For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -sin (x - 3π/4)

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 \(y = -\sin(x - \frac{3\pi}{4})\) to the general form. Here, \(A = -1\), \(B = 1\), \(C = \frac{3\pi}{4}\), and \(D = 0\).
3. Determine the amplitude by taking the absolute value of \(A\): \(\text{Amplitude} = |A| = |-1|\).
4. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B} = \frac{2\pi}{1}\).
5. Identify the phase shift as \(C = \frac{3\pi}{4}\) (shift to the right) and note that the vertical translation \(D = 0\) means there is no vertical shift.

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 = -sin(x - 3π/4), the amplitude is the absolute value of the coefficient before sine, which is 1, indicating the wave oscillates between -1 and 1.

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 = -sin(x - 3π/4), the period remains 2π, meaning the function repeats every 2π units.

Phase Shift and Vertical Translation

Phase shift is the horizontal shift of the sine curve, found by setting the inside of the function's argument to zero and solving for x. Here, x - 3π/4 = 0 gives a phase shift of 3π/4 to the right. Vertical translation moves the graph up or down, but since there is no added constant outside the sine, vertical translation is zero.