Problem 5 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=2 \csc x$$

Step-by-Step Solution

Answer

Period: \( 2\pi \). Asymptotes at \( x = n\pi \) for integer \( n \).

Step 1: Understand the Function

The given function is \( y = 2 \csc x \). The cosecant function, \( \csc x \), is the reciprocal of the sine function, \( \csc x = \frac{1}{\sin x} \). The graph of \( \csc x \) has vertical asymptotes where the sine function is zero.

Step 2: Identify Asymptotes

The sine function, \( \sin x \), equals zero at integer multiples of \( \pi \), i.e., \( x = n\pi \) where \( n \) is an integer. Therefore, the graph of \( \csc x \), and consequently \( 2\csc x \), has vertical asymptotes at these points.

Step 3: Determine the Period

The period of \( \csc x \) is the same as the sine function, which is \( 2\pi \). Thus, the period of \( 2 \csc x \) is also \( 2\pi \).

Step 4: Sketch the Graph

In the interval from \( 0 \) to \( 2\pi \), sketch the reciprocal of the sine function. At \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \), the sine function reaches its maximum and minimum, and the cosecant will have corresponding curves peaking at \( y = 2 \) or dipping to \( y = -2 \). Draw vertical asymptotes at \( x = 0 \), \( x = \pi \), and \( x = 2\pi \). Repeat the pattern for additional periods.

Key Concepts

Vertical Asymptotes

In trigonometric functions, especially when dealing with reciprocal functions like the cosecant, vertical asymptotes are a key characteristic. Vertical asymptotes occur where the original trigonometric function is undefined. For the cosecant function, which is defined as the reciprocal of the sine function, asymptotes arise where the sine function is equal to zero as these are points where division by zero would occur.In the function \(y = 2 \csc x\), the vertical asymptotes occur at the same places as for standard cosecant because multiplying by a constant (in this case, 2) doesn't affect the location of the asymptotes. The sine function is zero at integer multiples of \(\pi\), specifically \(x = n\pi\) where \(n\) is an integer. Therefore, the graph of \(2\csc x\) has vertical asymptotes at these values:

• \(x = 0\)
• \(x = \pi\)
• \(x = 2\pi\)
• And continuing at every integer multiple of \(\pi\)

These asymptotes divide the graph into segments where the function behaves reasonably—each section representing a full cycle of the cosecant function. Recognizing and plotting these asymptotes helps establish the boundaries of the function's unbounded behavior in its graph.

Period of Trigonometric Function

The period of a trigonometric function is the length required for the function to complete one full cycle of its pattern and begin repeating its values. Understanding the period is crucial for graphing trigonometric functions correctly.For the sine function, which forms the basis of the cosecant function, the period is \(2\pi\). This means every \(2\pi\) units along the x-axis, the sine function repeats its pattern. Since the cosecant is the reciprocal of the sine function, it naturally inherits this period. Therefore, the period of \(\csc x\) and consequently \(2\csc x\) remains \(2\pi\).For \(y = 2\csc x\), although the vertical stretch factor of 2 amplifies the peaks and troughs, the length of one complete cycle remains the same at \(2\pi\). This constant period ensures that when sketching its graph, the repeating pattern of peaks and troughs is spaced consistently over each interval of \(2\pi\). By marking out these intervals, you can accurately represent the periodic nature of the function on a graph.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived by taking the reciprocal (or inverse) of the basic trigonometric functions (sine, cosine, and tangent). The cosecant, secant, and cotangent functions are the three primary reciprocal trigonometric functions:

• Cosecant (\( \csc x = \frac{1}{\sin x} \))
• Secant (\( \sec x = \frac{1}{\cos x} \))
• Cotangent (\( \cot x = \frac{1}{\tan x} \))

Each of these is defined where the corresponding basic function is nonzero. In the case of the function \(y = 2\csc x\), since \(\csc x\) is the reciprocal of \(\sin x\), it implies that wherever \(\sin x\) reaches zero, \(\csc x\) (and thus \(2\csc x\)) will be undefined and have a vertical asymptote. The effect of the coefficient "2" is just to scale the graph vertically, altering the height of peaks and depth of troughs but leaving the asymptotes and zeros unchanged.Understanding reciprocal functions highlights how they differ visually and behaviorally from their base functions, due to their tendencies to approach infinity as their base functions approach zero.