Problem 9 - A Graphical Approach to College Algebra

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$\frac{x^{2}}{4}-\frac{y^{2}}{16}=1$$

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Identify the General Form

The given equation is \(\frac{x^{2}}{4}-\frac{y^{2}}{16}=1\). This equation resembles the standard form of a conic section: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 2: Determine the Conic Type

In the general form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the presence of a negative sign between the terms indicates a hyperbola. Thus, the type of conic section for the given equation is a hyperbola.

Key Concepts

Hyperbola

A hyperbola is a type of conic section, which is a curve formed by the intersection of a plane and a double cone. Unlike ellipses or circles, hyperbolas have two separate, mirror-image curves known as branches. These branches open either horizontally or vertically, depending on the equation of the hyperbola.

Hyperbolas have distinctive features: they have two foci, two vertices, and an asymptote for each branch. The asymptotes are straight lines that the hyperbola's branches approach but never touch. These lines help in sketching the graph of a hyperbola because they indicate the direction in which the branches open.

When analyzing conic sections like hyperbolas, it's important to note the placement of the terms in an equation which helps us determine the graph's orientation. If the equation has the form \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\), the hyperbola opens horizontally. Conversely, if it is \(\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1\), the branches open vertically.

General Form of Equation

In the realm of conic sections, each type has a specific general form of equation that helps in identifying it quickly without plotting.

The general form of a hyperbola's equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\). The distinguishing feature of hyperbolas, when compared to other conics like parabolas and ellipses, is the minus sign usage. This indicates a subtraction between the terms, which is key to recognizing the equation of a hyperbola.

Here, \(a\) and \(b\) are constants that determine the hyperbola's shape and orientation. The relationship between these constants and the terms of the equation influences whether the hyperbola opens horizontally or vertically. It is also crucial in defining the distance between the foci and vertices.

Algebraic Identification

The process of algebraic identification helps us recognize the type of conic section an equation represents. It involves analyzing the equation’s structure and its terms to determine the conic’s identity without needing to graph it.

For hyperbolas, this technique focuses on identifying the negative sign between squared terms in equations like \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). This distinction is crucial because it differentiates hyperbolas from other conic sections, such as ellipses, which have addition between the squared terms.

To perform algebraic identification:

• Observe the equation’s structure.
• Check for subtraction between the squared variables.
• Match the equation to its general form to confirm the presence of a hyperbola.

This method enables a quick and effective way to classify equations in the context of conic sections, especially helpful in mathematical exercises and problem-solving. By recognizing the defining characteristics found in the equation, you can efficiently identify the conic section type, which saves time and enhances understanding.