Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=2 \csc \frac{1}{2} x$$

Step-by-Step Solution

Answer

(a) Period: \(4\pi\) (b) Phase Shift: 0 (c) Range: \((-\infty, -2] \cup [2, \infty)\)

Step 1: Identify the Function

The function we have is \( y = 2 \csc \frac{1}{2} x \). This is a cosecant function of the form \( y = a \csc(bx + c) + d \), where \( a = 2 \), \( b = \frac{1}{2} \), \( c = 0 \), and \( d = 0 \). We will use these values to find the period, phase shift, and range.

Step 2: Calculate the Period

The period of a cosecant function \( y = a \csc(bx + c) + d \) is given by \( \frac{2\pi}{|b|} \). Here, \( b = \frac{1}{2} \), so the period is \( \frac{2\pi}{\left| \frac{1}{2} \right|} = 4\pi \).

Step 3: Determine the Phase Shift

The phase shift of the function is determined by the formula \( -\frac{c}{b} \). In this case, since \( c = 0 \), the phase shift is 0, meaning there is no horizontal shift in the function.

Step 4: Find the Range

The range of the cosecant function \( y = a \csc(bx + c) + d \) depends on the amplitude \( a \). Since \( \csc(x) = \frac{1}{\sin(x)} \), the function is undefined at integer multiples of \( \pi \), corresponding to sine function zeros. Hence, the range is \(( -\infty, -|a| ] \cup [ |a|, \infty )\). Here, \( a = 2 \), so the range is \(( -\infty, -2 ] \cup [ 2, \infty )\).

Key Concepts

Cosecant Function

The cosecant function is related to the basic sine function. It's defined as the reciprocal, or inverse, of the sine function, represented as \( \csc(x) = \frac{1}{\sin(x)} \). This means wherever the sine function has a value, the cosecant function will have its reciprocal.

Cosecant functions are unique because they have vertical asymptotes at the points where the sine function is zero. This makes them particularly interesting as they have undefined values, causing these so-called "gaps" in their graphs at integer multiples of \( \pi \). The general form of the cosecant function is given by \( y = a \csc(bx + c) + d \), which incorporates various transformations:

• \( a \) affects the vertical stretch or compression.
• \( b \) affects the horizontal stretch or compression.
• \( c \) results in horizontal shifts, known as phase shifts.
• \( d \) causes vertical shifts.

Understanding these transformations helps in sketching and analyzing the behavior of the function.

Period of Function

The period of a trigonometric function refers to the length of the smallest interval over which the function repeats itself.

For the cosecant function given by \( y = a \csc(bx + c) + d \), the period can be determined using the formula \( \frac{2\pi}{|b|} \). In our specific problem, given the function \( y = 2 \csc \frac{1}{2}x \), \( b = \frac{1}{2} \).

So, applying the formula, we find the period to be \( \frac{2\pi}{\left| \frac{1}{2} \right|} = 4\pi \). This tells us that the graph of the function repeats every \( 4\pi \) units along the x-axis. This predictable repetition is a key feature that assists in graphing and understanding the behavior across its domain.

Phase Shift

Phase shift in a trigonometric function describes any horizontal shift attributed to the function's graph along the x-axis.

For a function of the form \( y = a \csc(bx + c) + d \), the phase shift can be calculated using the formula \( -\frac{c}{b} \). A positive value indicates a shift to the right, while a negative value indicates a shift to the left.

Looking at our function \( y = 2 \csc \frac{1}{2}x \), we find \( c = 0 \), so the formula gives \( -\frac{c}{b} = 0 \). Hence, there is no phase shift, meaning the graph begins its cycle exactly where the origin would be expected.

By identifying phase shifts, we can anticipate how the position of the cycle changes with respect to the y-axis, but in this scenario, the absence of a phase shift simplifies analysis.

Range of Function

The range of a trigonometric function identifies all possible output values. For the cosecant function \( y = a \csc(bx + c) + d \), the range is affected primarily by the amplitude \( a \), which determines the magnitude of the wave's maximum and minimum values.

Since the cosecant function is the reciprocal of sine, it will be undefined wherever the sine function equals zero, leading to gaps in the values it takes. This results in a range for the cosecant function expressed as \( (-\infty, -|a|] \cup [|a|, \infty) \).

In the given example \( y = 2 \csc \frac{1}{2}x \), the amplitude \( a \) is 2. Hence, the range becomes \( (-\infty, -2] \cup [2, \infty) \).

Understanding the range helps in predicting not only the graph's vertical extent but also anticipating where the function's values can lie within the context of real applications.