Problem 40 - Beginning and Intermediate Algebra

Question

Identify the center of each hyperbola and graph the equation. $$x^{2}-y^{2}=25$$

Step-by-Step Solution

Answer

The center of the hyperbola is at (0,0) with a major radius (a) of 5 and a minor radius (b) of 5. The slopes of the asymptotes are 1 and -1. To graph the hyperbola, plot the center at the origin, draw a rectangle with dimensions 2a by 2b, and draw the asymptotes through the rectangle corners. Sketch the hyperbola using the center point, vertex points, and asymptote lines as guidelines.

Step 1: Determine the center of the hyperbola

The given equation is already simplified, so the center (h,k) of the hyperbola is at (0,0), which is the origin.

Step 2: Identify the standard form of the hyperbola

Comparing our given equation \(x^{2}-y^{2}=25\) with the standard form, we get the following: \[\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\] Since the right side of the equation equals 1, we can identify that our a and b values can be determined by the equation \(\frac{x^2}{25}-\frac{y^2}{25}=1\), thus obtaining: \[a^{2}=25\] \[b^{2}=25\] Solving for a and b: \[a = 5\] \[b = 5\]

Step 3: Determine the slopes of the asymptotes

To find the slopes of the asymptotes, use the ratio of b to a: \[m_1=\frac{b}{a}=\frac{5}{5}=1\] and, \[m_2=-\frac{b}{a}=-\frac{5}{5}=-1\] The slopes of the asymptotes are 1 and -1.

Step 4: Graph the hyperbola

To graph the hyperbola, follow these steps: 1. Plot the center at the origin (0,0) 2. Draw a rectangle with dimensions 2a by 2b and center at the origin 3. Draw the asymptotes through the corners of the rectangle with slopes 1 and -1 4. Sketch the hyperbola using the center point, vertex points, and asymptote lines as guidelines After completing these steps, the graph of the hyperbola equation \(x^{2}-y^{2}=25\) will be created.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. These curves are fundamental in geometry and include circles, ellipses, parabolas, and hyperbolas. Each type has its own set of characteristics and equations that define its shape on a graph.

The hyperbola is one of these conic sections. It consists of two separate curves called branches. Unlike ellipses and circles, which are rounded, hyperbolas have a more open, elongated shape. Understanding conic sections helps us grasp the broader picture of geometric properties and their applications in real-world scenarios.

Hyperbolas arise in different contexts, such as navigation systems and the path of celestial bodies. They are important in fields like physics and engineering, illustrating their practical significance.

Graphing Equations

Graphing equations involves plotting points on a coordinate plane to visualize mathematical relationships. For a hyperbola, graphing is a step-by-step process that transforms an equation into its geometric representation.

Here are key steps to graph a hyperbola:

• Identify the center, which is the point (h, k). For the equation \(x^{2} - y^{2} = 25\), the center is at the origin (0, 0).
• Find the vertices and axes lengths using the values of \(a\) and \(b\).
• Draw a rectangle centered at (0,0) using 2a and 2b as dimensions.
• Plot the asymptotes which pass through the corners of the rectangle.

By following these steps, we effectively illustrate the hyperbola. Visualizing equations on a graph makes mathematical relationships more tangible, aiding in understanding key concepts.

Asymptotes

Asymptotes are critical lines that a curve approaches but never touches. In hyperbolas, they dictate the orientation and the spread of the branches.

For the equation \(x^{2} - y^{2} = 25\), the asymptotes have slopes of 1 and -1, determined by the ratio \(\frac{b}{a}\). Here's how they are identified:

• Calculate the slopes using \(m_1 = \frac{b}{a} = 1\) and \(m_2 = -\frac{b}{a} = -1\).
• Draw these lines through the center of the hyperbola.
• These lines provide a guide for sketching the curve's branches.

Asymptotes are crucial for shaping the hyperbola since they define its behavior at infinity. Graphically, they provide a frame within which the curve sits.

Standard Form of Hyperbola

The standard form of a hyperbola's equation is a template that helps in easily identifying its characteristics. A hyperbola's equation is typically given as:\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]This formula allows us to determine vital features like the center, vertices, and asymptotes.

For the specific example \(x^{2} - y^{2} = 25\), we compare it to the standard form:

• Here \(h = 0\) and \(k = 0\), placing the center at the origin.
• \(a^2 = 25\), meaning \(a = 5\).
• Similarly, \(b^2 = 25\), implying \(b = 5\).

Using the standard form simplifies the process of analyzing and graphing hyperbolas. It gives us a foundation to explore these fascinating curves further.