Problem 31 - Precalculus: Mathematics for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus: \(F(0,6)\)

Step-by-Step Solution

Answer

The equation is \( x^2 = 24y \).

Step 1: Understanding the Parabola

A parabola with a vertex at the origin (0, 0) and a focus at (0,6) is oriented vertically since the focus lies on the y-axis. Therefore, the equation of the parabola will be in the form (x^2 = 4py), where p is the distance from the vertex to the focus.

Step 2: Identifying the Value of p

The focus is located at (0,6), so the distance (p) from the vertex to the focus is 6. Therefore, we have p = 6. This value will help us determine the exact equation of the parabola.

Step 3: Substituting the Value of p

Now that we have p = 6, we substitute this into the standard parabola equation form (x^2 = 4py) to get:\[ x^2 = 4(6)y \]or simplified as:\[ x^2 = 24y \].

Key Concepts

Vertex Form of a Parabola

When dealing with parabolas, the vertex form of a parabola is a convenient way to express its equation. It can be helpful for easily identifying key characteristics such as the vertex and orientation. The vertex form of a parabola's equation is generally written as

• \( y = a(x-h)^2 + k \)

where

• \((h, k)\) are the coordinates of the vertex of the parabola,
• \(a\) is a coefficient that affects the direction and width of the parabola.

In vertex form, if \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
Since our particular problem involves a parabola with a vertex at the origin \((0,0)\), the vertex form simplifies to \(y = ax^2\).
This makes it easier to compute other related features of the parabola, such as its focus or direction.

Focus of a Parabola

A parabola is a symmetrical open curve with an important point called the focus. The focus is a point from which distances are measured in a parabola so that it maintains its unique shape.
For a parabola oriented vertically (opening either up or down), the standard equation is

• \(x^2 = 4py\)

where \(p\) represents the distance from the vertex to the focus, and it is measured along the axis of symmetry.
In the original problem, our parabola has its vertex at the origin \((0,0)\) and its focus at \((0,6)\). This indicates that the parabola opens upwards along the y-axis, with a focal distance, \(p\), of 6.
The presence of the focus not only defines the shape but also guides us in writing the equation of the parabola. By knowing the focus, we can determine how the parabola will open and how stretched or compressed it will appear.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane at different angles. They include various shapes such as circles, ellipses, hyperbolas, and parabolas. Among these, the parabola is a unique conic section that exhibits distinct symmetry around a central axis.
The general parabola appears as a 'U' or an inverted 'U' shape depending on its orientation.

• It can open either vertically (up or down) or horizontally (left or right).
• The parabola intersects a plane such that it produces a curve with one main axis of symmetry and this symmetry is defined by the location of its focal point.

In our specific example, we covered a parabola where the focus determines how steep or wide the parabola will be. The focal point is crucial as it guides and dictates the parabola's geometry.
Understanding conic sections is fundamental to analyzing and understanding the properties of a parabola. This knowledge assists in visually and mathematically comprehending not only parabolas but other conic shapes in the field of mathematics and physics.