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The amplitude of a trigonometric function like the cosine function is a measure of its maximum height from the middle value, also known as the equilibrium position.

In the function provided: \( y = \frac{1}{2} \, \cos(x) \), the amplitude is the absolute value of the coefficient in front of the cosine term. Here, the coefficient is \( \frac{1}{2} \).

Therefore, the amplitude \( A \) is:
\( \begin{align*} A & = \left| \frac{1}{2} \right| \ & = \frac{1}{2} \end{align*} \).
This means the graph of \( y = \frac{1}{2} \, \cos(x) \) will peak at \( \frac{1}{2} \) and bottom out at \( -\frac{1}{2} \). Understanding the amplitude helps you visualize how far the graph deviates from its centerline.

The period of a trigonometric function like the cosine function is the length of one complete cycle of the wave. It tells you how frequently the pattern repeats along the x-axis.

For the general cosine function, \( y = A \, \cos(Bx - C) + D \), the period \( T \) can be found using the formula:
\( T = \frac{2\pi}{|B|} \).
In the function provided: \( y = \frac{1}{2} \, \cos(x) \), \( B = 1 \). Therefore, the period is:
\( T = \frac{2\pi}{1} = 2\pi \).
This means the graph completes one full cycle every \( 2\pi \) units along the x-axis. By knowing the period, you can accurately sketch and predict the behavior of the function over any interval.

The phase shift of a trigonometric function like the cosine function is a horizontal shift along the x-axis. It tells us how the graph is translated left or right from its usual position.

For the general cosine function, \( y = A \, \cos(Bx - C) + D \), the phase shift is calculated as \( \frac{C}{B} \).
In the function provided: \( y = \frac{1}{2} \, \cos(x) \), there is no \( C \) term, which means \( C = 0 \). So the phase shift is:
\( \frac{C}{B} = \frac{0}{1} = 0 \).
This indicates that there is no horizontal shift, and the graph of \( y = \frac{1}{2} \, \cos(x) \) will start at the usual position of the cosine function. Understanding the phase shift helps correctly position the graph on the x-axis.