Problem 30 - Algebra and Trigonometry

Question

\(29-40\) 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 of the parabola is \( x^2 = -2y \).

Step 1: Understanding the Relation between Vertex, Focus, and Parabola

For a parabola with its vertex at the origin (0,0), the equation of a parabola can be expressed in the form \[ x^2 = 4py \]where the focus is given as \( F(0, p) \). In the given problem, the focus is \( F\left(0, -\frac{1}{2}\right) \). Thus, \( p = -\frac{1}{2} \).

Step 2: Substitute the Focus into Parabola Equation

Given the focus \( p = -\frac{1}{2} \), substitute \( p \) in the standard form equation of a parabola:\[ x^2 = 4py \]Therefore, substituting the value of \( p \), we have:\[ x^2 = 4(-\frac{1}{2})y \]

Step 3: Simplify the Equation

Simplify the equation derived:\[ x^2 = 4 \times -\frac{1}{2} \times y \]\[ x^2 = -2y \]This is the equation of the parabola with the vertex at the origin and focus at \( (0, -\frac{1}{2}) \).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial concept in understanding the overall shape and orientation of the curve. In simple terms, the vertex is the point where the parabola changes direction. For parabolas that open upwards or downwards, this point represents the minimum or maximum height of the curve, respectively.

When the vertex is at the origin,

• The vertex coordinates are (0, 0).
• It simplifies the equation of the parabola since the terms involving the vertex's position drop out.

In the equation of a parabola \[ x^2 = 4py \] or \[ y^2 = 4px \], setting the vertex at the origin means:

• The horizontal and vertical shifts (h, k) are zero.
• The parabola's axis of symmetry will go through the origin.

Focus of a Parabola

The focus of a parabola is a fixed point used to define the curve more precisely. It is important in helping determine how the parabola opens and how it behaves.

Key characteristics of the focus include:

• The distance from the vertex determines the parabola's width and the direction it opens.
• A parabola will always "open" towards its focus (upward, downward, left, or right).

For a parabola with its vertex at the origin,

• If the focus is at \( F(0, p) \), then the parabola opens upwards or downwards.
• If the focus is at \( F(p, 0) \), then the parabola opens left or right.

When you have a specific focus point like \( F(0, -\frac{1}{2}) \), it tells us:

• The parabola opens downward because \( p < 0 \).
• The equation will take the form \[ x^2 = 4py \].

Standard Form of a Parabola

The standard form of a parabola's equation is used to make calculations simple and the graph easy to visualize. The general forms are:

• \[ x^2 = 4py \] for a parabola that opens vertically.
• \[ y^2 = 4px \] for one that opens horizontally.

These equations show a direct link between the vertex and the focus:

• \( p \)ds the distance between the vertex and the focus.
• "4p" indicates how stretched or compressed the parabola is.

In our example, given the vertex at the origin and focus at \( F(0, -\frac{1}{2}) \), the standard form becomes: \[ x^2 = -2y \]. This equation confirms:

• The parabola opens downward, as the negative sign indicates.
• The vertex is at the origin (0,0), simplifying graph plotting.

By understanding these components, it becomes easier to transition from vertex and focus information to the parabola’s full equation.