Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −3 cos (x + π)

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Identify the general form of the cosine function: \(y = A \cos(B(x - C))\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, and \(C\) is the phase shift.

Rewrite the given function \(y = -3 \cos(x + \pi)\) in the form \(y = A \cos(B(x - C))\). Notice that \(x + \pi\) can be written as \(x - (-\pi)\), so \(C = -\pi\).

Determine the amplitude \(A\) by taking the absolute value of the coefficient in front of the cosine: \(A = | -3 | = 3\).

Find the period using the formula \(\text{Period} = \frac{2\pi}{B}\). Here, \(B\) is the coefficient of \(x\), which is 1, so the period is \(2\pi\).

Identify the phase shift \(C\), which is \(-\pi\). This means the graph is shifted \(\pi\) units to the left. Use this information to sketch one full period of the function starting from \(x = -\pi\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. For y = -3 cos(x + π), the amplitude is |−3| = 3, indicating the graph oscillates 3 units above and below its midline.

Period of a Cosine Function

The period is the length of one complete cycle of the function. For y = cos(bx), the period is calculated as 2π/|b|. In y = −3 cos(x + π), since b = 1, the period is 2π, meaning the function repeats every 2π units along the x-axis.

Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, determined by solving inside the function for zero: x + π = 0 gives x = −π. This means the graph shifts π units to the left, affecting where the cosine wave starts its cycle.

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