In Exercises 1–6, determine the amplitude and period of each function. Then graph one period of the function.y = 3 sin 4x

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1. Identify the general form of the sine function: \( y = a \sin(bx) \), where \( a \) is the amplitude and \( \frac{2\pi}{b} \) is the period.
2. For the given function \( y = 3 \sin 4x \), compare it with the general form to find \( a = 3 \) and \( b = 4 \).
3. Determine the amplitude: The amplitude is the absolute value of \( a \), so the amplitude is \( |3| = 3 \).
4. Calculate the period: Use the formula \( \frac{2\pi}{b} \) to find the period. Substitute \( b = 4 \) to get \( \frac{2\pi}{4} = \frac{\pi}{2} \).
5. Graph one period of the function: Start at \( x = 0 \), and plot the sine wave over the interval \( [0, \frac{\pi}{2}] \), marking key points at \( x = 0, \frac{\pi}{8}, \frac{\pi}{4}, \frac{3\pi}{8}, \frac{\pi}{2} \) with corresponding \( y \)-values based on the amplitude.

Key Concepts

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position. In the context of sine functions, it is determined by the coefficient in front of the sine term. For the function y = 3 sin 4x, the amplitude is 3, indicating that the graph will oscillate between 3 and -3.

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 the function y = 3 sin 4x, the period is P = 2π / 4 = π, meaning the function completes one full cycle over the interval from 0 to π.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the function's values over a specified interval. For y = 3 sin 4x, one period can be graphed from 0 to π, showing the wave's oscillation between its amplitude limits. Understanding the amplitude and period is crucial for accurately representing the function's behavior on a graph.