Problem 14 - Precalculus

Question

In Exercises \(11-20,\) state the amplitude and period of each function. $$y=-\cos (7 x)$$

Step-by-Step Solution

Answer

The amplitude is 1, and the period is \(\frac{2\pi}{7}\).

Step 1: Identify the Amplitude Formula

The standard form of a cosine function is given by \(y = a \cdot \cos(bx)\). In this formula, \(a\) represents the amplitude of the function. The amplitude is the absolute value of \(a\), so the amplitude is \(|a|\).

Step 2: Determine the Amplitude

In the function \(y = -\cos(7x)\), the value of \(a\) is \(-1\). The amplitude is the absolute value of this coefficient, which is \(|-1| = 1\). Therefore, the amplitude of the function is 1.

Step 3: Identify the Period Formula

The period of a cosine function is determined by the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) inside the cosine function.

Step 4: Calculate the Period

For the function \(y = -\cos(7x)\), the coefficient \(b\) is 7. Using the period formula \(\frac{2\pi}{b}\), we calculate the period as \(\frac{2\pi}{7}\). Therefore, the period of the function is \(\frac{2\pi}{7}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are a type of mathematical function that relate angles of a triangle to the lengths of the sides of the triangle. They are fundamental in studying periodic phenomena such as sound waves, light waves, and tides. The most common trigonometric functions are sine, cosine, and tangent. These functions have special properties and regular patterns which make them unique in the world of mathematics.

For instance, the cosine function, often denoted as \(y = \cos(x)\), shows how the x-coordinate of a point on a unit circle changes as we move around the circle. Trigonometric functions are used widely in various fields such as engineering, physics, and even in finance to model cyclic patterns. Understanding the basic concepts of amplitude and period is crucial in mastering these functions.

Amplitude Formula

The amplitude of a trigonometric function refers to the peak height of the wave, measured from its equilibrium position (mean value). For the cosine and sine functions, amplitude is represented by the letter \(a\) in the equation \(y = a \cos(bx)\) or \(y = a \sin(bx)\). It tells you how far the wave travels above and below its midpoint.

To find the amplitude, take the absolute value of \(a\), which is given by the formula \(|a|\). This makes sense because amplitude is a distance and thus should always be positive. For example, in the function \(y = -\cos(7x)\), the value of \(a\) is -1. Therefore, the amplitude is \(|-1| = 1\).

• Amplitude indicates the wave's maximum displacement.
• It provides information on the strength or intensity of the wave.

Period Formula

The period of a trigonometric function is the distance over which the wave's pattern repeats itself. This is an important property as it defines a full cycle of the wave. For functions like sine and cosine, the period can be determined using the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) in the equations \(y = a \cos(bx)\) or \(y = a \sin(bx)\).

Calculating the period helps in understanding how long it takes the function to complete one full wave cycle. In the example of the function \(y = -\cos(7x)\), the coefficient \(b\) is 7. Plugging into the formula yields \(\frac{2\pi}{7}\), meaning the cycle repeats every \(\frac{2\pi}{7}\) units on the x-axis.

• Period defines the horizontal length of a full wave cycle.
• A short period indicates that the waves will be tightly packed.
• It is crucial for predicting and modeling periodic behavior.