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 general form of the sine function: \(y = A \sin(B(x - C))\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, and \(C\) is the phase shift.
2. Rewrite the given function \(y = \frac{1}{2} \sin(x + \pi)\) in the form \(y = A \sin(B(x - C))\). Notice that \(x + \pi\) can be written as \(x - (-\pi)\), so here \(A = \frac{1}{2}\), \(B = 1\), and \(C = -\pi\).
3. Determine the amplitude \(A\), which is the absolute value of the coefficient in front of the sine function: \(A = \left| \frac{1}{2} \right|\).
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 \(C\), which is the horizontal shift of the graph. Since \(C = -\pi\), the graph shifts \(\pi\) units to the left. Use this information to sketch one full period of the sine wave starting from \(x = -\pi\) to \(x = -\pi + 2\pi = \pi\).

Key Concepts

Amplitude of a Sine Function

Amplitude is the maximum absolute value of the sine function's output, representing the height from the midline to the peak. For y = (1/2) sin(x + π), the amplitude is 1/2, indicating the wave oscillates between -1/2 and 1/2.

Period of a Sine Function

The period is the length of one complete cycle of the sine wave, calculated as 2π divided by the coefficient of x inside the function. Since the coefficient of x is 1 in y = (1/2) sin(x + π), the period remains 2π.

Phase Shift of a Sine Function

Phase shift is the horizontal translation of the sine curve, determined by solving (x + π) = 0, giving a shift of -π. This means the graph is shifted π units to the left compared to the standard sine function.