Problem 8 - College Algebra

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

The equation is not a parabola; it is a hyperbola in the form \( \frac{x^2}{4} - \frac{y^2}{2} = 1 \).

Step 1: Identify Equation Type

The given equation is\[ 3x^2 - 6y^2 = 12. \] We need to check if it is a parabola. Parabolas typically have only one squared variable when in standard form, such as \( ax^2 + bx + c = y \) or \( ay^2 + by + c = x \). This equation has both \(x^2\) and \(y^2\), so it is not a parabola.

Step 2: Recognize the Equation Type

The presence of both \(x^2\) and \(y^2\) with opposite signs in \(3x^2 - 6y^2 = 12\) suggests the equation is a hyperbola, not a parabola.

Step 3: Convert to Standard Form of a Hyperbola

First, divide the entire equation \[ 3x^2 - 6y^2 = 12 \] by 12 to simplify. This gives us \[ \frac{3x^2}{12} - \frac{6y^2}{12} = 1. \] Simplifying further, we get \[ \frac{x^2}{4} - \frac{y^2}{2} = 1. \] This is the standard form of a hyperbola.

Key Concepts

Parabola

A parabola is a curve where any point is at an equal distance from a fixed point, called the focus, and a straight line, called the directrix. Parabolas have a distinct U-shaped form which can open either upwards or downwards (for parabolas in the xy-plane).
In mathematics, a parabola is represented by a quadratic equation that includes either an \(x^2\) term or a \(y^2\) term, but not both. When put into standard form, the equation appears as:

• \(y = ax^2 + bx + c\) if the parabola opens vertically
• \(x = ay^2 + by + c\) if the parabola opens horizontally

This standard form helps in identifying the direction of the parabola and its vertex, which is the highest or lowest point in the curve. In a coordinate plane, the vertex form of a parabola \((x-h)^2 = 4p(y-k)\) or \((y-k)^2 = 4p(x-h)\) can also greatly help in visualizing and graphing it.

Hyperbola

A hyperbola consists of two separate curves called branches, which look like mirrored parabolas set apart. A hyperbola represents the set of all points for which the absolute difference of the distances to two fixed points, called foci, is constant.
The standard form for the equation of a hyperbola depends on how it is oriented:

• If the transverse axis (the axis along the centers of the two branches) is horizontal, the equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• If the transverse axis is vertical, the equation is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)

Both versions include terms with \(x^2\) and \(y^2\), but one is positive and the other negative. This distinguishes a hyperbola from a circle or an ellipse, which would have positive terms.

Equation Standard Form

Standard form equations provide a clear and structured way to represent geometric shapes like parabolas and hyperbolas. It helps in identifying and graphing these shapes easily by clearly showing the relationship between the x and y coordinates.
For conic sections, converting an equation to standard form involves:

• Identifying the type of conic section by checking the equation's structure
• Rearranging the equation by completing the square, if necessary
• Ensuring the equation matches one of the standard forms

Using standard form, you can readily find key features like the vertex, axes, and foci, which are essential for understanding and plotting the conic section.

Conic Sections

Conic sections are shapes created by intersecting a plane with a double-napped cone. These shapes include parabolas, hyperbolas, ellipses, and circles, each having unique properties and equations.
The characteristics of conic sections are defined by:

• Parabolas with one squared term
• Ellipses with positive squared terms of the same sign but different coefficients
• Hyperbolas with squared terms of opposite signs
• Circles as special ellipses with equal coefficients for squared terms

Understanding these basic patterns helps students distinguish between conic sections and simplify their equations to the standard form, aiding in precise graphing and analysis.