Problem 46 - Algebra and Trigonometry

Question

Find the period and graph the function. $$ y=2 \sec \left(\frac{1}{2} x-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

The period of the function is \(4\pi\). Graph involves a horizontal stretch and a phase shift.

Step 1: Identify the General Form

The general form of the secant function is \( y = a \sec(bx - c) + d \). Here, the given function can be compared against this form to identify constants: \( a = 2 \), \( b = \frac{1}{2} \), \( c = \frac{\pi}{3} \), and \( d = 0 \).

Step 2: Determine the Period of the Secant Function

The period of a secant function is given by \( \frac{2\pi}{|b|} \). Substituting \( b = \frac{1}{2} \), the period becomes \( \frac{2\pi}{|\frac{1}{2}|} = 4\pi \).

Step 3: Graph Pre-Transformation Secant Function

Consider the basic form secant function graph, \( y = 2 \sec(x) \). For \( y = \sec(x) \), the function has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) for integers \( n \). As a result, the graph repeats every \( 2\pi \).

Step 4: Apply Phase Shift and Period Changes to the Graph

The phase shift is determined by \( \frac{c}{b} \), here it is \( \frac{\pi}{3} \div \frac{1}{2} = \frac{2\pi}{3} \) to the right. The graph of \( y = 2 \sec(x) \) is shifted to the right by \( \frac{2\pi}{3} \) and stretched horizontally to account for the new period \( 4\pi \).

Step 5: Plot the Final Transformed Graph

Start with plotting key points from the secant curve after the transformations, knowing that every \( 4\pi \) the curve will repeat. Place vertical asymptotes at \( x=\frac{\pi}{3} + 2n\pi \) for integers \( n \), ensuring the shifts and stretches match the calculated transformations.

Key Concepts

Secant Function

The secant function is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, defined as \( \sec(x) = \frac{1}{\cos(x)} \). This means that wherever cosine is equal to zero, the secant function will have vertical asymptotes because division by zero is undefined.

Here are some key points about secant functions:

• The secant function, like cosine, is periodic and repeats its pattern over intervals.
• It features vertical asymptotes, where the cosine function equals zero, which occur at \( x = \frac{\pi}{2} + n\pi \), with \( n \) being an integer.
• The secant graph displays a series of repeating curves, resembling waves that are interrupted by these vertical asymptotes.

When graphing a transformed secant function, such as in the exercise above, it helps to think of how these features change based on the transformations involved.

Period of Trigonometric Functions

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 and cosine functions, this period is \( 2\pi \). However, transformations can alter this period.

In the function \( y = 2 \sec\left(\frac{1}{2}x - \frac{\pi}{3}\right) \), the transformed period is calculated using the formula \( \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \).

Here's how transformations affect the period:

• When \( b \) is greater than 1, the period decreases and the function oscillates more frequently.
• When \( 0 < b < 1 \), as in our problem where \( b = \frac{1}{2} \), the period increases, lengthening the distance over which the function repeats. Hence, the calculated period is \( 4\pi \).

This understanding allows for determining where vertical asymptotes and critical points such as peaks and troughs will occur on the transformed graph.

Phase Shift

Phase shift refers to the horizontal translation of a trigonometric graph. It occurs when a constant is subtracted or added inside the argument of the trigonometric function. In the formula \( y = a \sec(bx - c) + d \), phase shift is indicated by \( \frac{c}{b} \), shifting the graph either left or right.

For \( y = 2 \sec\left(\frac{1}{2}x - \frac{\pi}{3}\right) \), the phase shift is calculated as:

• Determine \( c \): \( c = \frac{\pi}{3} \).
• Calculate \( \frac{c}{b} \): \( \frac{\pi}{3} \div \frac{1}{2} = \frac{2\pi}{3} \).
• This results in a phase shift of \( \frac{2\pi}{3} \) to the right.

Recognizing the phase shift is essential for accurately plotting the graph, as it moves the entire pattern of the graph horizontally along the x-axis.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting a graph. Understanding these transformations helps in sketching the graph accurately.

In the given secant function \( y = 2 \sec\left(\frac{1}{2}x - \frac{\pi}{3}\right) \), several transformations are at play:

• Vertical Stretch: The multiplier \( a = 2 \) causes the graph to stretch vertically compared to \( y = \sec(x) \), moving curves further from the x-axis.
• Horizontal Stretch: The period changes to \( 4\pi \) as calculated, resulting in a horizontal stretch of the basic curve.
• Phase Shift: The phase shift of \( \frac{2\pi}{3} \) to the right moves the graph horizontally.

The combination of these transformations determines the position and shape of the graph, especially where the curves and asymptotes are placed on the coordinate plane. Recognizing each transformation makes plotting and interpreting the graph clearer.