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In a trigonometric context, the amplitude is a vital parameter of a cosine function. It essentially describes how "tall" or "short" the wave is. For the cosine function given by the equation \( y = 5 \cos(2x - \frac{\pi}{2}) \), the amplitude is determined by the absolute value of the coefficient \( a \), which is in front of the cosine. This coefficient, \( a \), is equal to 5 in this case.

The amplitude is therefore \(|a| = 5\). This tells us that the maximum value the function can reach vertically from the middle or equilibrium line (which is 0 here, because there is no vertical shift) is 5 units upwards. Conversely, it can reach 5 units downwards from the equilibrium line, resulting in the minimum value of the function being -5. This amplitude gives the cosine wave its shape in terms of height.

The period of a cosine function is the length of one complete wave cycle. It indicates how often the wave pattern repeats over the x-axis. In mathematical terms for \( y = a \cos(bx + c) + d \), the period is calculated as \( \frac{2\pi}{|b|} \).

For the function \( y = 5 \cos(2x - \frac{\pi}{2}) \), we have \( b = 2 \). Thus, the period is \( \frac{2\pi}{2} = \pi \). This means that every \( \pi \) units along the x-axis, the wave pattern will start anew from a peak position, spanning through a complete maximum, equilibrium, and minimum before returning to the peak. Understanding the period helps to visualize how compact or spread out the waves on a graph will be.

Phase shift is the horizontal displacement of the cosine wave along the x-axis. It shifts the entire wave left or right in comparison to a regular cosine function, \( y = \cos(x) \), which starts at its maximum at \( x = 0 \). The phase shift is computed by \( -\frac{c}{b} \) from the function form \( y = a \cos(bx + c) + d \).

In this instance, \( c = -\frac{\pi}{2} \) and \( b = 2 \). Substituting these into the formula, the phase shift amounts to \( \frac{\pi}{4} \). This implies the entire wave is shifted \( \frac{\pi}{4} \) units to the right. Consequently, instead of beginning at a maximum at \( x = 0 \), our specific cosine function begins at a maximum peak when \( x = \frac{\pi}{4} \). Recognizing this shift is crucial for plotting or interpreting the graph correctly.

The maximum and minimum values of a cosine function describe the vertical extremes the wave will touch. These values can be easily identified by utilizing the amplitude and vertical shift © from the equation in the form \( y = a \cos(bx + c) + d \).

For our function \( y = 5 \cos(2x - \frac{\pi}{2}) \), the amplitude is 5 and there is no vertical shift (\( d = 0 \)). Therefore, the maximum value this function can reach is the amplitude itself, which is 5, while the minimum value is \(-5\).

These values occur alternately at specific x-values due to phase shifts and periodic repeats. The maximum value of 5 occurs when the cosine function is at its peak, here calculated at \( x = \frac{\pi}{4} \). Halfway through the period, at \( x = \frac{5\pi}{4} \), the function attains its minimum value \(-5\). These clear-cut values make it easy to anticipate the behavior of the cosine function across its cycle.