Problem 45 - Precalculus Mathematics for Calculus

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 of the function is 2; the graph is shifted right by \( \frac{1}{3} \).

Step 1: Identify the Basic Cosecant Function Parameters

The basic cosecant function is \( y = a \csc(bx - c) \). For the function given, \( y = 2 \csc(\pi x - \frac{\pi}{3}) \), we can identify \( a = 2 \), \( b = \pi \), and \( c = \frac{\pi}{3} \).

Step 2: Determine the Period of the Function

The period of a cosecant function is determined by the formula \( \frac{2\pi}{b} \). Substituting \( b = \pi \) into this formula gives:\[ \text{Period} = \frac{2\pi}{\pi} = 2 \].

Step 3: Identify the Phase Shift

The phase shift of the function is determined by \( \frac{c}{b} \). For \( c = \frac{\pi}{3} \) and \( b = \pi \), the phase shift is:\[ \text{Phase Shift} = \frac{\pi/3}{\pi} = \frac{1}{3} \].

Step 4: Sketch the Graph of the Function

1. Mark the phase shift on the x-axis: start the graph at \( x = \frac{1}{3} \).2. Use the period to find important points. Since the period is 2, full cycles occur at intervals of 2.3. Plot key points where the cosecant function is undefined (asymptotes) such as where the sine base equals zero.4. Plot maximum and minimum points using amplitude \( a = 2 \).5. Repeat the pattern for adjacent cycles by adding/subtracting the period length.

Key Concepts

Trigonometric Functions

The cosecant function, denoted as \( \csc(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function:

• \( \csc(x) = \frac{1}{\sin(x)} \) when \( \sin(x) eq 0 \).

This means that the cosecant function is undefined whenever \( \sin(x) = 0 \). Typically, the basic cosecant function has vertical asymptotes where the sine equals zero, causing discontinuities at those points. In terms of graphing, these asymptotes significantly impact the appearance of the cosecant curve, which resembles a series of alternating arcs. Therefore, understanding the relationship between the sine and cosecant functions is crucial for analyzing their behaviors in any given function.

Phase Shift

The phase shift in trigonometric functions refers to a horizontal translation of the graph along the x-axis. It indicates how far the graph is shifted from the basic position of the standard trigonometric function. For the function \( y = 2 \csc(\pi x - \frac{\pi}{3}) \), the phase shift is calculated by the formula:

• \( \text{Phase Shift} = \frac{c}{b} \)

Given \( c = \frac{\pi}{3} \) and \( b = \pi \), the phase shift equals \( \frac{1}{3} \). This means the entire graph is shifted to the right by \( \frac{1}{3} \) units. When graphing, you'd begin plotting points at \( x = \frac{1}{3} \) instead of the origin. This calculation is vital for correctly positioning the graph's features, especially when examining sine, cosine, and their related functions.

Period of Functions

The period of a trigonometric function is the interval length after which the function's values repeat. For the cosecant function and most other trigonometric functions, the period can be determined using the parameter \( b \) in the general formula \( y = a \csc(bx - c) \). The period is calculated as:

• \( \text{Period} = \frac{2\pi}{b} \)

For our given function, \( b = \pi \), so the period is \( 2 \). This deduction means the graph's repeating unit length is 2 units along the x-axis. Knowing the period is essential for predicting and accurately marking the cycles of the trigonometric graphs, which aids in sketching precise and informative visuals of these functions.

Graphing Trigonometric Functions

Graphing a trigonometric function such as \( y = 2 \csc(\pi x - \frac{\pi}{3}) \) involves following a systematic approach:

• Identify the function's key parameters: amplitude \( a \), period, and phase shift to guide the plotting.
• Start by marking the phase shift on the x-axis as the initial point for plotting.
• Determine the positions of the vertical asymptotes, typically where the sine base equals zero, as these are points of discontinuity.
• Recognize the maximum and minimum values, impacted by the amplitude.
• Use the period to mark the span of one full cycle and replicate it to observe repeated patterns along the x-axis.

By paying attention to these elements, plotting becomes more manageable, and the graph's intricate patterns become clearer. This structured approach is especially useful in educational settings where understanding the visual aspects of trigonometric functions is necessary for deeper comprehension.