Which Is An Asymptote Of The Graph Of The Function Y=\tan \left(\frac{3}{4} X\right ]?A. X = − 4 Π 3 X=-\frac{4 \pi}{3} X = − 3 4 Π ​ B. X = − 2 Π 3 X=-\frac{2 \pi}{3} X = − 3 2 Π ​ C. X = 3 Π 4 X=\frac{3 \pi}{4} X = 4 3 Π ​ D. X = 3 Π 2 X=\frac{3 \pi}{2} X = 2 3 Π ​

Introduction to Asymptotes

In mathematics, an asymptote is a line that a function approaches but never touches. It is a horizontal, vertical, or slanted line that the graph of a function gets arbitrarily close to as the input values get arbitrarily large. Asymptotes are an essential concept in understanding the behavior of functions, especially in trigonometry.

Asymptotes of Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, have asymptotes that occur when the denominator of the function is equal to zero. For the tangent function, the asymptotes occur when the cosine function is equal to zero. The general form of the tangent function is $y = \tan(x)$, and its asymptotes occur at $x = \frac{\pi}{2} + k\pi$, where $k$ is an integer.

Asymptotes of the Function $y=\tan \left(\frac{3}{4} x\right)$

The given function is $y=\tan \left(\frac{3}{4} x\right)$. To find the asymptotes of this function, we need to find the values of $x$ that make the cosine function equal to zero. The cosine function is equal to zero when its argument is an odd multiple of $\frac{\pi}{2}$. Therefore, we need to find the values of $x$ that satisfy the equation $\frac{3}{4} x = \frac{\pi}{2} + k\pi$.

Solving for $x$

To solve for $x$, we can multiply both sides of the equation by $\frac{4}{3}$, which gives us $x = \frac{4}{3} \left(\frac{\pi}{2} + k\pi\right)$. Simplifying this expression, we get $x = \frac{2\pi}{3} + \frac{4k\pi}{3}$.

Finding the Asymptotes

Now that we have the general form of the asymptotes, we can find the specific values of $x$ that correspond to each option. Let's evaluate each option:

• Option A: $x = -\frac{4\pi}{3}$
• Option B: $x = -\frac{2\pi}{3}$
• Option C: $x = \frac{3\pi}{4}$
• Option D: $x = \frac{3\pi}{2}$

Evaluating Each Option

To determine which option is correct, we need to substitute each value of $x$ into the equation $x = \frac{2\pi}{3} + \frac{4k\pi}{3}$ and check if it satisfies the equation.

• Option A: $-\frac{4\pi}{3} = \frac{2\pi}{3} + \frac{4k\pi}{3}$. Solving for $k$, we get $k = -2$. This is a valid solution, so option A is a possible asymptote.
• Option B: $-\frac{2\pi}{3} = \frac{2\pi}{3} + \frac{4k\pi}{3}$. Solving for $k$, we get $k = -1$. This is a valid solution, so option B is a possible asymptote.
• Option C: $\frac{3\pi}{4} = \frac{2\pi}{3} + \frac{4k\pi}{3}$. Solving for $k$, we get $k = \frac{1}{4}$. This is not a valid solution, so option C is not an asymptote.
• Option D: $\frac{3\pi}{2} = \frac{2\pi}{3} + \frac{4k\pi}{3}$. Solving for $k$, we get $k = \frac{5}{4}$. This is not a valid solution, so option D is not an asymptote.

Conclusion

Based on our analysis, the asymptotes of the function $y=\tan \left(\frac{3}{4} x\right)$ are $x = -\frac{4\pi}{3}$ and $x = -\frac{2\pi}{3}$. Therefore, the correct options are A and B.

Final Answer

The final answer is $\boxed{A, B}$.

Introduction

Asymptotes are an essential concept in mathematics, particularly in trigonometry. In this article, we will delve into the world of asymptotes and explore the concept in detail. We will also provide a comprehensive guide to help you understand and identify asymptotes of trigonometric functions.

Frequently Asked Questions (FAQs)

Q: What is an asymptote?

A: An asymptote is a line that a function approaches but never touches. It is a horizontal, vertical, or slanted line that the graph of a function gets arbitrarily close to as the input values get arbitrarily large.

Q: What are the types of asymptotes?

A: There are three types of asymptotes: horizontal, vertical, and slanted. Horizontal asymptotes occur when the function approaches a constant value as the input values get arbitrarily large. Vertical asymptotes occur when the function approaches infinity as the input values get arbitrarily close to a certain point. Slanted asymptotes occur when the function approaches a linear function as the input values get arbitrarily large.

Q: How do I find the asymptotes of a trigonometric function?

A: To find the asymptotes of a trigonometric function, you need to find the values of the input that make the denominator of the function equal to zero. For the tangent function, the asymptotes occur when the cosine function is equal to zero.

Q: What is the general form of the asymptotes of the tangent function?

A: The general form of the asymptotes of the tangent function is $x = \frac{\pi}{2} + k\pi$, where $k$ is an integer.

Q: How do I find the asymptotes of the function $y=\tan \left(\frac{3}{4} x\right)$?

A: To find the asymptotes of the function $y=\tan \left(\frac{3}{4} x\right)$, you need to find the values of $x$ that make the cosine function equal to zero. The cosine function is equal to zero when its argument is an odd multiple of $\frac{\pi}{2}$. Therefore, you need to find the values of $x$ that satisfy the equation $\frac{3}{4} x = \frac{\pi}{2} + k\pi$.

Q: What is the final answer for the asymptotes of the function $y=\tan \left(\frac{3}{4} x\right)$?

A: The final answer for the asymptotes of the function $y=\tan \left(\frac{3}{4} x\right)$ is $x = -\frac{4\pi}{3}$ and $x = -\frac{2\pi}{3}$.

Q: What are the correct options for the asymptotes of the function $y=\tan \left(\frac{3}{4} x\right)$?

A: The correct options for the asymptotes of the function $y=\tan \left(\frac{3}{4} x\right)$ are A and B.

Q: How do I determine which option is correct?

A: To determine which option is correct, you need to substitute each value of $x$ into the equation $x = \frac{2\pi}{3} + \frac{4k\pi}{3}$ and check if it satisfies the equation.

Q: What is the significance of asymptotes in mathematics?

A: Asymptotes are significant in mathematics because they help us understand the behavior of functions, particularly in trigonometry. They provide valuable information about the function's limits, continuity, and differentiability.

Q: How do I apply asymptotes in real-world problems?

A: Asymptotes can be applied in real-world problems to model and analyze complex systems. For example, in physics, asymptotes can be used to model the behavior of particles in a system, while in engineering, asymptotes can be used to design and optimize systems.

Conclusion

Asymptotes are a fundamental concept in mathematics, particularly in trigonometry. By understanding the concept of asymptotes, you can better analyze and model complex systems. In this article, we have provided a comprehensive guide to help you understand and identify asymptotes of trigonometric functions. We have also answered frequently asked questions to help you better grasp the concept.

Final Answer

The final answer is $\boxed{A, B}$.