Problem 35 - Precalculus: Mathematics for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus: \(F\left(0,-\frac{3}{4}\right)\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y = -\frac{1}{3}x^2\).

Step 1: Understand the properties of a Parabola

A parabola with its vertex at the origin can have its equation in the form \(y = ax^2\) when it opens vertically. Since the focus is below the vertex, this parabola opens downward.

Step 2: Calculate the distance between the vertex and focus

The distance between the vertex \((0,0)\) and the focus \(F(0,-\frac{3}{4})\) is \(\frac{3}{4}\). For a parabola that opens vertically, this distance is related to the parameter \(p\) where the vertex is at \((h,k)\), and the equation can be \(x^2 = 4py\), with \(p = -\frac{3}{4}\) because the parabola opens downwards.

Step 3: Substitute the parameter into the standard equation

The standard form equation is \(x^2 = 4py\), where \(p = -\frac{3}{4}\). Thus, the equation becomes \(x^2 = 4(-\frac{3}{4})y = -3y\).

Step 4: Rearrange to derive the equation

Rearrange \(x^2 = -3y\) to find the form that can be easily equated to \(y\): \[x^2 = -3y \implies y = -\frac{1}{3}x^2\]. This is the equation of the parabola.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point that defines its shape and position. It is essentially the "tip" or the "turning point" of the parabola. When a parabola is at "rest," it can open upward or downward when its vertex is at the origin (0,0). But what does this mean regarding its geometrical representation and equation?
When dealing with a parabola whose vertex is at the origin, you often use the simplest form of a quadratic equation:

• For an upward or downward opening parabola: \(y = ax^2\).
• For a parabola that opens sideways: \(x = ay^2\).

In our original exercise, the parabola has its vertex at the origin and a focus below it, indicating that the parabola opens downward. The vertex is pivotal because it sets the groundwork for where the parabola will sit in the coordinate plane. Moreover, by understanding the vertex, it allows easier modifications and translations to the parabola as needed.

Focus of a Parabola

The focus of a parabola is a unique point that helps in defining the parabola's shape and position along with the vertex. The official definition says that any point on the parabola is equidistant from the focus and a line called the directrix.
With the focus at \(F(0, -\frac{3}{4})\), it is vital to understand how this point positions the parabola. A crucial parameter involved is \(p\), which is the distance between the vertex and the focus. This parameter helps in forming the equation of the parabola.

• If the focus is above the vertex, the parabola opens upward.
• If the focus is below, it opens downward.

For the given problem, since the focus \(F\) is at \(0, -\frac{3}{4}\), the parabola opens downward, and \(p = -\frac{3}{4}\). The negative sign denotes the downward direction. This focus ensures that the shape and curvature of the parabola are distinct and consistently formed.

Standard Form of a Parabola

The standard form for the equation of a parabola helps in understanding its orientation and size. With the vertex at the origin, the standard equation for a parabola can either be \(x^2 = 4py\) or \(y^2 = 4px\).
In our exercise, the focus is placed vertically from the vertex, which positions the parabola to open vertically. This positions the standard form in the format \(x^2 = 4py\), where \(p\) is the distance from the vertex to the focus. Substituting \(p = -\frac{3}{4}\) from our problem gives:

• \(x^2 = 4(-\frac{3}{4})y\)
• which simplifies to \(x^2 = -3y\).

This is then rearranged to give the equation \(y = -\frac{1}{3}x^2\), making it easier to sketch and analyze. The standard form is essential when translating the geometric idea into a solvable mathematical equation and forms the backbone of variety in custom problems based around parabolas.