Problem 49 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\tan \frac{\pi}{2} x$$

Step-by-Step Solution

Answer

Period is 2, asymptotes at \( x = 1 + 2k \), where \( k \) is an integer.

Step 1: Understanding the Basic Tangent Function

The basic tangent function is given by the equation \( y = \tan x \), which has a period of \( \pi \). This means that every \( \pi \) units, the function repeats itself. The asymptotes occur wherever \( \tan x \) is undefined, i.e., at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.

Step 2: Transforming the Argument of the Tangent Function

The given function is \( y = \tan \left( \frac{\pi}{2} x \right) \). In this function, the argument of the tangent is \( \frac{\pi}{2} x \) instead of just \( x \). This transformation affects the period of the tangent function.

Step 3: Calculating the Modified Period

The period of a tangent function of the form \( y = \tan(bx) \) is \( \frac{\pi}{|b|} \). For our function, \( b = \frac{\pi}{2} \). The period is, therefore, \( \frac{\pi}{\frac{\pi}{2}} = 2 \).

Step 4: Identifying Asymptotes

Asymptotes occur where the argument of the tangent function makes it undefined. For \( \tan(x) \), these are at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. Plugging in \( bx = \frac{\pi}{2} x \), the asymptotes for \( y = \tan \left( \frac{\pi}{2} x \right) \) will occur at \( \frac{\pi}{2} x = \frac{\pi}{2} + k\pi \). Solving for \( x \), we get \( x = 1 + 2k \), where \( k \) is an integer.

Step 5: Sketching the Graph

Using the calculated period of 2 and the asymptotes at \( x = 1 + 2k \), we can sketch the graph. Between each set of asymptotes, the tangent curve starts from negative infinity, crosses through the x-axis at the midpoint of each interval, and approaches positive infinity as it nears the next asymptote. This process repeats every 2 units on the x-axis.

Key Concepts

Tangent Function

The tangent function, denoted as \( y = \tan x \), is a fundamental trigonometric function that describes the ratio between the opposite and adjacent sides of a right triangle. Unlike sine and cosine functions, the tangent function can take any value from negative to positive infinity. This unique property stems from the fact that tangent involves division by the cosine function, becoming undefined when cosine equals zero. For the basic tangent function, these points of undefined value appear at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is any integer. At these points, the function presents vertical asymptotes, which are lines where the function's value shoots up towards infinity. In graphs, these asymptotes help us identify intervals where the tangent curve exists.

Graphing Functions

Graphing the tangent function involves plotting its values against x-coordinates on a graph to visualize its behavior. The regular \( y = \tan x \) has a distinct "zig-zag" shape, repeating after each period, which is \( \pi \) units. To sketch this function, it's crucial to mark its asymptotes accurately. These asymptotes are vertical lines at \( x = \frac{\pi}{2} + k\pi \), indicating where the curve becomes undefined. As you move along the graph, the tangent function starts at negative infinity and ascends rapidly, crossing through the x-axis before veering off to positive infinity, repeating this pattern across each interval. With a modified function like \( y = \tan \left( \frac{\pi}{2} x \right) \), the characteristics remain consistent, but the periodic behavior changes, which we'll explore deeper under periodicity.

Periodicity

Periodicity refers to how often a trigonometric function repeats its values over specific intervals. The basic tangent function has a period of \( \pi \), meaning it resets every \( \pi \) units along the x-axis. For the function \( y = \tan \left( \frac{\pi}{2} x \right) \), the periodicity changes due to the manipulation of its variable. When the function argument changes from \( x \) to \( \frac{\pi}{2} x \), we re-calculate its period by determining \( \frac{\pi}{|b|} \), where \( b \) is the coefficient of \( x \). Here, \( b = \frac{\pi}{2} \), giving a new period of 2 units. This adjustment means that the tangent function here completes a full cycle every 2 units along the x-axis, which impacts how you will draw its repeating patterns on the graph.

Asymptotes

Asymptotes are essential in graphing functions, especially the tangent function, as they represent where the function does not exist. For \( y = \tan \left( \frac{\pi}{2} x \right) \), determining the location of its asymptotes involves solving for \( x \) in the equation \( \frac{\pi}{2} x = \frac{\pi}{2} + k\pi \). This leads to solutions \( x = 1 + 2k \), marking the position of vertical asymptotes along the x-axis. These lines signify boundaries where the tangent curve transitions from negative to positive infinity. Recognizing these points is crucial when sketching the function, as they establish the repeated cycles of the graph and help avoid incorrect interpretations. Understanding the role of asymptotes aids in recognizing how the tangent's unique peaks and troughs form around these invisible barriers, providing clarity on the function's dynamic behavior.