Problem 21 - Algebra and Trigonometry

Question

Find the amplitude and period of the function, and sketch its graph. $$ y=-\frac{1}{3} \cos \frac{1}{3} x $$

Step-by-Step Solution

Answer

Amplitude is \( \frac{1}{3} \) and period is \( 6\pi \). The graph is a reflected cosine wave.

Step 1: Identify the Amplitude

The amplitude of a cosine function in the form \( y = a \, \cos(bx) \) is given by the absolute value of \( a \). In this function, \( a = -\frac{1}{3} \). The amplitude is therefore \( \left| -\frac{1}{3} \right| = \frac{1}{3} \).

Step 2: Determine the Period

The period of a cosine function in the form \( y = a \, \cos(bx) \) can be computed using the formula \( \frac{2\pi}{b} \). Here, \( b = \frac{1}{3} \). Substitute \( b \) into the formula: \[ \text{Period} = \frac{2\pi}{\frac{1}{3}} = 6\pi \]

Step 3: Understand the Negative Amplitude

The negative sign in \( a = -\frac{1}{3} \) indicates that the cosine function is reflected over the x-axis. This will affect the sketch of the graph by turning it upside down compared to a regular cosine function.

Step 4: Sketch the Graph

To sketch the graph of \( y = -\frac{1}{3} \, \cos \left(\frac{1}{3}x\right) \), start by noting the amplitude and period. The function oscillates between \( y = \frac{1}{3} \) and \( y = -\frac{1}{3} \) with a period of \( 6\pi \). Given the reflection, the function begins at its minimum point (-1/3) when \( x = 0 \), reaches the maximum (1/3) at \( x = 3\pi \), and returns to minimum at \( x = 6\pi \). Repeat this pattern for further cycles.

Key Concepts

Amplitude

In trigonometric functions, the amplitude is a measure that indicates how far the waves stretch above and below the center line, also known as the axis. It is the maximum height from the baseline of the wave to its peak or the minimum depth to its trough.
To find the amplitude of a cosine function like \( y = a \cos(bx) \), we simply take the absolute value of the coefficient \( a \). In the function \( y = -\frac{1}{3} \cos\left(\frac{1}{3}x\right) \):

• The value of \( a \) is \(-\frac{1}{3}\),
• Therefore, the amplitude is \( \left| -\frac{1}{3} \right| = \frac{1}{3} \).

This means that the cosine wave will oscillate between \( \frac{1}{3} \) and \(-\frac{1}{3}\) on the y-axis. Even with a negative sign, amplitude is always considered as a positive value because it represents a distance.

Period

The period of a trigonometric function is the horizontal length required for one complete cycle of the wave to repeat. For a cosine function like \( y = a \cos(bx) \), the period is calculated using the formula \( \frac{2\pi}{b} \).
In the given function, the value of \( b \) is \( \frac{1}{3} \), leading to:

• Plug \( b \) into the period formula: \( \frac{2\pi}{\frac{1}{3}} \).
• Calculate the period: \( 6\pi \).

This indicates that the wave completes one full cycle over a length of \( 6\pi \) on the x-axis. As you visualize the graph, the wave pattern repeats every \( 6\pi \) units horizontally.

Cosine Function

The cosine function is a fundamental aspect of trigonometry, commonly represented as \( \cos \theta \), where \( \theta \) is the angle. In its standard form \( y = a \cos(bx) \), the graph of the function illustrates a repeating wave pattern.
Key aspects of the cosine function:

• The coefficient \( a \) affects the amplitude of the wave.
• The factor \( b \) affects the period of the wave.
• A negative sign in front of \( a \), like in our function, reflects the wave over the x-axis, flipping it upside down.

For the given function \( y = -\frac{1}{3} \cos\left(\frac{1}{3}x\right) \), the wave starts at its lowest point at \( x = 0 \) due to the negative amplitude. Rather than cresting at \( x = 0 \) as a typical cosine function does, it starts at a trough, reaches its peak midpoint through the cycle at \( x = 3\pi \), and completes the cycle back at a trough by \( x = 6\pi \). By understanding these components, you can sketch the graph by marking the key points and connecting them smoothly to capture the oscillating nature of the cosine.