For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin (x + π)

Verified step by step guidance

1. Identify the general form of the sine function: \(y = A \sin(B(x - C)) + D\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, \(C\) is the phase shift, and \(D\) is the vertical translation.
2. Compare the given function \(y = 2 \sin(x + \pi)\) to the general form. Notice that \(A = 2\), \(B = 1\), and the inside of the sine function is \((x + \pi)\), which can be rewritten as \(x - (-\pi)\).
3. Determine the amplitude: it is the absolute value of \(A\), so amplitude = \(|2|\).
4. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B}\), where \(B = 1\) in this case.
5. Find the phase shift by identifying \(C\) in the expression \(x - C\). Since the function is \(x + \pi\), the phase shift is \(-\pi\). The vertical translation \(D\) is \(0\) because there is no constant added outside the sine function.

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 = a sin(x), the amplitude is |a|. In this case, the amplitude is 2, indicating the wave oscillates between -2 and 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. For y = sin(bx), the period is 2π/b. Here, since the coefficient of x is 1, the period remains 2π.

Phase Shift and Vertical Translation

Phase shift is the horizontal shift of the sine curve, found by solving (x + c) = 0, giving a shift of -c. Vertical translation moves the graph up or down by a constant d in y = sin(x) + d. In y = 2 sin(x + π), the phase shift is -π, and there is no vertical translation.