Question

The function \(y=-3 \cos (6 x)\) has amplitude ________ and period _______.

Short answer

The amplitude is 3 and the period is \( \frac{\pi}{3} \).

Step-by-step solution

Identify the amplitude

For a function of the form \( y = a \, \text{cos}(bx) \), the amplitude is given by the absolute value of the coefficient \( a \). In this case, \( a = -3 \). The amplitude is \( | -3 | = 3 \).

Identify the period

For a function of the form \( y = a \, \text{cos}(bx) \), the period is given by \( \frac{2\pi}{b} \). In this case, \( b = 6 \). So the period is \( \frac{2\pi}{6} = \frac{\pi}{3} \).

Key concepts

amplitude

The amplitude of a trigonometric function determines the height of its peaks and the depths of its troughs.
It is always a positive value and gives us insight into how much the function oscillates above and below its central axis.
For a cosine function of the form \( y = a \, \text{cos}(bx) \), the amplitude is given by the absolute value of the coefficient \( a \).
In our example function, \( y = -3 \, \text{cos}(6x) \), the coefficient \( a \) is -3.
Hence, the amplitude is \( | -3 | = 3 \).
This means the function oscillates 3 units above and below its central horizontal axis.
When studying trigonometric functions, identifying the amplitude helps us visualize the overall shape and range of the graph.

period

The period of a trigonometric function is the length of the interval over which the function completes one full cycle of its pattern.
This concept explains how often the function repeats itself along the x-axis.
The period of a basic cosine function, \( \text{cos}(x) \), is \( 2\pi \).
For a scaled cosine function of the form \( y = a \, \text{cos}(bx) \), the period is given by \( \frac{2\pi}{b} \).
In our specific function, \( y = -3 \, \text{cos}(6x) \), the value of \( b \) is 6.
Therefore, the period is \( \frac{2\pi}{6} = \frac{\pi}{3} \).
This means the function completes one full cycle every \( \frac{\pi}{3} \) units along the x-axis.
Recognizing the period is crucial for analyzing the frequency and behavior of the trigonometric function over a given interval.

cosine function

The cosine function, denoted as \( \text{cos}(x) \), is a fundamental trigonometric function that describes the relationship between the angle and the adjacent side of a right triangle.
It is periodic and oscillatory, meaning it repeats its pattern over regular intervals.
In its basic form, the cosine function starts at its maximum value when \( x = 0 \) and follows a wave-like pattern.
The general form of a cosine function can be expressed as \( y = a \, \text{cos}(bx) \), where:

• \( a \) represents the amplitude (height of waves).
• \( b \) affects the period (length of one complete cycle).

In our example, \( y = -3 \, \text{cos}(6x) \), the negative sign indicates that the graph is reflected over the x-axis.
This means it starts at its minimum value instead of its maximum.
By understanding these parameters, you can predict and graph the behavior of cosine functions in various contexts.