In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −2 sin(2πx + 4π)

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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 \(C\) affects the phase shift.

Find the amplitude by taking the absolute value of the coefficient in front of the sine function: \(\text{Amplitude} = |A| = |-2|\).

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 \(2\pi\).

Determine the phase shift using the formula \(\text{Phase shift} = -\frac{C}{B}\), where \(C\) is the constant added inside the sine function. In this case, \(C = 4\pi\) and \(B = 2\pi\).

Use the amplitude, period, and phase shift to sketch one full cycle of the sine function, starting from the phase shift and covering one period along the x-axis.

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주요 개념

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

Amplitude of a Trigonometric Function

Amplitude is the maximum absolute value of the 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 2, indicating the wave oscillates 2 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) / |b| for y = sin(bx + c). Here, b = 2π, so the period is (2π) / (2π) = 1, meaning the function repeats every 1 unit along the x-axis.

Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, found by solving bx + c = 0 for x, giving x = -c/b. For y = −2 sin(2πx + 4π), the phase shift is -4π / (2π) = -2, indicating the graph shifts 2 units to the left.