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The amplitude of a function refers to the maximum vertical distance from the midpoint of the wave to its peak or trough. It is a crucial aspect of trigonometric functions like the cosine and sine functions. For any cosine function of the form:

• \( y = a \cdot \cos(b(x - c)) + d \)

the amplitude is determined by the absolute value of the coefficient \( a \). This value represents how "tall" or "short" the wave appears. If \( |a| \) is greater than 1, the wave becomes taller; if less than 1, it is compressed.

In the given function \( y = \cos(x - \frac{\pi}{2}) \), \( a = 1 \). Thus, the amplitude is \( |1| = 1 \). This means the wave reaches a maximum height of 1 unit above, and a minimum of 1 unit below, the horizontal axis. Understanding this concept helps in visually interpreting the graph and predicting how it will look.

In trigonometry, the period of a function is the length, horizontally, over which the wave pattern repeats itself. It's like knowing how long a loop of the wave will take before it starts again. For the standard cosine function:

• \( y = \cos(b(x - c)) \)

the period is calculated using the formula:\[ \frac{2\pi}{|b|} \] Here, \( b \) determines how many cycles occur in a typical duration. If \( |b| \) is large, the function oscillates more rapidly, and if small, it stretches over a longer span.

For our specific problem with \( y = \cos(x - \frac{\pi}{2}) \), the \( b \) value is 1, which simplifies the period to:\[ \frac{2\pi}{1} = 2\pi \]This means that every \( 2\pi \) units, or one complete cycle on the x-axis, the wave pattern repeats itself. Knowing the period helps in accurately sketching one complete cycle of the wave on a graph and understanding how the function behaves over different intervals.

Phase shift refers to the horizontal displacement of a trigonometric function. It shows how much the basic wave has been shifted along the x-axis from its usual starting position. In a cosine function of the form:

• \( y = a \cdot \cos(b(x - c)) + d \)

the phase shift is determined by the value of \( c \), and the function is shifted to the right if \( c \) is positive and to the left if it is negative.

In our scenario with the function \( y = \cos(x - \frac{\pi}{2}) \), \( c \) is \( \frac{\pi}{2} \). This indicates a shift of:

• \( \frac{\pi}{2} \) units to the right

Phase shifts are important when graphing because they show how the function aligns with other shifts in a wave pattern, aiding in understanding intersections or overlaps with other mathematical functions or data plotted on the same graph.