The Graph Of Y 2 Csc ⁡ ( X − Π 4 ) − 3 Y^2 \csc \left(x-\frac{\pi}{4}\right)-3 Y 2 Csc ( X − 4 Π ​ ) − 3 Is Shown. 1. What Is The Period Of The Function? - $4 \pi$2. Where Are The Asymptotes Of The Function? - $\square$3. What Is The Range Of The Function? - $y

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Understanding the Graph

The given graph represents the function $y^2 \csc \left(x-\frac{\pi}{4}\right)-3$. This function involves the cosecant function, which is a periodic trigonometric function. To analyze the graph, we need to understand the properties of the cosecant function and how it is affected by the given function.

Period of the Function

The period of a function is the distance between two consecutive points on the graph that have the same value. In the case of the cosecant function, the period is $2\pi$. However, in the given function, the argument of the cosecant function is $x-\frac{\pi}{4}$.

To find the period of the given function, we need to consider the shift in the argument. The shift in the argument is $\frac{\pi}{4}$, which means that the graph of the given function is shifted to the right by $\frac{\pi}{4}$.

The period of the given function is still $2\pi$, but it is shifted to the right by $\frac{\pi}{4}$. Therefore, the period of the function is $2\pi - \frac{\pi}{4} = \frac{7\pi}{4}$.

Asymptotes of the Function

The asymptotes of a function are the lines that the graph approaches as the input values approach infinity or negative infinity. In the case of the cosecant function, the asymptotes are the lines $y=0$ and $x=n\pi$, where $n$ is an integer.

However, in the given function, the argument of the cosecant function is $x-\frac{\pi}{4}$. This means that the asymptotes of the given function are shifted to the right by $\frac{\pi}{4}$.

The asymptotes of the given function are the lines $y=0$ and $x=n\pi + \frac{\pi}{4}$, where $n$ is an integer.

Range of the Function

The range of a function is the set of all possible output values. In the case of the cosecant function, the range is all real numbers except for zero.

However, in the given function, the argument of the cosecant function is $x-\frac{\pi}{4}$. This means that the range of the given function is affected by the shift in the argument.

The range of the given function is all real numbers except for zero, but it is shifted to the right by $\frac{\pi}{4}$. Therefore, the range of the function is $y \in (-\infty, -3] \cup [3, \infty)$.

Conclusion

In conclusion, the graph of the function $y^2 \csc \left(x-\frac{\pi}{4}\right)-3$ has a period of $\frac{7\pi}{4}$, asymptotes of $y=0$ and $x=n\pi + \frac{\pi}{4}$, and a range of $y \in (-\infty, -3] \cup [3, \infty)$.

Step-by-Step Solution

Step 1: Understand the Properties of the Cosecant Function

The cosecant function is a periodic trigonometric function with a period of $2\pi$. It has asymptotes at $y=0$ and $x=n\pi$, where $n$ is an integer.

Step 2: Analyze the Given Function

The given function is $y^2 \csc \left(x-\frac{\pi}{4}\right)-3$. This function involves the cosecant function with an argument of $x-\frac{\pi}{4}$.

Step 3: Find the Period of the Function

The period of the given function is still $2\pi$, but it is shifted to the right by $\frac{\pi}{4}$. Therefore, the period of the function is $2\pi - \frac{\pi}{4} = \frac{7\pi}{4}$.

Step 4: Find the Asymptotes of the Function

The asymptotes of the given function are the lines $y=0$ and $x=n\pi + \frac{\pi}{4}$, where $n$ is an integer.

Step 5: Find the Range of the Function

The range of the given function is all real numbers except for zero, but it is shifted to the right by $\frac{\pi}{4}$. Therefore, the range of the function is $y \in (-\infty, -3] \cup [3, \infty)$.

Final Answer

The final answer is:

• Period: $\frac{7\pi}{4}$
• Asymptotes: $y=0$ and $x=n\pi + \frac{\pi}{4}$
• Range: $y \in (-\infty, -3] \cup [3, \infty)$
  

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Frequently Asked Questions

Q1: What is the period of the function $y^2 \csc \left(x-\frac{\pi}{4}\right)-3$?

