In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = 4 cos 2πx

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Identify the general form of the cosine function, which is \(y = A \cos(Bx)\), where \(A\) represents the amplitude and \(B\) affects the period of the function.

Determine the amplitude \(A\) by taking the absolute value of the coefficient in front of the cosine function. In this case, \(A = |4|\).

Find the period of the function using the formula \(\text{Period} = \frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the cosine function. Here, \(B = 2\pi\).

Calculate the period by substituting \(B = 2\pi\) into the formula: \(\text{Period} = \frac{2\pi}{2\pi}\).

To graph one period of the function, plot the cosine curve starting at \(x = 0\) and ending at \(x\) equal to the period found in the previous step, using the amplitude to mark the maximum and minimum values on the \(y\)-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 a trigonometric function from its midline. For functions like y = a cos bx, the amplitude is |a|, representing the peak height of the wave. In this question, the amplitude determines how far the graph stretches vertically from the center line.

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), the period is calculated as (2π) / |b|. It tells us how frequently the function repeats its pattern, which is essential for graphing one full wave.

Graphing One Period of a Cosine Function

Graphing one period involves plotting the function from x = 0 to x = period, showing key points like maxima, minima, and zeros. Understanding amplitude and period helps accurately sketch the wave's shape and length, providing a visual representation of the function's behavior.