Problem 13 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\tan \frac{1}{4} x$$

Step-by-Step Solution

Answer

Period: \(4\pi\). Asymptotes at \(x = 2\pi + 4n\pi\).

Step 1: Identify the Function

The function given is \( y = \tan \left( \frac{1}{4}x \right) \), which is a tangent function with a horizontal stretch by a factor of \( \frac{1}{4} \).

Step 2: Determine the Period of the Tangent Function

The standard tangent function \( y = \tan x \) has a period of \( \pi \) because it repeats every \( \pi \). For the function \( y = \tan \left( \frac{1}{4}x \right) \), we find the period by using the formula: \( \text{Period} = \frac{\pi}{|b|} \) where \( b = \frac{1}{4} \). Thus, the period is \( \text{Period} = \frac{\pi}{\frac{1}{4}} = 4\pi \).

Step 3: Determine the Asymptotes

Tangent functions have vertical asymptotes where the function is undefined. For the standard \( \tan x \), these asymptotes occur at \( x = \frac{\pi}{2} + n\pi \). For \( y = \tan \left( \frac{1}{4}x \right) \), adjust the formula to find the asymptotes as \( \frac{1}{4}x = \frac{\pi}{2} + n\pi \). Solving for \( x \), we get \( x = 2\pi + 4n\pi \).

Step 4: Sketch the Graph

Start by plotting the basic points of the graph such as the zero crossings, which occur at the same period intervals \( 2n\pi \) for \( y = \tan \left( \frac{1}{4}x \right) \). Draw vertical dashed lines at the asymptotes \( x = 2\pi + 4n\pi \). Ensure that between each pair of asymptotes, the function starts from a negative infinity, crosses zero in the middle, and ends positive infinity, repeating every \( 4\pi \).

Key Concepts

Period of Trigonometric Functions

Understanding the period of trigonometric functions is essential for predicting their behavior over repeated intervals. The period is the length of one complete cycle before the function starts to repeat itself. For the standard tangent function, represented by \( y = \tan x \), this cycle or period is \( \pi \). This means every \( \pi \) units on the x-axis, the pattern restarts.

When transformations such as horizontal stretches or compressions are applied, the period can change. For instance, the function \( y = \tan \left( \frac{1}{4}x \right) \) involves a horizontal stretch by a factor of 4. To find its period, use the formula:

• \( \text{Period} = \frac{\pi}{|b|} \)

where \( b \) is the coefficient of \( x \). For our function, \( b = \frac{1}{4} \), leading to a period of \( 4\pi \).
This new period indicates that the function repeats itself every \( 4\pi \) units on the x-axis, instead of every \( \pi \) units.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their behavior over their defined periods. The graph of a function like \( y = \tan \left( \frac{1}{4}x \right) \) reveals essential features such as zero crossings, peak values, and asymptotes.

It's important to start by identifying key points. For a tangent function, the graph crosses zero at regular intervals. For \( y = \tan \left( \frac{1}{4}x \right) \), these zero crossings repeat every \( 2n\pi \), and the pattern will reset every \( 4\pi \).

• Plot the zero crossings at regular interval of \( 2n\pi \).
• Between these points, observe how the tangent function rises steeply from negative infinity, crosses the x-axis at zero, and climbs to positive infinity.

Remember, the tangent function is indefinite at its asymptotes which journey regularly along the x-axis.

Vertical Asymptotes in Trigonometry

Vertical asymptotes are lines where the function tends to infinity and the behavior of the function appears undefined. For tangent functions, such as \( y = \tan x \), these asymptotes occur every half-period at \( x = \frac{\pi}{2} + n\pi \).

In \( y = \tan \left( \frac{1}{4}x \right) \), the transformation affects where these asymptotes appear. To find their positions, set \( \frac{1}{4}x = \frac{\pi}{2} + n\pi \) and solve for \( x \):

• This gives \( x = 2\pi + 4n\pi \) as the asymptotes.

The asymptotes are crucial for sketching the graph, as they highlight intervals where the function is undefined.
Imagine placing vertical dashed lines on a graph at these points to accurately portray the function's behavior. In every cycle between two asymptotes, the tangent function transitions smoothly from negative to positive infinity.