Problem 34 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\sec 3 x$$

Step-by-Step Solution

Answer

Period: \( \frac{2\pi}{3} \), asymptotes at \( x = \frac{\pi}{6} + \frac{n\pi}{3} \).

Step 1: Identify the Base Function

The function given is based on the secant function, which is the reciprocal of the cosine function. Recall that the period of \( \sec(x) \) is the same as the period of \( \cos(x) \), which is \( 2\pi \).

Step 2: Determine the New Period

The general formula for finding the period of \( \sec(bx) \) is \( \frac{2\pi}{b} \). In this function, \( b = 3 \). Substitute this into the formula to find the period: \( \text{Period} = \frac{2\pi}{3} \).

Step 3: Find Vertical Asymptotes

Vertical asymptotes for \( \sec(x) \) occur where \( \cos(x) = 0 \). For \( \sec(3x) \), set \( \cos(3x) = 0 \). Solving \( 3x = \frac{\pi}{2} + n\pi \) (where \( n \) is an integer) gives the asymptotes at \( x = \frac{\pi}{6} + \frac{n\pi}{3} \).

Step 4: Sketch the Graph

Start by indicating the period on the x-axis. In one period from \( 0 \) to \( \frac{2\pi}{3} \), place vertical asymptotes at \( x = \frac{\pi}{6}, \frac{\pi}{2}, \) and so on following the pattern. Plot typical secant function values, and draw the upward and downward branches between the vertical asymptotes.

Key Concepts

Secant Function

The secant function, denoted as \( \sec(x) \), is a trigonometric function that is the reciprocal of the cosine function, meaning \( \sec(x) = \frac{1}{\cos(x)} \). It does not have values wherever cosine equals zero, as division by zero is undefined. Therefore, the secant function will have vertical asymptotes at these points. Understanding the secant function is key to solving various problems related to its graph and behavior.

• Core Property: Secant takes its root function from cosine. This makes it essential to understand cosine, as secant will mirror its periodicity but flip where cosine reaches zero.
• Use in equations: It's often found in more complex trigonometric expressions where reciprocal relationships are present.

Period of Trigonometric Functions

The period of a trigonometric function is the length of one complete cycle on a graph. For the cosine function, the period is \(2\pi\), as cosine returns to its starting point over this interval. The secant function inherits its period from cosine. When transformations like horizontal stretches or compressions occur, the period changes accordingly.

• For \( \sec(bx) \): The formula to calculate the new period is \( \frac{2\pi}{b} \). Therefore, for \( \sec(3x) \), the period is \( \frac{2\pi}{3} \).
• Effects of period: A shorter period compresses the graph horizontally, resulting in more frequent cycles within the same horizontal space.

Graphing Trigonometric Functions

Graphing trigonometric functions such as secant involves understanding their periodicity, symmetry, and asymptotes. To graph \( y=\sec(3x) \):

• Mark Asymptotes: Identify where the cosine function, the reciprocal base, is zero. For \( \sec(3x) \), this gives asymptotes at \( x=\frac{\pi}{6}, \frac{\pi}{2}, .. \).
• Identify One Period: From \(0\) to \(\frac{2\pi}{3}\), sketch one complete cycle by plotting typical points of secant and showcasing symmetry.
• Pattern Repetition: Use the aforementioned characteristics to repeat the graph beyond the initial period.

Vertical Asymptotes

Vertical asymptotes represent values where the trigonometric function is undefined. For secant, these occur at angles where cosine equals zero. Secant cannot be calculated at these points, as it would require division by zero.

• Locating Asymptotes: Solve \( \cos(3x)=0 \) to find the locations of vertical asymptotes. For \( \sec(3x) \), these are at \( x = \frac{\pi}{6} + \frac{n\pi}{3} \), where \( n \) is any integer.
• Drawing on Graphs: In practice, draw dashed lines to represent the asymptotes to show the boundaries where the function doesn't exist.
• Role in Graphing: These asymptotes help define the sections of valid plotting and demonstrate periodicity and break points.