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Understanding the amplitude of trigonometric functions is essential for analyzing their graphs. Amplitude, by definition, measures the height of the wave from the center line to its peak or trough. Specifically, for the function \(y = \frac{1}{2} \cos \frac{2x}{3}\), the coefficient in front of the cosine, which is \(\frac{1}{2}\), determines the amplitude.

This coefficient, often denoted by 'A' in the standard form \(y = A \cos (Bx + C) + D\), directly scales the output values of the cosine function. Therefore, the graph of our function will oscillate with a maximum height of 0.5 units above and below the horizontal axis, or the midline. The midline is the horizontal line situated midway between the peak and trough of the function. For a positive amplitude value like 0.5, the graph remains the same as the standard cosine function, just reduced in size, without being flipped upside down.

The period of a trigonometric function refers to the length of one full cycle of the wave—the distance on the horizontal axis after which the function starts repeating its values. The cosine function, which is periodic by nature, has a standard period of \(2\pi\) radians. However, when there's a coefficient attached to the \(x\)-variable inside the function, the period will change accordingly.

As per our function \(y = \frac{1}{2} \cos \frac{2x}{3}\), the coefficient of \(x\), denoted here as 'B' in \(y = A \cos (Bx + C) + D\), is \(\frac{2}{3}\). The period of the function can be found using the formula \(\frac{2\pi}{|B|}\). Applying this to our function yields \(\frac{2\pi}{|\frac{2}{3}|} = 3\pi\) as the period. Thus, the graph of our function will complete one full oscillation over the span of \(3\pi\) radians on the \(x\)-axis before repeating again.

Graphing trigonometric functions allows us to visually interpret their behavior and identify properties such as amplitude, period, phase shift, and vertical shift. When graphing \(y = \frac{1}{2} \cos \frac{2x}{3}\), aside from noting the amplitude and period as discussed earlier, we also observe that there is no horizontal or vertical shift since there are no additional terms inside the cosine or outside the entire function.

To graph, start by marking the amplitude on the vertical axis with points at +0.5 and -0.5. The function will oscillate between these two values. Then, plot the period along the horizontal axis, ensuring that one full wave spans \(3\pi\) radians. For cosine, the wave typically starts at the maximum amplitude when \(x = 0\). However, without phase shifts, the function would continue to repeat this pattern every \(3\pi\) units horizontally. The process of graphing illustrates the function's cyclic nature and visually communicates how the function would behave over any interval of the \(x\)-axis.