Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\frac{1}{3} \cot x $$

Step-by-Step Solution

Answer

The period is \( \pi \), and the graph has vertical asymptotes at \( x = n\pi \).

Step 1: Understanding the Cotangent Function

The cotangent function, denoted by \( \cot(x) \), is the reciprocal of the tangent function: \( \cot(x) = \frac{1}{\tan(x)} \). The general shape of the cotangent graph has vertical asymptotes where \( \tan(x) = 0 \), which occurs at \( x = n\pi \), where \( n \) is an integer.

Step 2: Determine the Basic Period of \( \cot(x) \)

The cotangent function \( \cot(x) \) has a fundamental period of \( \pi \). This means that the pattern of the graph repeats every \( \pi \) units.

Step 3: Calculate the Period of \( y = \frac{1}{3}\cot(x) \)

The given function \( y = \frac{1}{3}\cot(x) \) does not have any changes that affect the period of the cotangent function, such as multiplying the \( x \) argument by a constant. Therefore, the period remains \( \pi \).

Step 4: Identify Asymptotes

The vertical asymptotes of \( y = \frac{1}{3}\cot(x) \) occur at \( x = n\pi \), where \( n \) is an integer. These are the same points where \( \cot(x) \) is undefined because the tangent function is zero, resulting in division by zero.

Step 5: Sketch the Graph

In sketching the graph of \( y = \frac{1}{3}\cot(x) \), start by plotting the asymptotes at \( x = n\pi \). Between these asymptotes, sketch a regular cotangent curve, but scaled vertically by \( \frac{1}{3} \), since the coefficient \( \frac{1}{3} \) affects only the amplitude, not the period or position of the asymptotes. The graph crosses the \( x \)-axis at \( x = (n + \frac{1}{2})\pi \), where \( n \) is an integer.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is quite unique because it is the reciprocal of the tangent function. This means that \( \cot(x) = \frac{1}{\tan(x)} \), and it offers an interesting insight into trigonometry as the reciprocal relationship results in characteristics not shared by the sine, cosine, or tangent functions.

• Unlike sine and cosine, which oscillate between -1 and 1, cotangent extends from negative to positive infinity.
• Its graph is periodic, with a predictable repetitive pattern every \( \pi \) units.

The cotangent function is undefined where the tangent equals zero. Hence, you will notice vertical asymptotes at these points when graphing \( \cot(x) \). Understanding these foundational characteristics of the cotangent function is vital, as they directly impact how we approach graphing or analyzing the function in various contexts.

Vertical Asymptotes

Vertical asymptotes are critical when discussing rational functions like cotangent. When you see \( \tan(x) = 0 \), it indicates a point where the cotangent function is undefined, causing a vertical asymptote. For \( \cot(x) \), this occurs exactly at integer multiples of \( \pi \), making the asymptotes occur at \( x = n\pi \) for any integer \( n \).

• These points create a kind of 'barrier' on the graph where the function shoots towards infinity on both sides.
• It's essential to mark these on your graph to ensure accuracy in your sketch.

The concept of vertical asymptotes can initially seem confusing, but they are simply areas where the input values cause division by zero, which provides insights into the behavior and limitations of functions like cotangent.

Period of a Function

The period of a function is a very important property, especially in the realm of trigonometry. It tells us how often the function repeats itself over a specific interval. For the cotangent function \( \cot(x) \), the period is \( \pi \). Simply put, this means every \( \pi \) units, the graph looks the same. When you have an equation like \( y = \frac{1}{3}\cot(x) \), the multiplication factor (\( \frac{1}{3} \)) does not affect the period because it's applied to the function as a whole and not directly to the \( x \) inside the function.

• It remains \( \pi \) regardless of such coefficients.
• Knowing the period allows you to predict and model behavior over a set interval.

Grasping the concept of a period helps in determining how often vertical asymptotes and zeros occur within intervals, a key step in sketching any trigonometric graph.

Graph Sketching

Sketching graphs of trigonometric functions involves understanding several components. For \( y = \frac{1}{3}\cot(x) \), knowing the period and vertical asymptotes is critical. You'll want to

• Plot vertical asymptotes first, marking them at \( x = n\pi \).
• Remember that the cotangent curve passes through the x-axis at \( x = (n + \frac{1}{2})\pi \).
• Since the factor \( \frac{1}{3} \) scales the graph, shape your cotangent curve closer to the x-axis; it affects the steepness but not the zeros or asymptotes.

Using these steps and points, you gain a solid visual guide, making it easier to represent the intricate behavior of the function. Regular practice with graph sketching empowers you to fluently interpret and work with any trigonometric function.