How to Find the Equation of a Parabola with Vertex at the Origin and Focus at (0, -5/4)

Understanding the relationship between the vertex and the focus of a parabola can be crucial in determining its equation. This guide will walk through the steps required to find the equation of a parabola with a vertex at the origin (0, 0) and a focus at (0, -5/4).

Understanding the Basics

A parabola is a conic section defined by specific geometric properties. Its equation in standard form is often given as (y ax^2). Thevertex of a parabola, which is the point where the parabola changes its direction, can be located at the origin (0, 0) in some cases. The focus, on the other hand, is a fixed point from which the distances to the points on the parabola are measured according to specific rules.

The Relationship Between Vertex and Focus

Given that the vertex of the parabola is at the origin (0, 0), and the focus is at (0, -5/4), we need to determine the value of the parameter (a) (which determines the shape and orientation of the parabola).

Deriving the Equation

The standard form of a parabola with a vertical axis (the y-axis in this case) is given by the equation:

(y ax^2)

The vertex form of a parabola is:

(y a(x - h)^2 k)

Where ((h, k)) is the vertex of the parabola. Since the vertex is at (0, 0), this simplifies to:

(y ax^2)

The focus of a parabola with a vertex at the origin and a vertical axis is located at ((0, p)), where (p) is the distance from the vertex to the focus. For a parabola (y ax^2), the relationship between (a) and (p) is given by:

(p frac{1}{4a})

In this case, the focus is at (0, -5/4), so:

(p -frac{5}{4})

Substituting (p -frac{5}{4}) into the equation (p frac{1}{4a}) gives:

(-frac{5}{4} frac{1}{4a})

Solving for (a):

(a -frac{1}{5})

Thus, the equation of the parabola is:

(y -frac{5}{4}x^2)

Explanation

The process involves using the known coordinates of the focus to find the value of (a), which then helps in determining the equation of the parabola. The derived equation (y -frac{5}{4}x^2) represents a parabola that opens downwards, as the coefficient of (x^2) is negative.

Additional Information

It's important to note that the orientation of the parabola and the position of the focus determine the form of the equation. The vertex form (y a(x - h)^2 k) is particularly useful for understanding the location of the vertex and the orientation of the parabola.

Conclusion

By carefully analyzing the given information and using the relationship between the vertex and the focus, we can determine the equation of a parabola. This process is crucial for solving various problems in geometry and calculus.