Problem 47 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=\tan \left(\frac{2}{3} x-\frac{\pi}{6}\right)$$

Step-by-Step Solution

Answer

The period of the function is \(\frac{3\pi}{2}\).

Step 1: Understanding the Periodicity of Tangent

The tangent function has a typical period of \(\pi\). Given a tangent function in the form \(y = \tan(bx - c)\), the period is given by \(\frac{\pi}{|b|}\). In this function, \(b = \frac{2}{3}\).

Step 2: Calculate the Period

Use the formula \(\frac{\pi}{|b|}\) to find the period. Substitute \(b = \frac{2}{3}\) into the formula: \[ \text{Period} = \frac{\pi}{\left|\frac{2}{3}\right|} = \frac{\pi}{\frac{2}{3}} = \frac{3\pi}{2}. \]

Step 3: Identify Key Characteristics for Graphing

The function \(y = \tan\left(\frac{2}{3}x - \frac{\pi}{6}\right)\) will have vertical asymptotes where the inside of the tangent function equals \(\frac{\pi}{2} + k\pi\), where \(k\) is an integer. Solve for \(x\) to find the location of these asymptotes.

Step 4: Solve for Vertical Asymptotes

Set \(\frac{2}{3}x - \frac{\pi}{6} = \frac{\pi}{2} + k\pi\). Solve for \(x\):1. Multiply through by 3 to clear fractions: \(2x - \frac{\pi}{2} = \frac{3\pi}{2} + 3k\pi\),2. Add \(\frac{\pi}{2}\) to both sides: \(2x = 2\pi + 3k\pi\),3. Divide by 2: \(x = \pi + \frac{3k\pi}{2}.\)These are the values of \(x\) where vertical asymptotes occur.

Step 5: Sketch the Graph with Period and Asymptotes

Using the period \(\frac{3\pi}{2}\) and the calculated asymptotes, the tangent graph will repeat every \(\frac{3\pi}{2}\) and have vertical asymptotes at \(x = \pi + \frac{3k\pi}{2}\). Draw one period between two consecutive asymptotes. The graph should pass through the midpoint between asymptotes.

Key Concepts

Graphing Trigonometric Functions

When graphing trigonometric functions, it is essential to understand their shape and key characteristics like period, amplitude, and phase shift. For the tangent function, it has a unique property: its graph repeats every period, and it has no amplitude because it extends infinitely in the vertical direction.
Despite these unique features, there are several common steps in graphing any trigonometric function:

• Determine the period: This is the length over which the function completes one cycle.
• Identify any phase shifts and vertical shifts.
• Locate key points: For tangent, these include places where the graph intersects with the x-axis, as well as points where there are vertical asymptotes.

After sequencing these insights, plot the function with consideration of these points and trends, ensuring to represent the function's repeating nature accurately.

Tangent Function Periodicity

The periodicity of the tangent function is one of its defining characteristics. For a standard tangent function, the period is \[\pi \]and determines how often the pattern of the function repeats.
When the tangent function is modified, such as in \(y = \tan\left( \frac{2}{3}x - \frac{\pi}{6} \right)\), the period is affected by the coefficient in front of \(x\).The formula used to find the period for the tangent function \(y = \tan(bx - c)\)is \[\text{Period} = \frac{\pi}{|b|}.\]In our problem, this gives us a modified period of \(\frac{3\pi}{2}\).
This shorter period signifies the function will repeat its pattern more quickly along the x-axis compared to the standard period \(\pi\). This is why understanding the period is crucial for predicting the behavior of the function.

Vertical Asymptotes in Trigonometric Graphs

Vertical asymptotes are lines that the graph of a function approaches but never touches. The tangent function typically has vertical asymptotes where it is undefined. For the general tangent function, those vertical asymptotes occur where the function's angle equals \(\frac{\pi}{2} + k\pi\), where \(k\) is an integer.
In the function \(y = \tan\left( \frac{2}{3}x - \frac{\pi}{6} \right)\),to locate asymptotes, we solve for \(x\) using:\[\frac{2}{3}x - \frac{\pi}{6} = \frac{\pi}{2} + k\pi.\]Upon simplifying, we find vertical asymptotes at\(x = \pi + \frac{3k\pi}{2}\).These asymptotes play a pivotal role in defining the intervals within which each repeating section of the graph will be plotted, as the graph approaches these asymptotes without ever crossing them.
Understanding where they lie is crucial for accurately sketching the tangent function.