91Ó°ÊÓ

The amplitude of a trigonometric function, such as the sine function, defines how tall or short the waves are. It's found by taking the absolute value of the coefficient in front of the sine. In the equation \(y = -3 \sin \frac{x}{3}\), the coefficient is \(-3\). The absolute value is used because amplitude is always positive, describing the distance from the midline (or horizontal center) of the wave to the peak or trough. So, the amplitude here is \(3\).

The higher the amplitude, the taller the wave, which means the function will stretch vertically. If the amplitude were smaller, the wave would be shorter and closer to the midline. Remember, amplitude affects the vertical height of the waves, but not their width or how spread out they are.

The period of a trigonometric function, like the sine function, describes how long it takes for the function to complete one full cycle before repeating itself. For the sine function, this is determined by the equation \(\frac{2\pi}{|b|}\), where \(b\) represents the coefficient of \(x\).

In our example, \(y = -3 \sin \frac{x}{3}\), \(b\) is \(\frac{1}{3}\). By substituting into the period formula, it simplifies to \(\frac{2\pi}{|\frac{1}{3}|} = 6\pi\). This tells us that the wave pattern repeats every \(6\pi\) units along the x-axis.

A longer period means the wave takes more space along the x-axis to complete one full oscillation. Conversely, a shorter period indicates it completes the cycle more quickly.

The sine function is one of the foundational elements of trigonometry. It's often written as \(y = a \sin(bx + c) + d\). Here, \(a\), \(b\), \(c\), and \(d\) are parameters that modify the graph of the sine function.

In our specific case, \(y = -3 \sin \frac{x}{3}\), the sine function is modified as follows:

• \(a = -3\), which affects the amplitude and indicates a vertical reflection (flipping the graph upside down).
• \(b = \frac{1}{3}\), which affects the period of the function, making it longer than the normal \(2\pi\) cycle for a standard sine wave.
• There is no horizontal shift (\(c\)) or vertical shift (\(d\)) in this equation, meaning the sine wave starts at the origin and remains centered.

The sine function produces a wave-like curve, crucial for modeling periodic phenomena like sound waves, tides, and more.