Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = sin (x − π/2)

Verified step by step guidance

1. Identify the general form of the sine function: \(y = A \sin(B(x - C))\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, and \(C\) is the phase shift.
2. Compare the given function \(y = \sin(x - \frac{\pi}{2})\) to the general form. Here, \(A = 1\) (coefficient of sine), \(B = 1\) (coefficient of \(x\) inside the sine), and \(C = \frac{\pi}{2}\).
3. Calculate the amplitude as the absolute value of \(A\): \(\text{Amplitude} = |1| = 1\).
4. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B} = \frac{2\pi}{1} = 2\pi\).
5. Determine the phase shift, which is \(C = \frac{\pi}{2}\), meaning the graph shifts to the right by \(\frac{\pi}{2}\). Then, to graph one period, plot the sine curve starting at \(x = \frac{\pi}{2}\) and ending at \(x = \frac{\pi}{2} + 2\pi\).

Key Concepts

Amplitude of a Sine Function

Amplitude is the maximum value or height of the sine wave from its midline. For y = sin(x − π/2), the amplitude is 1, as the coefficient of the sine function is 1. It determines how far the graph stretches vertically.

Period of a Sine Function

The period is the length of one complete cycle of the sine wave. For y = sin(bx), the period is calculated as 2π divided by |b|. Since b = 1 here, the period is 2π, meaning the function repeats every 2π units along the x-axis.

Phase Shift of a Sine Function

Phase shift refers to the horizontal translation of the sine graph. It is found by solving (x − c) inside the function, where c is the phase shift. For y = sin(x − π/2), the graph shifts π/2 units to the right.