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

Verified step by step guidance

1. Step 1: Identify the amplitude. The amplitude of a sine function y = a \sin(bx + c) + d is given by the absolute value of 'a'. In this case, a = 2, so the amplitude is |2|.
2. Step 2: Determine the period. The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \), where 'b' is the coefficient of x. Here, b = 2, so the period is \( \frac{2\pi}{2} \).
3. Step 3: Check for vertical translation. The vertical translation is given by 'd' in the function y = a \sin(bx + c) + d. In this function, d = 0, so there is no vertical translation.
4. Step 4: Determine the phase shift. The phase shift is calculated using the formula \( -\frac{c}{b} \), where 'c' is the horizontal shift. In this function, c = 0, so the phase shift is \( -\frac{0}{2} \).
5. Step 5: Summarize the characteristics. The function y = 2 \sin(2x) has an amplitude of 2, a period of \( \pi \), no vertical translation, and no phase shift.

Key Concepts

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position. In the context of the sine function, it is determined by the coefficient in front of the sine term. For the function y = 2 sin 2x, the amplitude is 2, indicating that the wave oscillates 2 units above and below the horizontal axis.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For the sine function, the period can be calculated using the formula P = 2π/B, where B is the coefficient of x. In the function y = 2 sin 2x, B is 2, resulting in a period of π, meaning the wave completes one full cycle over the interval from 0 to π.

Phase Shift

Phase shift refers to the horizontal displacement of a wave from its standard position. It is determined by any horizontal translation in the function, typically represented as y = A sin(B(x - C)), where C indicates the shift. In the function y = 2 sin 2x, there is no horizontal translation, so the phase shift is 0, meaning the wave starts at the origin.