Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=3 \csc 2 x$$

Step-by-Step Solution

Answer

Period: \(\pi\), Phase shift: None, Range: \((- fty, -3] \cup [3, fty)\).

Step 1: Identify the Standard Form

The given function is \(y = 3 \csc 2x\). The standard form of a cosecant function is \(y = a \csc(bx + c) + d\). By comparing, we identify \(a = 3\), \(b = 2\), \(c = 0\), and \(d = 0\).

Step 2: Determine the Period

The period of a cosecant function \(y = a \csc(bx)\) is given by \(\frac{2\pi}{b}\). Substituting \(b = 2\) gives the period \(\frac{2\pi}{2} = \pi\).

Step 3: Calculate the Phase Shift

The phase shift is determined by \(-\frac{c}{b}\) for the function \(y = a \csc(bx + c)\). Substituting \(c = 0\) and \(b = 2\) gives the phase shift of \(-\frac{0}{2} = 0\). Hence, there is no phase shift.

Step 4: Determine the Range

The range of a cosecant function \(y = a \csc(bx)\) is \((-fty, -a] \cup [a, fty)\). For this function, \(a = 3\), so the range is \((-fty, -3] \cup [3, fty)\).

Key Concepts

Period of Cosecant Function

The period of a trigonometric function is a key aspect that tells us how long it takes for the function to repeat its pattern. For the cosecant function, denoted as \(y = a \csc(bx)\), the period can be found using the formula \(\frac{2\pi}{b}\). Let's break this down:

• The constant \(2\pi\) represents the standard period of the sine and cosecant functions.
• The variable \(b\) modifies this period. In our exercise, \(b = 2\), so the period becomes \(\frac{2\pi}{2} = \pi\).

This means that the pattern of the cosecant function will repeat every \(\pi\) units along the x-axis. It's important for students to understand that the period is crucial for graphing these functions accurately. Knowing this helps in predicting the wave-like motions and ensures you can replicate the graph over different intervals efficiently. Keep in mind that different values of \(b\) will adjust the period accordingly. However, the formula \(\frac{2\pi}{b}\) always applies, which simplifies finding these values.

Phase Shift

The phase shift in trigonometric functions shows how far to the left or right the graph of the function shifts from its standard position. For the general form \(y = a \csc(bx + c) + d\), the phase shift is calculated as \(-\frac{c}{b}\).In our specific function, \(y = 3 \csc(2x)\):

• We have \(c = 0\), meaning there is no shift in the x-direction since \(-\frac{0}{2}\) equals 0.

Understanding phase shifts is essential, especially when these expressions become part of larger, more complex equations involving multiple functions. Even though there's no phase shift in this case, being adept at calculating it means you can handle more challenging scenarios with ease. Remember, changes to the phase shift adjust the starting point of the function's graph but do not affect the overall shape or amplitude.

Range of Trigonometric Functions

The range of a function describes all possible output values (y-values) that the function can have. For trigonometric functions, especially the cosecant function, this range reflects their asymptotic nature. For a function \(y = a \csc(bx)\), the range is:\((-\infty, -a] \cup [a, \infty)\).Let's apply this to our given function, \(y = 3 \csc(2x)\):

• We have \(a = 3\), so the range becomes \((-\infty, -3] \cup [3, \infty)\).

This range implies that the output values of the function are never between -3 and 3. The reason behind this is that the cosecant function, defined as \(1/\sin(x)\), has vertical asymptotes at the points where the sine function equals zero. Consequently, the values are restricted to being either below -3 or above 3.Understanding the range is vital, especially when tackling real-world problems or technological applications where certain values might be out of feasible limits. Always check the values of \(a\) to help determine this range quickly and they ensure correct graphing and practical interpretations.