Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\csc 2 \pi x $$

Step-by-Step Solution

Answer

Period is 1; asymptotes at \( x = \frac{n}{2} \).

Step 1: Understand the Parent Function

The cosecant function, denoted as \( y = \csc x \), is the reciprocal of the sine function \( y = \sin x \). Therefore, \( \csc x = \frac{1}{\sin x} \) and it is undefined whenever \( \sin x = 0 \). This causes vertical asymptotes at those points.

Step 2: Determine the Period of \( \csc 2\pi x \)

The general form for the period of \( \csc(bx) \) is \( \frac{2\pi}{b} \). For \( \csc 2\pi x \), \( b = 2\pi \). Thus, the period is:\[\frac{2\pi}{2\pi} = 1\]This means the function repeats every 1 unit on the x-axis.

Step 3: Identify the Asymptotes

Since \( \csc x \) has vertical asymptotes at \( x = n\pi \) (where \( n \) is an integer), for \( \csc 2\pi x \), set \( 2\pi x = n\pi \). Solving for \( x \):\[x = \frac{n}{2}\]Thus, vertical asymptotes occur at \( x = \frac{n}{2} \) for every half integer (e.g., \( \pm \frac{1}{2}, \pm 1 \), etc.).

Step 4: Sketch the Graph

1. Plot the vertical asymptotes at \( x = \frac{n}{2} \).2. Note that \( \csc 2\pi x = \frac{1}{\sin 2\pi x} \) indicates local maximum and minimum at points where \( \sin 2\pi x \) is 1 or -1.3. Across each interval \( \left( n-\frac{1}{2}, n \right) \), draw the cosecant curves resembling U and upside down U-shapes stretched between asymptotes.

Key Concepts

Period of Trig Functions

In trigonometry, understanding the period of a function is crucial for graphing trigonometric functions accurately. The period of a function is the interval after which the function starts to repeat its values. For the standard sine and cosine functions, both have a period of \(2\pi\).

For functions like \(y = \csc(bx)\), where the function is the reciprocal of sine, the period is determined by the value of \(b\) in the term \(bx\). The formula to find the period is \(\frac{2\pi}{b}\). In our specific function \(\csc 2\pi x\):

• The variable \(b\) is \(2\pi\).
• Applying the formula, the period is \(\frac{2\pi}{2\pi} = 1\).
• This indicates that the function repeats its pattern every 1 unit along the x-axis.

Understanding the period helps you not only know over what interval to graph but also ensures that calculations for other features like asymptotes are accurate as well.

Vertical Asymptotes

Vertical asymptotes occur in functions where there are restrictions or undefined points. In the case of the cosecant function \(y = \csc x\), the vertical asymptotes arise from the points where the sine function (its reciprocal) equals zero.

For \(\csc x\):

• Vertical asymptotes are at \(x = n\pi\), where \(n\) is an integer.

When shifting to \(y = \csc 2\pi x\), the equation for vertical asymptotes changes slightly. Solving \(2\pi x = n\pi\):

• Gives us \(x = \frac{n}{2}\).
• This means vertical asymptotes happen at half-integer points, like \(x = \pm \frac{1}{2}, \pm 1\), and so on.

These asymptotes are crucial when graphing because they indicate the points where the function cannot have values, resulting in breaks in the graph.

Graphing Trigonometric Functions

Graphing trigonometric functions requires understanding not just their shape but their behavior over an interval. The cosecant function \(y = \csc 2\pi x\) is a periodic function, meaning it shows repeated patterns.

To graph \(y = \csc 2\pi x\):

• Mark the vertical asymptotes, as the function cannot be defined at these points.
• Since the function is \(\frac{1}{\sin 2\pi x}\), look for the sine function's peaks and valleys, where it equals 1 or -1.
• Between these asymptotes, draw the segments of the cosecant function, which appear as U-shaped curves stretching towards infinity.
• These segments occur in intervals like \((n-\frac{1}{2}, n)\), tracking the periodic nature closely.

Understanding this pattern allows you to sketch an accurate graph that captures the essence of the function and its properties.

Cosecant Function Properties

The cosecant function has unique properties because it is the reciprocal of the sine function. These properties are essential for understanding how the function behaves and for graphing it efficiently.

Key properties for \(y = \csc x\):

• It is undefined at points where \(\sin x = 0\), creating the characteristic vertical asymptotes.
• It has a periodic nature, similar to the sine function, repeating every \(2\pi\).
• Graphically, it consists of alternating arcs, forming U-shaped and inverted U-shaped sections.

The specific form \(y = \csc 2\pi x\) compresses the period to 1, but keeps these fundamental properties intact.

Knowing these properties is critical to accurately predicting where the function rises, falls, or becomes undefined, which are all vital features when sketching or analyzing the graph.