Problem 30 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Vertex Form of Parabola

Since the vertex of the parabola is at the origin (0,0), the general equation of a vertical parabola is \( y = ax^2 \). In this case, the parabola is vertical because the focus is above or below the vertex.

Step 2: Identify the Focus and its Relationship to Directrix

The focus of the parabola is given as \( F(0, -\frac{1}{2}) \). For a parabola with vertex at the origin and focus at \( (0, p) \), the equation is \( y = \frac{1}{4p}x^2 \). Thus, in this case, \( p = -\frac{1}{2} \) since the focus is directly on the y-axis below the origin.

Step 3: Substitute p into the Parabola Equation

We substitute \( p = -\frac{1}{2} \) into the equation \( y = \frac{1}{4p}x^2 \). This gives us: \[y = \frac{1}{4(-\frac{1}{2})}x^2 = \frac{1}{-2}x^2 = -\frac{1}{2}x^2.\]

Step 4: Write the Final Equation

After substitution, the final equation of the parabola becomes:\[y = -\frac{1}{2}x^2.\]

Key Concepts

Vertex Form of Parabola

When we talk about parabolas, the vertex form is a lovely way to express their equation. It centers around the vertex, which is the peak or the lowest point of the parabola depending on its orientation. The vertex form of a parabola is given by:

• \( y = a(x-h)^2 + k \)

where

• \((h, k)\) are the coordinates of the vertex, and
• \(a\) is a coefficient that determines the parabola's width and direction (upward if positive, downward if negative).

In many exercises, especially when working with simple vertical parabolas, the vertex might be at the origin, \((0, 0)\). In these instances, the equation simplifies to \( y = ax^2 \). This makes it easy to see how 'a' impacts the shape and orientation of the parabola.

Focus and Directrix

The focus and directrix play a pivotal role in defining a parabola. They highlight why a parabola looks the way it does:

• The **focus** is a point from which all points on the parabola are equidistant when considered alongside the directrix.
• The **directrix** is a line perpendicular to the axis of symmetry of the parabola.

For vertical parabolas with a vertex at the origin, the relationship becomes simple. If the vertex is at \((0,0)\) and the focus is \((0, p)\), the equation of the parabola is \( y = \frac{1}{4p}x^2 \). Just remember:

• Focusing further from the vertex makes the parabola wider.
• The directrix for a parabola with focus at \( (0, p) \) is \( y = -p \).

Knowing the focus and directrix allows us to understand the parabola's full geometric definition.

Vertical Parabolas

Vertical parabolas are simple as they have a direct relationship with the y-axis:

• They open either upwards or downwards.
• The vertex form directly shows the axis of symmetry as the y-axis.

The key equation, \( y = ax^2 \), is simple yet powerful. Here's what to remember:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

What really gives them a distinct behavior is the same value 'a'. It determines the "stretch" or "compression" of the parabola around its axis. By tinkering with 'a', one can control how open or narrow the parabola becomes. Therefore, understanding the vertex form helps in visualizing and graphically representing these vertical parabolas efficiently.