In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 2 cos (2πx + 8π)

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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, \(A = 2\), so the amplitude is \(|2| = 2\).

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

Determine the phase shift using the formula \(\text{Phase shift} = -\frac{C}{B}\). Given \(C = 8\pi\) and \(B = 2\pi\), substitute to find the phase shift.

To graph one period of the function, start at the phase shift on the x-axis, then mark points at intervals of \(\frac{\text{Period}}{4}\) to capture key points of the cosine wave (maximum, zero crossing, minimum, zero crossing, maximum). Use the amplitude to determine the y-values at these points.

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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 functions like y = a cos(bx + c), the amplitude is the absolute value of 'a'. In this example, the amplitude is |2| = 2, indicating the wave oscillates 2 units above and below the midline.

Period of a Trigonometric Function

The period is the length of one complete cycle of the wave, calculated as (2π) divided by the absolute value of the coefficient 'b' in y = a cos(bx + c). Here, with b = 2π, the period is 2π / 2π = 1. This means 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. It indicates how far the graph shifts left or right from the origin. For y = 2 cos(2πx + 8π), the phase shift is -8π / 2π = -4, meaning the graph shifts 4 units to the left.