Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=-\frac{1}{3} \csc \left(\frac{1}{2} x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

Period: \(4\pi\); Phase shift: \(\pi\) (right); Range: \((-\infty, -\frac{1}{3}] \cup [\frac{1}{3}, \infty)\).

Step 1: Identify the standard form of the cosecant function

The function to analyze is given as \( y=-\frac{1}{3} \csc \left(\frac{1}{2} x-\frac{\pi}{2}\right) \). The standard form of a cosecant function is \( y = A \csc(Bx - C) \), where \( A \) affects the range, \( B \) determines the period, and \( C/B \) indicates the phase shift.

Step 2: Calculate the period

The period of a cosecant function, derived from \( \csc(x) \), is given by \( \frac{2\pi}{B} \). For this function, \( B = \frac{1}{2} \). Therefore, the period is given by: \[\text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi \].

Step 3: Determine the phase shift

The phase shift of the function is determined by the formula \( \frac{C}{B} \). In this case, \( C = \frac{\pi}{2} \) and \( B = \frac{1}{2} \). Therefore, the phase shift is: \[\text{Phase shift} = \frac{\frac{\pi}{2}}{\frac{1}{2}} = \pi \]. Since the function is \( \frac{1}{2} x - \frac{\pi}{2} \), the shift is to the right by \( \pi \).

Step 4: Find the range

The range of a cosecant function \( y = A \csc(Bx - C) \) without vertical displacement is typically \( (-\infty, -|A|] \cup [|A|, \infty) \). For \( y=-\frac{1}{3} \csc \left(\frac{1}{2} x-\frac{\pi}{2}\right) \), the amplitude \(|A| = \frac{1}{3}\), so the range is: \[(-\infty, -\frac{1}{3}] \cup [\frac{1}{3}, \infty)\].

Key Concepts

Period of Trigonometric Functions

When approaching trigonometric functions, understanding the concept of a "period" is essential. The "period" of a function is the interval over which the function repeats itself. It's like the rhythm or cycle for the function.
For basic trigonometric functions like sine, cosine, and cosecant, the default period is \(2\pi\). This means that every \(2\pi\), the function will start to repeat its values.

• For a modified function like \( y = A \csc(Bx - C) \), the period is calculated as \(\frac{2\pi}{B}\).
• In our function, \( B = \frac{1}{2} \), making the period \(\frac{2\pi}{\frac{1}{2}} = 4\pi\). This means every \(4\pi\), this particular function will repeat its pattern.

Understanding periods helps predict the behavior of trigonometric functions, making analyzing and graphing them much more manageable.

Phase Shift

The "phase shift" of a trigonometric function tells you how the function is horizontally shifted from the usual position. Imagine dragging the function along the x-axis either to the left or to the right.
For a pattern expressed in the form \( y = A \csc(Bx - C) \), the phase shift is computed as \( \frac{C}{B} \).

• In this exercise, \( C = \frac{\pi}{2} \) and \( B = \frac{1}{2} \). Hence, the phase shift is \( \frac{\frac{\pi}{2}}{\frac{1}{2}} = \pi \).
• This result illustrates a shift to the right by \( \pi \).

Recognizing the phase shift is vital because it influences how the function's graph aligns with the traditional axis setup. A rightward shift may mean the function intersects common points later along the x-axis.

Range of Functions

The range of a function refers to all possible values the function can produce. For trigonometric functions, especially the cosecant \( \csc(x) \), we often encounter infinite limits.
With a generic cosecant function expressed as \( y = A \csc(Bx - C) \), the range, assuming no vertical shifts or transformations, is typically \((-\infty, -|A|] \cup [|A|, \infty)\).

• For our function, \( y = -\frac{1}{3} \csc\left(\frac{1}{2}x - \frac{\pi}{2}\right) \), \(|A|\) equals \(\frac{1}{3}\).
• Despite the coefficient being negative, the absolute value \(|A|\) means the range becomes \((-\infty, -\frac{1}{3}] \cup [\frac{1}{3}, \infty)\).

Understanding the range helps clarify the full spectrum of output values the function encompasses, especially where limitations or asymptotes might present in graphs.