Problem 8 - 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. $$x^{2}-y^{2}=1$$

Step-by-Step Solution

Answer

The graph is a hyperbola.

Step 1: Recognize the Form of the Equation

Examine the given equation, \( x^2 - y^2 = 1 \). Notice that both \( x^2 \) and \( y^2 \) are present, and they have different signs, meaning one is positive and one is negative.

Step 2: Identify Differences in Squared Terms

When comparing \( x^2 \) and \( y^2 \) in the equation, the difference in signs (one positive and one negative) indicates the format matching the standard form for a hyperbola \( x^2 - y^2 = 1 \).

Step 3: Use the Standard Form of Conics

Recall from the summary chart on conic sections: - A circle has the form \( x^2 + y^2 = r^2 \).- An ellipse has the form \( rac{x^2}{a^2} + rac{y^2}{b^2} = 1 \).- A hyperbola has the form \( rac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \).The given equation \( x^2 - y^2 = 1 \) directly matches the form of a hyperbola, where \( a = b = 1 \) (so \( a^2, b^2 = 1 \)).

Step 4: Conclude the Conic Type

Since \( x^2 - y^2 = 1 \) fits the standard form of a hyperbola, and it is not transformed (\( a^2 = b^2 \)), the conic section is confirmed to be a hyperbola.

Key Concepts

Hyperbola

The hyperbola is a fascinating type of conic section that is defined by its distinct properties. The hallmark of a hyperbola is its two mirrored curves that open either along the x-axis or the y-axis. These curves are the result of the difference in squared terms in its standard equation, such as having one positive and one negative term. This gives the hyperbola its characteristic shape.

A hyperbola looks like two opposing curves that can either open horizontally or vertically. These curves move outwards from a shared center point and never touch each other.

• A hyperbola's shape is mirrored along its central axis.
• Each curve is called a "branch."
• The branches are symmetrical to each other.

In real-world applications, hyperbolas are used in fields like astronomy and navigation, where they help in calculations for satellite positioning and radio wave transmissions, among other uses.

Equation Identification

Identifying the type of conic section from an equation means analyzing the form of the equation. It's all about noticing specific patterns in the terms present in the equation. In this process, you do not have to graph the equation to know which conic section type it matches.

For example, in the equation like \( x^2 - y^2 = 1 \), you would look at the appearance of both \( x^2 \) and \( y^2 \) and their respective signs:

• Observe if both terms are squared (as in \( x^2 \) and \( y^2 \)).
• Check whether the squared terms have different signs (one positive and one negative) which indicates a hyperbola.

This method offers a quick way to identify without plotting a graph or using a calculator, as the specific arrangement of terms visually cues us to the type of conic section.

Standard Form of Conics

The standard form of conics refers to the specific equations that represent different conic sections such as circles, ellipses, parabolas, and hyperbolas. Recognizing these forms is crucial for identifying the type of conic section an equation represents.

Here's a cheat sheet for quick reference:

• Circle: \( x^2 + y^2 = r^2 \), where every term is squared and added together with equal coefficients.
• Ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where the sum of the two squared terms equals one.
• Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \), where the difference of two squared terms equals one.

By knowing these forms, students can easily classify conic sections without confusion. It allows for quick identification and analysis of the equation type being dealt with in mathematical problems.