For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 3 cos [π/2 (x - ½)]

Verified step by step guidance

1. Identify the general form of the cosine function: \(y = A \cos(B(x - C)) + D\), where \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical translation.
2. Find the amplitude \(A\) by looking at the coefficient in front of the cosine function. In this case, \(A = 3\), so the amplitude is \(3\).
3. Determine the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B = \frac{\pi}{2}\), so substitute this value to find the period.
4. Identify the phase shift by looking at the expression inside the cosine function: \(x - \frac{1}{2}\). The phase shift is \(C = \frac{1}{2}\), which means the graph shifts to the right by \(\frac{1}{2}\) units.
5. Check for vertical translation \(D\). Since there is no constant added or subtracted outside the cosine function, the vertical translation is \(0\).

Key Concepts

Amplitude of a Trigonometric Function

Amplitude is the maximum absolute value of a trigonometric function from its midline. For functions like y = a cos(bx + c), the amplitude is |a|, representing the height of the wave peaks above or below the central axis.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function. For y = cos(bx), the period is calculated as (2π) / |b|. It determines how frequently the wave repeats along the x-axis.

Phase Shift and Vertical Translation

Phase shift is the horizontal shift of the function, found by solving (bx + c) = 0 for x, often expressed as -c/b. Vertical translation is the upward or downward shift of the midline, represented by a constant added or subtracted outside the function.