In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 1/2 cos (3x + π/2)

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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 \(C\) affects the phase shift.

Find the amplitude by taking the absolute value of \(A\). In this case, \(A = \frac{1}{2}\), so the amplitude is \(|\frac{1}{2}|\).

Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B = 3\), so substitute to find the period.

Determine the phase shift using the formula \(\text{Phase shift} = -\frac{C}{B}\). Given \(C = \frac{\pi}{2}\) and \(B = 3\), substitute these values to find the phase shift.

To graph one period of the function, start at the phase shift on the x-axis, then plot points over one full period length, using the amplitude to determine the maximum and minimum values of the cosine wave.

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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 a trigonometric function, representing the height from the midline to the peak. For functions like y = A cos(Bx + C), the amplitude is |A|. In this example, the amplitude is 1/2, indicating the graph oscillates between 1/2 and -1/2.

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 divided by |B|. Here, B = 3, so the period is 2π/3, meaning the function repeats every 2π/3 units.

Phase Shift of a Trigonometric Function

Phase shift refers to the horizontal translation of the graph, determined by solving Bx + C = 0 for x. It is given by -C/B. In this function, the phase shift is -π/2 divided by 3, or -π/6, indicating the graph shifts π/6 units to the left.