Determine the amplitude and period of each function. Then graph one period of the function. y = -3 sin 2πx

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. For \(y = -3 \sin 2\pi x\), the amplitude is \(| -3 | = 3\).
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, \(B = 2\pi\), so the period is \(\frac{2\pi}{2\pi}\).
4. Simplify the period expression to find the length of one complete cycle of the sine wave.
5. To graph one period of the function, plot points starting from \(x = 0\) to \(x = \) period, using the sine values scaled by the amplitude and reflected if the amplitude is negative, then connect these points smoothly to form the wave.

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 or trough of the wave. For y = -3 sin 2πx, the amplitude is |−3| = 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 as 2π divided by the coefficient of x inside the sine function. For y = -3 sin 2πx, the period is 2π / (2π) = 1.

Graphing One Period of a Sine Function

Graphing one period involves plotting the sine wave from 0 to the period length on the x-axis, showing key points such as the start, maximum, zero crossing, minimum, and end. The amplitude determines the height, and the period determines the width of the wave.