Problem 48 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=\tan \frac{1}{2}\left(x+\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The period of the function is \(2\pi\).

Step 1: Understand the General Form of Tangent Function

The general form of a tangent function is \( y = \tan(bx + c) \). To find the period of this function, you need to identify the coefficient \( b \) that multiplies with \( x \).

Step 2: Identify the Coefficient

For the given function \( y = \tan\left(\frac{1}{2}(x + \frac{\pi}{4})\right) \), rewrite it as \( y = \tan\left(\frac{1}{2}x + \frac{\pi}{8}\right) \). Here, \( b = \frac{1}{2} \).

Step 3: Calculate the Period

The period of a tangent function is given by \( \frac{\pi}{|b|} \). Substitute \( b = \frac{1}{2} \) into the formula: \[ \text{Period} = \frac{\pi}{\frac{1}{2}} = 2\pi. \]

Step 4: Graph the Function

When graphing \( y = \tan\left(\frac{1}{2}x + \frac{\pi}{8}\right) \), note that it has a period of \( 2\pi \). This means the graph of the function repeats every \( 2\pi \) units. Plot the asymptotes at every \( n\pi \), where \( n \) is an integer, and the function itself between these asymptotes.

Key Concepts

Period of a Function

The period of a function is a fundamental concept in trigonometry, especially for periodic functions like sine, cosine, and tangent. The period of a function tells us how often the function repeats itself over a certain interval.

For the tangent function, the standard period is \( \pi \), meaning \( \tan(x) \) repeats every \( \pi \) units along the \( x \)-axis. However, when a function is transformed, particularly by a coefficient \( b \) inside the angle (\( bx + c \)), the period changes. The formula to find the period of a tangent function is \( \frac{\pi}{|b|} \).

In our original exercise, \( b = \frac{1}{2} \). Plugging this into our formula, we find that the period becomes \( 2\pi \). This means our function repeats every \( 2\pi \) units. Understanding this concept is crucial because it influences how the function behaves across different intervals.

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding their unique characteristics like amplitude, period, phase shift, and asymptotes.

For the tangent function \( y = \tan(bx + c) \), the graph is shaped by the coefficient \( b \) which affects the period, and the constant \( c \) which determines the horizontal shift. The function's graph is composed of repeating units or 'cycles' that extend from negative to positive infinity but are bounded vertically by asymptotes.

In the graph of \( y = \tan\left(\frac{1}{2}x + \frac{\pi}{8}\right) \), the period is \( 2\pi \), and the phase shift is affected by \( \frac{\pi}{8} \). To graph this function:

• Draw vertical asymptotes at every interval of \( n\pi \), where each repetition, or cycle, completes.
• The curve starts at the origin of each cycle and stretches to infinity as it approaches each asymptote.
• Keep in mind the horizontal shift; in this case, the graph shifts by \( -\frac{\pi}{8} \).

This art of graphing requires visualizing how these features interact and the impact of transformations made to the base function.

Asymptotes in Graphs

Asymptotes are invisible lines that a graph approaches but never touches. They signify points of undefined value for the function, indicating where the function tends toward positive or negative infinity.

For the tangent function, vertical asymptotes occur at points where \( bx + c = k\pi + \frac{\pi}{2} \) for any integer \( k \). These are the points where the function is undefined and where it crosses from positive to negative infinity or vice versa.

In the exercise, having aligned \( y = \tan \left( \frac{1}{2}x + \frac{\pi}{8} \right) \), the asymptotes repeat every \( \pi \) apart, but due to the \( \frac{1}{2} \) factor, they occur at intervals of \( n2\pi \).

• For plotting, first find the x-values where these vertical asymptotes occur.
• In between these lines, the tangent curve will stretch from negative infinity to positive infinity.

Understanding asymptotes help to determine the structure and the 'shape' of the function's graph, guiding us on where the function will stretch infinitely and reset.