91Ó°ÊÓ

Trigonometric functions have a characteristic repetition over intervals. This repeating span is called the period. For the cosecant function, given by \[y = a \, \text{csc}(b x - c)\], the period is derived based on the value of 'b'. The formula to determine the period of such a function is \[ \text{Period} = \frac{2\pi}{b} \].

In our specific function \[ y = -2 \, \text{csc}(\pi x - \pi) \], \( b = \pi \). Plugging this into our formula gives us the period: \[ \frac{2\pi}{\pi} = 2 \]. Thus, the function repeats every 2 units along the x-axis.
Understanding the period is crucial for sketching the graph, as it helps us know where the function cycles and repeats.

The range of a trigonometric function is the set of all possible y-values it can take. Cosecant, being the reciprocal of the sine function, has a unique range. While sine values lie between -1 and 1, the cosecant values lie outside this interval. The general form for the range of a cosecant function \[ y = a \, \text{csc}(bx - c) \] is: \[ (-\infty, -a] \cup [a, \infty) \].

For our function \[ y = -2 \, \text{csc}(\pi x - \pi) \], \(a = -2\), hence \(|a| = 2\). This results in a range: \[ (-\infty, -2] \cup [2, \infty) \]. Thus, the y-values never lie between -2 and 2 but take values smaller than -2 or greater than 2.

To sketch the graph of a trigonometric function, first find its critical points. For the cosecant function \[ y = -2 \, \text{csc}(\pi x - \pi) \], identify the values where the sine function (sine is the reciprocal to cosecant) is zero, because these will be our vertical asymptotes.

Solve \[ \sin(\pi x - \pi) = 0\]: \[\sin(\pi (x-1)) = 0 \], giving vertical asymptotes at x-values where \( x = 1, 3, 5, 7, ... \).

Between these asymptotes, to plot the curve, identify points where the function reaches its peaks and troughs. The sine-based values defining these mid-points where cosecant approaches its max/min are \[ x=2, 4, 6, ... \]. The negative sign in \[ y = -2 \, \text{csc}(\pi x - \pi) \] indicates a reflection downwards. Draw a smooth curve approaching asymptotes but never touching them. Ensure it moves between the points identified.

Cosecant functions, similar to their sine-based origins, have distinct characteristics:

• **Periodicity**: They repeat over a defined interval, typically derived from \[ \frac{2\pi}{b} \]. For \[ y = -2 \, \text{csc}(\pi x - \pi)\], period is 2.
• **Range**: They have ranges that exclude certain middle values and extend infinitely out, in this case: \[ (-\infty, -2] \cup [2, \infty) \].
• **Asymptotes**: Cosecant functions have vertical asymptotes where the corresponding sine function equals zero. These serve as bounds in the sketch and never get intersected.
• **Reflections and Amplitude**: The given function \[ y = -2 \, \text{csc}(\pi x - \pi) \] is reflected downwards due to the negative sign. Amplitude is influenced by the coefficient 2, magnifying the y-values proportionally.

Understanding these characteristics forms the backbone of effectively analyzing and graphing cosine and its reciprocal function, cosecant.