Problem 80 - Algebra and Trigonometry

Question

In Exercises 77–82, use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$ y=\cot \frac{x}{2} $$

Step-by-Step Solution

Answer

The graph of the function \(y=\cot \frac{x}{2}\) will have vertical asymptotes at \(x = 2k\pi\) and a period of \(2\pi\). The actual graph of the function can be obtained using any standard graphing utility.

Step 1 - Understand the Base Function

Initially, reflect on the base function which is the cotangent. In essence, the cotangent is the reciprocal of the tangent function. It has vertical asymptotes where the function is undefined which occur at \(x = k\pi\), where \(k\) is an integer, and has a periodicity of \(\pi\).

Step 2 - Apply the Transformation

The function provided is a transformation of the cotangent function where the \(x\) value is divided by 2. This has the effect of doubling the period of the function. So, the function \(y=\cot \frac{x}{2}\) will have vertical asymptotes at \(x = 2k\pi\), where \(k\) is an integer, and has a periodicity of \(2\pi\).

Step 3 - Graph the Function

Now, graph the function \(y=\cot \frac{x}{2}\) using a graphing utility. Remember to set the viewing rectangle to display at least two periods of the function. This would require setting the x-axis range to at least \(4\pi\), and reasonable range on y-axis.

Key Concepts

Cotangent Transformations

Transforming trigonometric functions, such as the cotangent, involves altering its basic graph in various ways. With cotangent transformations, you often see changes in the function's period, phase shift, vertical shift, and even its orientation (reflection). When working with the formula for a transformed cotangent function, like \(y = \text{cot} \frac{x}{2}\), it's essential to understand these transformations step by step.

Firstly, dividing the variable \(x\) by a number, here 2, affects the period of the function. The cotangent function normally has a period of \(\text{π}\). However, by dividing \(x\), the period is elongated; specifically, it's multiplied by the same number. Hence, the period of \(y = \text{cot} \frac{x}{2}\) becomes \(2\text{π}\), which is double the original period.

Understanding this transformation is crucial for graphing the function accurately and for grasping the behavior of the graph. Keep in mind, a period change does not affect the cotangent function's range; it still varies from negative to positive infinity, but the horizontal stretch will change the location of its vertical asymptotes.

Periodicity of Trigonometric Functions

Periodicity is a foundational concept when dealing with trigonometric functions like sine, cosine, tangent, and cotangent. It refers to the feature that these functions repeat their values in regular intervals along the x-axis. For the cotangent function, the basic period is \(\text{π}\).

What does this mean for graphing and calculations? When you encounter a function such as \(y = \text{cot} \frac{x}{2}\), you must adjust your thinking to consider that the interval over which the function repeats its pattern has changed. This function will repeat its values every \(2\text{π}\) instead of every \(\text{π}\). Recognizing the periodicity helps in predicting the function's behavior without having to graph every single point.

This concept is particularly handy when working with trigonometric equations and modeling real-world phenomena where periodic behavior is a key feature, like waves and oscillations. By harnessing the power of periodicity, you can better understand and manipulate these functions to suit various contexts.

Graphing Trigonometric Functions

Graphing trigonometric functions such as the cotangent can seem daunting initially, but with a step-by-step approach, it is quite manageable. You need to be aware of the function's characteristics, such as its periodicity, amplitude, and asymptotes. For the cotangent, unlike sine or cosine, amplitude is not relevant because its range is all real numbers.

When graphing \(y = \text{cot} \frac{x}{2}\), start by identifying the periodicity, which we've established is \(2\text{π}\). Next, determine the positions of the vertical asymptotes; these occur at points where the tangent function (which cotangent is reciprocated) is zero, hence, where the cotangent is undefined. For the transformation given, asymptotes occur at \(x = 2k\text{π}\), where \(k\) is an integer.

Set up your graph with a clear scale to cover at least two periods to observe the function’s behavior over a significant interval. Plot the asymptotes and then carefully sketch the function's curve between them. The curve approaches infinity near the vertical asymptotes and crosses through zero at the midpoint of each period. With practice, graphing cotangent functions with transformations can become a systematic task that unveils the beauty of trigonometry's repetitive nature.