Problem 36 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=5 \sec 2 \pi x$$

Step-by-Step Solution

Answer

The period of the function is 1.

Step 1: Understand the Function Form

The given function is of the form \( y = a \sec(bx + c) + d \). For the function \( y = 5 \sec(2\pi x) \), the values are \( a = 5 \), \( b = 2\pi \), \( c = 0 \), and \( d = 0 \). This is a transformation of the secant function \( \sec x \).

Step 2: Determine the Period of \( y = \sec(bx) \)

The general period of the secant function \( \sec x \) is \( 2\pi \). For \( \sec(bx) \), the period is \( \frac{2\pi}{|b|} \). Here, \( b = 2\pi \), so the period is \( \frac{2\pi}{2\pi} = 1 \).

Step 3: Identify Critical Points for Graphing

The secant function \( \sec x \) has vertical asymptotes where the cosine function is zero. For \( \sec(2\pi x) \), these points occur at \( 2\pi x = \frac{\pi}{2} + k\pi \) where \( k \) is an integer. Solving for \( x \), we have \( x = \frac{1}{4} + \frac{k}{2} \). These x-values indicate vertical asymptotes.

Step 4: Sketch the Secant Function

Plot the function considering its period \( 1 \) and vertical asymptotes. Within one period \( [0, 1] \), asymptotes occur at \( x = \frac{1}{4} \) and \( x = \frac{3}{4} \). Between these points, sketch the characteristic ups and downs of the secant curve, avoiding where \( \cos(2\pi x) = 0 \). Scale by the factor \( 5 \), which stretches the curve vertically by a factor of 5.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec x \), is a trigonometric function related to the cosine function. Specifically, it is the reciprocal of the cosine function, which means it is defined as \( \sec x = \frac{1}{\cos x} \). Understanding this relationship is crucial because it helps to identify the behavior and characteristics of the secant function.

Here are some key points to remember about the secant function:

• The secant function is undefined wherever the cosine function is zero. This is because division by zero is undefined.
• Vertical asymptotes appear in the graph of \( \sec x \) at these undefined points, where \( \cos x = 0 \).
• The secant function has periodic vertical asymptotes, which contribute to its unique shape on the graph.

The secant function undergoes transformations similar to other trigonometric functions. In the given function \( y = 5 \sec(2\pi x) \), the transformation involves a vertical stretch by a factor of 5, affecting the amplitude of the graph.

Period of a Function

One of the essential characteristics of trigonometric functions, such as the secant function, is their periodic nature. The period of a function refers to the interval after which the function repeats its values. For a basic secant function, \( \sec x \), the period is \( 2\pi \), meaning the function repeats every \( 2\pi \) units.

When the function undergoes transformations, the period can change. Specifically, for a transformed secant function of the form \( \sec(bx) \), the period will be \( \frac{2\pi}{|b|} \). This is crucial for graphing, as it determines the horizontal stretch or compression of the curve. For the function \( y = 5 \sec(2\pi x) \):

• The value of \( b \) is \( 2\pi \).
• The period is calculated as \( \frac{2\pi}{2\pi} = 1 \).

Thus, the secant function in this context repeats every 1 unit along the x-axis. Understanding and identifying the period helps when you wish to sketch or interpret the graph of trigonometric functions.

Graphing Trigonometric Functions

Graphing trigonometric functions, such as the secant function, involves understanding both their mathematical definitions and transformations. For the function \( y = 5 \sec(2\pi x) \), sketching its graph accurately requires identifying key features like its period, vertical asymptotes, and any transformations applied to it.

• Vertical Asymptotes: First, determine where the cosine part of the function equals zero, creating vertical asymptotes. For this function, vertical asymptotes occur when \( \cos(2\pi x) = 0 \), resulting in asymptotes at points \( x = \frac{1}{4} + \frac{k}{2} \) where \( k \) is an integer.
• Plotting the Curve: Between each set of asymptotes, plot the characteristic ups and downs of the secant curve. Ensure you avoid crossing the vertical asymptotes as the function is not defined there.
• Transformation Effects: The multiplication factor of 5 stretches the curve vertically by a factor of 5, amplifying the maximum and minimum points of the secant curve within each period interval.

Through careful consideration of these factors, you can successfully graph transformed trigonometric functions, capturing their periodic properties and transformations.