Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=\frac{5}{2} \cot \left[\frac{1}{3}\left(x-\frac{\pi}{2}\right)\right]$$

Step-by-Step Solution

Answer

(a) Period: \(3\pi\); (b) Phase shift: \(\frac{\pi}{6}\) right; (c) Range: \((-\infty, \infty)\).

Step 1: Determine the Period of the Cotangent Function

The general form of the cotangent function is \( y = a \cot(bx - c) \). The period of a cotangent function is calculated using the formula \( \frac{\pi}{b} \). In our function, \( b = \frac{1}{3} \), so the period is \( \frac{\pi}{\frac{1}{3}} = 3\pi \).

Step 2: Calculate the Phase Shift

The phase shift of a trigonometric function in the form \( y = a \cot(bx - c) \) is found using \( \frac{c}{b} \). In this case, \( c = \frac{1}{3} \cdot \left(-\frac{\pi}{2}\right) = -\frac{\pi}{6} \), so the phase shift is right by \( \frac{\pi}{6} \).

Step 3: Determine the Range of the Function

The cotangent function \( \cot(x) \) has a range of \((-\infty, \infty)\). Since the function \( y = \frac{5}{2} \cot\left(\frac{1}{3}(x-\frac{\pi}{2})\right) \) only involves a multiplication by \( \frac{5}{2} \), it scales the amplitude but does not affect the range. Thus, the range of this function is also \((-\infty, \infty)\).

Key Concepts

Period

In the world of trigonometric functions, the period represents the length of one complete cycle of the function before it begins to repeat. For a cotangent function, the period is calculated differently than for regular sine or cosine functions. Specifically, the formula used is \( \frac{\pi}{b} \), where \( b \) is a coefficient of \( x \) inside the cotangent expression. In our given function, \( y = \frac{5}{2} \cot\left[\frac{1}{3}\left(x-\frac{\pi}{2}\right)\right] \), the value of \( b \) is \( \frac{1}{3} \).
To find the period, divide \( \pi \) by \( \frac{1}{3} \), as follows:

• Calculate \( \frac{\pi}{\frac{1}{3}} \)
• Multiplying by the reciprocal, \( \pi \times 3 \)
• Results in a period of \( 3\pi \)

Thus, this cotangent function repeats its pattern every \( 3\pi \) units along the x-axis. Calculating periods helps in predicting the behavior of trigonometric graphs over various intervals.

Phase Shift

Phase shift in trigonometric functions indicates how far the graph of the function is horizontally shifted from its usual position. It is particularly noted in functions of the form \( y = a \cot(bx - c) \). To determine the phase shift, the formula \( \frac{c}{b} \) is employed.
For our function, it is first important to interpret \( c \) correctly from the expression. Here, the term \( \left( bx - c \right) \) results in \( bx = \frac{1}{3}(x - \frac{\pi}{2}) \), revealing \( c = -\frac{1}{6}\pi \).
After identifying \( c \), follow these steps:

Recognize \( b = \frac{1}{3} \) and \( c = -\frac{\pi}{6} \)

Compute the phase shift: \( \frac{-\frac{\pi}{6}}{\frac{1}{3}} = -\frac{\pi}{6} \times 3 \)

Calculate to get a rightward shift \( \frac{\pi}{6} \)

This implies that the basic shape of the cotangent function moves \( \frac{\pi}{6} \) units to the right. Recognizing phase shifts aids in aligning graphs with other functions or data sets accurately.

Range

The range in trigonometric functions denotes all possible values that the output, or \( y \)-values, of the function can take. With the cotangent function \( \cot(x) \), the range is particularly widespread:

• The cotangent's range is \((-\infty, \infty)\)

For the exercise's function \( y = \frac{5}{2}\cot\left(\frac{1}{3}(x-\frac{\pi}{2})\right) \), even though it multiplies by \( \frac{5}{2} \), the inherent range remains unchanged. Multiplying by a constant affects amplitude, enlarging the output without restricting or modifying the boundaries of the range. Thus, the whole range from negative to positive infinity stays the same, \((-\infty, \infty)\).
Understanding the range helps in grasping how input changes affect the heights and depths of the function's output in a graph.