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The amplitude of a trigonometric function, like a cosine or sine function, is a measure of its maximum displacement from the mean value or equilibrium position. Think of it as the height of the wave from the center line to its peak (or trough). Amplitude tells you how tall or short the waves are.
For a cosine function in the standard form \( y = a \cos(bx + c) \), the amplitude is determined by the absolute value of \( a \). It's important because it affects the vertical stretch or shrinkage of the wave. In our case, the function given is \( y = 4 \cos(x + \pi) \). Here, \( a = 4 \). Thus, the amplitude is \(|4| = 4\).
Not only does this tell us that each peak reaches up to 4 units above and below the center line (y=0), but it also makes the wave quite pronounced, providing a clear visual of the wave's motion.

In trigonometry, the period of a function describes how long it takes for the function to repeat its pattern. It tells us the distance along the x-axis (or time) required for one full cycle of the wave.
With a cosine function, such as \( y = a \cos(bx + c) \), the standard formula to determine the period is \( \frac{2\pi}{|b|} \). Here, \( b \) is the multiplier of the x-variable, and it affects the wave's frequency or how often it repeats over a given interval.
For the given function \( y = 4 \cos(x + \pi) \), \( b = 1 \). This means that the period is \( \frac{2\pi}{1} = 2\pi \). That is, after every \( 2\pi \) units, the wave returns to its initial position and pattern. This period is standard for many sine and cosine functions, indicating a steady and consistent cyclic behavior of the graph.

The phase shift of a trigonometric function describes the horizontal shift along the x-axis. It tells us how the wave is translated left or right from its usual position.
In a function \( y = a \cos(bx + c) \), the phase shift is computed as \( -\frac{c}{b} \). This calculation indicates the starting point of the wave. A negative sign in front of the result indicates a shift to the left, while a positive sign indicates a shift to the right.
For the function \( y = 4 \cos(x + \pi) \), \( c = \pi \) and \( b = 1 \). Plugging these values into our formula, we find the phase shift to be \(-\frac{\pi}{1} = -\pi\). This means that the entire graph of the cosine wave is shifted \( \pi \) units to the left.
Understanding phase shift is crucial as it helps us determine the starting point of the wave on the coordinate plane, allowing for precise plotting and analysis of the wave's behavior over time.