In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function.y = 1/2 cos (3x + 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, \( a = \frac{1}{2} \), so the amplitude is \( \frac{1}{2} \).
3. Calculate the period using the formula \( \frac{2\pi}{b} \). Here, \( b = 3 \), so the period is \( \frac{2\pi}{3} \).
4. Find the phase shift by solving \( bx + c = 0 \) for \( x \). Here, \( 3x + \frac{\pi}{2} = 0 \), so the phase shift is \( -\frac{\pi}{6} \).
5. Graph one period of the function by starting at the phase shift \( -\frac{\pi}{6} \) and using the period \( \frac{2\pi}{3} \) to determine the end of one cycle, plotting key points such as maximum, minimum, and intercepts based on the amplitude.

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its midline. In the context of trigonometric functions like cosine, it is determined by the coefficient in front of the cosine term. For the function y = 1/2 cos(3x + π/2), the amplitude is 1/2, indicating that the graph will oscillate between 1/2 and -1/2.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. It can be calculated using the formula Period = 2π / |B|, where B is the coefficient of x in the function. For the given function y = 1/2 cos(3x + π/2), the period is 2π / 3, meaning the function completes one full cycle over this interval.

Phase Shift

Phase shift refers to the horizontal shift of the graph of a trigonometric function. It is determined by the expression inside the cosine function. For y = 1/2 cos(3x + π/2), we can rewrite it as y = 1/2 cos(3(x + π/6)), indicating a phase shift of -π/6 to the left, which affects the starting point of the wave on the x-axis.