Problem 34 - Precalculus with Calculus Previews

Question

Find the period, \(x\) -intercepts, and the vertical asymptotes of the given function. Sketch at least one cycle of the graph. $$ y=\frac{1}{4} \cot \left(x-\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

Period: \(\pi\); x-intercepts: \(x=\frac{\pi}{2}, \frac{3\pi}{2}, \ldots\); Vertical asymptotes: \(x=0, \pi, \ldots\).

Step 1: Identify the General Form

The given function is similar to the general form of the cotangent function: \[ y = a \cot(b(x - c)) + d \] In this case, the function is \( y = \frac{1}{4} \cot \left(x - \frac{\pi}{2}\right) \), so we have: \( a = \frac{1}{4} \), \( b = 1 \), \( c = \frac{\pi}{2} \), and \( d = 0 \). These coefficients will help us determine the period, x-intercepts, and vertical asymptotes of the function.

Step 2: Determine the Period

The period of the cotangent function is given by \( \frac{\pi}{b} \). Since \( b = 1 \) in our function, the period is:\[ \text{Period} = \frac{\pi}{1} = \pi \] This tells us that the function completes one full cycle every \( \pi \) units along the x-axis.

Step 3: Find the x-intercepts

The x-intercepts of the cotangent function \( \cot(x) \) occur at points where \( \tan(x) = 0 \), which are at multiples of \( \pi \). Adjusting for the shift in the given function, the x-intercepts occur at points that satisfy:\[ x - \frac{\pi}{2} = k\pi, \quad k \in \mathbb{Z} \]Solving for \( x \), we have:\[ x = k\pi + \frac{\pi}{2} \] Thus, the x-intercepts are at \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \)

Step 4: Identify the Vertical Asymptotes

Vertical asymptotes for \( \cot(x) \) occur where \( \sin(x) = 0 \) because \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) becomes undefined. These points occur at \( k\pi \). Adjusting for the given function, the vertical asymptotes satisfy:\[ x - \frac{\pi}{2} = (k + \frac{1}{2})\pi \]Solving for \( x \) results in:\[ x = k\pi \]Thus, the vertical asymptotes are at \( x = 0, \pi, 2\pi, \ldots \)

Step 5: Sketch One Cycle of the Graph

To sketch one cycle of the graph, plot the region from \( x = \frac{-\pi}{2} \) to \( x = \frac{\pi}{2} \):- Mark vertical asymptotes at \( x = 0 \) and \( x = \pi \).- Mark x-intercepts at \( x = \frac{\pi}{2} \).- The function is decreasing between vertical asymptotes, typical for a cotangent graph.- Max/min points or other markings are not required as they are at infinity for cotangent. Graph the general behavior and label it.

Key Concepts

Understanding Vertical Asymptotes

The concept of vertical asymptotes is vital when working with trigonometric functions like the cotangent. Vertical asymptotes occur where the function approaches infinity or negative infinity, essentially where it becomes undefined.
The standard cotangent function, \( \cot(x) \), has vertical asymptotes at every multiple of \( \pi \): \( x = k\pi \), because \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) becomes undefined when \( \sin(x) = 0 \). This is why these points appear at integers of \( \pi \).
For the given function, \( y = \frac{1}{4} \cot\left(x-\frac{\pi}{2}\right) \), the vertical asymptotes are where the inner part, \( x - \frac{\pi}{2} = (k + \frac{1}{2})\pi \), is satisfied. Solving for \( x \) gives us \( x = k\pi \), indicating the asymptotes occur at \( x = 0, \pi, 2\pi, \ldots \)

• These vertical lines are critical as they separate the repeating cycles or the periodic sections of the graph.
• They help identify where the function will display vertical directional behavior as it tends to infinity.

Recognizing these asymptotes is a key part of sketching graphs involving cotangent functions.

Uncovering the X-Intercepts

X-intercepts in a graph signify where the function crosses the x-axis. In trigonometric functions, these points are usually evenly spaced depending on the function's periodicity and phase shift.
For \( \cot(x) \), the x-intercepts occur at the points where \( \tan(x) = 0 \). These intercepts happen at points that are multiples of \( \pi \), such as \( x = m\pi \).
With the given function, \( y = \frac{1}{4} \cot\left(x-\frac{\pi}{2}\right) \), we adjust for the horizontal shift. The x-intercepts are determined by the condition \( x - \frac{\pi}{2} = m\pi \), leading to \( x = m\pi + \frac{\pi}{2} \).
This alteration tells us about the movement along the x-axis due to the phase shift:

• The x-intercepts then occur at \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \).
• These points give the exact location where the curve crosses the x-axis, informing us of periods of function zero-output in one cycle.
• Understanding and calculating these values helps in graph sketching and in understanding the function's periodic behavior.

Recognizing these intercepts is crucial for precise function graphing.

Graphing Trigonometric Functions

Graphing trigonometric functions like the cotangent involves representing periodic curves on a coordinate plane. The cotangent function differs from sine and cosine in that it has undefined values creating vertical asymptotes within the period.
To graph \( y = \frac{1}{4} \cot(x-\frac{\pi}{2}) \), we visualize one complete cycle:

• The period for the cotangent, calculated using \( \frac{\pi}{b} \) formula, equals \( \pi \) given \( b = 1 \).
• When graphing, observe one full cycle from \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \).
• Mark vertical asymptotes at \( x = 0 \) and \( x = \pi \), where the function approaches infinity.
• Identify x-intercepts, such as \( x = \frac{\pi}{2} \), where the graph crosses the x-axis.
• The cotangent function appears as a decreasing curve between these asymptotes, a characteristic of cotangent due to its reciprocal relationship.

Graphing this trigonometric function successfully relies on correctly identifying its periodicity, vertical asymptotes, and x-intercepts, ensuring a clear and representative visualization.