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Magazine Chapter 5: Problem 15
In Exercises \(9-28,\) graph the functions over the indicated intervals. $$y=-\frac{1}{4} \cot \left(\frac{x}{2}\right),-2 \pi \leq x \leq 2 \pi$$
Short Answer
Expert verified
Graph the transformed cotangent function with asymptotes at \(-2\pi, 0, 2\pi\) and zeros at \(-\pi, \pi\). Reflect over the x-axis and compress vertically by \(\frac{1}{4}\).
Step by step solution
01
Understand the Cotangent Function
The function given is a variation of the cotangent function, \(y = -\frac{1}{4} \cot \left(\frac{x}{2}\right)\). The basic cotangent function, \(y = \cot(x)\), has vertical asymptotes and repeats every \(\pi\). Understanding how transformations affect this function will guide us in graphing it.
02
Identify Transformations
Notice the transformations applied to the cotangent function:1. **Vertical Compression and Reflection**: The coefficient \(-\frac{1}{4}\) indicates a vertical compression by a factor of \(\frac{1}{4}\) and reflection over the x-axis.2. **Horizontal Stretch**: The argument \(\frac{x}{2}\) results in a horizontal stretching, elongating the period of the cotangent function from \(\pi\) to \(2\pi\).
03
Determine Asymptotes and Vertices
Determine where the vertical asymptotes and zeros of the function will occur within the given interval \(-2\pi \leq x \leq 2\pi\).- The vertical asymptotes for a basic cotangent function \(\cot(cx)\) are at integers of \(\frac{\pi}{c}\), where cotangent is undefined.- For \(\cot\left(\frac{x}{2}\right)\), asymptotes occur at \(x = -4\pi, -2\pi, 0, 2\pi, 4\pi\).- Zeros of the function \(\cot(cx)\) are at \(\left(2n+1\right)\frac{\pi}{c}\).- Here, zeros occur at \(x = -3\pi, -\pi, \pi, 3\pi\).
04
Sketch the Graph
Plot key points and asymptotes within the interval \(-2\pi \leq x \leq 2\pi\):- Draw vertical asymptotes at \(x = -2\pi, 0, 2\pi\).- Place zeros at \(x = -\pi, \pi\).- The function oscillates between vertical asymptotes with a reduced amplitude due to the factor \(-\frac{1}{4}\), producing a flatter curve.- Reflect over the x-axis considering the negative sign.- Ensure the graph repeats its pattern every \(2\pi\).
05
Verify the Interval
Ensure the graph only spans the given interval \(-2\pi \leq x \leq 2\pi\). The key features鈥攕uch as zeros, asymptotes, and inflections鈥攕hould all fit within these bounds, illustrating the behavior of the function accurately up to the edges of the interval.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Transformations
Trigonometric transformations are techniques that allow us to modify trigonometric functions in various ways. These transformations typically include translations, reflections, and stretches or compressions.
In this exercise, we are dealing with a cotangent function, which will undergo several transformations. By applying a vertical compression and horizontal stretch, the function will change its appearance significantly compared to the basic cotangent graph. Understanding these transformations helps in grasping why a graph looks the way it does.
- **Translations**: These can be vertical (up-down shifts) or horizontal (left-right shifts).
- **Reflections**: These flip the graph over the x-axis or the y-axis depending on the transformation.
- **Stretches and Compressions**: These alter the size of the wave either vertically or horizontally.
In this exercise, we are dealing with a cotangent function, which will undergo several transformations. By applying a vertical compression and horizontal stretch, the function will change its appearance significantly compared to the basic cotangent graph. Understanding these transformations helps in grasping why a graph looks the way it does.
Vertical Compression
When we apply vertical compression to a trigonometric function, it affects the amplitude, or height, of the wave. In the function given, the factor \(-\frac{1}{4}\) affects the cotangent function.
This means the peaks and troughs of the cotangent curve are less pronounced, and every point along the y-axis is closer to the x-axis than in the untransformed \(\cot(x)\) graph. The reflection flips the graph, showing that what was previously above the x-axis is now inverted below, and vice versa.
- A coefficient between -1 and 1 compresses the graph vertically.
- The negative sign before \(-\frac{1}{4}\) indicates that the graph is also reflected over the x-axis.
This means the peaks and troughs of the cotangent curve are less pronounced, and every point along the y-axis is closer to the x-axis than in the untransformed \(\cot(x)\) graph. The reflection flips the graph, showing that what was previously above the x-axis is now inverted below, and vice versa.
Horizontal Stretch
The horizontal stretch of a function modifies its period, which is the length of one complete cycle of the graph. Here, the argument \(\frac{x}{2}\) in the function \(\cot\left(\frac{x}{2}\right)\) stretches the graph horizontally.
Thus, stretching horizontally means that while the frequency of cycles decreases, each cycle covers more of the x-axis, changing how we might perceive the overall shape of the wave.
- Normally, the basic cotangent function, \(\cot(x)\), has vertical asymptotes every \(\pi\).
- With the horizontal stretch, these occur less frequently, or every \(2\pi\), for \(\cot\left(\frac{x}{2}\right)\).
- This elongation affects the zeros and asymptotes positions, making the cycle of the cotangent function longer.
Thus, stretching horizontally means that while the frequency of cycles decreases, each cycle covers more of the x-axis, changing how we might perceive the overall shape of the wave.
Graphing Asymptotes
Understanding asymptotes is vital when graphing cotangent functions. Asymptotes are lines that a graph approaches but never quite touches, showing where the function becomes undefined.
When sketching the graph, it's essential to mark these asymptotes accurately. They dictate the regions where the graph can exist and help in understanding the wave behavior of the function, showing peaks in the graphing wave between each pair of asymptotes.
- The basic cotangent function \(\cot(x)\) has vertical asymptotes where \(x\) is an integer multiple of \(\pi\).
- For our transformed function, \(\cot\left(\frac{x}{2}\right)\), these asymptotes occur at \(x = -4\pi, -2\pi, 0, 2\pi, 4\pi\).
When sketching the graph, it's essential to mark these asymptotes accurately. They dictate the regions where the graph can exist and help in understanding the wave behavior of the function, showing peaks in the graphing wave between each pair of asymptotes.
