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 standard form of the sine function: \( y = a \sin(bx - c) + d \).
2. Determine the amplitude by identifying the coefficient \( a \). In this case, \( a = 1 \), so the amplitude is 1.
3. Find the period of the function using the formula \( \frac{2\pi}{b} \). Here, \( b = 1 \), so the period is \( 2\pi \).
4. Calculate the phase shift using \( \frac{c}{b} \). Here, \( c = \pi/2 \) and \( b = 1 \), so the phase shift is \( \frac{\pi}{2} \) to the right.
5. Graph one period of the function by starting at the phase shift \( \frac{\pi}{2} \) and plotting points over the interval \( \left[ \frac{\pi}{2}, \frac{\pi}{2} + 2\pi \right] \).

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its midline. For the sine function, it is determined by the coefficient in front of the sine term. In the function y = sin(x - π/2), the amplitude is 1, indicating that the wave oscillates between 1 and -1.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For the sine function, the standard period is 2π. In the function y = sin(x - π/2), there is no coefficient affecting the x term, so the period remains 2π.

Phase Shift

Phase shift indicates how much the graph of a function is horizontally shifted from its standard position. In the function y = sin(x - π/2), the phase shift is π/2 units to the right, as indicated by the subtraction of π/2 from x. This shift affects the starting point of the sine wave.