Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = 4 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 by taking the absolute value of \(A\). For the function \(y = 4 \cos(2x - \pi)\), the amplitude is \(|4|\).

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

Determine the phase shift using the formula \(\text{Phase shift} = \frac{C}{B}\). Since the function is \(y = 4 \cos(2x - \pi)\), rewrite the inside as \(2x - \pi = 2(x - \frac{\pi}{2})\), so \(C = \pi\) and \(B = 2\).

To graph one period, start at the phase shift on the x-axis, then plot points over one full period length calculated in step 3, using the amplitude to mark 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 the given function y = 4 cos(2x − π), the amplitude is 4, indicating the graph oscillates between -4 and 4.

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|. In y = 4 cos(2x − π), b = 2, so the period is π, meaning the function 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 = 4 cos(2x − π), the phase shift is (π/2) units to the right, indicating the graph is shifted right by π/2 compared to the standard cosine function.

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.y = 4 cos(2x − π)