Problem 2 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The period is \( \pi \), with asymptotes at \( x = \frac{\pi}{2} + n\pi \).

Step 1: Identify the Standard Form

The standard form of the tangent function is \( y = a \tan(bx + c) + d \). In this function, \( y = \frac{1}{4} \tan x \), \( a = \frac{1}{4} \), \( b = 1 \), \( c = 0 \), and \( d = 0 \).

Step 2: Calculate the Period

The period of the tangent function in its standard form is \( \frac{\pi}{|b|} \). Here, \( b = 1 \), so the period is \( \frac{\pi}{1} = \pi \).

Step 3: Determine the Asymptotes

The asymptotes for the tangent function \( y = \tan x \) occur where \( \tan x \) is undefined, i.e., at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \). Since our function is \( y = \frac{1}{4} \tan x \), the positions of asymptotes remain unchanged.

Step 4: Sketch the Graph

- The graph of \( y = \frac{1}{4} \tan x \) will have a period of \( \pi \), meaning that it will repeat every \( \pi \) units.- It will have vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \).- The function passes through the origin (0,0).- The amplitude \( \frac{1}{4} \) compresses the graph vertically, making it less steep compared to \( y = \tan x \).

Key Concepts

Period of Trigonometric Functions

The period of a trigonometric function is the length of one complete cycle of the wave before it starts repeating. For periodic functions like sine, cosine, and tangent, the period determines how often the wave repeats itself along the x-axis.

In the standard form of the tangent function, given by \( y = a \tan(bx + c) + d \), the period is calculated using the formula \( \frac{\pi}{|b|} \). This is because the graph of the tangent function repeats every \( \pi \) units when \( b = 1 \).

For the given function \( y = \frac{1}{4} \tan x \), the parameter \( b \) is equal to 1, which means the period is also \( \pi \). Hence, every \( \pi \) units along the x-axis, the graph will look the same.

It’s essential to understand the period of trigonometric functions as it helps set the framework for graphing them. Knowing when the function repeats can simplify sketching the graph and analyzing its behavior over an interval.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their respective values through their defined periods, while taking into consideration important features like amplitude, period, phase shift, and vertical shifts.

When graphing \( y = \frac{1}{4} \tan x \), the following characteristics are critical:

• The function has a vertical stretch factor of \( \frac{1}{4} \) which affects its steepness.
• The period is \( \pi \), meaning the graph repeats itself every \( \pi \) units.
• The graph passes through origin (0,0) and continues having the recurrent pattern determined by the period.

To sketch the graph:

• Plot the points that the function will pass through within one period (e.g., from \( 0 \) to \( \pi \)).
• Include the vertical asymptotes which occur when the tangent function is undefined (the graph will never touch these asymptotes).
• Repeat the basic pattern identified within the period at intervals of \( \pi \), extending the sketch in both positive and negative directions.

Graphing trigonometric functions effectively helps visualize the behavior of the function, which is crucial for solving equations or understanding oscillatory motion illustrated by these functions.

Vertical Asymptotes

Vertical asymptotes are lines that the graph of the function approaches but never actually touches or crosses. They occur at the values of x where the function becomes undefined.

For the tangent function \( y = \tan x \), vertical asymptotes are noted at \( x = \frac{\pi}{2} + n\pi \) where \( n \) is an integer. These are the points where the tangent value approaches infinity as it is undefined. Consequently, the graph shoots up or down infinitely at these points, making the lines act like invisible barriers.

In \( y = \frac{1}{4} \tan x \), the positions of the vertical asymptotes do not change since the parameter \( b \) does not alter them. Thus, vertical asymptotes for this function are also at \( x = \frac{\pi}{2} + n\pi \).

Visualizing where these asymptotes lie helps in properly sketching the graph, ensuring you're aware of where the tangent expression can become undefined. Understanding the placement and role of vertical asymptotes is essential when interpreting and graphing trigonometric functions.

Amplitude in Trigonometric Functions

Though the notion of amplitude is primarily tied to the sine and cosine functions, it does play a role in understanding the vertical stretch or compression for the tangent function's graph.

In the equation \( y = a \tan(bx + c) + d \), the \( a \) value denotes the vertical stretch or compression. It affects how steep or flat the graph appears but does not affect the amplitude in the traditional sense since the tangent function is not bounded.

• For \( y = \frac{1}{4} \tan x \), the factor \( \frac{1}{4} \) compresses the graph vertically, making it less steep.
• This compression means the graph increases and decreases more gradually compared to \( y = \tan x \).

Recognizing the role of the coefficient \( a \) in the tangent function aids one in sketching the graph accurately and understanding how it impacts the graph's steepness.