Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. $$\text { Focus: }\left(-\frac{3}{2}, 0\right)$$

Step-by-Step Solution

Answer

The standard form of the equation of the parabola with the given characteristics is \(y = -\frac{1}{6}x^2\).

Step 1: Identify the Given Values

Starting with the given focus, it is located at \(-\frac{3}{2}, 0\). The x-coordinate of the focus is negative, which suggests that the parabola opens towards the negative x direction.

Step 2: Identify the value of p

The value of \(p\) is given by the x-coordinate of the focus, which is \(-\frac{3}{2}\). This is the distance from the vertex at the origin (0, 0) to the focus.

Step 3: Substitute the value of p into the standard parabolic equation

Use the standard equation for a parabola which is \(4py = x^2\), and substitute the value of \(p\), we get \(4(-\frac{3}{2})y = x^2\), or simply, \(-6y = x^2\). After rearranging the equation to the standard form, we get \(y = -\frac{1}{6}x^2\).

Key Concepts

Standard Form Equation

A parabola's standard form equation is a specific algebraic representation of its curve. This form helps us define the parabola using constants and variables, making it easier to understand the parabola's properties.
Different orientations of the parabola (vertical or horizontal) have distinct standard form equations. For a vertically oriented parabola opening upward or downward, the standard form is:

• \( y = ax^2 + bx + c \)
• \( y = a(x - h)^2 + k \), more specifically for parabolas with vertex form

For a horizontally oriented parabola, which opens to the left or right, the standard form would involve the variable \( y \) isolated:

• \( x = ay^2 + by + c \)

For the problem at hand, the equation \(-6y = x^2\) simplifies to \(y = -\frac{1}{6}x^2\), indicating the parabola opens in the negative y-direction. Understanding the standard form's role is key to solving related problems.

Focus of a Parabola

The focus is a significant point that is located inside the parabola. It plays a crucial role in defining the parabolic shape along with the directrix. The focus, together with the directrix, ensures that every point on the parabola is equidistant from both, hence creating the parabolic path.
In this specific case, the given focus is \((-\frac{3}{2}, 0)\). Since it's on the x-axis left of the origin, it suggests that the parabola opens sideways.
A general rule is when the parabola's vertex is at the origin and its focus is \((h, 0)\) or \((0, k)\), it means the distance \(p\) (from vertex to focus) directly influences the direction and shape of the parabola. Placing the focus determines the direction the parabola's arms extend.

Vertex of a Parabola

The vertex is a pivotal part of any parabola, as it represents the point where the parabola changes direction. It's the highest or lowest point on the graph, depending on how the parabola opens. In mathematical terms, the vertex can be noted as the coordinate, most often expressed as \((h, k)\).
For the parabola in question, the vertex is located at the origin, making it \((0, 0)\). This means that it serves as the midpoint of the parabola, dividing it symmetrically. A vertex at the origin simplifies many calculations because the axis of symmetry is the y-axis for vertical parabolas or the x-axis for horizontal ones.
Understanding the significance of the vertex aids in quickly identifying the shape and direction of the parabola.

Conic Sections

Conic sections are curves obtained as intersections of the surface of a cone with a plane. They are fundamental in mathematics and are classified into parabolas, ellipses, circles, and hyperbolas. Each section represents a different curve type with distinctive properties.
Parabolas are unique among conic sections because of their characteristic symmetry and defined focal point and directrix. They have a direct application in various fields such as physics, engineering, and even economics.
In the context of the exercise, the parabola is defined using its specific focal point and vertex, illustrating how these key components relate to the broader family of conic sections. Recognizing a parabola within conic sections helps build a holistic understanding of its properties and applications.