Problem 17 - Algebra and Trigonometry

Question

Find the amplitude and period of the function, and sketch its graph. $$ y=-3 \sin 3 x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Amplitude

The amplitude of a sinusoidal function of the form \( y = a \sin bx \) is given by the absolute value of \( a \). For the function \( y = -3 \sin 3x \), the amplitude is \( |a| = |-3| = 3 \).

Step 2: Determine the Period

The period of a sine function \( y = a \sin bx \) is calculated using the formula \( \frac{2\pi}{b} \). For the given function \( y = -3 \sin 3x \), the value of \( b \) is 3, so the period is \( \frac{2\pi}{3} \).

Step 3: Sketch the Graph

To sketch the graph of \( y = -3 \sin 3x \), start by noting that the amplitude is 3 and the period is \( \frac{2\pi}{3} \). This means each cycle completes in \( \frac{2\pi}{3} \) units on the x-axis. The graph of \( -\sin \) is reflected over the x-axis compared to \( \sin \), so it starts at the origin, goes downward to its minimum at \( -3 \), returns to zero, goes up to \( 3 \), and back to zero in one period. Plot these points and continue the pattern for more cycles if needed.

Key Concepts

Amplitude

When examining sinusoidal functions, the amplitude is one of the most crucial attributes to understand. The amplitude of a function like

• \( y = a \sin bx \)

is the distance from the midline of the wave to its peak (maximum) or trough (minimum). This gives us a measure of how "tall" or "short" the wave appears on a graph.

The amplitude is determined by taking the absolute value of the coefficient \( a \). In simpler terms, it's always a positive number, showing the size of the largest oscillation from the equilibrium line.

In our case with the function \( y = -3 \sin 3x \), the coefficient \( a \) is \(-3\). So, the amplitude is calculated as:

• |a| = |-3| = 3

In terms of graphing, this means the wave of the sine function will graze up to 3 units above and dive 3 units below the midline.

Period of a Function

The period of a function is the horizontal length it takes for the function to complete one full cycle. For sinusoidal functions like sine and cosine, recognizing the period is key to plotting accurate graphs and understanding the wave's behavior.

The period formula for a sine function \( y = a \sin bx \) is:

• \( \frac{2\pi}{b} \)

Here, \( b \) decides how "stretched" or "compressed" the wave is horizontally. In the function \( y = -3 \sin 3x \), \( b = 3 \). Therefore, the period is determined as follows:

• Period = \( \frac{2\pi}{3} \)

This means the sine function completes one full wave cycle in an interval of \( \frac{2\pi}{3} \) on the x-axis. This knowledge helps in graphing, showing us the frequency of the waves within a specific range.

Graphing Sinusoidal Functions

Graphing sinusoidal functions might seem daunting, but once you understand key attributes like amplitude and period, it becomes more intuitive. For the given function \( y = -3 \sin 3x \), follow these steps:

• Identify key points: Start plotting where the function starts, hits its maximum, goes back through zero, hits the minimum, and returns to zero. These landmarks help shape your graph.
• Remember the reflection: Since our function is \( - \sin \), it reflects over the x-axis compared to the typical \( \sin \) graph. This inversion means it starts by heading downward.
• Use the amplitude: The function will reach a peak of 3 and a trough of -3 as it completes its cycles.
• Apply the period: Each cycle of \( y = -3 \sin 3x \) covers an x interval of \( \frac{2\pi}{3} \), allowing us to follow a consistent pattern across the entire x-axis.

By maintaining a structured approach, you become confident in waveform plotting, turning abstract mathematical functions into clear visual representations.