Problem 24 - Algebra 2

Question

Identify the period for each tangent function. Then graph each function in the interval from \(-2 \pi\) to 2\(\pi .\) $$ y=\tan 2.5 \theta $$

Step-by-Step Solution

Answer

The period of the function \( y = \tan(2.5\theta) \) is \( \frac{2\pi}{5} \). The graph would display repeated patterns every \( \frac{2\pi}{5} \) with vertical asymptotes at odd multiples of \( \frac{\pi}{5} \) within the interval \( -2\pi \) to \( 2\pi \).

Step 1: Identify the Period of the Tangent Function

The period of a standard tangent function, \( y = \tan(\theta) \), is \( \pi \). For the function \( y = \tan(k\theta) \), the period is \( \frac{\pi}{|k|} \). In this case, \( k = 2.5 \), so the period is \( \frac{\pi}{2.5} \) or \( \frac{2\pi}{5} \).

Step 2: Plot the Tangent Function

To plot \( y = \tan(2.5\theta) \), start at \( \theta = -2\pi \) and make points at intervals of the period until \( \theta = 2\pi \). Remember the tangent has asymptotes where the function is undefined, which happens at \( \theta = \frac{\pi}{2k} + \frac{n\pi}{k} \) where \( n \) is an integer. For \( y = \tan(2.5\theta) \), the asymptotes occur at \( \theta = \frac{n\pi}{2.5} \) for odd integers \( n \).

Step 3: Draw the Graph in the Given Interval

Using the asymptotes and periodicity determined in steps 1 and 2, draw the continuous curves between the asymptotes that represent the tangent function in the interval from \( -2\pi \) to \( 2\pi \). The curve should cycle every \( \frac{2\pi}{5} \), with asymptotes occurring at each odd multiple of \( \frac{\pi}{5} \).

Key Concepts

Period of Tangent Function

Understanding the period of a tangent function is crucial when graphing it. The period refers to the distance along the horizontal axis (\theta) over which the function repeats its shape. For a standard tangent function, given as \( y = \tan(\theta) \), this period is \( \pi \).

However, when the function includes a coefficient, such as \( k \) in \( y = \tan(k\theta) \), the period is adjusted to \( \frac{\pi}{|k|} \). This adjustment is important because it directly influences where the function begins and ends one complete cycle. In the example of \( y = \tan(2.5\theta) \), the period is found to be \( \frac{2\pi}{5} \). Grasping this concept of period modification allows students to graph a tangent function accurately over any interval.

Tangent Asymptotes

Asymptotes play a vital role when dealing with the graph of tangent functions. They are the lines that the graph approaches but never touches or crosses. This is because tangent functions have points where they become undefined, leading to these asymptotes.

Specifically, for the function \( y = \tan(k\theta) \), asymptotes occur at \( \theta = \frac{\pi}{2k} + \frac{n\pi}{k} \), where \( n \) is an odd integer. Knowing this, we can pinpoint the exact location of the asymptotes for \( y = \tan(2.5\theta) \), occurring at \( \theta = \frac{n\pi}{2.5} \). By identifying these asymptotes before graphing, students avoid errors and understand where the function will infinitely increase or decrease.

Plotting Trigonometric Functions

Plotting trigonometric functions like tangent involves marking points on a graph at specific intervals and considering the function's unique properties, such as its period and asymptotes. To start, choose an appropriate scale for your horizontal (\theta) axis to accommodate the periodicity of the function.

For the given function \( y = \tan(2.5\theta) \), you plot from \( \theta = -2\pi \) to \( \theta = 2\pi \) using the identified period of \( \frac{2\pi}{5} \) to determine the distance between repeating patterns of the graph. Additionally, those asymptotes found previously will guide you in drawing the curves of the graph, ensuring you do not cross these invisible boundaries. This systematic approach to plotting ensures the correct representation of the function's behavior.

Trigonometry in Algebra

Incorporating trigonometry into algebra often involves analyzing and manipulating trigonometric functions, like tangent, within an algebraic context. This interplay is evident when we graph functions that are algebraically transformed through processes like stretching, compressing, or translating.

For instance, the exercise \( y = \tan(2.5\theta) \) represents an algebraic modification where the constant 2.5 alters the function's period and asymptotes. By mastering the relationship between trigonometric functions and algebra, students can tackle more complex scenarios, adapt to variations, and understand the graphical implications of algebraic changes to trigonometric equations.