Problem 13 - Algebra 2

Question

Identify the period and tell where two asymptotes occur for each function. $$ y=\tan 4 \theta $$

Step-by-Step Solution

Answer

The period of the function \(y = \tan 4\theta\) is \(\pi/4\). The asymptotes of the function occur at \(\theta = \frac{\pi}{8} + \frac{k\pi}{4}\), where k is an integer.

Step 1: Determine the period

We determine the period by using the modified formula for the period, \(\pi/B\). Given our function \(y = \tan 4\theta\), B is 4. Therefore the period of \(y = \tan 4\theta\) is \(\pi/4\).

Step 2: Find the Asymptotes

Asymptotes for \(\tan x\) occur when \(\cos x = 0\), which translates to \( x = \frac{\pi}{2} + k\pi \), where k is an integer. With \(x = 4\theta\), we have to adjust this formula as well. The general formula for the location of the asymptotes becomes \(4\theta = \frac{\pi}{2} + k\pi \), and solving for \(\theta\) gives \(\theta = \frac{\pi}{8} + \frac{k\pi}{4}\), where k is an integer.

Key Concepts

Period of a Function

In mathematics, the period of a function defines the interval after which it repeats its values. For trigonometric functions like sine, cosine, and tangent, understanding the period is essential because these functions are inherently cyclical.

When dealing with the tangent function, the standard function \(y = \tan x\) has a period of \(\pi\). However, when the function is modified, such as \(y = \tan(B\theta)\), the period changes. The formula to find the new period is \(\pi/B\), with \(B\) being the coefficient of \(\theta\).

• For example, in the function \(y = \tan 4\theta\), identify \(B = 4\).
• Applying the formula, the period therefore is \(\pi/4\).

This means that the function \(y = \tan 4\theta\) will repeat its cycle every \(\pi/4\) units along the \(\theta\) axis.

Knowing the period helps in predicting the behavior of the function over different intervals, essential for graphing and applying the function effectively.

Asymptotes

Asymptotes are lines that a curve approaches but never touches. In the context of the tangent function, it's crucial to know where these lines lie, as they indicate points where the function is undefined.

For a basic tangent function, \(y = \tan x\), vertical asymptotes occur when \(\cos x = 0\), because tangent is defined as the ratio \(\sin x / \cos x\). The typical equation for the asymptotes is therefore \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer.

• Alteration in the function, such as \(y = \tan 4\theta\), modifies the positions of these asymptotes.
• Substituting \(x = 4\theta\) into the asymptote formula leads to \(4\theta = \frac{\pi}{2} + k\pi\).
• Solving for \(\theta\) gives the formula \(\theta = \frac{\pi}{8} + \frac{k\pi}{4}\).

This shows the positions where the asymptotes occur for the modified function, marking points on the graph where the tangent function has breaks and is not defined.

Tangent Function

The tangent function is one of the fundamental trigonometric functions, often represented as \(\tan(\theta)\). It describes the ratio of the opposite side to the adjacent side in a right-angled triangle.

Graphically, the tangent function has a distinct and repeating pattern, characterized by its vertical asymptotes and periodic nature. Unlike sine and cosine functions, tangent's graph extends indefinitely vertically.

• The standard tangent function \(y = \tan x\) has asymptotes at \(x = \frac{\pi}{2} + k\pi\), separating each period.
• The function's graph moves from negative infinity to positive infinity through zero, repeating every \(\pi\) units.
• In modified tangent functions like \(y = \tan 4\theta\), the periodicity and asymptote positions change, as demonstrated above.

The tangent function's unique properties make it integral to trigonometry, especially in situations requiring angle measurement and analysis of cyclical phenomena.