Problem 64 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi .\) Describe any phase shift and vertical shift in the graph. $$ y=\csc 2\left(\theta-\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

The given function \(y=\csc 2(\theta-\frac{\pi}{2})\) has a phase shift of \(\frac{\pi}{2}\) to the right and no vertical shift. The graph shows it as a reflection of the basic cosecant function \(\csc\theta\) over vertical line \(x=\frac{\pi}{2}\), and is compressed horizontally by a factor of 2 in the interval from 0 to 2\(\pi\).

Step 1: Understanding the Function

The given function is \(y=\csc 2(\theta-\frac{\pi}{2})\). The basic function here is \(\csc\theta\), which is reciprocal of sine function. In \(\csc 2(\theta-\frac{\pi}{2})\), the '2' is multiplying with \(\theta-\frac{\pi}{2}\), implying that the period of the function is half of the basic function. Similarly, \(\theta-\frac{\pi}{2}\) indicates a phase shift to the right.

Step 2: Identify the Phase and Vertical Shifts

The phase shift can be identified directly from the function as \(\frac{\pi}{2}\) to the right. There's no coefficient added or subtracted to the function, hence there's no vertical shift.

Step 3: Graphing the Function

On a graph, represent angles from 0 to 2\(\pi\). Now, apply the phase shift of \(\frac{\pi}{2}\) to the right. Then plot the values of \(\csc 2(\theta-\frac{\pi}{2})\) in the range 0 to 2\(\pi\). The graph appears as a reflection of the basic cosecant function \(\csc\theta\) over vertical line \(x=\frac{\pi}{2}\) and compressed horizontally by a factor of 2.

Key Concepts

Phase Shift

The concept of phase shift in trigonometric functions refers to the horizontal movement of the function graph along the x-axis. In the function \( y = \csc 2(\theta - \frac{\pi}{2}) \), the expression inside the parenthesis, \( \theta - \frac{\pi}{2} \), indicates the amount of horizontal shift.
Breaking this down:

• The \( \theta \) represents the variable of the function.
• The \( -\frac{\pi}{2} \) signifies a shift to the right.

Therefore, the graph of the function \( y = \csc 2\theta \) is shifted \( \frac{\pi}{2} \) units to the right.
To visualize this, consider how each point of the basic cosecant function becomes adjusted horizontally by \( \frac{\pi}{2} \) towards positive x-values.

Vertical Shift

In trigonometric functions, a vertical shift is determined by what is added to or subtracted from the entire function. It moves the graph up or down along the y-axis. However, in the function \( y = \csc 2(\theta - \frac{\pi}{2}) \), no constant term is added or subtracted outside of the cosecant function, indicating there is no vertical shift.
This means the graph maintains its baseline position vertically, with no adjustments up or down. When plotting the function, all transformations are merely horizontal, focusing on phase shifts and compressions, not vertical displacements.

Period of Trigonometric Functions

The period of a trigonometric function is the length of one complete cycle of the graph. For the basic cosecant function \( \csc \theta \), the standard period is \( 2\pi \) because its graph completes a cycle in this interval. However, any transformation, like the coefficient before \( \theta \), affects the period.
In \( y = \csc 2(\theta - \frac{\pi}{2}) \), the coefficient \( 2 \) acts as a frequency multiplier.

• This effectively halves the period of the basic cosecant function.
• The new period becomes \( \frac{2\pi}{2} = \pi \).

Thus, the function completes its cycle from one point to another in only \( \pi \), causing it to compress horizontally. This compression makes the graph narrower over the x-axis.