Problem 44 - Precalculus: A Concise Course

Question

Use a graphing utility to graph the function. Include two full periods. $$ y=\frac{1}{4} \cot \left(x-\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

The graph of the function \(\frac{1}{4} \cot(x - \frac{\pi}{2})\) will be a vertically shrunk and horizontally shifted version of the base cotangent function. It is vertically shrunk by a factor of \(\frac{1}{4}\) and horizontally shifted to the right by \(\frac{\pi}{2}\) units. It has vertical asymptotes at integer multiples of \(\pi\) plus \(\frac{\pi}{2}\) (the phase shift), and crosses the \(x\)-axis at every \(\pi\) plus \(\frac{\pi}{2}\)

Step 1: Understand the base cotangent function

The cotangent function can be described as the reciprocal of the tangent function, that is, \(\cot(x) = \frac{1}{\tan(x)}\). The cotangent function has a period of \(\pi\) radians, and is undefined at integer multiples of \(\pi\) where it has vertical asymptotes. It is symmetric at its mid points, i.e, \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), where it equals 0.

Step 2: Apply the amplitude and phase shift

The given function is \(\frac{1}{4} \cot(x - \frac{\pi}{2})\). In this function, the amplitude is \(\frac{1}{4}\) which will vertically shrink the graph. Also, the phase shift is \(\frac{\pi}{2}\) which means the graph will be shifted to the right by \(\frac{\pi}{2}\) units.

Step 3: Draw the graph

The two full periods of the function \(\frac{1}{4} \cot(x - \frac{\pi}{2})\) would range from \(\frac{\pi}{2}\) to \(\frac{5\pi}{2}\), and from \(\frac{5\pi}{2}\) to \(\frac{9\pi}{2}\). Around each asymptote, the function decreases from positive infinity on the left to negative infinity on the right. And at mid points, it crosses the \(x\)-axis. Don't forget to shrink the graph vertically by a factor of \(\frac{1}{4}\).

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is an essential trigonometric function. It's closely related to tangent, as it represents the reciprocal: \( \cot(x) = \frac{1}{\tan(x)} \). This makes the cotangent function undefined at points where tangent is zero, leading to vertical asymptotes at integer multiples of \( \pi \). You'll notice that for \( \cot(x) \), the points where it crosses the \( x \)-axis are at odd multiples of \( \frac{\pi}{2} \), such as \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). This function is periodic with the wave pattern repeating every \( \pi \) units.

Phase Shift

A phase shift refers to the horizontal movement of a trigonometric graph left or right. For the function \( y=\frac{1}{4} \cot \left(x-\frac{\pi}{2}\right) \), there's a phase shift to the right. The \( x-\frac{\pi}{2} \) inside the cotangent's argument indicates this translation: the graph shifts by \( \frac{\pi}{2} \) units to the right. The phase shift changes the starting point of the period, effectively repositioning the entire pattern along the \( x \)-axis. If you've tracked cotangent before, you'll notice its zeros now occur at \( \pi \) and \( 2\pi \) due to the right shift.

Vertical Shrinkage

Vertical shrinkage occurs when the amplitude of a trigonometric function is altered. The amplitude fluctuates how far the curve moves above or below the main axis. With the function \( y=\frac{1}{4} \cot(x-\frac{\pi}{2}) \), the coefficient \( \frac{1}{4} \) causes a vertical shrinkage.

• This change affects how steep the curve is between its asymptotes.
• For a standard cotangent function, the amplitude ranges have more extreme values between peaks and valleys.
• Here, the graph's height decreases to just \( \frac{1}{4} \) of its usual value.

As you graph, note this reduction means the distances from peak-to-midaxis and midaxis-to-valley are scaled down uniformly.

Graphing Utility

A graphing utility is a tool used to visualize mathematical functions. These calculators come in handy especially when plotting complex functions like trigonometric graphs. They allow you to enter equations and automatically generate visual graphs.

• When graphing \( y=\frac{1}{4} \cot(x-\frac{\pi}{2}) \), a graphing utility will help plot the function over specified intervals.
• Such software can handle phase shifts, and amplitude adjustments effortlessly.
• Choose settings that clearly display asymptotes, crucial for interpreting the cotangent function.

Graphing utilities improve our understanding by providing immediate visual feedback, enhancing the learning experience.

Period of a Function

The period of a function defines the interval after which a trigonometric wave pattern repeats itself. In trigonometry, each primary function—sine, cosine, tangent, and cotangent—has its unique period. For cotangent, the period is \( \pi \), meaning the pattern starts anew every \( \pi \) units along the \( x \)-axis.In our function \( y=\frac{1}{4} \cot(x-\frac{\pi}{2}) \), the phase shift doesn't affect the period, so it remains \( \pi \), but it alters where these periods start. When using a graphing utility, ensure you display at least two full periods to capture the oscillation effect: here, that's from \( \frac{\pi}{2} \) to \( \frac{9\pi}{2} \). Such coverage ensures you witness the complete behavior of the function.