In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 3 cos(2x − π)

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

Find the amplitude \(A\) by taking the absolute value of the coefficient in front of the cosine: \(A = |3|\).

Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\). Here, \(B = 2\), so substitute to get the period.

Determine the phase shift by solving \(Bx - C = 0\) for \(x\), which gives \(x = \frac{C}{B}\). Here, \(C = \pi\), so calculate the phase shift as \(\frac{\pi}{2}\).

Interpret the phase shift direction: since the function is \(\cos(2x - \pi)\), the phase shift is to the right by \(\frac{\pi}{2}\). Then, use these values to sketch one period of the cosine function starting from the phase shift.

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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 of the wave. For y = a cos(bx − c), the amplitude is |a|. In this case, the amplitude is 3, indicating the wave oscillates 3 units above and below the midline.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function along the x-axis. For y = cos(bx − c), the period is calculated as (2π) / |b|. Here, with b = 2, the period is π, meaning the cosine wave repeats every π units.

Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, determined by solving bx − c = 0 for x. It is given by c / b. For y = 3 cos(2x − π), the phase shift is (π) / 2 units to the right, indicating the graph is shifted right by π/2 from the standard cosine curve.