Problem 76 - Precalculus

Question

For what values of \(n\) do \(y=\csc x\) and \(y=\csc (x-n \pi)\) have the same graph?

Step-by-Step Solution

Answer

The graphs are identical for even integer values of \( n \).

Step 1: Understanding the Problem

We are tasked with finding values of \( n \) for which the graphs of \( y = \csc x \) and \( y = \csc (x - n\pi) \) are identical. Both are related to the cosecant function, which is the reciprocal of the sine function.

Step 2: Analyzing the Cosecant Function

The cosecant function, \( y = \csc x \), has vertical asymptotes where \( \sin x = 0 \), that is, at \( x = k\pi \) for integer values of \( k \). Its period is \( 2\pi \).

Step 3: Condition for Same Graphs

For \( y = \csc x \) and \( y = \csc(x - n\pi) \) to have the same graph, the shift \( n\pi \) must represent a full period (or a multiple of full periods) of \( \csc x \), which means \( n\pi = 2k\pi \) for some integer \( k \).

Step 4: Solving for n

From the condition \( n\pi = 2k\pi \), divide both sides by \( \pi \) to get \( n = 2k \). Thus, \( n \) must be an even integer for the graphs to coincide.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( y = \csc x \), is the reciprocal of the sine function. It is defined as \( \csc x = \frac{1}{\sin x} \). This means wherever the sine function is zero, the cosecant function will have issues because dividing by zero is undefined. A few crucial characteristics of the cosecant function include:

• It is undefined at integer multiples of \( \pi \) (i.e., \( x = k\pi \) for any integer \( k \)), which corresponds to the nodes of the sine function.
• As a result, the graph displays vertical asymptotes at these points, where the function heads to infinity or negative infinity.
• The function is symmetric with respect to the origin, which infers that it is an odd function: \( \csc(-x) = -\csc x \).

Understanding these properties helps in analyzing how transformations of the cosecant function behave graphically. When you shift the graph left or right, affected aspects include its periodicity and the location of vertical asymptotes.

Periodicity

Periodicity is a fundamental concept in understanding trigonometric functions like the cosecant. A function is periodic if it repeats its values in regular intervals, known as periods. For the cosecant function \( y = \csc x \), the fundamental period is \( 2\pi \). This means the function's values repeat every \( 2\pi \) units along the x-axis. Translated into more intuitive terms:

• If you observe the graph across any stretch the length of \( 2\pi \), the portion you observe will mirror wherever else the same stretch occurs along the x-axis.
• So, shifting this function graph horizontally by multiples of \( 2\pi \) doesn't alter its appearance.

This repetition ensures that transformations involving integer multiples of the function’s period result in graphs that look identical, which is key in our understanding when functions like \( y = \csc(x - n\pi) \) coincide with \( y = \csc x \). The shift must be in multiples of the period for the graph to remain unchanged.

Vertical Asymptotes

Vertical asymptotes are a telling feature of the cosecant function graph. They occur at places where the function values grow indefinitely, that is, as vertical lines the function approaches but never actually reaches, representing undefined points.For \( y = \csc x \), these vertical asymptotes are located where the sine function, \( \sin x \), equals zero. This occurs at integer multiples of \( \pi \), as \( \sin x \) crosses the x-axis, which happens at \( x = k\pi \) for any integer \( k \). When you perform horizontal transformations on the function, such as shifting it by \( n\pi \), these vertical asymptotes shift along the x-axis accordingly. For instance:

• If you consider \( y = \csc (x - n\pi) \), the asymptotes will now be at \( x = n\pi + k\pi \).
• This movement doesn't change the graph's general shape but rather shifts where those undefined points occur.

Recognizing the position and behavior of these vertical asymptotes is essential for graphing transformations of the cosecant function accurately and understanding when two such graphs coincide. As seen in the solution, the condition for sharing the same graph involves ensuring these transformations align over one or more full periods.