Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=2 \cot \left(2 x+\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

Period: \( \frac{\pi}{2} \), shift: left \( \frac{\pi}{4} \). Asymptotes at \( x = \frac{n\pi}{2} - \frac{\pi}{4} \).

Step 1: Identify the basic function

The given function is a transformation of the basic cotangent function \( y = \cot(x) \). The cotangent function has vertical asymptotes where the tangent function is zero, that is, at \( x = n\pi \) for integer \( n \). Its period is \( \pi \).

Step 2: Determine the transformation effect

The function provided is \( y = 2 \cot(2x + \frac{\pi}{2}) \). The coefficient \( 2 \) outside the cotangent affects only the amplitude, not the period. The expression inside the cotangent, \( 2x + \frac{\pi}{2} \), will affect the period and shift.

Step 3: Calculate the new period

The period of the function \( y = \cot(bx) \) is \( \frac{\pi}{b} \). Here, \( b = 2 \), so the period becomes \( \frac{\pi}{2} \).

Step 4: Identify the phase shift

The phase shift can be found by setting \( 2x + \frac{\pi}{2} = 0 \) and solving for \( x \). Solving gives \( x = -\frac{\pi}{4} \). Thus, the graph is shifted to the left by \( \frac{\pi}{4} \).

Step 5: Determine the location of asymptotes

The asymptotes are at points where the argument of the cotangent makes it undefined. For \( y = 2 \cot(2x + \frac{\pi}{2}) \), the original asymptotes at \( 2x + \frac{\pi}{2} = n\pi \) imply \( 2x = n\pi - \frac{\pi}{2} \), thus \( x = \frac{n\pi}{2} - \frac{\pi}{4} \).

Step 6: Sketch the graph

To sketch the graph, draw vertical lines (asymptotes) at \( x = \frac{n\pi}{2} - \frac{\pi}{4} \) for integer \( n \). The function decreases from \( +\infty \) to \(-\infty\) between asymptotes over intervals of length \( \frac{\pi}{2} \). The graph is shifted left by \( \frac{\pi}{4} \), and the amplitude is multiplied by 2.

Key Concepts

Period of Trigonometric Functions

The period of a trigonometric function is a fundamental characteristic that indicates how often the function repeats itself over a particular interval. For many basic trigonometric functions, like sine and cosine, this period is usually around 2π. However, for the cotangent function, the period is simply π because it completes one full cycle within that span.
When dealing with transformations of these functions, the period might change depending on factors inside the function. For example, if you have a transformation such as \(y = \cot(bx)\), the period becomes \(\frac{\pi}{b}\), where \(b\) is a coefficient affecting the horizontal stretch or compression of the graph. In the case of \(y = 2\cot(2x + \frac{\pi}{2})\), since \(b = 2\), the new period is \(\frac{\pi}{2}\). This means every \(\frac{\pi}{2}\) units along the x-axis, the graph pattern repeats itself.

Cotangent Function

The cotangent function, symbolized as \(\cot(x)\), is the reciprocal of the tangent function. One of its distinct features is its behaviour as an odd function, meaning that \(\cot(-x) = -\cot(x)\). This characteristic results in a graph that is symmetric about the origin.
Unlike the sine and cosine functions, which have amplitudes describing their vertical stretches, the cotangent function does not have a traditional amplitude. Instead, it describes the rate at which the function approaches its asymptotes. Standard cotangent functions have a period of \(\pi\), with vertical asymptotes at integer multiples of \(\pi\), like \(x = n\pi\).
The function \(y = 2\cot(2x + \frac{\pi}{2})\) modifies this basic behaviour by introducing a vertical stretch with a coefficient of 2, scaling the steepness of the graph's descent between asymptotes.

Vertical Asymptotes

Vertical asymptotes occur in trigonometric functions like cotangent where the function is undefined and exhibits unbounded behaviour. For the basic \(\cot(x)\) function, these asymptotes are located where the tangent function equals zero, notably at every integer multiple of \(\pi\) (i.e., \(x = n\pi\)).
When transformations are applied to the cotangent function, such as \(y = 2\cot(2x + \frac{\pi}{2})\), the position of these asymptotes shifts. The expression within the cotangent sets the stage for where these undefined points will occur. By setting the expression \(2x + \frac{\pi}{2} = n\pi\) for integer \(n\), we can solve for \(x\) and determine the exact locations of asymptotes. Solving gives \(x = \frac{n\pi}{2} - \frac{\pi}{4}\), which leads to evenly spaced vertical lines representing sections where the function shoots to infinity.

Phase Shift

A phase shift describes a horizontal translation of the graph of a trigonometric function. It results from adjustments made to the function's argument within its trigonometric expression. For a function like \(y = \cot(bx + c)\), the phase shift can be found by setting the inside of the cotangent to zero: \(bx + c = 0\) and solving for \(x\).
In the case of \(y = 2\cot(2x + \frac{\pi}{2})\), solving \(2x + \frac{\pi}{2} = 0\) yields \(x = -\frac{\pi}{4}\). This means the entire graph shifts to the left by \(\frac{\pi}{4}\), moving every feature, including peaks, troughs, and asymptotes, in that direction.
This horizontal shift can significantly alter how the function's regular periodic pattern appears on the graph, making it essential to account for these shifts when sketching or interpreting trigonometric functions.