Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = 1/2 sin(x + π)

Verified step by step guidance

1. Identify the standard form of the sine function: \( y = a \sin(bx + c) + d \).
2. Determine the amplitude by identifying the coefficient \( a \). In this case, \( a = \frac{1}{2} \), so the amplitude is \( \frac{1}{2} \).
3. Find the period of the function using the formula \( \frac{2\pi}{b} \). Here, \( b = 1 \), so the period is \( 2\pi \).
4. Calculate the phase shift using \( -\frac{c}{b} \). With \( c = \pi \) and \( b = 1 \), the phase shift is \( -\pi \).
5. Graph one period of the function by starting at the phase shift \( -\pi \) and ending at \( -\pi + 2\pi = \pi \), marking key points at intervals of \( \frac{\pi}{2} \).

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its midline. In the context of the sine function, it is determined by the coefficient in front of the sine term. For the function y = 1/2 sin(x + π), the amplitude is 1/2, indicating that the wave oscillates between 1/2 and -1/2.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For the sine function, the standard period is 2π. Since there is no coefficient affecting the x variable in y = 1/2 sin(x + π), the period remains 2π, meaning the function will repeat every 2π units along the x-axis.

Phase Shift

Phase shift refers to the horizontal shift of the graph of a function. It is determined by the value added or subtracted from the x variable inside the function. In y = 1/2 sin(x + π), the phase shift is -π, indicating that the graph is shifted π units to the left along the x-axis.