For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 - sin(3x - π/5)

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, \(\frac{2\pi}{B}\) is the period, \(D\) is the vertical translation, and the phase shift is given by \(\frac{C}{B}\).
2. Rewrite the given function \(y = 2 - \sin(3x - \frac{\pi}{5})\) to match the general form. Notice that \(2 - \sin(3x - \frac{\pi}{5})\) can be seen as \(y = -\sin(3x - \frac{\pi}{5}) + 2\).
3. Determine the amplitude \(A\) by taking the absolute value of the coefficient in front of the sine function. Here, the coefficient is \(-1\), so \(A = | -1 | = 1\).
4. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the sine function. Here, \(B = 3\), so the period is \(\frac{2\pi}{3}\).
5. Find the vertical translation \(D\), which is the constant added outside the sine function. Here, \(D = 2\). Then, find the phase shift by dividing \(C\) by \(B\): \(\text{Phase shift} = \frac{\frac{\pi}{5}}{3} = \frac{\pi}{15}\). Since the function is \(\sin(3x - \frac{\pi}{5})\), the phase shift is to the right by \(\frac{\pi}{15}\).

Key Concepts

Amplitude of a Trigonometric Function

Amplitude measures the maximum distance a sine or cosine function's graph reaches from its midline. It is the absolute value of the coefficient before the sine or cosine term. For y = 2 - sin(3x - π/5), the amplitude is | -1 | = 1, since the sine coefficient is implicitly -1.

Period of a Sine Function

The period is the length of one complete cycle of the sine function. It is calculated as 2π divided by the absolute value of the coefficient of x inside the function. For y = 2 - sin(3x - π/5), the period is 2π/3, reflecting how the function compresses horizontally.

Vertical Translation and Phase Shift

Vertical translation shifts the graph up or down and is given by the constant added outside the sine function, here +2. Phase shift moves the graph horizontally and is found by solving the inside of the sine function for zero: 3x - π/5 = 0, so the phase shift is π/15 units to the right.