Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −3 sin(π/3 x − 3π)

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

Find the amplitude by taking the absolute value of the coefficient in front of the sine function: \(\text{Amplitude} = |A| = |-3|\).

Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B\) is the coefficient of \(x\) inside the sine function, which is \(\frac{\pi}{3}\).

Determine the phase shift using the formula \(\text{Phase shift} = \frac{C}{B}\), where \(C\) is the constant subtracted inside the sine function (note the sign inside the parentheses). In this case, \(C = 3\pi\) and \(B = \frac{\pi}{3}\).

To graph one period of the function, start at the phase shift on the x-axis, then plot points at intervals of \(\frac{\text{Period}}{4}\) to capture key points of the sine wave (maximum, zero crossing, minimum, zero crossing), and use the amplitude to determine the y-values.

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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 = a sin(bx + c), the amplitude is |a|. In this case, the amplitude is 3, indicating the graph oscillates 3 units above and below the midline.

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 absolute value of the coefficient b in y = a sin(bx + c). Here, with b = π/3, the period is 2π ÷ (π/3) = 6, meaning the function repeats every 6 units along the x-axis.

Phase Shift of a Sine Function

Phase shift is the horizontal translation of the sine graph, determined by solving bx + c = 0 for x. It equals -c/b, indicating how far the graph shifts left or right. For y = −3 sin(π/3 x − 3π), the phase shift is (3π) ÷ (π/3) = 9 units to the right.