In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = sin(x − π)

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, \( c = \pi \) and \( b = 1 \), so the phase shift is \( \pi \) to the right.
5. Graph one period of the function by starting at the phase shift \( \pi \) and plotting the sine wave over the interval \( [\pi, 3\pi] \).

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its midline. In the context of the sine function, it indicates how far the graph reaches above and below the horizontal axis. For the function y = sin(x - π), the amplitude is 1, as the coefficient of the sine function is 1, meaning the graph 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 given function y = sin(x - π), there is no coefficient affecting the x variable, so the period remains 2π, indicating that the function will repeat its values every 2π units along the x-axis.

Phase Shift

Phase shift refers to the horizontal displacement of a periodic function. It is determined by the value subtracted from the variable inside the function. In y = sin(x - π), the phase shift is π units to the right, as the function is shifted from the standard position of sin(x) to the right by π, affecting where the wave starts on the x-axis.