Problem 36 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The period is 2.

Step 1: Identify the Basic Tangent Function

The standard form of the tangent function is \( y = a \tan(bx) \), where \( a \) is the amplitude (affects the steepness but not the period), and \( b \) affects the period of the function.

Step 2: Determine the Period of the Tangent Function

The period of the tangent function \( \tan(bx) \) is given by \( \frac{\pi}{b} \). In our function, \( b = \frac{\pi}{2} \). So, calculate the period: \[ \text{Period} = \frac{\pi}{\frac{\pi}{2}} = 2. \]

Step 3: Graph the Basic Tangent Function

Graph the standard tangent function \( y = \tan(x) \), which repeats every \( \pi \) and has vertical asymptotes at \( x = \frac{\pi}{2} + k\pi \) for integer \( k \).

Step 4: Transform the Tangent Function for the Given period

Since \( b = \frac{\pi}{2} \), the transformed function \( y = 2 \tan \frac{\pi}{2}x \) has asymptotes at \( \frac{2}{2} + k \cdot 2 \), meaning \( x = 1 + 2k \), and it completes one cycle every \( 2 \).

Step 5: Sketch the Transformed Function

To graph \( y = 2 \tan \frac{\pi}{2} x \), sketch the graph of the basic tangent function and adjust for the new period. Increase the amplitude to \( 2 \); this affects the steepness of the graph. Every cycle spans \( 2 \) units along the x-axis, and the pattern repeats with new asymptotes at \( x = 1, 3, 5, \ldots \).

Key Concepts

Period of Tangent

The period of a trigonometric function describes the interval after which the function starts to repeat itself. For the tangent function, its typical period is different from that of sine and cosine. The standard period for a tangent function is found using the formula:

• \frac{\pi}{b}

Where \( b \) represents the coefficient of \( x \) in the function \( y = a \tan(bx) \). In the given function \( y = 2 \tan(\frac{\pi}{2}x) \), \( b \) is \( \frac{\pi}{2} \).
This affects the period because substituting \( b \) into the formula gives \( \frac{\pi}{\frac{\pi}{2}} = 2 \). As a result, the tangent graph completes one full cycle every 2 units on the \( x \)-axis.
This periodicity means that the function repeats its shape and behavior every 2 units, establishing new asymptotes and patterns as it progresses along the axis.

Graphing Trigonometric Functions

Graphing a trigonometric function involves understanding its basic shape and how transformations modify it. The tangent function has a characteristic shape with a repeating pattern every half turn or full period. It features vertical asymptotes where the function is undefined, which mark the endpoints of each repeated section.

• For \( y = \tan(x) \), vertical asymptotes are at \( x = \frac{\pi}{2} + k\pi \) for integers \( k \)

For our transformed tangent function \( y = 2 \tan(\frac{\pi}{2}x) \), the graph should be sketched with new asymptotes at integer values, such as \( x = 1, 3, 5, \ldots \).
These are established by calculating the positions based on the function's altered period. The graph does not have an amplitude as seen in sine and cosine, but the multiplier in front affects its steepness.
The steps in graphing are:

• Identify the function's period and calculate where vertical asymptotes occur.
• Plot these asymptotes to determine intervals.
• Sketch the tangent curve within each interval, ensuring it approaches asymptotes in the same crank-like shape.

Amplitude in Trigonometry

Amplitude in trigonometric functions typically refers to the maximum height from the central axis to the peak. However, it does not apply directly to tangent functions. Instead, the coefficient \( a \) in \( y = a \tan(bx) \) determines the steepness of the curve.

• In \( y = 2 \tan(\frac{\pi}{2}x) \), the number 2 doesn't limit the height like amplitude would in sine or cosine functions.

This value stretches the graph, making it steeper, which alters the visual appearance of the tangent wave without affecting repetitions or periodic behavior.
Unlike sine or cosine, which oscillate between certain values, the tangent's values grow infinitely larger as they approach each asymptote.

• The coefficient modifies how sharply the tangent curve climbs or descends.

Understanding that the tangent function isn't bounded by the same amplitude concept allows for better sketching and interpretation.