Question

Determine the amplitude and period of each function without graphing. $$ y=-3 \cos (3 x) $$

Short answer

Amplitude: 3, Period: \( \frac{2\pi}{3} \)

Step-by-step solution

Identify the Standard Form

The standard form of a cosine function is \( y = A \, \text{cos}(Bx) \). For the given function \( y = -3 \, \text{cos}(3x) \), identify the values of A and B.

Determine the Amplitude

The amplitude of a cosine function is the absolute value of A. In this case, \( A = -3 \). Therefore, the amplitude is \( | -3 | = 3 \).

Determine the Period

The period of a cosine function is given by \( \frac{2\pi}{B} \). For the given function, \( B = 3 \). Therefore, the period is \( \frac{2\pi}{3} \).

Key concepts

Amplitude Calculation

The amplitude of a trigonometric function like a cosine function is a measure of its 'height.' This tells us how far the function's values can reach from its middle or central axis. For cosine functions given in the form \( y = A \, \text{cos}(Bx) \), the amplitude is simply the absolute value of the coefficient \( A \).
In our exercise, the function is \( y = -3 \, \text{cos}(3x) \). Here, \( A = -3 \).
To find the amplitude, we use the absolute value: \( | A | = | -3 | = 3 \).
This means the function's peaks reach 3 units above and 3 units below its central axis. Knowing the amplitude helps us understand the function's vertical stretch and shrink, making it easier to sketch its graph and analyze its behavior.

Period Determination

The period of a trigonometric function indicates how long it takes for the function to complete one cycle. For cosine functions, the general form is \( y = A \, \text{cos}(Bx) \), and the period is calculated using the formula: \( \text{Period} = \frac{2\pi}{B} \).
In our example, the function is \( y = -3 \, \text{cos}(3x) \), where \( B = 3 \). So, we plug \( B \) into the period formula: \( \text{Period} = \frac{2\pi}{3} \).
This tells us that the function will repeat its cycle every \( \frac{2\pi}{3} \) radians. Understanding the period is crucial for predicting the function’s values and sketching its graph accurately.
If you think of the period as the 'frequency' of the function, you're on the right track. It tells you how 'frequent' the repetitions are.

Trigonometric Functions

Trigonometric functions like sine and cosine are crucial in understanding periodic phenomena. They help us model and understand patterns that repeat over time.
The cosine function, in particular, is often used to model wave-like phenomena, like sound waves or tides. The general form is \( y = A \, \text{cos}(Bx) \), where:

• \( A \) represents the amplitude, which affects the height of the wave.
• \( B \) determines the period, which tells us how often the wave repeats.

These parameters allow us to modify the wave and fit it to various scenarios.
The cosine and sine functions are foundational in trigonometry and calculus. They provide tools for analyzing angles, distances, and other geometric properties.
Understanding these functions is essential for students, as they are widely used in physics, engineering, and other sciences. The better you understand these, the more equipped you'll be to tackle complex problems in various fields.