Graph two periods of the given cosecant or secant function.


y = 3 sec x

Verified step by step guidance

1. Recall that the secant function is defined as the reciprocal of the cosine function: \(y = \sec x = \frac{1}{\cos x}\). Therefore, \(y = 3 \sec x\) can be written as \(y = \frac{3}{\cos x}\).
2. Identify the period of the basic secant function. Since \(\cos x\) has a period of \(2\pi\), the secant function also has a period of \(2\pi\). So, one period of \(y = 3 \sec x\) spans an interval of length \(2\pi\).
3. To graph two periods, determine the interval over which you will plot the function. For example, from \(x = 0\) to \(x = 4\pi\) covers two full periods of \(y = 3 \sec x\).
4. Find the vertical asymptotes of the function by locating where \(\cos x = 0\), because \(\sec x\) is undefined there. These occur at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is any integer. Mark these asymptotes within your chosen interval.
5. Plot key points by evaluating \(y = 3 \sec x\) at values where \(\cos x\) is \(\pm 1\) or other convenient points, such as \(x = 0, \pi, 2\pi, 3\pi, 4\pi\). Use these points and the asymptotes to sketch the two periods of the graph, noting that the graph will have branches opening upwards or downwards depending on the sign of \(\cos x\).

Key Concepts

Secant Function and Its Properties

The secant function, sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). It has vertical asymptotes where cos(x) = 0, causing the function to be undefined at these points. Understanding its periodicity and behavior near asymptotes is essential for graphing.

Amplitude and Vertical Stretch

In the function y = 3 sec x, the coefficient 3 represents a vertical stretch, scaling the secant graph by a factor of 3. This affects the distance of the graph's branches from the x-axis but does not change the location of asymptotes or the period.

Periodicity of Secant Function

The secant function has a fundamental period of 2π, meaning its pattern repeats every 2π units along the x-axis. Graphing two periods involves plotting the function from 0 to 4π (or an equivalent interval), including all asymptotes and key points within this range.