Problem 41 - Precalculus Mathematics for Calculus

Question

Find the period and graph the function. $$y=\tan 2(x-\pi)$$

Step-by-Step Solution

Answer

The period is \( \frac{\pi}{2} \) and the graph is a right-shifted compression of the tangent function.

Step 1: Understanding the Basic Function

The base function here is the tangent function, denoted as \( y = \tan(x) \). This function has a period of \( \pi \), meaning it repeats every \( \pi \) units.

Step 2: Analyzing the Coefficient

The function given is \( y = \tan(2(x - \pi)) \). The coefficient of \( x \) within the tangent function, which is 2, affects the period. The period of \( y = \tan(bx) \) is \( \frac{\pi}{b} \). Therefore, the period of \( \tan(2(x - \pi)) \) is \( \frac{\pi}{2} \).

Step 3: Understanding the Shift

The expression \( 2(x - \pi) \) indicates a horizontal shift. It can be rewritten as \( 2x - 2\pi \), shifting the graph by \( \pi \) units to the right.

Step 4: Sketching the Graph

Start with the basic tangent graph, which passes through the origin (0,0) and repeats every \( \pi \). Now compress the period from \( \pi \) to \( \frac{\pi}{2} \), then shift each point on the graph \( \pi \) units to the right. The new graph also has vertical asymptotes where the function is undefined, aligned with the compressed period and shift.

Key Concepts

Tangent Function

The tangent function, denoted as \( y = \tan(x) \), is a fundamental trigonometric function. It relates the angle of a right triangle to the ratio of the opposite side to the adjacent side. In terms of the unit circle, this function gives the slope of the terminal side of the angle \( x \). The tangent function is unique among trigonometric functions because it is undefined where the cosine function is zero, leading to vertical asymptotes.

Key characteristics of the tangent function:

• Repeats every \( \pi \) units - the distance between two consecutive asymptotes.
• Has vertical asymptotes at odd multiples of \( \frac{\pi}{2} \) (e.g., \( \frac{\pi}{2}, \frac{3\pi}{2} \), etc.).
• The graph increases from negative infinity to positive infinity between asymptotes.

These properties make the tangent function periodic and distinctively different from sine and cosine functions, which have a period of \( 2\pi \). Understanding the basic untampered pattern of the tangent function is crucial before applying transformations such as those in our original exercise.

Period of Trigonometric Functions

The period of a trigonometric function is the interval at which the function completes one full cycle and begins to repeat. For the standard tangent function \( y = \tan(x) \), this period is \( \pi \), due to its repeating cycle between each consecutive vertical asymptote.

In our exercise, the tangent function is altered to be \( y = \tan(2(x - \pi)) \). When you encounter a coefficient \( b \) in \( \tan(bx) \), the period changes to \( \frac{\pi}{b} \). Here, \( b = 2 \), so the period is \( \frac{\pi}{2} \).
Thus, this modification compresses the function's cycle, causing it to repeat twice as frequently as the standard tangent function. Understanding how these coefficients affect the period is a valuable tool when graphing or solving equations involving trigonometric functions.

Function Transformations

Transformations allow you to manipulate the base graph of a function in various ways - shifting, stretching, compressing, or reflecting it. For trigonometric functions, these transformations help model many real-world scenarios and can make problems appear more complex when presented in different forms.

In the function \( y = \tan(2(x - \pi)) \), two major transformations occur:

• Compression: The period compression, happening due to the coefficient 2, shrinks each cycle of the tangent function, making the graph repeat every \( \frac{\pi}{2} \).
• Translation: The \( (x - \pi) \) inside the function moves the entire graph \( \pi \) units to the right. This is a horizontal shift, effectively moving each vertical asymptote and intercept "\( \pi \) units to the right."

Combining these transformations, you see how \( y = \tan(x) \) morphs into \( y = \tan(2(x - \pi)) \), leading to a newly configured graph. Recognizing and mastering these transformations can simplify the process of graphing and analyzing trigonometric functions.