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The amplitude of a trigonometric function such as a cosine function is a measure of its height. It tells you how far the peaks of the wave are from the midline. If you imagine the wave as rising and falling above and below a horizontal line, the amplitude measures this rise and fall.

In the equation you've been given, which is \( y = \frac{1}{2} \cos \frac{\pi}{3} x \), the amplitude is represented by the coefficient of the cosine function, in this case \( \frac{1}{2} \).

Here's how to determine it:

• Identify the coefficient of the cosine term. This is the value that comes before the \( \cos \) term.
• The amplitude is simply the absolute value of this coefficient, ensuring it's always a positive quantity as it represents a distance.

You arrive at: \( |\frac{1}{2}| = \frac{1}{2} \), which means the graph reaches maximum and minimum distances of \( \frac{1}{2} \) units from the center line.

When graphing, this amplitude affects how "tall" or "short" the wave appears. A smaller amplitude results in a flatter wave, while a larger amplitude would stretch the wave vertically.

The period of a trigonometric function is the distance along the x-axis before the wave pattern repeats itself.

In the general form of a cosine function, \( y = a \cos(bx + c) \), the period is determined using the coefficient \( b \) of the \( x \) term.

The period is calculated as:

• \( \frac{2\pi}{|b|} \)

Applying this to \( y = \frac{1}{2} \cos \frac{\pi}{3} x \):

• Here, \( b = \frac{\pi}{3} \).
• The period then becomes \( \frac{2\pi}{\left| \frac{\pi}{3} \right|} = 6 \).

This means that one complete cycle of the cosine wave will occur every 6 units on the x-axis. When graphing the function, you'll notice that every 6 units, the wave will start to repeat its pattern.

Understanding the period helps in sketching the graph and recognizing how compressed or stretched the wave will be compared to a standard cosine function, which has a default period of \( 2\pi \).

Phase shift refers to the horizontal movement or shift of the graph of a trigonometric function along the x-axis.

In our general cosine equation \( y = a \cos(bx + c) \), the phase shift is determined by the constant \( c \). The formula used is:

• \( -\frac{c}{b} \)

For the equation \( y = \frac{1}{2} \cos \frac{\pi}{3}x \), there is no \( c \) term present, meaning \( c = 0 \).

• Thus, the phase shift calculation is \( -\frac{0}{\frac{\pi}{3}} = 0 \).

This signifies that there is no horizontal shift. The graph of the function starts its cycle from its usual starting point, without shifting left or right.

Understanding phase shifts is important when comparing two trigonometric functions, as even identical functions can appear quite different if one starts further along the x-axis than the other. However, in this case, there's no shift to consider.