Problem 24 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=3 \csc \left(x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The period is \(2\pi\) and the graph is a vertically stretched \(\csc\) function shifted left by \(\frac{\pi}{2}\).

Step 1: Understand the Cosecant Function

The function given is a transformation of the cosecant function, which is the reciprocal of the sine function. The formula for the basic cosecant function is \(y = rac{1}{ ext{sin}(x)}\). The function \(y=3 \csc\left(x+\frac{\pi}{2}\right)\) means it is vertically stretched by a factor of 3 and phase shifted left by \(\frac{\pi}{2}\).

Step 2: Determine the Period of the Cosecant Function

The period of the basic sine function \(\sin(x)\) is \(2\pi\). The cosecant function, being the reciprocal of the sine function, also has a period of \(2\pi\). Here, the function inside the cosecant is \(x + \frac{\pi}{2}\), which does not affect the period. Hence, the period remains \(2\pi\).

Step 3: Plot Key Points

To help graph the function, identify key points, especially the vertical asymptotes where the sine part is zero. For \(\csc(x)\), asymptotes occur at multiples of \(\pi\), but here the function inside changes those points to multiples of \(\pi\) minus \(\frac{\pi}{2}\). Thus, vertical asymptotes occur at \(x = -\frac{\pi}{2} + k\pi\).

Step 4: Sketch the Graph

Plot a vertical line (asymptote) at every point found in Step 3. Then, sketch the graph of \(\csc\) in each interval created between consecutive asymptotes. Ensure to apply the vertical stretch by 3, meaning the maximum and minimum distances of the plot will be 3 units from the x-axis.

Key Concepts

Periodicity of Trigonometric Functions

Trigonometric functions like sine, cosine, and cosecant have repetitive patterns, known as periods. These are the intervals over which the function's values repeat. For example, the sine function, forms the foundation for cosecant. The periodicity of these functions provides predictability and consistency.

The standard period of the sine function \( \sin(x) \) is \( 2\pi \). This means that every \( 2\pi \), the sine wave repeats itself. Since cosecant, \( \csc(x) \), is the inverse of sine, it shares the same period of \( 2\pi \). This remains true even when transformations occur, such as translations or reflections. In the function \( y=3 \csc\left(x+\frac{\pi}{2}\right) \), although there is a horizontal shift, the core periodicity remains \( 2\pi \). This understanding helps when graphing, as it tells us how often the key features of the function, like asymptotes, recur.

Phase Shift in Trigonometry

Phase shift refers to the horizontal movement of a trigonometric graph along the x-axis. When a function's argument is altered by addition or subtraction, a phase shift happens. This is key in creating different starting points for these otherwise repetitive functions.

In the context of \( y = 3 \csc \left(x + \frac{\pi}{2}\right) \), the term \(+ \frac{\pi}{2}\) results in a phase shift. This particular addition inside the cosecant function shifts the graph to the left by \( \frac{\pi}{2} \) units.

The basic sine graph shifts, meaning the vertical asymptotes for \( \csc(x) \) are also affected. Original sine function asymptotes occur at multiples of \( \pi \). With the shift, these are now located at \( x = -\frac{\pi}{2} + k\pi \), where \( k \) is an integer. Grasping phase shift is crucial for accurately plotting the function on a graph.

Graphing Trigonometric Functions

Graphing a function like \( y=3 \csc\left(x+\frac{\pi}{2}\right) \) involves several steps, emphasizing shifts, stretches, and key points of alteration. Begin by imagining the basic \( \csc(x) \) graph. This reciprocal of sine graph features repeating upward and downward arcs, with distinct vertical asymptotes where the sine part, \( \sin(x) \), is zero.

To graph the transformed function:

• Identify the vertical asymptotes, changed by the phase shift to \( x = -\frac{\pi}{2} + k\pi \).
• Place the asymptotes on the graph at these calculated points.
• Note the vertical stretch—each arc peaks and troughs are amplified or compressed by a factor of 3.

The graphing process involves plotting these stretched arcs between the asymptotes, maintaining the intervals of original sine behavior but with modified height and position due to the transformations. This careful placement demystifies the impact of shifts and stretches on how cosecant's peaks and valleys emerge.