In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = cos 2x

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Identify the general form of the cosine function: \(y = \cos(Bx)\), where \(B\) affects the period of the function.

Recall that the amplitude of a cosine function \(y = A \cos(Bx)\) is the absolute value of \(A\). In this problem, since the function is \(y = \cos 2x\), the amplitude \(A\) is 1.

Calculate the period of the function using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B = 2\), so the period is \(\frac{2\pi}{2} = \pi\).

To graph one period of the function, start by plotting key points over the interval from \(x = 0\) to \(x = \pi\), since this represents one full cycle of the function.

Mark the amplitude on the graph by noting the maximum value at 1 and the minimum value at -1, then sketch the cosine curve starting at \(y=1\) when \(x=0\), going down to \(y=-1\) at \(x=\frac{\pi}{2}\), and returning to \(y=1\) at \(x=\pi\).

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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 from its midline. For functions like y = cos(kx), the amplitude is the coefficient in front of the cosine, indicating the height of peaks and depth of troughs. In y = cos 2x, the amplitude is 1, meaning the graph oscillates between 1 and -1.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function along the x-axis. For y = cos(kx), the period is calculated as (2π) / |k|. In y = cos 2x, the period is π, meaning the cosine wave repeats every π units.

Graphing One Period of a Cosine Function

Graphing one period involves plotting the function from 0 to its period, marking key points such as maxima, minima, and zeros. For y = cos 2x, plot from 0 to π, noting that the function starts at 1, crosses zero at π/4 and 3π/4, and reaches -1 at π/2, then returns to 1 at π.