A1: The period of the function is $\frac{7\pi}{4}$. This is because the cosecant function has a period of $2\pi$, but the argument of the cosecant function is shifted to the right by $\frac{\pi}{4}$.

Q2: Where are the asymptotes of the function $y^2 \csc \left(x-\frac{\pi}{4}\right)-3$?

A2: The asymptotes of the function are the lines $y=0$ and $x=n\pi + \frac{\pi}{4}$, where $n$ is an integer. This is because the cosecant function has asymptotes at $y=0$ and $x=n\pi$, but the argument of the cosecant function is shifted to the right by $\frac{\pi}{4}$.

Q3: What is the range of the function $y^2 \csc \left(x-\frac{\pi}{4}\right)-3$?

A3: The range of the function is $y \in (-\infty, -3] \cup [3, \infty)$. This is because the cosecant function has a range of all real numbers except for zero, but the argument of the cosecant function is shifted to the right by $\frac{\pi}{4}$.

Q4: How does the shift in the argument affect the graph of the function?

A4: The shift in the argument affects the graph of the function by shifting it to the right by $\frac{\pi}{4}$. This means that the period, asymptotes, and range of the function are all affected by the shift.

Q5: What is the significance of the cosecant function in the given function?

A5: The cosecant function is significant in the given function because it is the primary function being manipulated. The shift in the argument of the cosecant function affects the period, asymptotes, and range of the function.

Q6: How does the given function compare to the standard cosecant function?

A6: The given function compares to the standard cosecant function in that it has the same period, asymptotes, and range, but it is shifted to the right by $\frac{\pi}{4}$. This means that the graph of the given function is identical to the graph of the standard cosecant function, but it is shifted to the right by $\frac{\pi}{4}$.

Q7: What are some real-world applications of the cosecant function?

A7: The cosecant function has many real-world applications, including modeling the behavior of sound waves, modeling the behavior of light waves, and modeling the behavior of electrical signals. The given function can be used to model the behavior of these waves or signals in a specific context.

Q8: How can the given function be used in mathematical modeling?

A8: The given function can be used in mathematical modeling to describe the behavior of a system or phenomenon that is periodic and has asymptotes. The shift in the argument of the cosecant function can be used to model the behavior of a system or phenomenon that is shifted in time or space.

Conclusion

In conclusion, the graph of the function $y^2 \csc \left(x-\frac{\pi}{4}\right)-3$ has a period of $\frac{7\pi}{4}$, asymptotes of $y=0$ and $x=n\pi + \frac{\pi}{4}$, and a range of $y \in (-\infty, -3] \cup [3, \infty)$. The shift in the argument of the cosecant function affects the period, asymptotes, and range of the function. The given function can be used in mathematical modeling to describe the behavior of a system or phenomenon that is periodic and has asymptotes.

Step-by-Step Solution

Step 1: Understand the Properties of the Cosecant Function

The cosecant function is a periodic trigonometric function with a period of $2\pi$. It has asymptotes at $y=0$ and $x=n\pi$, where $n$ is an integer.

Step 2: Analyze the Given Function

The given function is $y^2 \csc \left(x-\frac{\pi}{4}\right)-3$. This function involves the cosecant function with an argument of $x-\frac{\pi}{4}$.

Step 3: Find the Period of the Function

The period of the given function is still $2\pi$, but it is shifted to the right by $\frac{\pi}{4}$. Therefore, the period of the function is $2\pi - \frac{\pi}{4} = \frac{7\pi}{4}$.

Step 4: Find the Asymptotes of the Function

The asymptotes of the given function are the lines $y=0$ and $x=n\pi + \frac{\pi}{4}$, where $n$ is an integer.

Step 5: Find the Range of the Function

The range of the given function is all real numbers except for zero, but it is shifted to the right by $\frac{\pi}{4}$. Therefore, the range of the function is $y \in (-\infty, -3] \cup [3, \infty)$.

Final Answer

The final answer is:

• Period: $\frac{7\pi}{4}$
• Asymptotes: $y=0$ and $x=n\pi + \frac{\pi}{4}$
• Range: $y \in (-\infty, -3] \cup [3, \infty)$