Graphing The Hyperbola $\frac{x^2}{5^2}-\frac{y^2}{4^2}=1$ A Comprehensive Guide

The fascinating world of conic sections brings us the hyperbola, a curve defined by a specific equation and possessing unique graphical properties. Understanding hyperbolas is crucial in various fields, from physics (describing the paths of celestial objects) to engineering (designing cooling towers). In this article, we will delve into the hyperbola represented by the equation $\frac{x^2}{5^2}-\frac{y^2}{4^2}=1$ and explore its graphical representation. We will meticulously break down the key parameters, including the center, vertices, foci, and asymptotes, which will ultimately allow us to accurately sketch the hyperbola and choose the correct graph. This comprehensive guide aims to provide a clear and detailed explanation, making it easier to grasp the concepts and confidently identify the graphical representation of this hyperbola. The goal is to empower you with the knowledge and skills to analyze and graph hyperbolas with ease.

Deconstructing the Hyperbola Equation

The given equation, $\frac{x^2}{5^2}-\frac{y^2}{4^2}=1$, is in the standard form of a hyperbola centered at the origin. Let's dissect this equation to extract the vital information needed for graphing. The standard form equation for a hyperbola centered at the origin with a horizontal transverse axis is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. By comparing this general form with our specific equation, we can immediately identify the values of a and b. Here, $a^2 = 5^2$, so a = 5, and $b^2 = 4^2$, so b = 4. The value of a represents the distance from the center to the vertices along the transverse axis, while b is related to the conjugate axis. The transverse axis is the axis that passes through the foci and vertices, and in this case, since the $x^2$ term is positive, the transverse axis is horizontal. This tells us that the hyperbola will open to the left and right. Understanding the significance of a and b is paramount to accurately graphing the hyperbola, as they dictate the shape and orientation of the curve. Furthermore, these values are essential for determining other key parameters like the foci and asymptotes, which further define the hyperbola's characteristics. The relationship between a, b, and c (the distance from the center to each focus) is given by $c^2 = a^2 + b^2$, which we will use later to find the foci.

Locating the Center, Vertices, and Foci

With a and b determined, we can now pinpoint the center, vertices, and foci of the hyperbola. As mentioned earlier, the standard form equation $\frac{x^2}{5^2}-\frac{y^2}{4^2}=1$ indicates that the hyperbola is centered at the origin (0, 0). This is because there are no shifts in the x or y terms in the equation. The vertices are located at a distance of a from the center along the transverse axis. Since a = 5 and the transverse axis is horizontal, the vertices are at coordinates (-5, 0) and (5, 0). These points are the closest the hyperbola gets to its center and are crucial for sketching the basic shape of the curve. To find the foci, we need to calculate c using the formula $c^2 = a^2 + b^2$. Substituting a = 5 and b = 4, we get $c^2 = 5^2 + 4^2 = 25 + 16 = 41$. Therefore, $c = \sqrt{41}$. The foci are located at a distance of c from the center along the transverse axis. Thus, the foci are at coordinates $(-\sqrt{41}, 0)$ and $(\sqrt{41}, 0)$. The foci are essential elements of a hyperbola, as they are used in its geometric definition: a hyperbola is the set of all points such that the difference of the distances to the two foci is constant. Understanding the location of the center, vertices, and foci provides a solid foundation for accurately graphing the hyperbola and interpreting its properties.

Defining the Asymptotes

Asymptotes are lines that the hyperbola approaches as it extends infinitely. They serve as guidelines for sketching the hyperbola's branches and are crucial for accurately depicting its shape. For a hyperbola centered at the origin with a horizontal transverse axis, the equations of the asymptotes are given by $y = \pm \frac{b}{a}x$. In our case, a = 5 and b = 4, so the equations of the asymptotes are $y = \pm \frac{4}{5}x$. This means we have two asymptotes: $y = \frac{4}{5}x$ and $y = -\frac{4}{5}x$. These lines pass through the origin and have slopes of 4/5 and -4/5, respectively. To visualize the asymptotes, we can draw a rectangle centered at the origin with sides of length 2a and 2b. The vertices of this rectangle would be at (-5, -4), (-5, 4), (5, -4), and (5, 4). The asymptotes are the lines that pass through the corners of this rectangle. When sketching the hyperbola, the branches will approach these asymptotes without ever actually touching them. The asymptotes provide a framework for the hyperbola's shape, ensuring that it opens correctly and extends infinitely in the appropriate directions. Accurately determining and sketching the asymptotes is a critical step in graphing hyperbolas.

Sketching the Hyperbola

Now that we have determined the center, vertices, foci, and asymptotes, we can sketch the hyperbola $\frac{x^2}{5^2}-\frac{y^2}{4^2}=1$. Start by plotting the center at (0, 0). Then, plot the vertices at (-5, 0) and (5, 0). Next, draw the asymptotes, which are the lines $y = \frac{4}{5}x$ and $y = -\frac{4}{5}x$. You can visualize this by drawing the rectangle with vertices at (-5, -4), (-5, 4), (5, -4), and (5, 4) and drawing the lines through the corners of the rectangle. The branches of the hyperbola will open to the left and right, approaching the asymptotes as they extend infinitely. Starting at the vertices, sketch the curves that approach the asymptotes. The foci, located at $(-\sqrt{41}, 0)$ and $(\sqrt{41}, 0)$, lie on the transverse axis and are enclosed by the branches of the hyperbola. As you sketch the curves, ensure they are smooth and symmetrical about both the x-axis and the y-axis. The hyperbola should get closer and closer to the asymptotes without ever crossing them. By carefully plotting the key points and following the asymptotes, you can accurately sketch the hyperbola. The resulting graph will clearly show the hyperbola opening horizontally, centered at the origin, and guided by its asymptotes.

Identifying the Correct Graph

Finally, to identify the correct graph representing the hyperbola $\frac{x^2}{5^2}-\frac{y^2}{4^2}=1$, we need to look for a hyperbola with the following characteristics: 1. Center at the origin (0, 0) 2. Vertices at (-5, 0) and (5, 0) 3. Horizontal transverse axis (opens left and right) 4. Asymptotes with slopes of ${\pm \frac{4}{5}}$ The graph should show two branches opening horizontally away from the center, approaching the asymptotes as they extend infinitely. The vertices should be clearly marked at (-5, 0) and (5, 0). The asymptotes should pass through the origin and have the correct slopes. By visually inspecting the given graphs and comparing them to these characteristics, you can confidently select the one that accurately represents the hyperbola. Pay close attention to the orientation of the hyperbola, the location of the vertices, and the slopes of the asymptotes. A graph that matches all these criteria is the correct representation of the hyperbola $\frac{x^2}{5^2}-\frac{y^2}{4^2}=1$. By understanding these key features, you can quickly and accurately identify the correct graph among various options. This comprehensive approach ensures that you can confidently analyze and interpret the graphical representation of hyperbolas.