Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\cot \pi x $$

Step-by-Step Solution

Answer

The period is 1 with asymptotes at integer x-values.

Step 1: Understand the Cotangent Function

The cotangent function, denoted as \( y = \cot(x) \), is the reciprocal of the tangent function and has vertical asymptotes where the sine of the angle is zero. The basic period of the \( \cot(x) \) function is \( \pi \). For \( y = \cot(\pi x) \), the period will be adjusted by the coefficient of the variable inside the cotangent function.

Step 2: Calculate the Period

The period of the function \( y = \cot(kx) \) is given by \( \frac{\pi}{k} \). In this case, \( y = \cot(\pi x) \) means \( k = \pi \). Therefore, the period is \( \frac{\pi}{\pi} = 1 \). Thus, the period of the function is 1.

Step 3: Identify Asymptotes

The cotangent function has vertical asymptotes at intervals of its period, starting from the base asymptote where the angle is zero. For \( y = \cot(\pi x) \), this occurs when \( \pi x = n\pi \), where \( n \) is an integer. Solving gives asymptotes at \( x = n \), for \( n = 0, \pm 1, \pm 2, \ldots \).

Step 4: Sketch the Graph

\( y = \cot(\pi x) \) repeats every 1 unit along the x-axis. Its graph will have vertical asymptotes at \( x = n \) with \( n \) being all integers. Between asymptotes, the graph decreases from positive to negative infinity as the input increases. Plot these characteristics from \(-2\) to 2 for a representative sketch.

Key Concepts

Period of a Function

In mathematics, a periodic function is one that repeats its values in regular intervals or periods. Understanding the period of a function is crucial when dealing with trigonometric functions like the cotangent function. The period describes how frequently the function repeats itself. For a standard cotangent function, \( y = \cot(x) \), the period is \( \pi \). This means that after an interval of \( \pi \), the function values start repeating.
For functions of the form \( y = \cot(kx) \), the period changes based on the coefficient \( k \). The formula to find the period of such a function is \( \frac{\pi}{k} \).

• For \( y = \cot(\pi x) \), \( k \) is equal to \( \pi \).
• Applying the formula, the period is \( \frac{\pi}{\pi} = 1 \).

Thus, the function repeats every 1 unit along the x-axis, giving it a period of 1. Recognizing this period is essential when sketching the graph.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. They are significant in understanding the behavior of trigonometric functions, especially the cotangent function which can have vertical asymptotes.
In the context of \( y = \cot(\pi x) \), vertical asymptotes occur where the sine function is zero because the cotangent is undefined at those points. This happens at intervals where \( \pi x = n\pi \) for integers \( n \).

• Simplifying for \( x \), the asymptotes are at \( x = n \), where \( n \) is an integer (\(..., -2, -1, 0, 1, 2, ...\)).

These asymptotes visually separate each period of the function on the graph, and the curve will approach but never reach or cross these vertical lines between consecutive intervals. Understanding these intervals is key to sketching accurate graphs of trigonometric functions.

Cotangent Function

The cotangent function, represented as \( y = \cot(x) \), is one of the six trigonometric functions and is the reciprocal of the tangent function. Its properties and characteristics are crucial for solving and graphing equations that involve trigonometry. The classic form \( y = \cot(x) \) provides a foundational understanding.
Some important characteristics of the cotangent function include:

• It has a period of \( \pi \), signifying it repeats every \( \pi \) interval.
• The graph of the cotangent function descends from positive to negative infinity between each pair of asymptotes.
• For transformed functions like \( y = \cot(\pi x) \), the period is altered, this time to 1, which reflects more frequent repeating cycles.

Typically, the cotangent function has zero crossings halfway between the asymptotes, which gives it a distinctive decreasing curve. Recognizing these patterns can greatly aid in accurately plotting and interpreting the behavior of \( y = \cot(\pi x) \) and similar functions.