Problem 14 - Algebra 2

Question

Identify the period and tell where two asymptotes occur for each function. $$ y=\tan \frac{2}{3 \pi} \theta $$

Step-by-Step Solution

Answer

The period of the function \(y = \tan\left(\frac{2}{3 \pi}\theta\right)\) is \(\frac{3 \pi^2}{2}\), and the first two asymptotes are found at \(\theta = \frac{3 \pi^2}{4}\) and \(\theta = \frac{3 \pi^2}{4} + \frac{3 \pi^2}{2}\) respectively.

Step 1: Identifying the factor

First, we need to identify the factor by which \(x\) (in our case \(\theta\)) is multiplied inside the tangent function. Here, it is \(\frac{2}{3 \pi}\). Therefore, \(b = \frac{2}{3 \pi}\).

Step 2: The Period of the Function

We calculate the period of the function using the formula: \(\frac{\pi}{b}\). Substituting the value of \(b = \frac{2}{3 \pi}\) we calculated in the previous step, we get \(\frac{\pi}{\frac{2}{3 \pi}} = \frac{3 \pi^2}{2}\).

Step 3: The Asymptotes of the Function

The tangent function has asymptotes at \(x = \frac{\pi}{2} + n\pi\) (for \(y = \tan(x)\)). When there is a factor of \(b\), the locations of asymptotes change to \(x = \frac{\pi}{2b} + n\frac{\pi}{b}\). Substituting our calculated value of \(b = \frac{2}{3 \pi}\), we obtain \(x = \frac{\pi}{2*\frac{2}{3 \pi}} + n*\frac{\pi}{\frac{2}{3 \pi}}\), simplifying we get, \(\theta = \frac{3 \pi^2}{4} + n*\frac{3 \pi^2}{2}\) where \(n\) is an integer. So the first two asymptotes will occur at \(\theta = \frac{3 \pi^2}{4}\) and \(\theta = \frac{3 \pi^2}{4} + \frac{3 \pi^2}{2}\).

Key Concepts

Tangent Function

The tangent function, denoted as \(y = \tan(\theta)\), is one of the six primary trigonometric functions. It is defined as the ratio of the sine function to the cosine function: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). This means that the function is undefined whenever the cosine of \(\theta\) is zero, leading to vertical asymptotes at those points. The tangent function exhibits a periodic pattern, which means its values repeat at regular intervals along the angle \(\theta\). This function is fundamental in trigonometry and is often used in various fields such as physics, engineering, and geometry.

Asymptotes

Asymptotes are lines that a curve approaches as it heads towards infinity. In the case of the tangent function, which has the formula \(y = \tan(\theta)\), asymptotes arise where the cosine part of the tangent formula equals zero, causing the tangent function to become undefined.

For the standard tangent function \(y = \tan(x)\), vertical asymptotes occur at \(x = \frac{\pi}{2} + n\pi\), where \(n\) is an integer. The presence of a coefficient \(b\) in the equation \(y = \tan(b\theta)\) shifts these asymptotes to a new position. This adjustment is captured by the formula: \(\theta = \frac{\pi}{2b} + n\frac{\pi}{b}\), which adjusts the locations accordingly and crucially helps predict where the graph spikes between positive and negative infinity.

Trigonometric Period

The trigonometric period of a function refers to the interval over which its values repeat. For basic trigonometric functions like sine, cosine, and tangent, the period gives the length of one full cycle of the wave.

For the standard tangent function, the period is \(\pi\), meaning every \(\pi\) units along the \(x\)-axis, the function repeats its values. The period alters based on the coefficient \(b\) in the formula \(y = \tan(b\theta)\). The modified period becomes \(\frac{\pi}{b}\). For example, if \(b = \frac{2}{3\pi}\), then the period will extend to \(\frac{3\pi^2}{2}\), prolonging the interval between repeated segments. Understanding the trigonometric period is essential for graphing and analyzing the behavior of trigonometric functions in various applications.

Algebra 2 Concepts

At the Algebra 2 level, students are introduced to complex functions and trigonometric principles. Understanding how trigonometric transformations affect the graphs of functions is a key concept. Here, students learn how coefficients like \(a\), \(b\), \(h\), and \(k\) in the function \(y = a \tan(b\theta - h) + k\) alter the amplitude, period, and horizontal/vertical shifts of the function.

This exercise specifically focuses on how the coefficient \(b\) within the tangent function affects its period and the location of asymptotes. By mastering these transformations, students can predict and describe the graphical behavior of trigonometric functions, using skills vital to math courses beyond Algebra 2, including calculus and physics. These algebra skills form the groundwork for exploring more advanced mathematical concepts.