Problem 50 - Precalculus: Functions and Graphs

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 vertical asymptotes at each integer value of \(x\).

Step 1: Identify the Standard Form of the Cotangent Function

The given function is \(y = \cot(\pi x)\). The standard form of the cotangent function is \(y = \cot(bx)\). In our function, \(b = \pi\). The cotangent function in the standard form has a period \(\frac{\pi}{b}\).

Step 2: Calculate the Period

The period \(T\) of the function \(y = \cot(bx)\) is given by the formula:\[T = \frac{\pi}{b}\]Substitute \(b = \pi\) from our function:\[T = \frac{\pi}{\pi} = 1\]Thus, the period of \(y = \cot(\pi x)\) is 1.

Step 3: Determine Asymptotes

The standard cotangent function \(y = \cot(bx)\) has asymptotes where \(bx = n\pi\), for integer \(n\). Thus for \(y = \cot(\pi x)\), this means:\[\pi x = n\pi\]Simplifying gives \(x = n\). Therefore, vertical asymptotes occur at every integer value of \(x\).

Step 4: Sketch the Graph

Start by sketching a single period from \(x=0\) to \(x=1\). Since there are vertical asymptotes at \(x=0\) and \(x=1\), the cotangent curve begins near \(x=0\), decrements, and approaches another asymptote at \(x=1\). Match this cycle for consecutive periods, repeating every 1 unit along the x-axis.

Key Concepts

Period of a Function

The period of a function refers to the length of the interval over which the function repeats itself. For trigonometric functions, determining the period is crucial for understanding how the wave-like pattern repeats across the x-axis.
To find the period of the cotangent function, we use its general form: \(y = \cot(bx)\). The formula for the period \(T\) is \(T = \frac{\pi}{b}\), where \(b\) is a constant multiplying the variable \(x\).
In our exercise with the function \(y = \cot(\pi x)\), we substitute \(b = \pi\) into the formula:

• T = \(\frac{\pi}{\pi} = 1\)

This means the function completes a full cycle every 1 unit along the x-axis. Recognizing this pattern aids in predicting the function's behavior and sketching its graph accurately.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches or crosses. For trigonometric graphs like the cotangent function, vertical asymptotes occur periodically and are vital to graphing.
The standard cotangent function \(y = \cot(bx)\) has vertical asymptotes at \(bx = n\pi\), where \(n\) is an integer. This is because \(\cot(\theta)\) is undefined at these points, causing the graph to "shoot" towards infinity.
For the specific function \(y = \cot(\pi x)\), setting up the equation \(\pi x = n\pi\) gives:

• x = \(n\)

Thus, there are vertical asymptotes at every integer value of \(x\). These integers (0, 1, 2, ...) are locations where the function's graph cannot exist, creating breaks in the wave pattern. Recognizing and plotting these points helps prevent errors when drawing the graph.

Trigonometric Graph Sketching

Sketching the graph of a trigonometric function involves several steps, including identifying periods and locating asymptotes. For \(y = \cot(\pi x)\), understanding these features ensures an accurate representation.
To sketch the graph:

• Begin within one period, from \(x=0\) to \(x=1\).
• Mark the vertical asymptotes at \(x=0\) and \(x=1\). This will create a boundary for a single cycle within this interval.
• The cotangent curve starts near one asymptote, moves downwards, and approaches the next asymptote.
• Repeat this pattern for each additional period, as the cotangent function repeats every unit.

The asymptotic lines at each cycle remind us where the function is undefined, guiding the placement of the curve. Understanding these key ideas like the period and asymptotes simplifies the complex task of trigonometric graph sketching.