Problem 25 - Algebra 2

Question

Identify the period for each tangent function. Then graph each function in the interval from \(-2 \pi\) to 2\(\pi .\) $$ y=\tan \left(-\frac{3}{2 \pi} \theta\right) $$

Step-by-Step Solution

Answer

The period of the function \( y = \tan \left( - \frac{3}{2 \pi} \theta \right) \) is \( \frac{2\pi}{3} \). An accurate graph of the function can be obtained by plotting points at multiples of this period in the interval from \( -2\pi \) to \( 2\pi \).

Step 1: Determine the period

The period of a tangent function is given by \( \pi/b \), where \( b \) is the coefficient in front of the variable \( \theta \) in the function. For the given function, \( y = \tan \left( - \frac{3}{2 \pi} \theta \right) \), the coefficient \( b \) is \( - \frac{3}{2 \pi} \). Hence, the period of the function is \( \pi/ \left( - \frac{3}{2 \pi} \right) = - \frac{2\pi}{3} \). Since a period cannot be negative, we take the absolute value, making the period \( \frac{2\pi}{3} \).

Step 2: Create the graph

In order to graph the function in the interval from \( -2\pi \) to \( 2\pi \), plot points at each multiple of the period \( \frac{2\pi}{3} \) from \( -2\pi \) to \( 2\pi \). As the tangent function is undefined at the end points of each period, vertical asymptotes will be placed at these points. Then, draw one cycle of the tangent function in each period. The function decreases from \( \infty \) at \( x = - \frac{2\pi}{3} \) to \( -\infty \) at \( x = 0 \) then rises up from \( \infty \) at \( x = \frac{2\pi}{3} \), thus completing one cycle within the period. This pattern continues between each pair of vertical asymptotes.

Key Concepts

Tangent Function

The tangent function, often denoted as \( y = \tan(\theta) \), is one of the fundamental trigonometric functions. It links the angle to the ratio of the opposite side over the adjacent side in a right triangle. However, its function as a graph has a repeating pattern and specific characteristics.

The graph of the tangent function shows a repeating pattern called periodicity, characterized by vertical asymptotes where the function is undefined. It alternates between positive and negative infinity within each period, creating a pattern of upward and downward slopes.

Here's what you should know about tangent functions:

• They are undefined whenever the cosine equals zero.
• Each tangent cycle occurs between its vertical asymptotes.
• The basic form of the tangent graph ranges from \(-\infty\) to \( \infty\) within a single period.

Period of a Function

The period of a trigonometric function measures how often the function's pattern repeats along the x-axis. For the tangent function, the period is determined by the formula \( \pi / b \), where \( b \) is the coefficient of \( \theta \) in the expression \( y = \tan(b \theta) \).

A negative coefficient like \( b = -\frac{3}{2\pi} \) inverts the graph but does not affect the calculation of the period's absolute value. This results in a positive period of \( \frac{2\pi}{3} \), signifying that the graph's cycle repeats every \( \frac{2\pi}{3} \) units along the x-axis.

Key points to remember when considering function periods:

• A function's period dictates the spacing between repeating cycles.
• The period of a function indicates the distance on the x-axis after which the pattern repeats.
• When graphing, consistent increments of the period help to plot accurate cycles.

Graphing Trigonometric Functions

Graphing trigonometric functions involves identifying key characteristics like amplitude, period, phase shift, and vertical shift. For the tangent function, emphasis lies on the period and vertical asymptotes rather than amplitude, as the tangent doesn't have a fixed range.

To graph the tangent function over a given interval, such as from \(-2\pi\) to \(2\pi\), crucial steps include locating vertical asymptotes, which occur every half-period, and plotting points representing the function's behavior between these asymptotes.

• Vertical Asymptotes: Occur where the function is undefined, typically at the boundaries of each period.
• Slope Direction: A positive coefficient implies a positive slope, while a negative coefficient, as in this case, implies a negative slope between asymptotes.
• Plotting the Points: Each cycle starts from \(-\infty\) approaches zero, then rises to \( \infty \).

By understanding these elements, one can effectively sketch trigonometric functions, showing their full pattern of repetition over any specified interval.