Problem 33 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=2 \tan 3 \pi x$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Tangent Function

The function given is of the form \( y = a \tan(bx) \). In this case, \( a = 2 \) and \( b = 3\pi \). It's a transformation of the standard tangent function.

Step 2: Determine the Period of Tangent Function

The period of a tangent function \( y = a \tan(bx) \) is given by \( \frac{\pi}{|b|} \). Substitute \( b = 3\pi \) into the formula: \[ \text{Period} = \frac{\pi}{|3\pi|} = \frac{1}{3}. \] Thus, the period of the function is \( \frac{1}{3} \).

Step 3: Set Up the Graph

To graph \( y = 2 \tan 3\pi x \), start by plotting one period of the tangent function. Begin at \( x = 0 \) and end at \( x = \frac{1}{3} \). Identify the vertical asymptotes, which occur at the ends of each period.

Step 4: Find Asymptotes and Important Points

For \( y = 2 \tan 3\pi x \), vertical asymptotes occur where the argument of the tangent is an odd multiple of \( \frac{\pi}{2} \). Solve \( 3\pi x = (2n + 1)\frac{\pi}{2} \), giving asymptotes at \( x = \frac{(2n+1)}{6} \).

Step 5: Plot Key Points

The tangent function is zero at 0 for each period. Calculate a few more points; for example, \( \tan \left(\frac{\pi}{6} \right) = \frac{\sqrt{3}}{3} \), so at \( x = \frac{1}{18} \), \( y = 2\cdot\frac{\sqrt{3}}{3} \).

Step 6: Sketch the Graph

Draw the curve starting at the origin (after every period), passing smoothly through the calculated points, and heading towards the vertical asymptotes, repeating every \( \frac{1}{3} \) unit along the x-axis.

Key Concepts

Periodic Functions

Tangent functions, like many trigonometric functions, are periodic. This means they repeat their values in regular intervals, known as the periods of the function.
For the standard tangent function, the period is \(\pi\), which means it repeats every \(\pi\) units.
However, when you have a transformation, such as in the function \(y = 2 \tan 3\pi x\), the period changes.

• The general formula to find the period of the tangent function \(y = a \tan(bx)\) is \(\frac{\pi}{|b|}\).
• For \(y = 2 \tan 3\pi x\), substitute \(b = 3\pi\) into the formula, resulting in \(\frac{\pi}{|3\pi|} = \frac{1}{3}\).
• This tells us that the function repeats its values every \(\frac{1}{3}\) units along the x-axis.

Recognizing the periodic nature of trigonometric functions helps in graphing and analyzing their behavior over intervals. Understanding periodicity is crucial for applications like signal processing in engineering or analyzing cyclical patterns in nature.

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. For the tangent function, these asymptotes occur because tangent is undefined whenever the angle is an odd multiple of \(\frac{\pi}{2}\).
In the function \(y = 2 \tan 3\pi x\), vertical asymptotes are where \(3\pi x\) equals \((2n + 1)\frac{\pi}{2}\), where \(n\) is an integer.

• Solving \(3\pi x = (2n + 1)\frac{\pi}{2}\) gives us the location of the asymptotes at \(x = \frac{2n+1}{6}\).
• These points repeat with the same interval as the period, every \(\frac{1}{3}\) units.
• Vertical asymptotes help partition the graph into sections where the tangent function can be plotted smoothly from negative to positive infinity.

These asymptotes indicate positions where the function's value grows infinitely large, and they are a key feature of the graph of a tangent function.

Trigonometric Graphing

Graphing trigonometric functions involves understanding both their periodic nature and characteristics such as amplitude and transformations.
When graphing \(y = 2 \tan 3\pi x\), you need to consider both its period and where it is undefined, creating vertical asymptotes.

• Start by identifying a period, which is from \(x = 0\) to \(x = \frac{1}{3}\) in this case.
• Next, mark the vertical asymptotes at \(x = \frac{1}{6}\) and \(x = \frac{1}{2}\).
• Key points on the graph include where the function crosses the x-axis. The tangent function is zero at \(x = 0\) within each period.
• Additional points, such as \(y = 2\cdot\frac{\sqrt{3}}{3}\) at \(x = \frac{1}{18}\), can be plotted to better define the curve.

Draw the curve by starting at the origin for each repeating section. The curve should pass smoothly through calculated points, moving toward the asymptotes.
The graph then repeats this process every \(\frac{1}{3}\) units along the x-axis, illustrating the repeating nature and critical behavior of the tangent function.