Problem 52 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The period of the function is \( \frac{2\pi}{3} \).

Step 1: Identify the Standard Form

The standard form of the secant function is \( y = a \sec(bx + c) + d \). Here, \( a = 1 \), \( b = 3 \), \( c = \frac{\pi}{2} \), and \( d = 0 \). Since we are focusing on period determination, notice that \( b \) is essential to find the period.

Step 2: Calculate the Period

The period of the parent secant function \( y = \sec(x) \) is \( 2\pi \). To find the period of \( y = \sec(3x + \frac{\pi}{2}) \), divide the standard period by \( b \). Thus, the period is \( \frac{2\pi}{b} = \frac{2\pi}{3} \).

Step 3: Determine the Phase Shift

The phase shift can be calculated using the formula \( \text{phase shift} = -\frac{c}{b} \). Substituting the given values \( c = \frac{\pi}{2} \) and \( b = 3 \), we have \( \text{phase shift} = -\frac{\frac{\pi}{2}}{3} = -\frac{\pi}{6} \).

Step 4: Graph the Function

To graph \( y = \sec(3x + \frac{\pi}{2}) \), first sketch the graph of the corresponding cosine function \( y = \cos(3x + \frac{\pi}{2}) \). It has a period of \( \frac{2\pi}{3} \) and a phase shift of \(-\frac{\pi}{6}\). The secant function will have vertical asymptotes where the cosine function has zeros. Plot these asymptotes and draw the secant curve, which will touch the maximum and minimum points of the cosine curve.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and secant, are mathematical functions based on the angles of triangles, specifically right triangles. They are fundamental in expressing periodic phenomena and are frequently used in various branches of science and engineering. Trigonometric functions cyclically repeat values over an interval, making them ideal for modeling waves, oscillations, and rotations. The secant function, denoted as \(\sec(x)\), is derived from the cosine function and is defined as \(\sec(x) = \frac{1}{\cos(x)}\). As a reciprocal function, the secant has unique behaviors, such as having undefined values when the cosine is zero, leading to vertical asymptotes in its graph.
When working with functions like the secant, it's key to understand transformations based on parameters such as amplitude, period, phase shift, and vertical shift. The general form \(y = a\sec(bx + c) + d\) encompasses all these elements, allowing for comprehensive manipulation and understanding of the function's behavior.
Recognizing these parameters enables us to graph the function accurately and understand how changes in each parameter affect the overall shape and position of the trigonometric curve.

Period of a Function

The period of a trigonometric function is the length of the interval over which the function completes one full cycle before repeating. For the basic secant function, \(y = \sec(x)\), the period is \(2\pi\). This means every \(2\pi\) units along the x-axis, the secant function's pattern repeats itself.
In our exercise, we have a modified secant function represented by \(y = \sec(3x + \frac{\pi}{2})\). To determine the period of this altered function, we divide the basic period by \(b\) in the function formula \(y = a\sec(bx + c) + d\).
The formula to find the new period is \(\frac{2\pi}{b}\). In this case, \(b = 3\), leading to a period of \(\frac{2\pi}{3}\). This adjustment compresses the wave horizontally, meaning that the function repeats itself more frequently than the basic secant function.
Understanding the period is crucial for graphing, as it informs us how often the wave pattern appears within a given domain.

Phase Shift

Phase shift refers to the horizontal displacement of the function's graph along the x-axis. Shifting the function left or right affects where the cycle begins, although it doesn't alter the actual length of a cycle.
In the formula for a trigonometric function, \(y = a \sec(bx + c) + d\), the phase shift is calculated using the expression \(-\frac{c}{b}\). This shows how much the entire curve moves horizontally.
For the function \(y = \sec(3x + \frac{\pi}{2})\), with \(c = \frac{\pi}{2}\) and \(b = 3\), the phase shift is \(-\frac{\frac{\pi}{2}}{3} = -\frac{\pi}{6}\). This signifies that the curve is shifted \(\frac{\pi}{6}\) units to the left.
Accounting for the phase shift is especially important when graphing, as it impacts where key features, such as peaks, valleys, and asymptotes, appear on the graph. Mapping these correctly is essential to depicting the function's behavior accurately on a graph.