Determine the amplitude and period of each function. Then graph one period of the function. y = 3 sin (1/2) x

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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.

Determine the amplitude \(A\) by taking the absolute value of the coefficient in front of the sine function. In this case, \(A = |3|\).

Find the period of the function using the formula \(\text{Period} = \frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the sine function. Here, \(B = \frac{1}{2}\).

Calculate the period by substituting \(B = \frac{1}{2}\) into the formula: \(\text{Period} = \frac{2\pi}{\frac{1}{2}}\).

To graph one period of the function, start at \(x = 0\) and plot points up to \(x = \text{Period}\). Use the amplitude to determine the maximum and minimum values of the sine wave, which will be \(3\) and \(-3\) respectively.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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(1/2 x), 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 absolute value of the coefficient of x inside the sine function. For y = 3 sin(1/2 x), the period is 2π ÷ (1/2) = 4π, meaning the function repeats every 4π units.

Graphing One Period of a Sine Function

Graphing one period involves plotting the sine curve from 0 to the period length, marking key points such as the start, maximum, midline crossing, minimum, and end. For y = 3 sin(1/2 x), graph from x = 0 to x = 4π, showing the wave rising to 3, descending to -3, and returning to zero to complete one cycle.