Problem 49 - A Graphical Approach to College Algebra

Question

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$49 y^{2}-36 x^{2}=1764$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \((-\infty, -6] \cup [6, \infty)\).

Step 1: Rewrite the Equation in Standard Form

First, divide each term of the equation by 1764 to bring it to standard form: \[ \frac{49y^2}{1764} - \frac{36x^2}{1764} = 1 \] Simplifying, we get: \[ \frac{y^2}{36} - \frac{x^2}{49} = 1 \] This represents the standard form of a hyperbola centered at the origin.

Step 2: Identify the Equation Elements

Identify the components of the standard form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) for the given equation: - \(a^2 = 36\) so \(a = 6\) - \(b^2 = 49\) so \(b = 7\) - Since the \(y^2\) term is positive, the hyperbola opens vertically.

Step 3: Determine the Vertices

For vertical hyperbolas, vertices are at \((0, \pm a)\). Thus, the vertices are: \((0, 6)\) and \((0, -6)\).

Step 4: Determine the Asymptotes

The equations for the asymptotes of the hyperbola are given by:\[ y = \pm \frac{a}{b}x \] Substituting the values of \(a\) and \(b\): \[ y = \pm \frac{6}{7}x \].

Step 5: Graph the Hyperbola

1. Plot the vertices at \((0, 6)\) and \((0, -6)\).2. Draw the asymptotes with slopes \( \frac{6}{7} \) and \( -\frac{6}{7} \) passing through the origin.3. Sketch the hyperbola opening upwards and downwards, approaching the asymptotes.

Step 6: Find the Domain and Range

Since the hyperbola opens vertically:- **Domain**: All real numbers, \(( -\infty, \infty )\).- **Range**: \(( -\infty, -6] \cup [6, \infty )\).

Key Concepts

Equation of Hyperbola

The equation of a hyperbola is a mathematical expression that represents a symmetric curve on a graph. A hyperbola has two branches that mirror each other. For a hyperbola with a vertical or horizontal orientation, the equation can typically be written in the form:

• For a vertical hyperbola: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)
• For a horizontal hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

This form is known as the standard form of a hyperbola. In these equations, \(a\) and \(b\) are real numbers that influence the shape and orientation of the hyperbola. Each hyperbola will have distinct properties that define its vertices and asymptotes, which are lines that the hyperbola approaches but never touches. Understanding the base equation of a hyperbola allows you to determine these critical features easily.
Hyperbolas differ from other conic sections such as ellipses and circles primarily due to their transverse axis and how they open, either vertically or horizontally.

Standard Form of Hyperbola

The standard form of a hyperbola's equation is essential because it reveals the orientation and key features of the hyperbola. The standard form helps classify whether the hyperbola is centered around the origin or elsewhere. The equation might look different based on whether it is vertical or horizontal:
For the exercise given, the form \( \frac{y^2}{36} - \frac{x^2}{49} = 1 \) indicates a vertical hyperbola centered at the origin. This conclusion is arrived at because the \(y\) term is positive and comes first.

• \(a^2 = 36\) implies \(a = 6\): The distance from the center to each vertex along the y-axis.
• \(b^2 = 49\) implies \(b = 7\): Influences the slopes of the asymptotes which are found using \( \pm\frac{a}{b} \).

In this form, you can easily extract the vertices and sketch the asymptotes. The asymptotes are the guidelines that visually aid in drawing the curves of the hyperbola. The standard form is a powerful tool in graphing as it simplifies the analysis of the hyperbola's structure, making it less mystifying once decomposed into recognizable parts.

Domain and Range

When dealing with functions and their graphs, domain and range are vital concepts that describe the set of possible inputs (domain) and outputs (range).
For hyperbolas, these concepts express where the curves extend within the coordinate plane. Each piece of information tells you how broad and tall the hyperbola is. In a vertical hyperbola, like the one in the exercise, the domain includes all real numbers:

• **Domain**: \(( -\infty, \infty )\) - meaning the hyperbola extends indefinitely left and right.

The range is more restrictive, as it excludes a segment of the graphed plane, defined by the vertices:

• **Range**: \(( -\infty, -6] \cup [6, \infty )\) - indicating that the hyperbola branches open upwards and downward, starting at the vertices \((0, 6)\) and \((0, -6)\).

Understanding domain and range in detail lets you predict and interpret the behavior of the hyperbola. This understanding can heavily influence your ability to solve problems involving inequalities and other advanced mathematical topics that use these principles.