Question

Find the amplitude (if one exists), period, and phase shift of each function. Graph each function. Be sure to label key points. Show at least two periods. $$ y=\frac{1}{2} \sec (3 x-\pi) $$

Short answer

The function's period is \( \frac{2\pi}{3} \), phase shift is \( \frac{\pi}{3} \) to the right, and there is a vertical scaling factor of \( \frac{1}{2} \).

Step-by-step solution

Identify the components of the function

The given function is defined as: \[ y = \frac{1}{2} \sec (3x - \pi) \] This is a secant function, which is derived from the cosine function. Therefore, look for the form: \[ y = a \sec(bx - c) \] where \( a \) is the amplitude, \( b \) affects the period, and \( c \) affects the phase shift. In this case, \( a = \frac{1}{2} \), \( b = 3 \), and \( c = \pi \).

Determine the amplitude

Secant functions do not have an amplitude in the same way that sine and cosine functions do. Instead, the value \( a \) affects the vertical stretch of the graph. Therefore, we acknowledge \( \frac{1}{2} \) as a vertical scaling factor.

Calculate the period

The period of a secant function is influenced by the coefficient \( b \) in front of the variable \( x \). The period is given by: \[ \text{Period} = \frac{2\pi}{|b|} = \frac{2\pi}{3} \] So, the period of the function \( y = \frac{1}{2} \sec(3x - \pi) \) is \( \frac{2\pi}{3} \).

Determine the phase shift

The phase shift of the function is given by: \[ \text{Phase shift} = \frac{c}{b} = \frac{\pi}{3} \] Since \( c = \pi \) and \( b = 3 \), the phase shift is \( \frac{\pi}{3} \) to the right.

Graph the function

To graph \( y = \frac{1}{2} \sec(3x - \pi) \), follow these steps: 1. Start by graphing the related cosine function: \( y = \frac{1}{2} \cos(3x - \pi) \). 2. Identify and plot key points: maximum, minimum, and intercepts over one period. 3. Since the secant is the reciprocal of the cosine, draw vertical asymptotes at the points where \( \cos(3x - \pi) = 0 \). These are the points where cosine crosses the x-axis. 4. Plot secant values approaching infinity at these asymptotes. Show at least two periods, taking \( \frac{2\pi}{3} \) as one period.

Key concepts

secant function

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

The secant function \( y = \frac{1}{2} \text{sec}(3x - \pi) \) has a vertical stretch affected by the factor \( \frac{1}{2} \). This acts similarly to amplitude in sine and cosine functions but is more accurately described as altering the height of the secant graph. When graphing a secant function, you need to graph its corresponding cosine function first, and then use its reciprocal to find the points of the secant function. Key points for cosine, such as where it crosses the x-axis, become vertical asymptotes for the secant graph.

period of a function

The period of a trigonometric function is the length of one complete cycle of the wave. For secant functions, which are derived from cosine functions, the period is determined similarly.

In the given function \( y = \frac{1}{2} \text{sec}(3x - \pi) \), the coefficient \( b \) (which is 3) influences the period. The formula to calculate the period for secant (and cosine) functions is:

\[ \text{Period} = \frac{2\pi}{|b|} = \frac{2\pi}{3} \]

This means the graph of the function will repeat its pattern every \( \frac{2\pi}{3} \) units along the x-axis. To visualize the graph, it’s essential to plot at least two full periods, especially when identifying and labeling key points. Periodic functions like these help in understanding repeating phenomena in various physical and engineering applications.

phase shift

The phase shift of a trigonometric function represents a horizontal shift from its standard position. It is calculated using the parameter \( c \) in the function \( y = a \text{sec}(bx - c) \).

In the function \( y = \frac{1}{2} \text{sec}(3x - \pi) \), the phase shift is determined by:

\[ \text{Phase shift} = \frac{c}{b} = \frac{\pi}{3} \]

Since \( c = \pi \) and \( b = 3 \), the phase shift is \( \frac{\pi}{3} \) to the right. The phase shift moves the entire graph horizontally along the x-axis. When graphing, this shift must be taken into account to accurately reflect the position of key points and asymptotes.

In summary:

• Identify \( c \) and \( b \).
• Calculate \( \frac{c}{b} \).
• Shift the graph horizontally by \( \frac{\pi}{3} \) units to the right.

The phase shift is crucial for accurately graphing trigonometric functions and understanding their behavior in different contexts.