Problem 75 - Precalculus

Question

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes. $$ f(x)=2 \csc (x) $$

Step-by-Step Solution

Answer

The period is \( 2\pi \), the stretching factor is 2, and asymptotes are at \( x = n\pi \).

Step 1: Understand the Function

The function given is \( f(x) = 2 \csc(x) \). Recall that \( \csc(x) \) is the cosecant function, the reciprocal of \( \sin(x) \), thus: \( \csc(x) = \frac{1}{\sin(x)} \). This function will have vertical asymptotes where \( \sin(x) = 0 \) because division by zero is undefined.

Step 2: Identify the Period of the Function

The basic \( \csc(x) \) function has the same period as \( \sin(x) \), which is \( 2\pi \). Therefore, \( f(x) = 2 \csc(x) \) will also have a period of \( 2\pi \). This means the function repeats every \( 2\pi \) units.

Step 3: Determine the Stretching Factor

The expression \( 2 \csc(x) \) implies a vertical stretch by a factor of 2. This is because the coefficient 2 outside of \( \csc(x) \) multiplies the output values of \( \csc(x) \), effectively doubling the distance from the x-axis for each point.

Step 4: Find and Draw Asymptotes

Vertical asymptotes occur whenever \( \sin(x) = 0 \), which happen at \( x = n\pi \) for any integer \( n \). Within two periods (\( 0 \) to \( 4\pi \)), the asymptotes are at \( x = 0, \pi, 2\pi, 3\pi, \) and \( 4\pi \), although technically \( 0 \) and \( 4\pi \) are the same point on a circular graph.

Step 5: Sketch the Graph

Start by drawing vertical asymptotes at the positions identified in the previous step. Plot the function \( y = 2 \csc(x) \) by identifying key points such as the maxima and minima of the corresponding \( 2\sin(x) \), ensuring the graph approaches \( \pm \infty \) as it nears each asymptote. Complete this pattern twice to represent two periods from \( x = 0 \) to \( x = 4\pi \).

Key Concepts

Vertical asymptotes

Vertical asymptotes are a key feature of the cosecant function, since they occur where the associated sine function is equal to zero. For the function \( f(x) = 2 \csc(x) \), which is a transformation of the basic cosecant function \( \csc(x) = \frac{1}{\sin(x)} \), vertical asymptotes appear where \( \sin(x) = 0 \). The sine function is zero at integer multiples of \( \pi \), so the asymptotes for \( \csc(x) \) occur at \( x = n\pi \) for any integer \( n \).
Within a single period of \( 2\pi \), the vertical asymptotes are found at \( x = 0 \) and \( x = \pi \). When sketching two periods of the graph, you will include asymptotes at \( x = 0, \pi, 2\pi, 3\pi, \) and \( 4\pi \). Keep in mind that on a circular or repeating graph, \( x = 0 \) and \( x = 4\pi \) actually correspond to the same point, as the cycle begins anew.
This characteristic of having vertical asymptotes at regular intervals is crucial for accurately sketching the graph of the cosecant function.

Periodicity

Periodicity is an inherent trait of trigonometric functions, including the cosecant function. For the function \( f(x) = 2 \csc(x) \), the periodic nature means that the graph of the function repeats itself at regular intervals. Specifically, \( \csc(x) \) shares the period of the sine function, which is \( 2\pi \).

• This means that every \( 2\pi \) units, the graph looks identical. Whether you start at \( x = 0 \), \( x = 2\pi \), or any multiple of \( 2\pi \), the shape of the graph will remain unchanged.

When you're tasked with sketching two periods of \( f(x) = 2 \csc(x) \), you will need to replicate the graph over an interval of \( 0 \leq x < 4\pi \). This feature of periodicity is useful because it allows you to predict the behavior of the function outside any given interval, and it simplifies graphing by only requiring one cycle to be plotted initially and then repeated.

Vertical stretching

Vertical stretching in trigonometric functions occurs when a constant factor multiplies the function. In the case of \( f(x) = 2 \csc(x) \), the factor 2 introduces vertical stretching, altering how the graph looks compared to the basic \( \csc(x) \) function.

• Vertical stretching by a factor of 2 means that each point on the graph of \( \csc(x) \) is pushed twice as far from the x-axis.
• This not only makes high points (peaks) of the graph higher but also deepens the low points (troughs).

Vertical stretching does not affect the horizontal position of points, nor does it change when vertical asymptotes occur. Instead, it changes the amplitude of the function graph, making the range of \( y \) values extend further in both positive and negative directions.
It's essential to consider vertical stretching when plotting the function, as it gives the graph its proportionate look and ensures it accurately reflects the multiplicative factor involved with the cosecant function.