Problem 40 - Precalculus Mathematics for Calculus

Question

Find the period and graph the function. $$y=\csc 2\left(x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The period is \( \pi \). The graph is a translated and vertically stretched form of \( \csc x \).

Step 1: Identify the Parent Function and Transformation

The given function is \( y = \csc 2\left(x + \frac{\pi}{2}\right) \). The parent function is \( y = \csc x \), which has a period of \( 2\pi \). There is a horizontal compression and horizontal shift present in the function.

Step 2: Determine the Period

To find the period of \( y = \csc 2\left(x + \frac{\pi}{2}\right) \), we consider \( y = \csc(kx) \). The period of \( y = \csc kx \) is \( \frac{2\pi}{|k|} \). Here, \( k = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).

Step 3: Identify Horizontal Shift

The function is translated horizontally by \( -\frac{\pi}{2} \) units (since we adjust the input as \( x + \frac{\pi}{2} \)). This means the entire graph shifts to the left by \( \frac{\pi}{2} \).

Step 4: Plot the Key Points and Graph

Start graphing by identifying key points based on the period and shifts. Begin at \( -\frac{\pi}{2} \) because of the shift. Calculate values over one period \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). Pvts (pivots) occur just like cosine graph between \( \pi \) intervals.

Step 5: Graph Efficiently

Note that \( y = \csc 2\left(x + \frac{\pi}{2}\right) \) is the reciprocal of \( y = \sin 2\left(x + \frac{\pi}{2}\right) \). So divide traditional sine pivots at 0s create asymptotes. Between those points, the graph makes smooth hyperbolic curves undulating between pivots.

Key Concepts

Cosecant Function

The cosecant function is an often overlooked but essential trigonometric function. It is the reciprocal of the sine function, written as \( y = \csc x \), which means \( \csc x = \frac{1}{\sin x} \).

This function is undefined wherever the sine function equals zero, leading to vertical asymptotes at those points. For the parent cosecant function, these vertical asymptotes occur at integer multiples of \( \pi \) (e.g., \( 0, \pi, 2\pi \), etc).

When graphing the cosecant function, it is important to identify these asymptotes because they are critical traits of the graph. The curves of the cosecant graph rise and fall steeply as they approach these asymptotes, creating a series of U-shaped curves.

Period of Trigonometric Functions

The period of a trigonometric function is the smallest positive interval over which the function repeats itself. For standard trigonometric functions like sine, cosine, and cosecant, the period is traditionally \( 2\pi \).

To modify the period, you can alter the function with a coefficient within: \( y = \csc kx \). Here, the period P is calculated using the formula \( P = \frac{2\pi}{|k|} \). In our specific function, \( k = 2 \), so the period becomes \( \frac{2\pi}{2} = \pi \).

This means the graph of \( y = \csc 2x \) repeats every \( \pi \) units, making it more compressed compared to its parent counterpart.

Graphing Transformations

Graphing transformations involve shifting, stretching, or compressing the graph of a function from its original form.

In our exercise, two primary transformations are applied:

• Horizontal Compression: This occurs when the period \( k \) of the function changes, making the graph repeat more frequently. For example, \( y = \csc 2x \) compresses the usual \( 2\pi \) period to \( \pi \).
• Horizontal Shift: This involves moving the graph along the x-axis either to the left or right. In our specific function, \( y = \csc 2\left(x + \frac{\pi}{2}\right) \), the graph is shifted to the left by \( \frac{\pi}{2} \), as indicated by the plus sign inside the parentheses.

Each transformation changes the graphical output, affecting the location and shape of the cosecant's characteristic asymptotes and curves.

Horizontal Shifts in Functions

A horizontal shift in a function involves moving the graph along the x-axis. The shift is determined by changes in the function's argument, usually inside parentheses with variables.

For example, in \( y = \csc 2\left(x + \frac{\pi}{2}\right) \), the entire graph is shifted horizontally. The term \( x + \frac{\pi}{2} \) indicates a shift to the left by \( \frac{\pi}{2} \).

Understanding horizontal shifts is vital when predicting how the graph of a function will change because they alter the starting points for cycles and affect where asymptotes and intercepts occur.
This knowledge is essential for accurately plotting the graph based on modified functions.