Problem 8 - Algebra 2

Question

Write an equation of a parabola with a vertex at the origin and the given focus. focus at \((0,-4)\)

Step-by-Step Solution

Answer

The equation of the parabola with vertex at the origin and focus at (0, -4) is \(x^2 = -16y\).

Step 1: Determine the Focus

The focus of the parabola is given as (0, -4). The x-coordinate of the focus is 0 and the y-coordinate is -4. Therefore, the focal length 'p' is -4.

Step 2: Substitute 'p' into the equation

Substitute 'p' value into the standard equation \(x^2 = 4py\). Here, \(x^2 = 4*(-4)y\).

Step 3: Simplify the equation

After substituting 'p' into the equation, simplify it to get the final equation of the parabola. The equation simplifies to \(x^2 = -16y\).

Key Concepts

Vertex Form of Parabola

The vertex form of a parabola is an important way to express a quadratic equation. It focuses on the vertex of the parabola, which is its peak or lowest point. This form is given by \( y = a(x - h)^2 + k \), where:

• \( (h, k) \) is the vertex.
• \( a \) determines the parabola's width and direction (upwards if positive, downwards if negative).

For parabolas with a vertex at the origin, like in our example, this equation simplifies to \( y = ax^2 \). If the parabola opens vertically, like \( x^2 = 4py \), this is related to conic sections, which is a more advanced topic connecting geometry with algebra.
The vertex form helps to easily identify key features: direction, width, and position. This makes graphing and understanding the parabola's shape more intuitive.

Focus of a Parabola

The focus of a parabola is a fixed point that defines its shape and symmetry. For vertical parabolas, this point lies along the axis of symmetry.

• The standard form \( x^2 = 4py \) describes a parabola centered at the origin, with its focus either above or below the vertex.
• In this equation, \( p \) represents the distance from the vertex to the focus.

In our example, the focus is at \((0, -4)\), meaning the parabola opens downwards. The focal length \( p = -4 \) helps us find this using the formula. This distance is crucial for forming the precise shape and direction.
The focus isn't just a point; it is key to understanding how parabolas reflect and concentrate things like light or sound in real-world applications.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. These include circles, ellipses, hyperbolas, and parabolas. Each differs based on how the intersecting plane cuts through the cone.

• Parabolas form when the plane is parallel to the cone's side.
• They have unique properties, including a focus and a directrix.

In the equation \( x^2 = 4py \), the value of \( p \) not only tells us where the focus is but also helps in constructing tangents and understanding the parabola's overall curve.
Parabolas are common in nature and technology, seen in satellite dishes and car headlights due to their reflective properties. Understanding conic sections hones analytical skills and provides insights into both mathematics and its practical applications.