Problem 35 - Algebra and Trigonometry

Question

Find the period and graph the function. $$ y=2 \tan 3 \pi x $$

Step-by-Step Solution

Answer

The period of the function is \( \frac{1}{3} \).

Step 1: Identify the Standard Form

The standard form of the tangent function is given by \( y = A \tan(Bx) \), where \( A \) is the amplitude and \( B \) affects the period of the function. In this case, the function is \( y = 2 \tan(3\pi x) \). Here, \( A = 2 \) and \( B = 3\pi \).

Step 2: Calculate the Period

The period of the tangent function is given by \( \frac{\pi}{B} \). Substituting \( B = 3\pi \), we get:\[\text{Period} = \frac{\pi}{3\pi} = \frac{1}{3}\]This means the function repeats every \( \frac{1}{3} \) units along the \( x \)-axis.

Step 3: Graph the Basic Tangent Function

First, plot the basic \( \tan(x) \) function to get an idea of its shape. This function has vertical asymptotes at each odd multiple of \( \frac{\pi}{2} \). The function passes through the origin \((0, 0)\) and repeats every \( \pi \) along the \( x \)-axis.

Step 4: Modify for Amplitude and Period

Modify the graph of \( \tan(x) \) to reflect the amplitude and period changes due to \( A = 2 \) and \( B = 3\pi \). The graph will have vertical asymptotes at \( x = \pm \frac{1}{6} \) since the asymptotes occur at \( x = \frac{n}{6} \) where \( n \) is an odd integer. The graph stretches vertically by a factor of 2.

Step 5: Plot the Function

Using the transformed features, draw the graph of the function \( y = 2 \tan(3\pi x) \) showing one complete cycle between \( x = -\frac{1}{6} \) and \( x = \frac{1}{6} \). Confirm the vertical stretch and horizontal compression are correct.

Key Concepts

Tangent Function

The tangent function, commonly abbreviated as "tan," is one of the fundamental trigonometric functions. It is defined as the ratio between the sine and cosine of a given angle. Mathematically, this is expressed as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). The behavior of the tangent function is unique compared to other trig functions like sine and cosine.
It has several interesting properties:

• The tangent function has vertical asymptotes, which are lines the graph approaches but never touches. These occur at odd multiples of \( \frac{\pi}{2} \) because the cosine function is zero at these points, making the tangent undefined.
• The function is periodic, meaning it repeats its pattern over regular intervals.
• The graph of the tangent function passes through the origin \(0,0\), allowing us to easily recognize its characteristic S-shape.

These features make the tangent function distinctive and essential in understanding trigonometric relationships.

Period of Function

The period of a function is the horizontal length over which the function's shape repeats itself. For the standard tangent function, this period is typically \( \pi \).
The formula to find the period of a tangent function in the form \( y = A \tan(Bx) \) is given by \( \frac{\pi}{B} \). This means when the tangent function is affected by a factor \( B \), it alters the period.In the exercise given, the function is \( y = 2 \tan(3\pi x) \):

• The value of \( B \) is \( 3\pi \).
• Substituting \( B \) into the period formula, the period becomes \( \frac{\pi}{3\pi} = \frac{1}{3} \).

This result signifies that each complete wave of the tangent function is repeated every \( \frac{1}{3} \) units along the x-axis, distinguishing it from the regular tangent curve.

Graphing Functions

Graphing trigonometric functions involves understanding their key features such as amplitude, period, and asymptotes. To sketch the graph of \( y = 2 \tan(3\pi x) \), note these modifications:

• Vertical Stretch: The value of \( A \), which is 2, results in a vertical stretch of the graph by a factor of 2, meaning it will climb higher and dip lower than the standard tan(x) curve.
• Altered Period: The new period of \( \frac{1}{3} \) compresses the horizontal scale, ensuring the wave repeats more frequently over the x-axis.
• New Asymptotes: The vertical asymptotes will now occur at points \( x = \pm \frac{1}{6} \), since they follow the locations \( x = \frac{n}{6} \), where \( n \) is an odd integer, bridging the waves together.

By understanding and applying these changes, sketching the graph of the modified tangent function becomes a systematic process. Confirm each feature, ensuring the function's cyclic pattern and asymptote locations correctly reflect the described transformations.