Problem 45 - Algebra and Trigonometry

Question

Find the period and graph the function. $$ y=2 \csc \left(\pi x-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

The period is 2, with a phase shift of \( \frac{1}{3} \) units to the right.

Step 1: Rewrite in Standard Form

The standard form of the cosecant function is \( y = a \csc(bx - c) + d \). Here, \( a = 2 \), \( b = \pi \), \( c = \frac{\pi}{3} \), and \( d = 0 \).

Step 2: Determine the Period

The period \( T \) of a cosecant function \( y = a \csc(bx - c) \) is calculated using the formula \( T = \frac{2\pi}{|b|} \). Substitute \( b = \pi \) to get \( T = \frac{2\pi}{\pi} = 2 \).

Step 3: Find the Phase Shift

The phase shift \( \phi \) is given by \( \frac{c}{b} \), where \( c = \frac{\pi}{3} \) and \( b = \pi \). This simplifies to \( \frac{\frac{\pi}{3}}{\pi} = \frac{1}{3} \). So there is a phase shift of \( \frac{1}{3} \) units to the right.

Step 4: Graphing the Function

To graph \( y = 2 \csc(\pi x - \frac{\pi}{3}) \), first sketch the basic sine function \( y = 2 \sin(\pi x - \frac{\pi}{3}) \), since the reciprocal of the sine function gives the cosecant. Then, draw vertical asymptotes where the sine function crosses the x-axis, as the cosecant function is undefined there. Next, mark the maximum and minimum points of the sine function, and sketch the U-shaped perodic pattern of the cosecant function between asymptotes.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc(x) \), is the reciprocal of the sine function. This means that it is defined as \( \csc(x) = \frac{1}{\sin(x)} \).
When graphing the cosecant function, it's essential to realize that wherever the sine function is zero, the cosecant function is undefined, resulting in vertical asymptotes in its graph.
Thus, these occur at multiples of \( \pi \) where the sine function crosses the x-axis.

• Value Range: The range of the cosecant function includes all values \( y \) such that \( y \leq -1 \) or \( y \geq 1 \), since it is the reciprocal of values in the range of sine, \([-1, 1]\).
• Graph Characteristics: The graph of \( \csc(x) \) consists of repeating U-shaped branches between consecutive vertical asymptotes.
• Behavior: As the sine function approaches zero, \( \csc(x) \) tends towards positive or negative infinity, creating the vertical lines that denote its undefined nature at those points.

Function Period

The period of a trigonometric function is the interval it takes for the function to complete one full cycle and begin repeating. For the standard sine and cosine functions, the period is \( 2\pi \). However, when coefficients are introduced in front of the variable \( x \), this affects the calculation of the period.
For functions in the form \( y = a \csc(bx - c) \), the period \( T \) is calculated as:

• \( T = \frac{2\pi}{|b|} \)

In our given function, \( y = 2 \csc(\pi x - \frac{\pi}{3}) \), we have \( b = \pi \). Therefore:

• \( T = \frac{2\pi}{\pi} = 2 \)

This means that the function repeats its pattern every 2 units along the x-axis.

• Practical Implication: Knowing the period is critical for graphing the function as it tells us where to place the repeating structures of the graph such as the U-shaped branches between vertical asymptotes.

Phase Shift

The phase shift of a trigonometric function determines how the graph of the function is horizontally shifted from its standard position.
For the cosecant function given in the form \( y = a \csc(bx - c) \), the phase shift is given by \( \phi = \frac{c}{b} \).
This tells us how much the function is moved along the x-axis.

• Calculation: In the example \( y = 2 \csc(\pi x - \frac{\pi}{3}) \), where \( c = \frac{\pi}{3} \) and \( b = \pi \), the phase shift \( \phi \) is calculated as \( \phi = \frac{\frac{\pi}{3}}{\pi} = \frac{1}{3} \).
• Direction: A positive phase shift means that the graph is shifted to the right, and a negative shift would move it to the left.

Understanding phase shift is crucial because it helps to accurately plot the basic features of the function such as where the vertical asymptotes and peaks of the U-shapes should appear on the graph.
It basically influences where the whole cycle of the function begins on the graph.