91Ó°ÊÓ

Understanding the period of a trigonometric function is fundamental in graphing and analyzing its behavior. The period refers to the length of one complete cycle of the function. For the sine and cosecant (csc) functions, this is typically 2Ï€ radians, meaning that the functions repeat their values every 2Ï€ radians.

The function given, \(y = 3 \csc x\), has the same period as the sine function, 2Ï€, because the cosecant is the reciprocal of sine. This doesn't change unless the function includes a horizontal scaling factor. In the absence of such a factor, each cycle of \(3 \csc x\) completes a graph repeating every 2Ï€ radians, resulting in a consistent wave pattern on the x-axis.

When graphing a function, it is essential to identify its critical points because these are where the function's graph has a peak, a valley, or changes its direction of concavity. Critical points occur where the derivative of the function is either zero or undefined. In the case of the cosecant function, \(y = 3 \csc x\), critical points are where the sine function, which is the reciprocal of cosecant, is zero.

Since \(\sin x = 0\) at \(x = n\pi\), where \(n\) is an integer, these are the values at which \(3 \csc x\) is undefined, and we encounter vertical asymptotes. Understanding the location of these asymptotes is crucial because they divide the graph into its repeating intervals.

Points of inflection on a graph are where the concavity of the function changes—from concave up to concave down, or vice versa. These points reveal the dynamic nature of the function's curvature. For the function \(y = 3 \csc x\), points of inflection correspond to where the sine function reaches its maximum and minimum values.

Since the \(\sin x\) achieves maximum and minimum at \(x = \pi/2 + n\pi\), where \(n\) is an integer, these are the locations where \(3 \csc x\) changes its concavity. These points are crucial for accurately sketching the bends and turns of the graph as it moves between its peaks and valleys.

Sketching trigonometric graphs like \(y = 3 \csc x\) involves several key steps. First, identify the period and plot the vertical asymptotes where the function is undefined—these typically occur at integer multiples of π for csc. Next, determine the critical points and points of inflection to know where peaks, valleys, and concavity changes happen.

In the case of \(y = 3 \csc x\), you would mark the x-values at intervals of \(\pi/2 + n\pi\), alternating between the maximum value of 3 and the minimum value of -3. By connecting these critical points smoothly, approaching but never reaching the vertical asymptotes, you create an accurate graph of the cosecant function repeating every period of 2Ï€ radians. Proper graphing reflects the function's behavior and its important features, aiding in a deeper comprehension of trigonometric concepts.