Problem 30 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=5 \csc 3 x$$

Step-by-Step Solution

Answer

Period: \(\frac{2\pi}{3}\); graph requires plotting asymptotes at \( x = \frac{n\pi}{3} \) and stretching waves vertically by 5.

Step 1: Identify the Parent Function

The given function is \( y = 5 \csc 3x \). The parent function is \( \csc x \), which is the reciprocal of \( \sin x \). It has vertical asymptotes where \( \sin x = 0 \).

Step 2: Determine the Coefficient Effect

In the function \( y = 5 \csc 3x \), the coefficient 5 implies an amplitude stretch for the cosecant waves. The 3 multiplies x, affecting the period of the function.

Step 3: Calculate the Period

The period of a \( \csc bx \) function is \( \frac{2\pi}{b} \). Here, \( b = 3 \), so the period is \( \frac{2\pi}{3} \).

Step 4: Identify Key Points and Asymptotes

Cosecant has asymptotes where the sine function is zero. In \( y = \csc 3x \), asymptotes occur at \( 3x = n\pi \), where \( n \) is an integer, hence at \( x = \frac{n\pi}{3} \).

Step 5: Graph the Function

Plot the vertical asymptotes at every \( \frac{n\pi}{3} \) on the x-axis. Sketch the basic shape of \( \csc x \) between these asymptotes to represent one cycle over \( \frac{2\pi}{3} \). The waves peak outwards from these asymptotes and stretch vertically by a factor of 5.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is a trigonometric function that is the reciprocal of the sine function. Mathematically, this means \( \csc x = \frac{1}{\sin x} \). As it is derived from the sine function, the cosecant function shares some properties but also has some unique characteristics.

• Domain: The domain of \( \csc x \) consists of all real numbers except where \( \sin x = 0 \), because division by zero is undefined.
• Range: The range of \( \csc x \) is all real numbers greater than or equal to 1 and less than or equal to -1, because these are the values achievable by the reciprocal of the sine function.
• Period: The parent function \( \csc x \) has a period of \( 2\pi \), meaning it repeats itself every \( 2\pi \) units along the x-axis.

Understanding these properties helps in graphing the function more intuitively, especially when modifying the function with coefficients and different periods.

Function Graphing

Graphing functions like \( y = 5 \csc 3x \) involves a few key steps. To accurately sketch these transformations, you should recognize the effect of different coefficients in the function's equation.

• Amplitude and Vertical Stretch: The coefficient multiplied by the \( \csc x \) function affects how the graph stretches vertically. In \( y = 5 \csc 3x \), the '5' makes the "waves" of the graph stretch outwards from the x-axis by a factor of 5. This doesn't affect asymptotes but alters the height of peaks.
• Period Adjustment: The term '3x' in \( \csc 3x \) compresses the period to \( \frac{2\pi}{3} \). This means the function completes one full cycle every \( \frac{2\pi}{3} \) units, as opposed to the parent function's \( 2\pi \).

When graphing the function, plot vertical asymptotes accordingly and then sketch the characteristic "arches" of the cosecant, peaking at each cycle's midpoint.

Vertical Asymptotes

Vertical asymptotes are a significant characteristic of the cosecant function. They occur at values where the sine function equals zero since \( \csc x \) involves taking the reciprocal of \( \sin x \). This results in divisions by zero, which are undefined, hence vertical asymptotes.

• Identifying Asymptotes: For the generic \( \csc bx \) function, vertical asymptotes occur where \( b \cdot x = n\pi \) for integer \( n \), equivalent to the sine function's zeros.
• Specific Example: For \( y = 5 \csc 3x \), the vertical asymptotes are at \( x = \frac{n\pi}{3} \) because these are the points where \( \sin 3x = 0 \). Thus, the function is undefined at these x-values.

Understanding where these asymptotes occur is crucial as they guide the placement and "peaking" of the cosecant's graph. This ensures the function's structure is accurately represented on a graph.