Problem 28 - Precalculus Mathematics for Calculus

Question

Find the period and graph the function. $$y=\tan \frac{1}{2} x$$

Step-by-Step Solution

Answer

The period of \( y = \tan \frac{1}{2} x \) is \( 2\pi \).

Step 1: Understand the Basic Period of the Tangent Function

The basic tangent function, \( y = \tan x \), has a period of \( \pi \). This means it repeats its values every \( \pi \) units.

Step 2: Determine the Period of Transformed Tangent Function

The function given is \( y = \tan \left( \frac{1}{2} x \right) \). The period of the tangent function can be found using the formula \( \frac{\pi}{k} \), where \( k \) is the coefficient of \( x \). Here, \( k = \frac{1}{2} \).

Step 3: Calculate the New Period

Substitute \( k = \frac{1}{2} \) into the period formula: \[ \text{Period} = \frac{\pi}{\frac{1}{2}} = 2\pi. \] This means the function \( y = \tan \frac{1}{2} x \) has a period of \( 2\pi \).

Step 4: Plot the Key Points and Asymptotes

To graph \( y = \tan \frac{1}{2} x \), note that asymptotes occur where \( \frac{1}{2}x = \frac{\pi}{2} + n\pi \) (where \( n \) is an integer). Solving gives \( x = \pi + 2n\pi \), thus vertical asymptotes are at \( x = \pi, 3\pi, 5\pi, \ldots \).

Step 5: Sketch the Graph Based on Period and Asymptotes

Between each pair of asymptotes, the graph of \( \tan \frac{1}{2} x \) will pass through the origin and have similar shape to the standard tangent curve. It will go from \(-\infty\) to \(+\infty\) between asymptotes \( \pi \) units apart, repeating every \( 2\pi \).

Key Concepts

Tangent Function Transformation

Transformations of the tangent function involve changes to its standard formula, which can modify its period and orientation. The standard tangent function is represented as \( y = \tan x \). In transformations, the formula is adjusted to \( y = \tan(kx) \), where \( k \) is a scaling factor that modifies the period. The period of the tangent function is normally \( \pi \), but with transformations, the period can be recalculated using the formula \( \frac{\pi}{k} \). For example, in the function \( y = \tan(\frac{1}{2} x) \), \( k \) is \( \frac{1}{2} \), leading to a new period of \( 2\pi \). This means the function takes \( 2\pi \) units instead of \( \pi \) to complete one cycle. Transforming the tangent function alters how frequently the function repeats its wave-like pattern on a graph. This concept is vital for understanding how multipliers in the function equation affect its behavior and representation.

Asymptotes in Trigonometric Functions

Asymptotes are lines that a graph approaches but never touches. In trigonometric functions like the tangent, asymptotes occur where the function is undefined. For the basic tangent function \( y = \tan x \), these vertical asymptotes occur at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. Asymptotes are crucial for understanding the tangent graph's behavior, as the function approaches infinity near these points. When transforming the tangent function, such as in \( y = \tan(\frac{1}{2} x) \), the asymptotes shift according to the transformation. After applying a transformation, the new locations for asymptotes can be found. In this case, by solving \( \frac{1}{2}x = \frac{\pi}{2} + n\pi \) for \( x \), we find them at \( x = \pi + 2n\pi \), i.e., at intervals of \( 2\pi \). Understanding asymptotes helps in graphing functions accurately as they influence the shape and direction of curves in trigonometric graphs.

Graphing Trigonometric Functions

Graphing a transformed trigonometric function begins with understanding its period and asymptotes. For the function \( y = \tan(\frac{1}{2} x) \), the period is \( 2\pi \), and the asymptotes are located at \( x = \pi, 3\pi, 5\pi, \ldots \). These asymptotes act as guides to avoid while sketching the graph. In the interval between any two consecutive asymptotes, the graph will complete one full cycle, starting from the origin, crossing the horizontal axis, and sweeping from \(-\infty\) to \(+\infty\). When setting up the graph, mark the asymptotes first. Then, pinpoint the central point of each cycle, which here is the origin due to the absence of any vertical or horizontal shifts. The graph will exhibit the characteristic S-shape of a tangent function, repeating every \( 2\pi \) units. Graphing involves repetitive patterns, and knowing the period and asymptote position helps make this process systematic and accurate, capturing the nature of trigonometric waves effectively.