Problem 14 - 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. $$\frac{x^{2}}{4}=1+\frac{y^{2}}{9}$$

Step-by-Step Solution

Answer

The graph represents a hyperbola.

Step 1: Recognize the Standard Form

The given equation is \( \frac{x^{2}}{4} = 1 + \frac{y^{2}}{9} \). Start by recognizing the form of this equation to see if it matches any standard conic sections forms.

Step 2: Rearrange the Equation

Reorganize the given equation into a form more similar to the standard forms of conic sections. To do this, subtract \( \frac{y^{2}}{9} \) from both sides to get: \[ \frac{x^{2}}{4} - \frac{y^{2}}{9} = 1. \]

Step 3: Identify the Standard Form

Now, the equation \( \frac{x^{2}}{4} - \frac{y^{2}}{9} = 1 \) resembles the standard form of a hyperbola. A hyperbola in standard form is written as \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) or \( \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 \).

Step 4: Determine the Type of Conic Section

Since the equation \( \frac{x^{2}}{4} - \frac{y^{2}}{9} = 1 \) has both \( x^{2} \) and \( y^{2} \) terms with opposite signs, it represents a hyperbola where \( a^{2} = 4 \) and \( b^{2} = 9 \).

Key Concepts

Hyperbola

A hyperbola is a type of conic section that is distinct from ellipses, circles, and parabolas. It occurs when the plane cuts through both nappes of the cone,
creating two symmetrical open curves. Hyperbolas have the unique property of having a center, two foci, two vertices, and two asymptotes.

Think of a hyperbola as a pair of mirrored arches. These curves open either horizontally or vertically around a center point depending on the arrangement of the variables in its standard form equation.

• A hyperbola's standard equation has two squared terms with opposite signs.
• Such uniqueness differentiates it from other conic sections.

Hyperbolas have practical applications in many fields such as astronomy and physics, particularly where properties of distant objects need precise measurement.

Standard Form

The standard form of a hyperbola's equation helps identify and work with these curves effectively.
Standard forms are established equations used to recognize the type of curve without graphing it directly.

For hyperbolas, there are two variations:

• The equation can follow the form: \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), which describes a hyperbola opening left and right.
• Alternatively, it can be \( \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 \), indicating a hyperbola opening up and down.

In both cases, the terms \(a^{2}\) and \(b^{2}\) are the denominators,determining the width and height of the hyperbola's branches respectively.

Understanding the standard form is critical in graphing hyperbolas and identifying their properties without visual aids.

Equation Rearrangement

Equation rearrangement is an essential skill to match given equations with their standard forms for easier identification. It involves algebraic manipulation to align your given equation to look like classic representations of mathematical concepts.
This process is particularly crucial in conic sections where standard forms provide immediate insights into the graph's characteristics.

Let's break down how this works:

• Given an equation like \( \frac{x^{2}}{4} = 1 + \frac{y^{2}}{9} \), rearrangement is necessary.
• Subtract \( \frac{y^{2}}{9} \) from both sides to reformat it as \( \frac{x^{2}}{4} - \frac{y^{2}}{9} = 1 \).
• Now, it follows the standard hyperbola form \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \).

This approach is a primary step for students learning to identify conic sections, greatly aiding comprehension and solving efficiency.