91Ó°ÊÓ

The amplitude of a trigonometric function such as the sine function is essentially the peak value of the wave above its central axis. For the function \( f(x) = \sin(x - \frac{\pi}{6}) \), the amplitude can be directly determined by the absolute value of the coefficient before the sine function, denoted by \( a \) in the standard form \( y = a\sin(bx - c) + d \).
Since \( a = 1 \), the amplitude is \( 1 \).
This means that the function will oscillate 1 unit above and below the horizontal axis at its highest and lowest points:

• The maximum point is at \( y = 1 \)
• The minimum point is at \( y = -1 \)

Remember, the amplitude is always a positive number, showing us how "tall" the wave is from the center line to the peak. In the graph, you can see the smooth, consistent rise and fall from the center that creates the wave-like pattern.

Periodicity refers to the length of one complete cycle of a trigonometric function before it starts repeating. For sine functions, the standard period is \( 2\pi \).
To calculate the period of a sine function formatted as \( y = a\sin(bx - c) + d \), we use the formula \( \frac{2\pi}{b} \).
In our function \( f(x) = \sin(x - \frac{\pi}{6}) \), the coefficient \( b \) is \( 1 \). Thus, the period is:

• \( \frac{2\pi}{1} = 2\pi \)

That means, after every \( 2\pi \) units along the x-axis, the sine wave pattern repeats.
This repeating nature makes trigonometric functions particularly useful in modeling real-world phenomena, like sound waves or tidal movements, that also exhibit periodic behavior.

Phase shift describes how far the graph of a trigonometric function has been shifted horizontally from its original position.
In the function \( f(x) = \sin(x - \frac{\pi}{6}) \), we determine the phase shift using \( \frac{c}{b} \), where \( c \) is the horizontal shift component and \( b \) alters the frequency.

• Given \( c = \frac{\pi}{6} \) and \( b = 1 \), the phase shift is \( \frac{\pi}{6} \) to the right.

This means that every x-coordinate on the graph is translated \( \frac{\pi}{6} \) units to the right compared to the basic sine function \( \sin(x) \).
Visually, the phase shift can be seen as the starting point of one cycle of the sine wave is no longer at \( x = 0 \), but is instead at \( x = \frac{\pi}{6} \). The oscillations remain the same; only the starting point on the x-axis has changed.