Problem 25 - Precalculus: A Concise Course

Question

Sketch the graph of the function. Include two full periods. $$ y=\csc \frac{x}{2} $$

Step-by-Step Solution

Answer

The graph of the function \(y = \csc (\frac{x}{2})\) has a period of \(4\pi\) with vertical asymptotes at \(0, 2\pi, 4\pi, -2\pi\), etc. The function manifests as 'U' shapes between these asymptotes as peaks of the function coincide with the 'valleys' of the sine function and vice versa.

Step 1: Draw period markers

Firstly, as the period of the function is modified to \(4\pi\), we mark the x-axis with multiple period markers that span least two full periods. Here, we can choose 0, \(4\pi\), \(8\pi\) and \(-4\pi\) or a similar pattern to represent two full periods.

Step 2: Draw vertical asymptotes

The cosecant function has vertical asymptotes wherever the sine function is zero. The sine function is zero at \(0\), \(\pi\), \(2\pi\), etc. Due to the inside function being \(\frac{x}{2}\), the vertical asymptotes occur at \(0, 2\pi, 4\pi, -2\pi\), etc. Draw these asymptotes as dashed lines to lay the foundations for our graph.

Step 3: Draw the shape of the function

In each period, draw a U shape from maximum to minimum (or vice versa) following the asymptotes drawn. Visualization of the sine function can aid us here as the cosecant function is just reflection around the x-axis where \(y=1\) and \(y=1\), i.e., wherever sine has a peak, cosecant has its minimum, and where sine has its 'valley', cosecant has the maximum. Make also sure the function approaches the drawn asymptotes.

Step 4: Label the graph

Now, add the function name \(y = \csc (\frac{x}{2})\) and other important points and intervals on the graph. This proper labeling gives the viewer a clear understanding of the graph.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. Among them, the cosecant function (\f\(\text{csc}\f\)) is one of the lesser-known, yet equally important functions. It is the reciprocal of the sine function, such that \f\(\text{csc}(\theta) = \frac{1}{\text{sin}(\theta)}\f\). For angles where the sine value is zero, the cosecant function is undefined, leading to vertical asymptotes in its graph.

In the context of graphing \f\(y = \text{csc}\f\) functions, understanding the behavior and transformation of their parent function, the sine wave, is essential. For example, when we consider \f\(y = \text{csc}\f\) of \f\(\frac{x}{2}\f\), it signifies a horizontal stretch, meaning the period is altered from the usual \f\(2\text{π}\f\) to \f\(4\text{π}\f\). The resulting graph is a series of repeating curves that never intersect the x-axis, where each 'curve' spans from negative to positive infinity, only defined at certain intervals based on its relationship to the sine function.

Vertical Asymptotes

Vertical asymptotes are lines that a function approaches but never touches or crosses. They represent the values at which a function is undefined and serve as a boundary of sorts, dictating the shape of the graph. For the cosecant function, vertical asymptotes occur at the x-values where its counterpart, the sine function, equals zero because division by zero is undefined.

Hence, in graphing \f\(y = \text{csc}\f\) of \f\(\frac{x}{2}\f\), we first determine the sine function's zeros which are at integer multiples of \f\(2\text{π}\f\); however, since we have the inner function \f\(\frac{x}{2}\f\), these zeros are 'stretched' to the multiples of \f\(4\text{π}\f\) for our graph. Each vertical asymptote is a crucial navigation point, providing the framework to draw the U-shaped curves of the cosecant function, ensuring they approach these asymptotes while never crossing them.

Periodicity in Trigonometry

Periodicity is the characteristic of functions to repeat their values at regular intervals. It's essential to trigonometry as many trigonometric functions are periodic. The period is the length of the smallest interval over which the function repeats itself. For \f\(y = \text{csc}\f\) of \f\(\frac{x}{2}\f\), the period is \f\(4\text{π}\f\), meaning that every \f\(4\text{π}\f\) units along the x-axis, the graph of the cosecant function will repeat its shape.

When graphing the cosecant function, it's beneficial to first mark the period on the x-axis to guide the plotting of the function's shape. In our example, two full periods are equivalent to a length of \f\(8\text{π}\f\) along the x-axis with period markers. This periodic behavior not only helps to predict the behavior of the function beyond the plotted points but also aids in understanding the symmetry and patterns inherent in trigonometric graphs.