Determine the amplitude and period of each function. Then graph one period of the function. y = 3 sin 4x

Verified step by step guidance

1. Identify the general form of the sine function, which is \(y = A \sin(Bx)\), where \(A\) represents the amplitude and \(B\) affects the period of the function.
2. Determine the amplitude by taking the absolute value of the coefficient in front of the sine function. In this case, the amplitude is \(|3|\).
3. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the sine function. Here, \(B = 4\).
4. Substitute \(B = 4\) into the period formula to express the period as \(\frac{2\pi}{4}\), which can be simplified but do not calculate the final value yet.
5. To graph one period of the function, plot the sine curve starting from \(x = 0\) to \(x = \frac{2\pi}{4}\), marking key points such as the maximum, minimum, and zeros based on the amplitude and period.

Key Concepts

Amplitude of a Sine Function

The amplitude of a sine function is the absolute value of the coefficient in front of the sine term. It represents the maximum vertical distance from the midline (usually the x-axis) to the peak of the wave. For y = 3 sin 4x, the amplitude is 3, indicating the wave oscillates between -3 and 3.

Period of a Sine Function

The period of a sine function is the length of one complete cycle along the x-axis. It is calculated by dividing 2π by the coefficient of x inside the sine function. For y = 3 sin 4x, the period is 2π/4 = π/2, meaning the function repeats every π/2 units.

Graphing One Period of a Sine Function

Graphing one period involves plotting the sine curve from 0 to the period length on the x-axis, showing key points such as the start, maximum, zero crossing, minimum, and end. For y = 3 sin 4x, graphing from 0 to π/2 captures one full wave oscillation between -3 and 3.