Problem 28 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

Period: \(\frac{\pi}{2}\), Phase Shift: \(\frac{\pi}{6}\) right, Range: \((-\infty, \infty)\).

Step 1: Identify the standard form

The standard form for the cotangent function is \(y = a + b \cdot \cot(c(x - d))\), where \(a\) is the vertical shift, \(b\) is the amplitude (not applicable for cotangent functions as they don't have a maximum or minimum), \(c\) affects the period, and \(d\) affects the phase shift.

Step 2: Determine the period

The period of a cotangent function is calculated using the formula \( \frac{\pi}{|c|} \). In our equation, we have \(c = 2\). Therefore, the period is \( \frac{\pi}{2} \).

Step 3: Calculate the phase shift

The phase shift (horizontal shift) of the cotangent function can be found from the term \(c(x-d)\) in the standard form. Rearrange the equation inside the cotangent to match \(c(x-d)\). We have: \(2x - \frac{\pi}{3} = 2(x - \frac{\pi}{6})\). Thus, the phase shift is \(\frac{\pi}{6}\) units to the right.

Step 4: Determine the range

The range of a cotangent function is always \((-\infty, \infty)\) since cotangent functions continue indefinitely in both directions along the y-axis without bound. The vertical shift does not affect the range.

Key Concepts

Period of a Function

When working with trigonometric functions like cotangent, understanding the concept of the period is essential. The period of a function is the length of the smallest interval over which the function repeats its values. For a basic cotangent function, \( ext{cot}(x)\), the period is usually \( ext{π}\).
However, modifications to the function such as multiplication by a constant can change this period.
For instance, in the expression \(y = 2 + \cot\left(2x - \frac{\pi}{3}\right)\), the term that affects the period is the multiplier before \(x\), which is 2 in this case.
Using the formula \( \frac{\pi}{|c|} \), where \(c\) is the multiplier, we find that the period of this function becomes \( \frac{\pi}{2} \). Thus, the function completes one cycle every \( \frac{\pi}{2} \).

• The general formula for the period of a cotangent function: \( \frac{\pi}{|c|} \)
• Input from our specific function: \( \frac{\pi}{2} \)
• Why periods matter: They tell us how often the function repeats.

Phase Shift

Phase shift in trigonometric functions involves a horizontal shift along the x-axis. In terms of the function \(y = a + b \cdot \cot(c(x - d))\), this is determined by the term inside the function where \(d\) is directly indicative of the shift.
For the equation \(y = 2 + \cot\left(2x - \frac{\pi}{3}\right)\), we first rewrite the expression inside the cotangent function to reflect our standard form: \(2(x - \frac{\pi}{6})\).

• This rearrangement shows us that the expression shifts the function \(\frac{\pi}{6}\) units to the right.

It's important to note that positive \(d\) results in a shift to the right, and negative \(d\) results in a shift to the left.

• Key takeaway: The phase shift indicates where the start of a new cycle is located compared to the origin.
• The concept ensures we know how the graph's cycle is adjusted horizontally.

Range of Trigonometric Functions

Understanding the range of a function helps us know the possible values output by the function on the y-axis. Trigonometric functions have specific ranges.
For the cotangent function, \( \text{cot}(x) \), the output can extend indefinitely, meaning its range is \(( -\infty, \infty )\).
Even after accounting for vertical shifts like in \(y = 2 + \cot\left(2x - \frac{\pi}{3}\right)\), the range remains the same.

• Vertical shifts affect the position but not the range of cotangent.
• The y-values possible for a cotangent function will always cover all real numbers.

This is critical to remember when graphing or analyzing the behavior of these functions in various contexts.

• The independence of range from vertical shifts emphasizes the unbounded nature of cotangent outputs.