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

Verified step by step guidance

1. Identify the general form of the sine function: \(y = A \sin(Bx - C) + D\), where \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical translation.
2. Compare the given function \(y = 2 \sin 2x\) to the general form. Here, \(A = 2\), \(B = 2\), \(C = 0\), and \(D = 0\).
3. Calculate the amplitude, which is the absolute value of \(A\): \(\text{Amplitude} = |2|\).
4. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), so substitute \(B = 2\) to find the period.
5. Determine the phase shift using \(\text{Phase shift} = \frac{C}{B}\). Since \(C = 0\), the phase shift is zero. The vertical translation \(D\) is also zero, meaning no vertical shift.

Key Concepts

Amplitude of a Sine Function

Amplitude is the maximum value the sine function reaches from its midline. It is the absolute value of the coefficient before the sine term. For y = 2 sin 2x, the amplitude is 2, indicating the wave oscillates 2 units above and below the midline.

Period of a Sine Function

The period is the length of one complete cycle of the sine wave. It is calculated as 2π divided by the coefficient of x inside the sine function. For y = 2 sin 2x, the period is 2π/2 = π, meaning the wave repeats every π units.

Vertical Translation and Phase Shift

Vertical translation shifts the graph up or down and is determined by any constant added outside the sine function; here, it is zero. Phase shift moves the graph left or right and depends on horizontal shifts inside the function's argument; since there is no added or subtracted value inside sin(2x), the phase shift is zero.