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The amplitude in a sine function refers to the height of the wave from its middle point to its peak. It shows how far up or down the wave stretches from the center line, which runs horizontally through the middle of the wave. If the wave were music, the amplitude would be the volume level. The higher the amplitude, the louder the sound.

In the function given by the equation \( y = \sin(x + \frac{\pi}{6}) \), the amplitude is determined by the coefficient "a" in front of the sine. As there is no visible coefficient (other than 1), the amplitude is 1. This means that from the sine wave's central axis (often the x-axis), it reaches 1 unit up to its peak and 1 unit down to its lowest point. Here's why it matters: when you graph the sine wave, the highest point of your curve (the crest) is at \( y = 1 \), and the lowest point (the trough) is at \( y = -1 \).

Understanding amplitude helps us visualize how intense or mild the variations in the sine wave will appear. It's particularly important as it remains consistent throughout one cycle of the wave, making it easier to predict the shape of the graph.

The period of a sine function tells us how long it takes for the wave to complete one full cycle. It's like a repeating pattern. Once the function completes a cycle, it starts over. You can think of it like a looped track—once each complete loop of the track is done, another begins right after.

In the equation for a sine function, the period is derived from the expression \( \frac{2\pi}{|b|} \). In the function \( y = \sin(x + \frac{\pi}{6}) \), the value of "b" is 1. So, substituting into the formula, we find that our period is \( \frac{2\pi}{1} = 2\pi \).

This means that one full cycle of the sine wave spans a horizontal distance of \( 2\pi \) units. Along this distance, the wave starts at a middle position, climbs to a peak, descends through the middle, reaches its lowest dip, and returns back to the starting level—this entire sequence repeats every \( 2\pi \) units along the x-axis. A consistent period helps anticipate where the wave will be at any point, crucial for plotting the graph accurately.

Phase shift in a sine function is all about horizontal movement. It indicates how the entire wave shifts left or right along the x-axis. This shift is like moving the starting point of the wave without changing its shape.

The formula for determining phase shift is \( -\frac{c}{b} \). In our function \( y = \sin(x + \frac{\pi}{6}) \), we identify "c" as \( \frac{\pi}{6} \) and "b" as 1. Plugging these into the phase shift formula, we get:

• Phase Shift = \(-\frac{\pi}{6}\)

This negative sign indicates the wave shifts to the left along the x-axis by \( \frac{\pi}{6} \) units. In other words, the graph of the sine wave starts its cycle \( \frac{\pi}{6} \) units earlier than if there were no phase shift.

By understanding the phase shift of a sine function, you can accurately position your graph. This is essential as it significantly affects where the "starting" and "ending" of each cycle appear on the graph, emphasizing the importance of labeling when plotting.