Problem 11 - 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. $$\frac{x^{2}}{25}-\frac{y^{2}}{25}=1$$

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Recognize the Standard Form

The equation \(\frac{x^{2}}{25}-\frac{y^{2}}{25}=1\) is given. This equation resembles the standard form of a conic section, specifically the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for hyperbolas.

Step 2: Identify Key Characteristics

Note that the equation has both squared terms with opposite signs, i.e., a positive \(x^2\) term and a negative \(y^2\) term. Additionally, both terms are divided by the same constant \(25\), indicating equal semi-axis lengths for the conic section.

Step 3: Determine the Type of Conic Section

The presence of one positive and one negative squared term in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) means that this equation represents a hyperbola. Specifically, it is a horizontal hyperbola centered at the origin.

Key Concepts

Hyperbola

A hyperbola is a type of conic section that forms when a plane intersects both nappes (the two opposite cones) of a double cone. Unlike circles and ellipses, which are closed shapes, hyperbolas are open and consist of two separate branches. Each branch of a hyperbola reflects across the center point, known as the origin in this case.
Hyperbolas arise when the difference between two squared terms equals a constant, as seen in this specific format:

• The general equation follows the form: \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
• When the \( x^2 \) term is positive and the \( y^2 \) term is negative, this indicates a horizontal hyperbola.
• Conversely, if the \( y^2 \) term is positive, the hyperbola is vertical.

Hyperbolas feature asymptotes, lines which the branches approach but never touch. These lines cross at the hyperbola's center, guiding the overall shape of the curves.

Standard Form Equation

Identifying the standard form of a hyperbola's equation is crucial for recognizing its properties. The exercise's given equation \( \frac{x^2}{25} - \frac{y^2}{25} = 1 \) directly matches the general equation of a hyperbola.
Let's break down the components of the equation:

• \( x^2 \) and \( y^2 \): These indicate the squared variables of the equation. Each square is divided by a constant representing the square of a semi-axis length.
• \( 25 \): Both \( x^2 \) and \( y^2 \) are divided by this same constant, signifying that both semi-axis lengths \( a \) and \( b \) are equal and can further imply the shape's symmetry around both axes. The calculation \( a^2 = b^2 = 25 \) results in \( a = b = 5 \).
• \( 1 \): This constant shows that the equation describes a standard hyperbola centered at the origin (0,0).

Grasping the standard form helps determine the hyperbola's characteristics and shape without the need for sketching.

Graph Identification

Identifying the graph type based on the equation's structure can save time and effort when dealing with conic sections. For the equation \( \frac{x^2}{25} - \frac{y^2}{25} = 1 \), it's evident, through comparing the structure to the standard form of a hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), that this is a hyperbola.
Various key steps can be followed for such identification:

• Look at squared terms: If squared terms have opposite signs, it indicates a hyperbola.
• Compare constants: The same denominator for the squared terms suggests equal asymptotic behavior along both axes.
• Observe coefficient signs: With positive x-term and negative y-term, the hyperbola is horizontal.

All these factors collectively assist in quicker and accurate graph identification without sketching.

Algebraic Characteristics

Understanding a hyperbola's algebraic characteristics provides significant insight into its geometry and shape. Such characteristics can be identified from the equation \( \frac{x^2}{25} - \frac{y^2}{25} = 1 \).
Here are some notable traits:

• Center: Origin (0,0) as the equation equals 1, indicating symmetry about this point.
• Axes: The symmetry is equally along both the x- and y-axis due to equal denominators signifying equal semi-lengths.
• Asymptotes: Imaginary lines along the plane which for this equation are both \( y = \pm \frac{5}{5}x \) simplifying to \( y = \pm x \).
• Vertices: Located at \( (+5, 0) \) and \( (-5, 0) \) on the x-axis, providing physical points from which the branches start extending.

Grasping these algebraic details arms you with more information about the hyperbola without having to visualize or graph it physically, using purely algebraic analysis to deduce its intricacies.