Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Write an equation for each parabola with vertex at the origin. Focus \(\left(-\frac{1}{2}, 0\right)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Standard Form

A parabola with a horizontal axis of symmetry and a vertex at the origin is described by the equation \( y^2 = 4px \), where \( p \) is the distance from the vertex to the focus.

Step 2: Identify the Distance to the Focus

From the vertex (0,0) to the focus \((-\frac{1}{2}, 0)\), the distance \( p \) is \(-\frac{1}{2} \), implying the parabola opens to the left.

Step 3: Substitute the Distance into the Equation

Substitute \( p = -\frac{1}{2} \) into the standard equation to get \( y^2 = 4(-\frac{1}{2})x \). This simplifies to \( y^2 = -2x \).

Step 4: Double-Check the Equation Orientation

Since \( p = -\frac{1}{2} \) is negative, the parabola indeed opens to the left, which matches the equation \( y^2 = -2x \).

Key Concepts

Vertex

In a parabola, the vertex is one of the most crucial elements. It represents the point where the parabola changes direction.
For parabolas centered at the origin (0,0), the vertex indicates whether the parabola opens up, down, left, or right.
In the standard parabola equation forms:

• Vertical opening: \( x^2 = 4py \)
• Horizontal opening: \( y^2 = 4px \)

Since the vertex is at the origin, it simplifies the equations to focus on the parameter \( p \), the distance to the focus.Identifying the vertex's location helps you determine the parabola's direction simply from its open orientation without complicated calculations.

Focus

The focus of a parabola is a critical point that defines its shape and size. It is not on the curve itself but rather a point such that distances from it to any point on the parabola are equal to distances from the point on the parabola to a directrix (a not explicitly visible line).
A parabola is defined as the set of all points equidistant from the focus and the directrix.
For our given problem:

• The focus is \( \left(-\frac{1}{2}, 0\right) \).
• This specific placement of the focus means that the parabola will open in the horizontal direction, either left or right, contingent on the value of \( p \).

Understanding where the focus lies is crucial for writing the accurate equation of a parabola, especially when it contributes to understanding whether the parabola opens left or right based on its position left or right of the vertex.

Conic Sections

Parabolas are one of the four types of conic sections, which also include ellipses, circles, and hyperbolas. Conic sections arise by cutting a right circular cone with a plane. Parabolas occur when the plane is parallel to the cone's side.
Key features that are used to categorize and define conic sections include:

• The vertex - the point where the conic section is widest or narrowest.
• The focus - a point used in the definition and formation of the shape.
• The directrix - a line used to define the conic section along with the focus.

Parabolas stand out because their distinction stems from having only one focus and one directrix. Every point on a parabola is equidistant from the focus and its directrix, which entirely shapes its U-like structure.

Distance from Vertex to Focus

The distance from the vertex to the focus, usually denoted as \( p \), directly impacts the equation and direction of a parabola.
This distance can be positive or negative, indicating which way the parabola opens.
In the problem you encountered:

• \( p = -\frac{1}{2} \)
• This negative value means that the parabola opens to the left, as seen in the equation \( y^2 = 4px \), which becomes \( y^2 = -2x \).

The distance value defines not just the interaction with the other elements of the parabola, like the axis of symmetry, but its entire orientation. This shows how crucial understanding \( p \) is, making it a pivotal parameter in parabola equations.