Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = cos(x + π/2)

Verified step by step guidance

1. Identify the general form of the cosine function: \(y = A \cos(B(x - C))\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, and \(C\) is the phase shift.
2. Rewrite the given function \(y = \cos(x + \frac{\pi}{2})\) in the form \(y = \cos(x - (-\frac{\pi}{2}))\) to clearly identify the phase shift.
3. Determine the amplitude \(A\) by looking at the coefficient in front of the cosine function. Here, since there is no coefficient, \(A = 1\).
4. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B}\). Since the coefficient of \(x\) is 1, \(B = 1\), so the period is \(2\pi\).
5. Find the phase shift \(C\) from the expression inside the cosine. Since it is \(x - (-\frac{\pi}{2})\), the phase shift is \(-\frac{\pi}{2}\), meaning the graph shifts to the left by \(\frac{\pi}{2}\) units.

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 = cos(x), the amplitude is the coefficient before the cosine term. In y = cos(x + π/2), the amplitude is 1, since there is no coefficient other than 1.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function, calculated as 2π divided by the coefficient of x inside the function. For y = cos(bx + c), the period is 2π/|b|. In y = cos(x + π/2), since b = 1, the period remains 2π.

Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, determined by solving (bx + c) = 0 for x. It equals -c/b, indicating how far the graph shifts left or right. For y = cos(x + π/2), the phase shift is -π/2, meaning the graph shifts π/2 units to the left.