In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function.y = 3 sin(2x − π)

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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, \( a = 3 \), so the amplitude is \( |3| = 3 \).
3. Find the period of the function using the formula \( \frac{2\pi}{b} \). Here, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).
4. Calculate the phase shift using \( \frac{c}{b} \). Here, \( c = \pi \) and \( b = 2 \), so the phase shift is \( \frac{\pi}{2} \). Since the expression is \( bx - c \), the shift is to the right.
5. Graph one period of the function by starting at the phase shift \( \frac{\pi}{2} \) and using the period \( \pi \) to determine the end of one cycle. Plot key points such as maximum, minimum, and intercepts based on the amplitude and period.

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its midline. In the context of the sine function, it is determined by the coefficient in front of the sine term. For the function y = 3 sin(2x − π), the amplitude is 3, indicating that the wave oscillates 3 units above and below the midline.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For the sine function, the period can be calculated using the formula 2π divided by the coefficient of x inside the sine function. In this case, the period of y = 3 sin(2x − π) is π, meaning the function completes one full cycle over an interval of π units.

Phase Shift

Phase shift refers to the horizontal shift of the graph of a trigonometric function. It is determined by the constant added or subtracted from the x variable inside the function. For y = 3 sin(2x − π), the phase shift can be calculated as π/2, indicating that the graph is shifted π/2 units to the right from the origin.