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In trigonometry, the amplitude of a function refers to the height of its peaks from the centerline. For a cosine function, the amplitude is half the distance between the maximum and minimum values it can take. The standard form of a cosine function is given by:
\[ y = A \cos(Bx) \] Here, A represents the amplitude. In the exercise problem: \[ y = \frac{1}{2} \cos(\frac{\pi}{2} x) \] The amplitude (A) is \frac{1}{2}\. This means that the maximum value of the function is 0.5 and the minimum value is -0.5. The function oscillates between these two values. The amplitude is essential for understanding the range of the function and ensuring the accuracy of its graph.

The period of a trigonometric function is the distance it takes for the function to complete one full cycle before repeating. For the cosine function, the standard form is: \[ y = A \cos(Bx) \] The period (T) is calculated using the formula:
\[ T = \frac{2\pi}{B} \] For the given function:
\[ y = \frac{1}{2} \cos(\frac{\pi}{2} x) \] The coefficient of x (B) is \frac{\pi}{2}\. Substituting this into the formula gives: \[ T = \frac{2\pi}{\frac{\pi}{2}} = 4 \] This means that one full cycle of the cosine wave completes every 4 units. Over a two-period interval, the graph should be plotted from x=0 to x=8.

The cosine function is one of the fundamental trigonometric functions used to describe oscillatory behaviors, such as waves. It is generally expressed in its standard form as:
\[ y = A \cos(Bx) \] Key things to remember about cosine functions:

• They start at their maximum value when x=0.
• They are symmetric about the y-axis.
• They oscillate between positive and negative amplitudes.

In the exercise, the function is:
\[ y = \frac{1}{2} \cos(\frac{\pi}{2} x) \] To plot this function over a two-period interval, recognize its amplitude (\frac{1}{2}\right). By plotting key points at intervals of one-fourth the period (every 1 unit in this case), the function starts at its maximum value of \frac{1}{2}\right) at x=0. It then reaches zero at x=2, its minimum of -\frac{1}{2}\left) at x=4, and zero again at x=6. This pattern repeats in the second period from x=4 to x=8.