Problem 14 - A Graphical Approach to Precalculus with Limits

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. $$\frac{x^{2}}{4}=1+\frac{y^{2}}{9}$$

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Standard Form Recognition

First, rewrite the given equation \( \frac{x^2}{4} = 1 + \frac{y^2}{9} \). Before analyzing the conic section, remember that the standard forms are important: \( Ax^2 + By^2 + Cx + Dy + E = 0 \) for general conics and \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) for hyperbolas.

Step 2: Equation Rearrangement

Subtract 1 from both sides of the equation: \( \frac{x^2}{4} - 1 = \frac{y^2}{9} \). Then, bring all terms to one side of the equation and rewrite it as follows: \( \frac{x^2}{4} - \frac{y^2}{9} = 1 \).

Step 3: Identify the Conic Type

Compare the rearranged equation \( \frac{x^2}{4} - \frac{y^2}{9} = 1 \) with the standard form of a hyperbola equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Since the equation matches the hyperbola form with \( a^2 = 4 \) and \( b^2 = 9 \), the graph is identified as a hyperbola.

Key Concepts

Hyperbola

A hyperbola is one of the four main types of conic sections, the others being parabolas, ellipses, and circles. Think of a hyperbola as a set of all points in a plane, such that the absolute difference of the distances to two fixed points, known as foci, is constant. Unlike an ellipse, where this sum is constant, the key characteristic of a hyperbola is this *difference*.

Here are a few features of hyperbolas:

• They consist of two separate curves, called branches, which mirror each other.
• Each branch approaches but never meets its asymptotes. These are not axes but lines that the hyperbola gets closer to, endlessly extending outward.
• The center of a hyperbola is the midpoint between its foci, and it does not necessarily lie on the curve itself.

When dealing with conic sections, recognizing the equation form helps to quickly identify a graph as a hyperbola.

Standard Form

The standard form of a hyperbola is crucial for understanding and identifying it without graphing. The equation is typically \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]where:

• \(a^2\) and \(b^2\) are constants that determine the shape and orientation of the hyperbola.
• The term with \(x^2\) comes first, and its coefficient divides \(x^2\) in the standard form when dealing with this type.

In practice, understanding this format at a glance can save you a lot of time. For hyperbolas, you'll always subtract one term from the other; this is a notable giveaway.

Knowing the standard form allows mathematicians and students to quickly verify and solve equations without detailed graphing, by simple comparison.

Equation Rearrangement

Rearranging an equation to standard form is an essential skill when identifying conic sections. It's like repositioning a puzzle piece to see the whole picture. Let's dive into the given example where we start from \[\frac{x^2}{4} = 1 + \frac{y^2}{9}\]To begin, move all terms involving \(y\) to one side using subtraction, resulting in:\[\frac{x^2}{4} - 1 = \frac{y^2}{9}\]Further rearrangement involves ensuring that all terms are consolidated on one side:\[\frac{x^2}{4} - \frac{y^2}{9} = 1\]By doing so, the equation is reformatted into the standard hyperbola form, which, at a glance, reveals the characteristic structure. This step not only aids in clear problem-solving but is a valuable practice in algebra.

Graph Identification

Identifying a graph without plotting is like having a map with a clear legend. For conic sections, knowing the right clues helps you decode the type of graph you might encounter.

For example, by recognizing the equation\[\frac{x^2}{4} - \frac{y^2}{9} = 1\]you can identify it as a hyperbola. Here's how you make such an identification:

• You look for the subtraction between two squared terms and check if they equate to a constant like 1.
• Recognize that if the equation has the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), it's a hyperbola based on order and signs.

By using these examination techniques, you enhance your mathematical intuition and your ability to interpret equations rapidly. This skill is especially useful during exams or when you need quick checks on the nature of graphs.