For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -¼ cos (½ x + π/2)

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1. Identify the general form of the cosine function: \(y = A \cos(Bx + C) + D\), where \(A\) is the amplitude, \(B\) affects the period, \(C\) is related to the phase shift, and \(D\) is the vertical translation.
2. Find the amplitude by taking the absolute value of the coefficient in front of the cosine function: \(|A| = |-\frac{1}{4}|\).
3. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the cosine function. Here, \(B = \frac{1}{2}\).
4. Determine the phase shift using the formula \(\text{Phase shift} = -\frac{C}{B}\), where \(C\) is the constant added inside the cosine argument. Here, \(C = \frac{\pi}{2}\).
5. Identify the vertical translation \(D\), which is the constant added outside the cosine function. In this case, there is no constant added, so \(D = 0\).

Key Concepts

Amplitude of a Trigonometric Function

Amplitude measures the maximum distance of the function's values from its midline. For functions like y = a cos(bx + c), the amplitude is the absolute value of 'a'. It determines the height of the peaks and depths of the troughs in the graph.

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 divided by the absolute value of 'b'. It indicates how frequently the function repeats its pattern along the x-axis.

Phase Shift and Vertical Translation

Phase shift refers to the horizontal shift of the graph, found by solving (bx + c) = 0 for x, resulting in -c/b. Vertical translation is the upward or downward shift of the graph, represented by a constant added or subtracted outside the function. In this question, vertical translation is zero since no constant is added.