Understanding Hyperbolic Graphs

Oct 13, 2025 by windailys.com 44 views

Introduction

In mathematics, hyperbolic functions are used to describe the behavior of certain types of curves. These curves are characterized by their asymptotes and transverse axes. In this article, we will compare two hyperbolic graphs and determine which statement is true.

Understanding Hyperbolic Graphs

A hyperbolic graph is a type of curve that can be represented by the equation:

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

where $a$ and $b$ are the lengths of the transverse axes. The directrices of a hyperbolic graph are two lines that are perpendicular to the transverse axes and are located at a distance of $\frac{a^2}{e}$ and $\frac{b^2}{e}$ from the center of the graph.

Comparing the Given Graphs

The two given graphs are:

\frac{x^2}{6^2} - \frac{y^2}{8^2} = 1

and

\frac{x^2}{8^2} - \frac{y^2}{6^2} = 1

To compare these graphs, we need to determine the lengths of the transverse axes and the directrices of each graph.

Lengths of Transverse Axes

The length of the transverse axis of the first graph is $6$ , and the length of the transverse axis of the second graph is $8$ . Therefore, the lengths of the transverse axes are not the same.

Directrices of the Graphs

The directrices of the first graph are located at a distance of $\frac{6^2}{e}$ and $\frac{8^2}{e}$ from the center of the graph. The directrices of the second graph are located at a distance of $\frac{8^2}{e}$ and $\frac{6^2}{e}$ from the center of the graph.

Comparing the Directrices

Since the directrices of the two graphs are located at the same distances from the center of the graph, we can conclude that the directrices of the two graphs are the same.

Conclusion

Based on the analysis above, we can conclude that the statement "The directrices of the two graphs are the same" is true.

Answer

The correct answer is:

B. The directrices of the two graphs are the same.

Additional Information

Q: What is the difference between the two given hyperbolic graphs?

A: The two given hyperbolic graphs are:

\frac{x^2}{6^2} - \frac{y^2}{8^2} = 1

and

\frac{x^2}{8^2} - \frac{y^2}{6^2} = 1

The main difference between the two graphs is the length of the transverse axes. The length of the transverse axis of the first graph is $6$ , and the length of the transverse axis of the second graph is $8$ .

Q: What are the directrices of a hyperbolic graph?

A: The directrices of a hyperbolic graph are two lines that are perpendicular to the transverse axes and are located at a distance of $\frac{a^2}{e}$ and $\frac{b^2}{e}$ from the center of the graph, where $a$ and $b$ are the lengths of the transverse axes.

Q: How do the directrices of the two graphs compare?

A: The directrices of the two graphs are located at the same distances from the center of the graph. Therefore, the directrices of the two graphs are the same.

Q: What is the significance of the directrices of a hyperbolic graph?

A: The directrices of a hyperbolic graph are important because they help to define the shape and behavior of the graph. The directrices are used to determine the asymptotes of the graph, which are the lines that the graph approaches as $x$ or $y$ approaches infinity.

Q: How do the lengths of the transverse axes affect the shape of the graph?

A: The lengths of the transverse axes affect the shape of the graph by determining the width and height of the graph. A graph with a longer transverse axis will be wider and taller than a graph with a shorter transverse axis.

Q: Can you provide an example of a hyperbolic graph?

A: Yes, an example of a hyperbolic graph is the equation:

\frac{x^2}{4^2} - \frac{y^2}{3^2} = 1

This graph has a transverse axis of length $4$ and a directrix of distance $\frac{4^2}{e}$ from the center of the graph.

Q: How do you determine the length of the transverse axis of a hyperbolic graph?

A: To determine the length of the transverse axis of a hyperbolic graph, you need to look at the equation of the graph and identify the values of $a$ and $b$ . The length of the transverse axis is then given by $a$ or $b$ , depending on which value is larger.

Q: Can you provide a summary of the key points of this article?

A: Yes, the key points of this article are:

The two given hyperbolic graphs have different lengths of transverse axes.
The directrices of the two graphs are the same.
The directrices of a hyperbolic graph are important for defining the shape and behavior of the graph.
The lengths of the transverse axes affect the shape of the graph.
An example of a hyperbolic graph is the equation $\frac{x^2}{4^2} - \frac{y^2}{3^2} = 1$ .
To determine the length of the transverse axis of a hyperbolic graph, you need to look at the equation of the graph and identify the values of $a$ and $b$ .