Graph each function over a one-period interval.


y = - (1/2) csc (x + π/2)

Verified step by step guidance

1. Step 1: Recognize that the function is a transformation of the basic cosecant function, \( y = \csc(x) \). The given function is \( y = -\frac{1}{2} \csc(x + \frac{\pi}{2}) \).
2. Step 2: Identify the transformations: The \( +\frac{\pi}{2} \) inside the function indicates a horizontal shift to the left by \( \frac{\pi}{2} \) units. The \( -\frac{1}{2} \) outside the function indicates a vertical stretch by a factor of \( \frac{1}{2} \) and a reflection across the x-axis.
3. Step 3: Determine the period of the cosecant function. The period of \( \csc(x) \) is \( 2\pi \). Since there is no horizontal scaling factor, the period remains \( 2\pi \).
4. Step 4: Identify the vertical asymptotes of the function. For \( \csc(x) \), the vertical asymptotes occur where \( \sin(x) = 0 \). For \( \csc(x + \frac{\pi}{2}) \), this occurs at \( x = -\frac{\pi}{2}, \frac{3\pi}{2}, \ldots \).
5. Step 5: Sketch the graph over one period \([-\frac{\pi}{2}, \frac{3\pi}{2}]\), applying the transformations: shift the basic \( \csc(x) \) graph left by \( \frac{\pi}{2} \), vertically stretch by \( \frac{1}{2} \), and reflect it across the x-axis.

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 has a range of all real numbers except for values between -1 and 1, and it is undefined wherever the sine function is zero. Understanding the properties of the cosecant function is essential for graphing it accurately.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their values over a specified interval, typically one period. For the cosecant function, this includes identifying key points, asymptotes, and the overall shape of the graph. The period of the cosecant function is 2π, and transformations such as vertical shifts and reflections must be considered when graphing functions like y = - (1/2) csc(x + π/2).

Transformations of Functions

Transformations of functions refer to changes made to the basic function's graph, including shifts, stretches, and reflections. In the given function, y = - (1/2) csc(x + π/2), the term (x + π/2) indicates a horizontal shift to the left by π/2, while the negative sign reflects the graph across the x-axis, and the factor of -1/2 compresses the graph vertically. Understanding these transformations is crucial for accurately graphing the function.