Eve's Teacher Asked Her To Graph The Function $y = -\cot (x) - 1$ By Reflecting The Graph Of The Function $y = \cot (x$\] About The $x$-axis And Translating It Vertically. Does It Matter In Which Order Eve Does The

Introduction

Graphing trigonometric functions can be a complex task, especially when dealing with transformations such as reflection and translation. In this article, we will explore the process of graphing the function $y = -\cot (x) - 1$ by reflecting the graph of the function $y = \cot (x)$ about the $x$-axis and translating it vertically. We will also discuss whether the order of these transformations matters.

Understanding the Function

The function $y = \cot (x)$ is a trigonometric function that is defined as the ratio of the cosine of $x$ to the sine of $x$. The graph of this function is a periodic curve that has a period of $\pi$ and a range of $(-\infty, \infty)$.

Reflection About the $x$-Axis

When we reflect the graph of the function $y = \cot (x)$ about the $x$-axis, we are essentially multiplying the function by $-1$. This means that the new function will have the same shape as the original function, but it will be reflected about the $x$-axis.

Mathematical Representation

The reflection of the function $y = \cot (x)$ about the $x$-axis can be represented mathematically as:

$$y = -\cot (x)$$

This new function has the same period as the original function, but it has a range of $(-\infty, \infty)$.

Translation Vertically

When we translate the graph of the function $y = -\cot (x)$ vertically, we are essentially adding a constant value to the function. This means that the new function will have the same shape as the original function, but it will be shifted up or down by a certain amount.

Mathematical Representation

The translation of the function $y = -\cot (x)$ vertically can be represented mathematically as:

$$y = -\cot (x) + c$$

where $c$ is a constant value.

Order of Transformations

Now that we have discussed the reflection and translation of the function $y = \cot (x)$, we can ask whether the order of these transformations matters. In other words, does it matter whether we reflect the graph of the function about the $x$-axis first and then translate it vertically, or vice versa?

Mathematical Representation

Let's consider the two possible orders of transformations:

1. Reflect the graph of the function $y = \cot (x)$ about the $x$-axis first, and then translate it vertically:

$$y = -\cot (x) + c$$

2. Translate the graph of the function $y = \cot (x)$ vertically first, and then reflect it about the $x$-axis:

$$y = -(\cot (x) + c)$$

Comparison of Results

When we compare the two possible orders of transformations, we can see that they produce the same result:

$$y = -\cot (x) + c = -(\cot (x) + c)$$

This means that the order of the transformations does not matter. We can reflect the graph of the function about the $x$-axis first and then translate it vertically, or vice versa, and we will get the same result.

Conclusion

In conclusion, graphing the function $y = -\cot (x) - 1$ by reflecting the graph of the function $y = \cot (x)$ about the $x$-axis and translating it vertically is a complex task that requires a deep understanding of trigonometric functions and their transformations. We have discussed the reflection and translation of the function $y = \cot (x)$ and have shown that the order of these transformations does not matter. By following the steps outlined in this article, Eve can successfully graph the function $y = -\cot (x) - 1$ and gain a deeper understanding of trigonometric functions and their transformations.

Graphing the Function

To graph the function $y = -\cot (x) - 1$, we can follow these steps:

1. Reflect the graph of the function $y = \cot (x)$ about the $x$-axis by multiplying the function by $-1$:

$$y = -\cot (x)$$

2. Translate the graph of the function $y = -\cot (x)$ vertically by adding a constant value $c$:

$$y = -\cot (x) + c$$

3. Set the constant value $c$ to $-1$ to get the final function:

$$y = -\cot (x) - 1$$

Graph of the Function

The graph of the function $y = -\cot (x) - 1$ is a periodic curve that has a period of $\pi$ and a range of $(-\infty, \infty)$. The graph has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = -1$.

Final Thoughts

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about graphing trigonometric functions, including the function $y = -\cot (x) - 1$.

Q: What is the difference between reflecting and translating a graph?

A: Reflecting a graph about the $x$-axis means multiplying the function by $-1$, while translating a graph vertically means adding a constant value to the function.

Q: How do I graph the function $y = -\cot (x) - 1$?

A: To graph the function $y = -\cot (x) - 1$, you can follow these steps:

1. Reflect the graph of the function $y = \cot (x)$ about the $x$-axis by multiplying the function by $-1$:

$$y = -\cot (x)$$

2. Translate the graph of the function $y = -\cot (x)$ vertically by adding a constant value $c$:

$$y = -\cot (x) + c$$

3. Set the constant value $c$ to $-1$ to get the final function:

$$y = -\cot (x) - 1$$

Q: What is the period of the function $y = -\cot (x) - 1$?

A: The period of the function $y = -\cot (x) - 1$ is $\pi$, which means that the graph repeats itself every $\pi$ units.

Q: What is the range of the function $y = -\cot (x) - 1$?

A: The range of the function $y = -\cot (x) - 1$ is $(-\infty, \infty)$, which means that the graph can take on any real value.

Q: What is the vertical asymptote of the function $y = -\cot (x) - 1$?

A: The vertical asymptote of the function $y = -\cot (x) - 1$ is $x = 0$, which means that the graph approaches infinity as $x$ approaches $0$.

Q: What is the horizontal asymptote of the function $y = -\cot (x) - 1$?

A: The horizontal asymptote of the function $y = -\cot (x) - 1$ is $y = -1$, which means that the graph approaches $-1$ as $x$ approaches infinity.

Q: Can I graph the function $y = -\cot (x) - 1$ using a graphing calculator?

A: Yes, you can graph the function $y = -\cot (x) - 1$ using a graphing calculator. Simply enter the function into the calculator and adjust the window settings to view the graph.

Q: Can I graph the function $y = -\cot (x) - 1$ using a computer algebra system?

A: Yes, you can graph the function $y = -\cot (x) - 1$ using a computer algebra system. Simply enter the function into the system and adjust the graph settings to view the graph.

Conclusion

Graphing trigonometric functions can be a complex task, but with the right tools and techniques, it can be done easily. In this article, we have answered some of the most frequently asked questions about graphing trigonometric functions, including the function $y = -\cot (x) - 1$. By following the steps outlined in this article, you can successfully graph the function $y = -\cot (x) - 1$ and gain a deeper understanding of trigonometric functions and their transformations.