Problem 47 - Precalculus Mathematics for Calculus

Question

Find the period and graph the function. $$y=5 \sec \left(3 x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The period is \( \frac{2\pi}{3} \), with a phase shift of \( \frac{\pi}{6} \) right.

Step 1: Identify the Function Type

The given function is \[ y = 5 \sec \left(3x - \frac{\pi}{2}\right) \]It is a secant function, defined as \( y = A \sec(Bx - C) \), where \( A = 5 \), \( B = 3 \), and \( C = \frac{\pi}{2} \).

Step 2: Determine the Period of the Secant Function

The period of a secant function \( y = A \sec(Bx - C) \) is given by the formula:\[ \text{Period} = \frac{2\pi}{B} \]Plugging in \( B = 3 \), we get:\[ \text{Period} = \frac{2\pi}{3} \]

Step 3: Find the Phase Shift

To find the phase shift, use the formula:\[ \text{Phase Shift} = -\frac{C}{B} \]Substitute \( C = \frac{\pi}{2} \) and \( B = 3 \) into the formula:\[ \text{Phase Shift} = -\frac{\frac{\pi}{2}}{3} = -\frac{\pi}{6} \]This means the graph shifts \( \frac{\pi}{6} \) units to the right.

Step 4: Graph the Function

To graph \( y = 5 \sec(3x - \frac{\pi}{2}) \):1. First, graph the corresponding cosine function: \( y = 5 \cos(3x - \frac{\pi}{2}) \) as a guide.2. Notice the amplitude of \( 5 \), which affects the maximum and minimum values of the cosine.3. Mark the vertical asymptotes where the cosine is zero, i.e., when \( 3x - \frac{\pi}{2} = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.4. Plot the secant curve by drawing curves opening upward and downward between the asymptotes, reflecting the reciprocal of the cosine values.

Key Concepts

Function Period

The period of a function is an important concept when dealing with trigonometric functions like the secant function. It determines how often the function repeats itself along the x-axis. For our given function, \( y = 5 \sec(3x - \frac{\pi}{2}) \), the period is derived using the formula: \[ \text{Period} = \frac{2\pi}{B} \]where \( B \) is the coefficient of \( x \) in the angle part of the secant function. In this case, \( B = 3 \). Therefore, the period of the function is:\[ \text{Period} = \frac{2\pi}{3} \]This means that the secant function will complete one full cycle and start repeating its values every \( \frac{2\pi}{3} \) units along the x-axis. Understanding the period helps in graphing as it determines the spacing between repeating features of the graph, such as the locations of the vertical asymptotes.

Phase Shift

Phase shift is the horizontal shift that moves the graph of a trigonometric function left or right. It is crucial for accurately positioning the graph on the x-axis. For the function \( y = 5 \sec(3x - \frac{\pi}{2}) \), the phase shift is calculated as follows:\[ \text{Phase Shift} = -\frac{C}{B} \]where \( C \) is the constant added or subtracted in the angle part, and \( B \) is the coefficient of \( x \). Here, \( C = \frac{\pi}{2} \) and \( B = 3 \). Substituting these values gives:\[ \text{Phase Shift} = -\frac{\frac{\pi}{2}}{3} = -\frac{\pi}{6} \]This indicates that the graph of the secant function moves \( \frac{\pi}{6} \) units to the right. This shift is important to note when graphing the function, ensuring the secant curve aligns correctly with the phase-aligned positions of the cosine graph from which it derives.

Graphing Trigonometric Functions

Graphing trigonometric functions like the secant involves a few steps and requires understanding both the theoretical aspects and practical steps. To graph the function \( y = 5 \sec(3x - \frac{\pi}{2}) \), we first consider its reciprocal function, the cosine. Here's a step-by-step guide to creating the graph:

• Graph the Core Cosine Function: Plot \( y = 5 \cos(3x - \frac{\pi}{2}) \). This serves as a guide. The amplitude here is 5, indicating the highest and lowest points the cosine will reach.
• Identify Vertical Asymptotes: These occur where the cosine function equals zero because the secant function, being the reciprocal, becomes undefined (division by zero scenario). The points where \( 3x - \frac{\pi}{2} = \frac{\pi}{2} + k \pi \) determines these locations, where \( k \) is an integer.
• Plot the Secant Arcs: Between each pair of asymptotes, draw the secant arcs. These should curve upward and downward, reflecting the reciprocal nature of the cosine values.

By using these steps, you can accurately graph the secant function and visualize how its properties such as period and phase shift impact its graph. Understanding the relationship between trigonometric functions, their graphs, and how they transform is essential for mastering these concepts.