Problem 81 - 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. $$ f(x)=2 \csc \left(x+\frac{\pi}{4}\right)-1 $$

Step-by-Step Solution

Answer

1. Stretch: 2; 2. Period: \(2\pi\); 3. Asymptotes: \(x = n\pi - \frac{\pi}{4}\). Sketch it reflecting \(\csc(x)\) shift.

Step 1: Identify the base function

The base function for the given exercise is \( \csc(x) \), which is the cosecant function.

Step 2: Determine the transformation parameters

For the function \( f(x) = 2 \csc\left(x + \frac{\pi}{4}\right) - 1 \), we have:- A vertical stretch by a factor of 2.- A phase shift of \(-\frac{\pi}{4}\) (leftward shift by \(\frac{\pi}{4}\)).- A vertical translation downwards by 1 unit.

Step 3: Determine the period of the function

For \( \csc(x) \), the period is \( 2\pi \). Since the period isn't affected by horizontal scaling in the given function, the period remains \( 2\pi \).

Step 4: Determine asymptotes

\( \csc(x) \) has asymptotes where the sine function is zero. Normally, these occur at integer multiples of \( \pi \). The phase shift moves these asymptotes to \( x = n\pi - \frac{\pi}{4} \).

Step 5: Sketching the graph

Plot two periods starting from the phase-shifted asymptotes. First locate the vertical asymptotes, then sketch the sine curve transformed by stretching and translations, and reflect \( \sin(x) \) pattern to get \( \csc(x) \). Mark the amplitude as FACTOR = 2 and midline at \( y = -1 \).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc(x) \), is the reciprocal of the sine function. It is defined as \( \csc(x) = \frac{1}{\sin(x)} \) whenever \(\sin(x)\) is not zero.
The graph of the cosecant function is characterized by its distinctive set of curves and discontinuities called vertical asymptotes. These asymptotes occur at points where the sine function is zero, which are integer multiples of \( \pi \.\)

• Cosecant is undefined at these points because division by zero is undefined.
• It creates a series of vertical lines (asymptotes) at these undefined points.
• The function increases to positive or decreases to negative infinity near these points, resulting in "U" or inverted "U" shapes between asymptotes.

Understanding the shape and behavior of \( \csc(x) \) is crucial for sketching its graph and interpreting transformations.

Graph Transformations

Graph transformations involve altering the graph of a function in various ways, such as stretching, shifting, or translating it. For the given function \( f(x) = 2 \csc\left(x + \frac{\pi}{4}\right) - 1 \), several transformations are applied to the basic cosecant curve.

• Vertical Stretch: The factor of 2 multiplies the \( \csc(x) \) function, effectively stretching it vertically and doubling its amplitudes away from the midline.
• Phase Shift: Adding \( \frac{\pi}{4} \) inside the function results in a leftward (or negative) shift by \( \frac{\pi}{4} \) units. This change shifts the start of the pattern horizontally.
• Vertical Translation: The subtraction of 1 moves the entire graph downward by one unit, lowering the midline of the cosecant arches.

These transformations combine to alter the position, size, and height of the original \( \csc(x) \) function, leading to a distinctive new graph.

Periodicity

Periodicity refers to the repeating nature of a function over a regular interval. The cosecant function \( \csc(x) \) is inherently periodic, with a period of \( 2\pi \), meaning every \( 2\pi \) units, the graph pattern repeats itself.
For the function \( f(x) = 2 \csc\left(x + \frac{\pi}{4}\right) - 1 \), the period remains \( 2\pi \) given that there are no horizontal scaling transformations that affect the period.

• The consistent period allows us to sketch the graph over a fixed interval, such as [0, \( 2\pi \)], and then replicate this pattern.
• Despite the changes due to vertical stretch, phase shift, and translation, the period is not altered.
• Understanding periodicity is significant for predicting the graph's behavior over its domain and is essential when sketching multiple periods, as the exercise requires.

Vertical Asymptote

Vertical asymptotes are lines that a graph approaches but never actually touches or crosses. For the cosecant function \( \csc(x) \), vertical asymptotes occur at points where the sine function equals zero because \( \csc(x) = \frac{1}{\sin(x)} \,\) thus becoming undefined.
In the transformed graph of \( f(x) = 2 \csc\left(x + \frac{\pi}{4}\right) - 1 \), the vertical asymptotes are shifted due to the phase shift component of the function. These occur at:

• \( x = n\pi - \frac{\pi}{4} \) – indicating shifts of the asymptotes from \( x = n\pi \).
• \( n\) represents any integer, ensuring the asymptotes repeat in alignment with the function's periodicity.

These shifts are critical in determining where the graph exhibits its characteristic sharp drop or rise, setting clear boundaries for the oscillating curves of \( \csc(x) \). Recognizing vertical asymptotes ensures accurate graph sketching, avoiding misunderstandings of function behavior at these intersections.