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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Identify the general form of the sine function: \(y = A \sin(Bx)\), where \(A\) is the amplitude and \(B\) affects the period.

Determine the amplitude \(A\) by taking the absolute value of the coefficient in front of the sine function. Here, \(A = \left| \frac{1}{2} \right|\).

Find the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the sine function. In this case, \(B = \frac{\pi}{3}\).

Calculate the period by substituting \(B\) into the formula: \(\text{Period} = \frac{2\pi}{\frac{\pi}{3}}\) (do not simplify the fraction, just set up the expression).

To graph one period of the function, plot the sine curve starting at \(x=0\) and ending at \(x\) equal to the period found in the previous step, using the amplitude to mark the maximum and minimum values on the \(y\)-axis.

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주요 개념

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

Amplitude of a Sine Function

Amplitude is the maximum absolute value of the sine function's output, representing the height from the midline to the peak. For y = (1/2) sin(π/3 x), the amplitude is the coefficient 1/2, indicating the wave oscillates between -1/2 and 1/2.

Period of a Sine Function

The period is the length of one complete cycle of the sine wave. It is calculated as 2π divided by the coefficient of x inside the sine function. Here, the period is 2π ÷ (π/3) = 6, meaning the function repeats every 6 units along the x-axis.

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, minimum, and end. For y = (1/2) sin(π/3 x), plot from x = 0 to x = 6, using the amplitude and period to shape the wave accurately.