Problem 38 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The period is 1 for \( y = 5 \sec 2\pi x \).

Step 1: Understand the Function

The function given is \( y = 5 \sec 2\pi x \). This is a secant function, which is derived from the cosine function as \( \sec\theta = \frac{1}{\cos\theta} \). The coefficient 5 represents a vertical stretch, but it doesn't affect the period. The crucial part for the period is the term \( 2\pi x \) inside the secant.

Step 2: Identify the B Coefficient

In the general form of a trigonometric function \( y = A \sec(Bx + C) + D\), the value \( B \) affects the period of the function. Here, \( B = 2\pi \). In trigonometric functions, \( B \) is used to determine the period.

Step 3: Calculate the Period

The period of the secant function is given by \( \frac{2\pi}{B} \). For our function, \( B = 2\pi \), so the period is \( \frac{2\pi}{2\pi} = 1 \). This means that the function repeats its pattern every 1 unit along the x-axis.

Step 4: Determine Key Points for Graphing

The secant graph is undefined at the points where the corresponding cosine function is zero. For \( y = 5\sec(2\pi x) \), find key points by considering the cosine function \( \cos(2\pi x) = 0 \) at \( x = \frac{1}{4}, \frac{3}{4},... \) within a period. The secant graph will have vertical asymptotes at these undefined points.

Step 5: Graph the Secant Function

Plot the graph of \( y = 5\sec(2\pi x) \) by first sketching the cosine curve \( y = 5\cos(2\pi x) \) lightly, which helps locate the vertical asymptotes. Plot a curve above and below the x-axis, avoiding the undefined points where the cosine is zero, to complete the secant graph. Mark the cycle for one period which repeats every 1 unit on the x-axis.

Step 6: Verify Graph Features

Check that the secant graph has vertical asymptotes at \( x = \frac{1}{4}, \frac{3}{4} \) etc., with complete cycles every 1 unit, and has a maximum and minimum amplitude of 5 and -5, respectively, over each period.

Key Concepts

Secant Function

The secant function, represented as \( \sec \theta \), is the reciprocal of the cosine function. It can be expressed mathematically as \( \sec \theta = \frac{1}{\cos \theta} \). This relationship means that wherever the cosine function is zero, the secant function becomes undefined due to division by zero. In a graph, this is shown as vertical asymptotes.

The secant function inherits many characteristics from the cosine function but has its unique features. For instance, while the cosine function oscillates between -1 and 1, the secant function doesn't have such bounds. Instead, it's unbounded and forms hyperbolic curves for its graph.

• The secant curve has peaks and troughs where the cosine has maximum and minimum values.
• Points of vertical asymptotes occur at values that make the cosine function zero.
• It repeats the same pattern for every complete oscillation, which is known as the period.

Period of Function

Every trigonometric function has a period, which is the horizontal length over which the function repeats its shape. The period is crucial because it tells us just how frequently the function cycles through its pattern.

For basic trigonometric functions, the period of the secant function can be gleaned from the cosine function, as they share the same period since \( \sec \theta = \frac{1}{\cos \theta} \).However, when transforming with a coefficient in front of \( x \), such as \( B \) in \( \sec(Bx) \), the formula \( \text{Period} = \frac{2\pi}{B} \) is used.

• For \( y = 5\sec(2\pi x) \), \( B = 2\pi \).
• The period is \( \frac{2\pi}{2\pi} = 1 \), indicating the pattern repeats every 1 unit on the x-axis.

This concept of periodicity governs not just the secant function, but all trigonometric functions, making it a fundamental characteristic in understanding waves and oscillations.

Graphing Trigonometric Functions

When graphing trigonometric functions like the secant function, identifying key features such as periods and asymptotes is essential.

Begin by considering the corresponding cosine function, as it helps outline where the secant function will have its asymptotes:

• Sketch the cosine function first. For \( y = 5\sec(2\pi x) \), lightly draw \( y = 5\cos(2\pi x) \).
• The values where the cosine intersects the x-axis (i.e., where the cosine equals zero) become the vertical asymptotes on the secant graph.

These asymptotes are visually represented by dotted lines marking undefined points for the secant function:

• Place the secant curve in areas not interrupted by asymptotes, using smooth, unbounded hyperbolic curves above and below the x-axis.
• Each complete section of the graph corresponds to the period of the function, ensuring repetitive cycles.

With practice, graphing a secant function becomes an insightful way to visualize both its nature and relation to the cosine function.

Cosine Function

The cosine function is the foundation upon which the secant function is built. It is defined as \( \cos \theta \), representing the x-coordinate of a unit circle corresponding to a given angle \( \theta \). This function naturally oscillates between -1 and 1, shaping familiar wave patterns.

Understanding the cosine function is crucial for graphing the secant function effectively:

• The cosine function determines where secant is undefined and helps predict behaviors of related functions.
• It provides key intersections with the x-axis, essential for locating vertical asymptotes in the secant function.
• The regular alignment and periodicity of the cosine function further dictate the secant function's periodic behavior.

This interconnectedness highlights how mastering the cosine function lays the groundwork for tackling more complex trigonometric graphs like the secant function. Recognizing the cosine's pivotal role is an essential skill in trigonometry.