Problem 23 - Algebra 2

Question

Identify the period for each tangent function. Then graph each function in the interval from \(-2 \pi\) to 2\(\pi .\) $$ y=\tan \frac{\pi}{6} \theta $$

Step-by-Step Solution

Answer

The period of the function \(y=\tan \frac{\pi}{6} \theta\) is 6. The graph of this function repeats every 6 units within the interval from \(-2 \pi\) to \(2 \pi\).

Step 1: Determine the basic period

The period of the standard tangent function, \(\tan(\theta)\), is \(\pi\). Studying the function given, you can see it is a variation of the standard tangent function, but with a scale factor of \(\frac{\pi}{6}\) applied to \(\theta\). This will affect the period.

Step 2: Calculate the period for the given function

The period of the given function, \(y=\tan \frac{\pi}{6} \theta\), is found by dividing the basic period of \(\tan(\theta)\), which is \(\pi\), by the absolute value of the coefficient of \(\theta\), which is \(\frac{\pi}{6}\). Therefore, the period for \(y=\tan \frac{\pi}{6} \theta\) is \(\pi / |\frac{\pi}{6}|\), which simplifies to 6.

Step 3: Graph the function

When graphing \(y=\tan \frac{\pi}{6} \theta\), it’s important to remember that the graph repeats every 6 units (the period) and that tangent has asymptotes at odd multiples of \(\frac{1}{2}\) its period. So, the graph would have asymptotes at \(-3, 3\), and so on. With this in mind, sketch the graph from \(-2 \pi\) to \(2 \pi\). Note also that the range of the tangent function is \(-\infty, +\infty\) so there are no maximum or minimum values.

Key Concepts

Tangent Function

The tangent function is one of the three primary trigonometric functions, alongside sine and cosine. It is defined as the ratio of the sine and cosine functions. The tangent of an angle \(\theta\) within a right triangle is denoted as \(\tan(\theta)\) and can be described by the formula:

• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

This function is quite unique due to its periodic nature and vertical asymptotes. This means that its graph rises or falls without bound near certain vertical lines. This behavior occurs because the cosine, the denominator of the tangent function, approaches zero causing the ratio to skyrocket to infinity.
The standard tangent function \(\tan(\theta)\) is periodic, repeating its values in regular intervals, making its study vital for understanding trigonometric wave patterns. One thing to note is that because of its ratio form, the tangent function is undefined whenever cosine equals zero.

Period of a Function

Understanding the period of a function is crucial when dealing with trigonometric functions. The period of a function refers to the length of the smallest interval over which the graph of the function repeats itself.
For the standard tangent function \(\tan(\theta)\), the period is \(\pi\). This means that every \(\pi\) units, the function's graph will look identical. However, when a transformation is applied, like in the function \(y=\tan \frac{\pi}{6}\theta\), the period changes. To calculate the new period, the general rule is to divide the standard period by the absolute value of the coefficient multiplying \(\theta\).

• New period = \(\frac{\pi}{|\text{factor}|}\)

For our function \(y=\tan \frac{\pi}{6}\theta\), dividing \(\pi\) by \(\frac{\pi}{6}\) simplifies to 6, making the period 6. This means the function will repeat its values every 6 units.

Graphing Functions

Graphing trigonometric functions can initially seem daunting, but once you understand the key components, it becomes straightforward. When graphing a tangent function like \(y=\tan \frac{\pi}{6}\theta\), keep in mind the following:

• The calculated period, which in this case is 6, indicates how often the pattern repeats along the x-axis.
• Vertical asymptotes, which occur where the function is undefined, appear at odd multiples of half the period (e.g., \(-3, 3, 9\), etc.).
• The tangent function’s range is always from \(-\infty, +\infty\), indicating the graph extends infinitely upwards and downwards.

To graph \(y=\tan \frac{\pi}{6}\theta\), start by drawing vertical lines for the asymptotes. Then, sketch curves between these lines that approach the asymptotes without touching. This pattern repeats over the interval you consider, which in this case is from \(-2\pi\) to \(2\pi\), or more intuitively, from about \(-6.28\) to \(6.28\) on the x-axis. Knowing these characteristics makes the process less intimidating and enables more precise graphing.