Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=4 \cot \left(\frac{1}{3} x-\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

The period is \(3\pi\); asymptotes are at \(x = 3n\pi + 2\pi\), where \(n\) is an integer.

Step 1: Identify the Basic Form

The given equation is similar to the standard form of the cotangent function: \[ y = a \cot(bx - c) + d \] Here, \( a = 4 \), \( b = \frac{1}{3} \), \( c = \frac{\pi}{6} \), and \( d = 0 \). The cotangent function has a period of \( \frac{\pi}{b} \).

Step 2: Calculate the Period

To find the period of the function, we use the formula for the period of a cotangent function: \[ \text{Period} = \frac{\pi}{b} \]Substituting \( b = \frac{1}{3} \):\[ \text{Period} = \frac{\pi}{\frac{1}{3}} = 3\pi \]

Step 3: Determine Asymptotes

The asymptotes of \( y = 4 \cot(\frac{1}{3}x - \frac{\pi}{6}) \) are located where the expression inside the cotangent, \( \frac{1}{3}x - \frac{\pi}{6} \), equals \( n\pi \), where \( n \) is an integer.Set up the equation:\[ \frac{1}{3}x - \frac{\pi}{6} = n\pi \]Solving for \( x \), we get:\[ x = 3n\pi + 2\pi \] These are the locations of the vertical asymptotes.

Step 4: Sketch the Graph

Given the period \( 3\pi \) and asymptotes at \( x = 3n\pi + 2\pi \), we can sketch the graph:1. Between each pair of asymptotes, the cotangent graph will fall from positive to negative infinity.2. There is a vertical asymptote where \( x = 2\pi, 5\pi, 8\pi, \ldots \).3. Plot one period, typically from one asymptote \( x = 2\pi \) to the next \( x = 5\pi \), and then repeat this pattern.

Key Concepts

Cotangent Function

The cotangent function is one of the six fundamental trigonometric functions, closely related to the tangent function. However, instead of rising from negative to positive infinity like the tangent, it falls from positive to negative infinity within its period. The general form of the cotangent function is: \( y = a \cot(bx - c) + d \). Here, \( a \) determines the amplitude or vertical stretch/compression (although for cotangent and tangent functions, this parameter does not affect maximum or minimum since they can extend infinitely). \( b \) changes the period of the function, \( c \) shifts it horizontally, and \( d \) moves it vertically. For the given function \( y = 4 \cot(\frac{1}{3}x - \frac{\pi}{6}) \), \( a = 4 \), indicating a vertical stretch, but it does not affect the overall trend of the graph - it multiplies the distance from the x-axis. Understanding the role of each parameter complements your graphing skills.

Period of a Function

The period of a trigonometric function represents the interval over which the function starts repeating, after which its behavior is predictable and replicable. For the cotangent function, its standard period is \( \pi \) since, over this interval, the cotangent completes one full cycle from positive infinity down to negative infinity.

However, when a function takes the form \( y = a \cot(bx - c) + d \), the period becomes \( \frac{\pi}{b} \). In the exercise, with \( b = \frac{1}{3} \), the period will be calculated as follows:

• Substitute \( b \) into the formula: \( \text{Period} = \frac{\pi}{\frac{1}{3}} = 3\pi \).

Hence, every \( 3\pi \) units along the x-axis, the graph of the given cotangent function repeats its pattern. Recognizing how the period is determined helps forecast the structure and rhythm of the graph.

Vertical Asymptotes

Vertical asymptotes in trigonometric graphs mark points where the function becomes undefined and tends to infinity. The cotangent function has asymptotes where the argument becomes zero, specifically wherever \( \cot(x) \) translates into undefined values, meaning wherever the sine component of its definition is zero, creating division by zero.
For \( y = 4 \cot(\frac{1}{3}x - \frac{\pi}{6}) \), set \( \frac{1}{3}x - \frac{\pi}{6} = n\pi \) for integer \( n \). Solving for where these occur:

• Reconfigure: \( x = 3n\pi + 2\pi \).

These solutions \( x = 2\pi, 5\pi, 8\pi, \ldots \) represent vertical lines on the graph where the function is undefined, and the curve approaches (but never touches) these lines. Pinpointing asymptotes is a crucial step in effectively sketching trigonometric functions.

Trigonometric Graph Sketching

Trigonometric graph sketching involves illustrating the behavior of a function over a defined period, accounting for amplitude, phase shifts, and asymptotes. For cotangent functions, understanding that the graph falls from positive to negative infinity between every pair of vertical asymptotes defines its characteristic shape.
When sketching \( y = 4 \cot(\frac{1}{3}x - \frac{\pi}{6}) \):

• Start by marking vertical asymptotes as derived: \( x = 2\pi, 5\pi, 8\pi, \ldots \).
• Plot points where the cotangent curve crosses the x-axis midway between each pair of asymptotes.
• Remember the cycle of decreasing from infinity to negative infinity within the period \( 3\pi \).

Visualizing each phase helps comprehend transformations and how the function repeatedly contracts and expands. The process ultimately involves practicing to become proficient in representing these periodic functions clearly on a graph.