Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \(3 x^{2}-6 y^{2}=12\)

Step-by-Step Solution

Answer

No, it is not a parabola.

Step 1: Identify the Trigonometric Problem

We examine the given expression: \(3 x^{2}-6 y^{2}=12\).

Step 2: Apply Trigonometric Identities

We apply the relevant trigonometric identities, definitions, or formulas to solve the problem.

Step 3: Simplify and Compute

We simplify the expression using known values and identities.

Step 4: State the Result

No, it is not a parabola.

Key Concepts

Standard Form Conversion

When dealing with algebraic equations, especially those of conic sections, it's essential to rewrite them in their standard forms. Standard form conversion helps in easily identifying and analyzing their properties. However, the given exercise asks us to convert an equation to its standard form, assuming it is a parabola.To convert an equation to the standard form of a parabola, focus on rearranging the terms to fit the general form:

• For a vertical parabola: \(x^2 = 4py\)
• For a horizontal parabola: \(y^2 = 4px\)

In these forms, \(p\) represents the distance from the vertex to the focus. The standard form is achieved by isolating the square term and adjusting the coefficients accordingly. In our exercise, the equation \(3x^2 - 6y^2 = 12\) contains both \(x^2\) and \(y^2\), indicating it might not be a simple parabola, given that a parabola equation should have only one squared term.

Identifying Conic Sections

Determining the type of conic section represented by an equation requires a keen eye on the structure of the terms. The key to identifying whether an equation represents a circle, ellipse, parabola, or hyperbola is to observe the coefficients of the squared terms:- **Circle**: Both \(x^2\) and \(y^2\) terms are present, have the same coefficient, and the equation is in the form \(Ax^2 + Ay^2 = C\).- **Ellipse**: Similar to a circle but with different coefficients. The general form is \(Ax^2 + By^2 = C\) with \(A eq B\).- **Parabola**: Has only one squared term, either \(x^2\) or \(y^2\).- **Hyperbola**: Both \(x^2\) and \(y^2\) terms exist but with opposite signs, such as in \(Ax^2 - By^2 = C\).In the exercise, \(3x^2 - 6y^2 = 12\), we observe both squared terms with different coefficients and opposite signs, which tells us that the equation represents a hyperbola, not a parabola. This is an insight into how recognizing conic sections can help in correctly interpreting mathematical problems.

Algebraic Equations

Algebraic equations are the foundation of many mathematical concepts, including those of conic sections. Understanding how to manipulate and interpret these equations is crucial for solving complex problems. In algebra, especially when dealing with conic sections, we often encounter higher-order equations involving squares, cubes, and sometimes even variable interactions.Algebraic equations come in many forms:

• Linear equations: Typically of the form \(ax + b = 0\)
• Quadratic equations: Involves terms like \(ax^2 + bx + c = 0\)
• Cubic or higher-order equations: Incorporates terms like \(ax^3, ax^N\), etc.

In our example, \(3x^2 - 6y^2 = 12\), we see quadratic terms where both \(x^2\) and \(y^2\) terms appear. These square terms are characteristic of the types of algebraic equations used to describe conic sections. Mastery of algebraic manipulation is the key to simplifying such equations into their standard forms, enabling us to understand their geometric interpretations.