Problem 219 - Precalculus

Question

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline. $$ f(x)=5 \csc (3 x) $$

Step-by-Step Solution

Answer

Amplitude: None (cosecant function); Period: \(\frac{2\pi}{3}\); Midline: \(y=0\).

Step 1: Understand the Function

The given function is \( f(x) = 5 \csc(3x) \). It is a transformation of the cosecant function, which is the reciprocal of the sine function. Therefore, \( f(x) \) is undefined where \( \sin(3x) = 0 \).

Step 2: Identify the Amplitude

Cosecant functions do not have an amplitude in the traditional sense because they do not oscillate between a maximum and minimum value as sine and cosine do. However, the vertical dilation factor in front of the \( \csc \) function, 5, can be thought of as a factor that stretches the graph vertically.

Step 3: Determine the Period

The period of \( \csc(x) \) is \( 2\pi \). However, due to the coefficient 3 in \( \csc(3x) \), the period changes. The formula to find the period of a transformed cosecant function \( \csc(bx) \) is \( \frac{2\pi}{b} \). Here, \( b = 3 \), so the period is \( \frac{2\pi}{3} \).

Step 4: Determine the Midline Equation

The midline for a \( \csc \) function is a horizontal line. For the function \( f(x) = A \csc(bx) \), the midline is defined as \( y = D \) where does not directly change \( y = 0 \), unless a vertical shift \( D \) is present. Without a vertical translation, the midline remains \( y = 0 \).

Step 5: Sketch the Graph

To sketch two periods of the function, start from \( x = 0 \) and extend to \( \frac{4\pi}{3} \). The \( \csc \) function will have vertical asymptotes at points where \( 3x = k\pi \) (where \( k \) is an integer), starting at \( 0 \), \( \frac{\pi}{3} \), \( \frac{2\pi}{3} \), and \( \pi \). Between these asymptotes, sketch the curved branches of the cosecant, keeping in mind that they stretch around a virtual sine wave for visual guidance.

Key Concepts

Period of Trigonometric Functions

In trigonometry, the period of a function refers to the interval over which the function repeats itself. For basic trigonometric functions like sine and cosine, the standard period is often \(2\pi\). However, when these functions are transformed, their periods can change. This is crucial in graphing trigonometric functions because knowing the period helps you set up the x-axis properly.

To find the period of a transformed function such as \(\csc(bx)\), use the formula:

• Period = \( \frac{2\pi}{b} \)

This formula shows how the value of \(b\) affects the period. For example, in the function given as \(f(x) = 5\csc(3x)\), \(b = 3\). Thus, its period becomes \(\frac{2\pi}{3}\).

Graphically, this means the function completes one full wave between 0 and \(\frac{2\pi}{3}\). Understanding this helps in plotting two full periods simply by multiplying the period by two, allowing you to accurately capture the behavior of the entire function on the graph.

Vertical Asymptotes

Vertical asymptotes are lines where a function approaches infinity as the x-value gets closer but never actually touches or crosses. In the case of cosecant, there are specific x-values where the sine function equals zero. Since cosecant is the reciprocal of sine, these points make \(\csc(x)\) undefined, creating vertical asymptotes.

To determine where these asymptotes are in any function like \(f(x) = 5\csc(3x)\), solve the equation for the points where \( \sin(3x) = 0 \). This occurs at values where \(3x = k\pi\), with \(k\) being an integer.

• Find potential asymptotes by setting \(3x = k\pi\). To solve for \(x\), \(x = \frac{k\pi}{3}\).

For two periods, you'll calculate where these asymptotes occur in the interval from \(0\) to \(\frac{4\pi}{3}\). They will appear at \(x = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}\), strongly influencing the shape and form of the function's graph.

Transformation of Trigonometric Functions

Transformations alter the appearance of trigonometric functions by changing their amplitude, period, phase shift, or vertical displacement. This exercise focuses on the vertical dilation and period change. Transformations are integral for modifying a function to fit specific criteria or real-world data.

For \(f(x) = 5\csc(3x)\), transformations include:

• A vertical stretch by a factor of 5, seen in the coefficient before \(\csc\).
• A change in period from \(2\pi\) to \(\frac{2\pi}{3}\), influenced by the coefficient \(3\) in \(\csc(3x)\).

Essentially, you're looking at a graph that stretches vertically, making the peaks and valleys more pronounced while also having more oscillations within the same horizontal distance. These transformations do not change the function completely but adjust how it appears on the graph. Recognizing them allows you to modify and interpret different scenarios accurately, providing a deeper understanding of its functionality.