In Exercises 1–6, determine the amplitude and period of each function. Then graph one period of the function.y = 1/2 sin π/3 x

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1. Identify the standard form of the sine function, which is \( y = a \sin(bx + c) + d \). In this case, \( a = \frac{1}{2} \) and \( b = \frac{\pi}{3} \).
2. Determine the amplitude of the function. The amplitude is the absolute value of \( a \), so \( \text{Amplitude} = |\frac{1}{2}| = \frac{1}{2} \).
3. Calculate the period of the function. The period of a sine function is given by \( \frac{2\pi}{b} \). Substitute \( b = \frac{\pi}{3} \) to find the period: \( \text{Period} = \frac{2\pi}{\frac{\pi}{3}} \).
4. Simplify the expression for the period: \( \text{Period} = \frac{2\pi \times 3}{\pi} = 6 \).
5. Graph one period of the function. Start at \( x = 0 \), plot key points such as the maximum, minimum, and intercepts within one period, and complete the graph at \( x = 6 \).

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its central axis. In the context of sine functions, it is determined by the coefficient in front of the sine term. For the function y = 1/2 sin(π/3 x), the amplitude is 1/2, indicating that the wave oscillates between 1/2 and -1/2.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For sine functions, the period can be calculated using the formula P = 2π / |b|, where b is the coefficient of x. In this case, with b = π/3, the period is 2π / (π/3) = 6, meaning the function completes one full cycle every 6 units along the x-axis.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the function's values over a specified interval. For y = 1/2 sin(π/3 x), one period can be graphed from x = 0 to x = 6. The graph will show a smooth wave oscillating between 1/2 and -1/2, starting at the origin, reaching its maximum at x = 3, and returning to the axis at x = 6.