In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 2 sec x

Verified step by step guidance

1. Step 1: Understand the function. The function given is $y=2\sec x$. The secant function, $\sec x$, is the reciprocal of the cosine function, $\cos x$. Therefore, $\sec x=\frac{1}{\cos x}$.
2. Step 2: Identify the period of the secant function. The basic period of $\sec x$ is the same as $\cos x$, which is $2\pi$. Since there is no horizontal stretching or compression, the period of $y=2\sec x$ remains $2\pi$.
3. Step 3: Determine the vertical stretch. The coefficient 2 in $y=2\sec x$ indicates a vertical stretch by a factor of 2. This means the maximum and minimum values of the secant function will be multiplied by 2.
4. Step 4: Identify the asymptotes. The secant function has vertical asymptotes where the cosine function is zero. For $\cos x=0$, $x=\frac{\pi }{2}+n\pi$, where $n$ is an integer. These are the points where the secant function is undefined.
5. Step 5: Sketch the graph. Plot the basic shape of the secant function over two periods, $0$ to $4\pi$, considering the vertical stretch and asymptotes. The graph will have repeating U-shaped curves between the asymptotes, with peaks at $y=2$ and troughs at $y=-2$.

Key Concepts

Secant Function

The secant function, denoted as sec(x), is the reciprocal of the cosine function. It is defined as sec(x) = 1/cos(x). The secant function has a period of 2π, meaning it repeats its values every 2π units along the x-axis. Understanding the behavior of the secant function is crucial for graphing it accurately, especially its vertical asymptotes where cos(x) = 0.

Amplitude and Vertical Stretch

In the function y = 2 sec(x), the coefficient '2' indicates a vertical stretch of the secant function. This means that the values of the secant function are multiplied by 2, affecting the height of the graph. The amplitude in this context refers to how far the graph stretches away from the x-axis, which is important for visualizing the graph's peaks and troughs.

Graphing Periodic Functions

Graphing periodic functions like secant involves identifying key points, asymptotes, and the overall shape of the graph. For sec(x), vertical asymptotes occur at x = (2n + 1)π/2, where n is an integer, indicating where the function is undefined. To graph two periods, one must plot the function from 0 to 4π, ensuring to mark the asymptotes and the maximum and minimum values of the function.