Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\cot 3 x $$

Step-by-Step Solution

Answer

The period is \(\frac{\pi}{3}\), with asymptotes at \(x = \frac{n\pi}{3}\). Graph shows vertical asymptotes and cotangent cycles every \(\frac{\pi}{3}\).

Step 1: Identify the General Form

The equation for the cotangent function is generally given by \(y = \cot(bx)\). Here, we compare \(y = \cot(3x)\) and see that \(b = 3\). This value of \(b\) will be essential in determining the period of the cotangent function.

Step 2: Determine the Period

The period of a cotangent function \(y = \cot(bx)\) is given by \(\frac{\pi}{b}\). Since \(b = 3\), the period is \(\frac{\pi}{3}\). This means that the cotangent function repeats itself every \(\frac{\pi}{3}\) units along the x-axis.

Step 3: Identify the Asymptotes

The vertical asymptotes of a cotangent function occur where the function is undefined, i.e., where \(\cot(3x)\) goes to infinity. The general formula for the asymptotes of \(y = \cot(bx)\) is \(bx = n\pi\), where \(n\) is an integer. For \(y = \cot(3x)\), we have \(3x = n\pi\), implying \(x = \frac{n\pi}{3}\). Thus, vertical asymptotes are located at \(x = \frac{n\pi}{3}\) for all integers \(n\).

Step 4: Sketch the Graph

To sketch the graph of \(y = \cot(3x)\), mark the calculated period \(\frac{\pi}{3}\) on the x-axis along with key points and the asymptotes. The graph of \(\cot(3x)\) will have a typical cotangent shape, recurring every \(\frac{\pi}{3}\) units. The function will pass through the x-axis at points halfway between the asymptotes. Draw vertical dashed lines at x-values where the asymptotes occur \((x = \frac{n\pi}{3})\), create the fundamental period between two consecutive asymptotes, and mirror this shape for subsequent periods.

Key Concepts

Period of Trigonometric Functions

Trigonometric functions, like the cotangent function, have a property where they repeat their pattern over a specific interval called the period. For the cotangent function, the period tells us how far along the x-axis the function goes before starting to repeat itself. In general, for a function like \( y = \cot(bx) \), the period is determined by the formula \( \frac{\pi}{b} \).
When considering \( y = \cot(3x) \), \( b \) is 3, so the period becomes \( \frac{\pi}{3} \). This value means that every \( \frac{\pi}{3} \) units, the function repeats its entire cycle. Understanding and calculating the period is essential, as it helps in sketching the graph and predicting where the function behaves similarly. By dividing the x-axis into intervals of \( \frac{\pi}{3} \), you can see exactly how the function will behave after each complete cycle.

Vertical Asymptotes

In trigonometric functions like cotangent, vertical asymptotes are points where the function heads off to infinity, and hence, is undefined. These asymptotes are crucial for understanding the behavior of the function and are key features in its graph.
For the equation \( y = \cot(3x) \), the vertical asymptotes occur wherever the function has undefined values. For the cotangent function, it is undefined where the sine of the angle is zero. Using the formula \( bx = n\pi \), where \( n \) can be any integer, we find that \( x = \frac{n\pi}{3} \) defines the locations of the vertical asymptotes. This setup means there is an asymptote every \( \frac{\pi}{3} \) units. As these asymptotes are places where the function shoots off towards positive or negative infinity, sketching them as vertical dashed lines helps visualize how the function behaves in their vicinity.

Graphing Trigonometric Functions

Graphing the cotangent function, such as \( y = \cot(3x) \), involves understanding its periodicity, asymptotes, and points of zero (where the graph crosses the x-axis). When graphing, you need to mark these features:

• **Period:** Place key points on the x-axis at intervals of \( \frac{\pi}{3} \), identifying each full cycle of the function.


• **Vertical Asymptotes:** Draw vertical dashed lines at intervals \( x = \frac{n\pi}{3} \) where the function is undefined.


• **Zero crossings:** These occur at points halfway between each pair of consecutive asymptotes, indicating where the graph crosses the x-axis.

By systematically marking these points, you sketch the fundamental shape of the cotangent function, resembling a set of increasing or decreasing curves between asymptotes. Understanding the role of each feature, such as the periodic nature, helps in visualizing and sketching the graph accurately over the ranged defined by the given function.