Problem 30 - Precalculus : Building Concepts and Connections

Question

Graph at least two cycles of the given functions. $$g(x)=\frac{3}{2} \cos (2 x+\pi)$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=\frac{3}{2} \sec (3 x)\) has amplitude of 3/2, period of \(2\pi/3\), and oscillates between 3/2 and -3/2 with vertical asymptotes at every half period starting from zero. The graph is repeated for at least two cycles.

Step 1: Identify the basic properties

Begin by identifying the key properties of the provided function which are the amplitude and period. The amplitude of the secant function is given by the absolute value of the coefficient of the equation, in this case, 3/2 or 1.5. The period of the function can be found using the formula \(\frac{2\pi}{|B|}\), where B represents the frequency of the function. Here, B is 3, hence, the period will be \(\frac{2\pi}{3}\). Be aware that secant function has vertical asymptotes where the corresponding cosine function equals zero.

Step 2: Determine the critical points

Next, determine the critical points and asymptotes based on information from step 1. The secant function has vertical asymptotes in every half period. So, the asymptotes are given by the equation \(x = \frac{(2n+1)\pi}{6}\), where n is an integer. The maximum (3/2) and minimum (-3/2) points occur where the corresponding cosine function equals 1 or -1, respectively, which is at \(x = \frac{n\pi}{3}\), where n is an integer.

Step 3: Plot the function

With the critical points and asymptotes at hand, make a sketch of the function. Plot the determined critical points, ensuring that the curve approaches but does not touch the asymptotes. Additionally, the function should oscillate between the maximum and minimum points. Repeat the appearance of the cycle at least twice to meet the exercise's criteria.

Key Concepts

Amplitude of Trigonometric Functions

Understanding the amplitude of trigonometric functions is crucial when it comes to graphing them. Amplitude represents the function's maximum displacement from its central axis or equilibrium position. In simpler terms, it's the distance from the middle of the wave to its crest (the highest point) or trough (the lowest point).

For the secant function, which is the reciprocal of the cosine function, the amplitude is the absolute value of the coefficient in front of the sec function. This coefficient stretches or compresses the graph vertically. For instance, the function \(g(x) = \frac{3}{2} \sec(3x)\) has an amplitude of \(\frac{3}{2}\), meaning the peaks and troughs of the secant curve will reach \(\frac{3}{2}\) units from the centerline.

Period of Trigonometric Functions

The period of a trigonometric function is the length of one complete cycle of the curve. It indicates how often the function repeats its shape. For the basic cosine and sine functions, the period is \(2\pi\), stemming from their cyclical nature as they correspond to one complete rotation around a circle.

To find the period of a function like \(g(x) = \frac{3}{2} \sec(3x)\), we apply the formula \(\frac{2\pi}{|B|}\), where B is the number multiplying x, which represents the frequency of the function. If B is 3 in our case, the period is then \(\frac{2\pi}{3}\). This means that the secant function completes one cycle every \(\frac{2\pi}{3}\) units along the x-axis.

Vertical Asymptotes

Vertical asymptotes are lines where a function’s value grows without bound, typically occurring in rational functions and reciprocal trigonometric functions. These asymptotes represent the input values at which the function is undefined and the output heads towards positive or negative infinity.

In the context of secant functions, vertical asymptotes occur at points where the cosine function, which is the denominator when considering secant as a reciprocal, equals zero. For \(g(x) = \frac{3}{2} \sec(3x)\), the vertical asymptotes are found using the equation \(x = \frac{(2n+1)\pi}{6}\), with n being any integer. This formula essentially denotes all the points where cosine is zero, and thus the secant function becomes undefined. Graphing these will help create a proper visualization of the secant function, showing where it approaches infinity.