Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a one-period interval. $$y=\cot x$$

Step-by-Step Solution

Answer

Graph \( y = \cot x \) over \( [0, \pi] \) with vertical asymptotes at \( x = 0 \) and \( x = \pi \).

Step 1: Understanding the Function

The function given is \( y = \cot x \). The cotangent function is defined as the reciprocal of the tangent function, \( \cot x = \frac{1}{\tan x} \). It is important to note that \( \tan x \) is undefined at \( x = n\pi + \frac{\pi}{2} \) where \( n \) is an integer, and so the cotangent will also be undefined at these points.

Step 2: Identify One Period of \( \cot x \)

The cotangent function has a period of \( \pi \). This means that the function will complete one full cycle (beginning, repeating, and ending at the same values) over any interval of length \( \pi \). Therefore, we can consider the interval \( [0, \pi] \) to graph one period of \( y = \cot x \).

Step 3: Determine Key Points for One Period

Within the interval \( [0, \pi] \), note the key points where the cotangent is zero and where it is undefined: \( \cot x = 0 \) at \( x = \frac{\pi}{2} \) and is undefined at \( x = 0 \) and \( x = \pi \).

Step 4: Evaluate \( y = \cot x \) at Key Points

Let's evaluate \( y = \cot x \) at several critical points within the interval \( [0, \pi] \). We find that: \( \cot 0 = \text{undefined}, \) \( \cot \frac{\pi}{4} = 1, \) \( \cot \frac{\pi}{2} = 0, \) \( \cot \frac{3\pi}{4} = -1, \) and \( \cot \pi = \text{undefined}. \)

Step 5: Graph the Function over One Period

Begin to graph using the interval \( [0, \pi] \): plot points at \( (0, \text{undefined}), \left(\frac{\pi}{4}, 1\right), \left(\frac{\pi}{2}, 0 \right), \left(\frac{3\pi}{4}, -1\right), \text{ and } (\pi, \text{undefined}) \). Notice the curve decreases from infinity at \( x = 0 \) to zero at \( x = \frac{\pi}{2} \), and continues to decrease to negative infinity at \( x = \pi \). Connect these points with a smooth curve to complete the graph.

Step 6: Apply Asymptotes

Vertical asymptotes occur where the function is undefined. In the interval \( [0, \pi] \), vertical asymptotes are at \( x = 0 \) and \( x = \pi \). On the graph, draw dashed lines at these points to indicate the asymptotes where the function trends towards infinity or negative infinity.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( y = \cot x \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function, which means it is expressed as \( \cot x = \frac{1}{\tan x} \).

• Wherever the tangent function reaches zero, the cotangent function becomes undefined, leading to vertical asymptotes on its graph. This occurs at \( x = n\pi + \frac{\pi}{2} \) where \( n \) is an integer.


• In comparison to the sine and cosine functions, cotangent is periodic with different key points, zero crossings, and asymptotes.

Cotangent is part of the family of trigonometric functions useful for modeling periodic phenomena, often employed in higher mathematics and physics. Understanding its reciprocal nature assists in solving various trigonometric equations when combined with other functions.

Period of a Function

The period of a function is the interval over which it completes one full cycle of its pattern before repeating. For trigonometric functions, this concept is crucial, as it defines the length of the smallest interval that contains one complete set of values for the function.

• The cotangent function \( y = \cot x \) has a period of \( \pi \). This means every \( \pi \) units, the shape of the cotangent graph repeats itself.


• This period length contrasts with the sine and cosine functions, which have a period of \( 2\pi \).

In practical terms, when graphing or analyzing the cotangent function, you would examine an interval such as \([0, \pi]\) or any other segment of span \( \pi \) to visualize one complete cycle. Recognizing and utilizing the period helps in sketching graphs correctly and solving equations involving trigonometric functions.

Graphing Functions

Graphing trigonometric functions like \( y = \cot x \) begins with understanding key characteristics such as intercepts, asymptotes, and periodicity. For successful graphing, follow these steps:

• Identify the interval over which to graph one period; for cotangent, that's usually \([0, \pi]\).


• Note where the function is undefined. For cotangent, this occurs at \( x = 0 \) and \( x = \pi \), creating vertical asymptotes.


• Determine points where the function crosses the horizontal axis or other key values. \( \cot x \) is zero at \( x = \frac{\pi}{2} \) and positive or negative 1 at other critical points within the period.


• Plot these points and sketch a curve that decreases from infinity and trends towards negative infinity.

Graphing the cotangent function involves connecting the dots with care, keeping in mind the vertical asymptotes, where the graph approaches but never touches or crosses. These characteristics form the signature 'S' shape of the cotangent graph, essential for visualizing and understanding its behavior.