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In trigonometry, the period of a function is the distance on the x-axis after which the function starts to repeat itself. For the cosecant function, which is the reciprocal of the sine function, this period is determined by the coefficient in front of the variable inside the trigonometric function. The general formula for the period of the cosecant function is \( \frac{2\pi}{|b|} \), where \( y = \csc(bx) \).

For our exercise, \( y = \csc(\pi x) \), the value of \( b \) is \( \pi \). Applying the period formula, \( \frac{2\pi}{\pi} \) simplifies to 2. This means that every two units on the x-axis, the graph of \( y = \csc(\pi x) \) will repeat its pattern. Understanding this concept is critical as it helps students determine the intervals they should use when sketching the graph, ensuring that they include two full periods as requested.

Trigonometric functions like sine, cosine, tangent, and their reciprocals have specific points where they are undefined. For the cosecant function, which is the reciprocal of sine (\( \csc(x) = \frac{1}{\sin(x)} \)), it is undefined whenever the sine function's value is zero. This is because dividing by zero is not permissible in mathematics.

In the exercise \( y=\csc(\pi x) \), to find these undefined points, we look for values of \( x \) where \( \sin(\pi x) = 0 \). This occurs at all integer multiples of \( \pi \) for the sine function, which translates to every whole number for the variable \( x \) when \( b=\pi \). Therefore, the graph of the cosecant function will have vertical asymptotes (lines that the function approaches but never touches) at each integer value of \( x \) due to the function being undefined at these points. Recognizing these undefined points is vital for accurately drawing the graph and for understanding the behavior of the function near these points.

The process of sketching the graph of a function requires understanding of the function's behavior, its critical points such as maxima, minima, and points of inflection, and where the function is undefined. When sketching the graph of \( y=\csc(\pi x) \), we note its similarity to the sine graph, but with key differences due to it being the cosecant.

As \( y=\csc(\pi x) \) is undefined at every integer value of \( x \) and oscillates between negative and positive infinity at these points, we start by marking these undefined points with vertical dashed lines to indicate the asymptotes. The graph will approach these asymptotes but never cross them. Next, between these asymptotes, at half-integer values of \( x \), the graph reaches its minimum and maximum values. We can then sketch the curves connecting these peaks and troughs, making sure they do not touch or cross the vertical asymptotes.

Continuing this pattern will reveal a wave-like graph, with the curves alternating between concave up and concave down, mirroring across the x-axis. By including at least two full periods, the graph not only shows the local behavior of the function but also its repetitive nature, which is part of understanding how trigonometric functions operate over their domain.