Problem 26 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=-\frac{1}{3} \cot (3 x-\pi)$$

Step-by-Step Solution

Answer

The period is \(\frac{\pi}{3}\). Graph has vertical asymptotes at \(x = \frac{\pi}{3} + n\frac{\pi}{3}\).

Step 1: Identify Cotangent Function Properties

The function in question is of the form \(y = A \cot(Bx + C)\). Here, \(A = -\frac{1}{3}\), \(B = 3\), and \(C = -\pi\). The general properties of the cotangent function are:- It has a period \(\frac{\pi}{B}\).- The cotangent function has undefined values (asymptotes) where its argument is an integer multiple of \(\pi\).

Step 2: Calculate the Period

The period of the function \(y = -\frac{1}{3} \cot(3x - \pi)\) is calculated using the formula for the period of a cotangent function: \[\text{Period} = \frac{\pi}{B} = \frac{\pi}{3}\]This tells us that the function repeats itself every \(\frac{\pi}{3}\) units along the x-axis.

Step 3: Determine Asymptotes

For \(y = \cot(Bx + C)\), asymptotes occur when \(Bx + C\) equals any integer multiple of \(\pi\). Solving for the initial asymptote:\[3x - \pi = 0\quad \Rightarrow \quad x = \frac{\pi}{3}\]Using the period, the general equation for the asymptotes is:\[x = \frac{\pi}{3} + n\left(\frac{\pi}{3}\right), \quad n \in \mathbb{Z}\]

Step 4: Sketch One Period of the Graph

To sketch one period of the function:1. Note the vertical asymptotes at \(x = \frac{\pi}{3}\) and \(x = \frac{2\pi}{3}\).2. The function is undefined at these x-values.3. Plot the behavior between these asymptotes. The cotangent function decreases from positive infinity to negative infinity as it goes from the left asymptote to the right.

Step 5: Sketch Additional Periods

Repeat the graph pattern for subsequent periods using the general asymptote equation. Mark the next set of asymptotes at multiples of the period \(x = n\left(\frac{\pi}{3}\right), n \in \mathbb{Z}\) such as \(x = 0, \pi, \frac{4\pi}{3}, \frac{5\pi}{3}\), etc.

Key Concepts

Period of Trigonometric Functions

The period of a trigonometric function tells us how often the function repeats its pattern along the x-axis. For the cotangent function, like other trigonometric functions, the period is determined by a specific formula.
The formula for the period of a cotangent function, expressed as \(y = A \cot(Bx + C)\), is \(\frac{\pi}{B}\). Here, \(B\) controls how stretched or compressed the function is horizontally.

In our example equation, \(y = -\frac{1}{3} \cot(3x - \pi)\), \(B\) equals 3, giving the period as \(\frac{\pi}{3}\). This means the pattern of the cotangent function, with its peaks and valleys, repeats every \(\frac{\pi}{3}\) units along the x-axis.

Understanding periods is critical when graphing trigonometric functions, as it helps define where and how often the pattern of the graph repeats itself. Being aware of the period allows you to anticipate how the entire graph will look just by examining one complete cycle.

Graphing Trigonometric Functions

Graphing a trigonometric function involves plotting its values to visualize its behavior over one complete cycle and then extending this pattern. For the cotangent function, like \(y = -\frac{1}{3} \cot(3x - \pi)\), it's crucial to identify key features like period, amplitude, phase shift, and asymptotes to accurately sketch the graph.

To start sketching:

• Identify the period, which aids in knowing the extent of one cycle. For \(y = -\frac{1}{3} \cot(3x - \pi)\), the period is \(\frac{\pi}{3}\).
• Plot the vertical asymptotes, the lines where the function approaches infinity. These lines are found where the cotangent function is undefined.
• Check where the function equals zero within one period. For cotangent, this typically occurs at the midpoint between two asymptotes.

Between the asymptotes, the cotangent function generally decreases, transitioning from positive infinity to negative infinity. This general behavior helps describe one cycle's shape.

Once a single period is graphed, replicate this pattern along the x-axis to show how the function repeats. Graphing additional periods involves extending the drawn cycle by increments equivalent to the original period \(\frac{\pi}{3}\).

Asymptotes of Trigonometric Functions

Asymptotes are critical in understanding the nature of trigonometric functions, especially those like the cotangent. These are lines that a function approaches but never crosses. For the function \(y = -\frac{1}{3} \cot(3x - \pi)\), asymptotes indicate where the function becomes undefined, often resulting in vertical lines on the graph.

To find asymptotes within the cotangent function \(y = \cot(Bx + C)\), solve for points where \(Bx + C = n\pi\), where \(n\) is any integer. This means the argument inside the cotangent function equals an integer multiple of \(\pi\).

Following this approach:

• Substitute \(Bx + C\) with the expression from the equation, getting \(3x - \pi = n\pi\).
• Solve for \(x\) to obtain the general formula for the asymptotes \(x = \frac{\pi}{3} + n\left(\frac{\pi}{3}\right)\).

These asymptotes recur at regular intervals, spaced by the function's period. They help set boundaries for each period's sketched graph and are crucial in predicting where the function will rise and fall as it moves towards infinity.