Problem 33 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Sketch the secant function using the parent function \( \cos{x} \), stretch it vertically by a factor of 2, locate the asymptotes at \( x = \frac{\pi}{2} + n\pi \), and sketch two full periods.

Step 1 Graph the Parent Function and Asymptotes

First graph the parent function \(y=\cos{x}\). Remember that \(\sec{x} = \frac{1}{\cos{x}}\), so wherever the cosine function equals zero, the secant function will be undefined and have a vertical asymptote. Draw the vertical asymptotes at these x-values, which are \(x = \frac{\pi}{2} + n\pi\), where \(n\) is any integer.

Step 2 Apply the Vertical Stretch

The number in front of the secant function, 2, is a vertical stretch. This means you multiply the y-coordinates of each point on the parent function by 2. Remember that this does not affect the vertical asymptotes, which stay at the same x-values.

Step 3 Sketch Two Periods of the Function

Finally, sketch the graph using the asymptotes as guides. Remember, the graph will consist of an infinite series of U-shapes that get infinitely close to the asymptotes without touching. Sketch two full periods of these U-shapes.

Key Concepts

Vertical Asymptotes

Understanding vertical asymptotes is crucial when graphing functions like the secant, which may involve division by zero. A vertical asymptote is a vertical line that represents an x-value where the function tends to infinity or negative infinity. Essentially, the function will not have a value at that point and, graphically, the curve approaches but never touches the asymptote.

For the secant function, vertical asymptotes occur where the cosine function is zero, because secant is the reciprocal of cosine. In the given exercise, this happens at intervals of \(x = \frac{\pi}{2} + n\pi\), with \(n\) being any integer. When graphing \(y=2\sec x\), plotting these asymptotes helps to outline the shape of the function and guides where the U-shapes of the secant graph are formed.

Vertical Stretch

The concept of vertical stretch is an important transformation in graphing trigonometric functions. A vertical stretch scales the function in the y-direction. In the equation \(y=2\sec x\), the '2' indicates that every y-value of the parent cosine function is multiplied by 2. This does not affect the locations of the vertical asymptotes but does impact the amplitude and steepness of the curves between the asymptotes.

The vertical stretch gives the function a different appearance without altering its basic shape. For a clearer graph of the secant function, after identifying the asymptotes, apply the stretch to the parent cosine curve before considering its reciprocal, which in this case would amplify any \(\cos x\) value by a factor of two.

Parent Trigonometric Functions

Parent trigonometric functions are the simplest form of trigonometric functions and serve as the cornerstone for understanding their transformations. For the secant function, the parent is the cosine function \(y=\cos x\). Graphing these parent functions first provides a baseline which transformations like stretches, shifts, and reflections will adjust from.

In the provided exercise, graphing the parent function \(y=\cos x\) helps in understanding where the secant function will become undefined, thus identifying the vertical asymptotes. Since the secant function is defined as the reciprocal of the cosine function, knowing the behavior and properties of the cosine function is essential for accurately graphing the secant function.

Trigonometric Function Period

The period of a trigonometric function is the length of the smallest interval over which the function repeats its pattern. For the parent cosine and secant functions, the period is \(2\pi\), meaning after an interval of \(2\pi\), the functions start repeating their values.

When you graph \(y=2\sec x\), you are asked to graph two periods to showcase this repetitive nature. This requires plotting the function from \(x=0\) to \(x=4\pi\), thus covering two full cycles of the curve. The concept of period is particularly significant in trigonometric functions as it reflects their oscillating behavior, which is foundational for analyzing and understanding waves and harmonic motion in various scientific and engineering contexts.