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 standard form of the cosine function: \( y = a \cos(bx + c) + d \).
2. Determine the amplitude by identifying the coefficient \( a \). In this case, \( a = 3 \), so the amplitude is \( |3| = 3 \).
3. Find the period of the function using the formula \( \frac{2\pi}{b} \). Here, \( b = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \).
4. Identify the phase shift by solving \( bx + c = 0 \) for \( x \). Here, \( c = \frac{\pi}{2} \), so the phase shift is \( -\frac{\pi}{2} \) (shift to the left).
5. Determine the vertical translation by identifying the value of \( d \). In this function, \( d = 0 \), so there is no vertical translation.

Key Concepts

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position. In the context of trigonometric functions like cosine, it is determined by the coefficient in front of the cosine function. For the function y = 3 cos(x + π/2), the amplitude is 3, indicating that the wave oscillates between 3 and -3.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For the cosine function, the standard period is 2π. However, if the function includes a coefficient affecting the x variable, the period is adjusted accordingly. In this case, since there is no coefficient affecting x in y = 3 cos(x + π/2), the period remains 2π.

Phase Shift

Phase shift refers to the horizontal displacement of a trigonometric function along the x-axis. It is determined by the value added or subtracted from the x variable inside the function. In the function y = 3 cos(x + π/2), the phase shift is -π/2, indicating that the graph of the cosine function is shifted to the left by π/2 units.