In Exercises 45–52, graph two periods of each function. y = csc|x|

Verified step by step guidance

1. Step 1: Understand the function y = csc|x|. The cosecant function, csc(x), is the reciprocal of the sine function, so csc(x) = 1/sin(x). The absolute value in csc|x| means we consider the positive value of x for the sine function.
2. Step 2: Identify the domain of y = csc|x|. Since csc(x) is undefined where sin(x) = 0, the function y = csc|x| is undefined at x = nπ, where n is an integer. This means there will be vertical asymptotes at these points.
3. Step 3: Determine the range of y = csc|x|. Since csc(x) is the reciprocal of sin(x), and sin(x) ranges from -1 to 1, csc(x) will range from (-∞, -1] ∪ [1, ∞). The absolute value does not affect the range since it only affects the input to the sine function.
4. Step 4: Sketch the graph of y = csc|x|. Start by plotting the vertical asymptotes at x = nπ. Then, plot the basic shape of the csc(x) function between these asymptotes, keeping in mind that the graph will be symmetric about the y-axis due to the absolute value.
5. Step 5: Repeat the pattern to graph two periods of the function. Since the period of csc(x) is 2π, graph the function from -2π to 2π, ensuring to include the vertical asymptotes and the general shape of the csc(x) function in each period.

Key Concepts

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x). The cosecant function is undefined wherever the sine function is zero, leading to vertical asymptotes in its graph. Understanding the properties of the sine function is crucial for accurately graphing the cosecant function.

Graphing Periodic Functions

Periodic functions repeat their values in regular intervals, known as periods. For the cosecant function, the period is 2π, meaning the function's values repeat every 2π units along the x-axis. When graphing, it is essential to identify key points, asymptotes, and the overall shape of the function to accurately represent its behavior over two periods.

Absolute Value in Functions

The absolute value function, denoted as |x|, transforms all negative inputs into positive outputs. In the context of y = csc|x|, this means the graph will be symmetric about the y-axis, as the function behaves the same for both positive and negative x-values. Recognizing how absolute values affect the graph's symmetry and behavior is vital for accurate representation.