Problem 26 - Algebra and Trigonometry

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F\left(0,-\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The equation is \( x^2 = -2y \).

Step 1: Understanding the Problem

We need to find the equation of a parabola with the vertex at the origin (0,0) and focus at \( F(0, -\frac{1}{2}) \). For a parabola, the focus and vertex give us all the information necessary to write its equation.

Step 2: Choosing the Parabola Orientation

Since the vertex is at the origin and the focus \( F \) is below the vertex, the parabola opens downward. This suggests the equation is of the form \( x^2 = -4py \), where \( p \) is the distance from the vertex to the focus.

Step 3: Calculating the Distance \( p \)

The distance \( p \) from the vertex \((0,0)\) to the focus \( F(0, -\frac{1}{2}) \) is \( \frac{1}{2} \). Thus, \( p = \frac{1}{2} \).

Step 4: Substituting \( p \) into the Equation

Substitute \( p = \frac{1}{2} \) into the equation \( x^2 = -4py \) to get \( x^2 = -4 \left( \frac{1}{2} \right) y \), which simplifies to \( x^2 = -2y \).

Step 5: Finalizing the Parabola Equation

The equation of the parabola that satisfies the given conditions is \( x^2 = -2y \), which represents a parabola opening downward with its vertex at the origin and focus at \( F(0, -\frac{1}{2}) \).

Key Concepts

Vertex Form

The vertex form of a parabola provides an insightful way to understand and express the equation of a parabola, especially when you know the coordinates of its vertex. In this exercise, the vertex is located at the origin, specifically at

• (0,0).

Hence, the equation of the parabola in vertex form can be simplified greatly. The general vertex form of a parabola is \[ y = a(x-h)^2 + k \] where

• \( (h, k)\) represent the coordinates of the vertex,
• and \(a\) is a constant that affects the parabola's width and direction.

For a parabola with its vertex at the origin, the equation simplifies to \[ y = ax^2 \]. In this context, as the focus determines the opening direction and distance, the constant \(a\) will be influenced by these parameters. When the parabola opens downwards, as is the case in this problem, it signifies \(a < 0\).

Focus of a Parabola

The focus of a parabola is a crucial point that, together with the vertex, defines the unique shape and orientation of a parabola. In this problem, the focus is given at

• \( F(0, -\frac{1}{2})\).

The focus helps in determining how "wide" or "narrow" the parabola is, by providing the distance from the vertex that guides the curvature. Every point on the parabola is equidistant from the focus and a particular line called the directrix. Understanding the focus is essential because:

• The parabola will always curve around the focus.
• This point lies "inside" the parabola.
• The line of symmetry of the parabola passes through the vertex and the focus.

In this exercise, knowing the vertical location of the focus aids in deriving the equation by determining the value of \(p\) in the parabola's equation \( x^2 = -4py \).

Parabola Orientation

The orientation of a parabola indicates in which direction the parabola opens, whether it is upward, downward, leftward, or rightward. In this exercise, the parabola has its vertex at the origin,

• and the provided focus is below the vertex at \(F(0, -\frac{1}{2})\).

This orientation informs us that the parabola must open downwards. The direction of a parabola is determined by the position of the focus relative to the vertex.

• If the focus had been situated above the vertex, the parabola would have opened upwards.
• Here, since the focus is below the vertex, the parabola adopts a downward orientation noted in the equation \(x^2 = -4py\).

Thus, the equation unveiling the parabola's orientation is configured with a negative sign in front of \(4p\), indicating the downward pathway of the shape.

Distance to Focus

The distance to the focus, represented as \(p\), is a determining factor in forming the parabola's equation. It measures the length between the vertex and the focus and deepens understanding of how "spread out" or "tight" the parabola appears. In this example, the distance from the vertex

• \((0,0)\) to the focus \(F(0, -\frac{1}{2})\) is calculated as \(\frac{1}{2}\).

This value is essential for substituting into the standard form of the parabola equation \(x^2 = -4py\).Some key points regarding this distance:

• The sign of \( p\) informs the orientation of the parabola.
• A smaller \( p \) results in a narrower parabola.
• A larger \( p \) makes the parabola "wider."

Thus, when substituting \(p = \frac{1}{2}\) into \(x^2 = -4py\), we derive \(x^2 = -2y\), defining the parabolic curve with its unique spread and orientation. This showcases how crucial the distance to the focus is in defining the full attributes of the parabola.