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

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, \(\frac{2\pi}{B}\) is the period, \(C\) is the phase shift, and \(D\) is the vertical translation.
2. Compare the given function \(y = 3 \cos(x + \frac{\pi}{2})\) to the general form. Notice that \(A = 3\), \(B = 1\), and the inside of the cosine is \(x + \frac{\pi}{2}\), which can be rewritten as \(x - (-\frac{\pi}{2})\).
3. Determine the amplitude, which is the absolute value of \(A\). So, amplitude = \(|3|\).
4. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B}\). Since \(B = 1\), the period is \(2\pi\).
5. Find the phase shift by identifying \(C\) in the expression \(x - C\). Here, \(C = -\frac{\pi}{2}\), so the phase shift is \(-\frac{\pi}{2}\) (which means a shift to the left by \(\frac{\pi}{2}\)). The vertical translation \(D\) is \(0\) since there is no added constant outside the cosine.

Key Concepts

Amplitude of a Trigonometric Function

Amplitude is the maximum absolute value of a trigonometric function, representing the height from the midline to the peak. For functions like y = a cos(x), the amplitude is |a|. In the given function y = 3 cos(x + π/2), the amplitude is 3, indicating the wave oscillates 3 units above and below its midline.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function along the x-axis. For y = cos(bx), the period is calculated as 2π/|b|. Since the given function is y = 3 cos(x + π/2), where b = 1, the period remains 2π, meaning the function repeats every 2π units.

Phase Shift and Vertical Translation

Phase shift refers to the horizontal shift of the graph, determined by the inside addition or subtraction in the function's argument. For y = cos(x + π/2), the phase shift is -π/2 (shift left). Vertical translation moves the graph up or down, indicated by added constants outside the function; here, there is none, so vertical translation is zero.