real-world scenarios, making it a valuable skill.