In Exercises 7–11, determine the amplitude, period, and phase shift of each function. Then graph one period of the function.y = −3 sin(π/3 x − 3π)

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1. Identify the general form of the sine function: \( y = a \sin(bx - c) + d \).
2. Determine the amplitude by taking the absolute value of \( a \). Here, \( a = -3 \), so the amplitude is \( |a| = 3 \).
3. Calculate the period using the formula \( \frac{2\pi}{b} \). In this case, \( b = \frac{\pi}{3} \), so the period is \( \frac{2\pi}{\frac{\pi}{3}} = 6 \).
4. Find the phase shift by solving \( bx - c = 0 \). Here, \( \frac{\pi}{3}x - 3\pi = 0 \), which gives \( x = 9 \). The phase shift is \( 9 \) units to the right.
5. Graph one period of the function by starting at the phase shift \( x = 9 \) and using the amplitude and period to plot key points, such as the maximum, minimum, and intercepts.

Key Concepts

Amplitude

Amplitude refers to the maximum distance from the midline of a periodic function to its peak or trough. In the context of sine and cosine functions, it is determined by the coefficient in front of the sine or cosine term. For the function y = -3 sin(π/3 x - 3π), the amplitude is 3, indicating that the graph 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. It can be calculated using the formula 2π divided by the coefficient of x inside the sine or cosine function. For the function y = -3 sin(π/3 x - 3π), the period is 2π / (π/3) = 6, meaning the function repeats every 6 units along the x-axis.

Phase Shift

Phase shift refers to the horizontal shift of a periodic function along the x-axis. It is determined by the constant added or subtracted inside the function's argument. In the function y = -3 sin(π/3 x - 3π), the phase shift can be calculated by setting the inside of the sine function to zero, resulting in a shift of 9 units to the right, as derived from the equation π/3 x - 3π = 0.