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Amplitude in trigonometric functions determines how tall the wave peaks are from the midpoint, which is often the horizontal axis if no vertical shifts are involved. In simple terms, amplitude tells us how high and low the graph of the function goes.For the function \(y = \frac{5}{2} \cos \frac{x}{4}\), amplitude is determined by the coefficient in front of the cosine function.- To find amplitude, take the absolute value of this coefficient.- In our example, the coefficient is \(\frac{5}{2}\).- Therefore, the amplitude is \(\frac{5}{2}\).Remember, amplitude will always be a positive number because it represents a distance. Whether the graph stretches above or below its middle point, it extends equally in both directions.

The period of a trigonometric function is the length of one complete cycle of the wave. It tells you how much you have to move along the x-axis before the wave pattern repeats.To determine the period of a cosine function, use the formula:- \[ \text{Period} = \frac{2\pi}{B} \]where \(B\) is the coefficient of \(x\) inside the cosine.In our function \(y = \frac{5}{2} \cos \frac{x}{4}\):- The coefficient \(B = \frac{1}{4}\).- Plugging \(B\) into the period formula gives:- \[ \frac{2\pi}{\frac{1}{4}} = 8\pi \]So, the period of this function is \(8\pi\). This means the cosine wave takes a distance of \(8\pi\) to start repeating its shape.

The cosine function, part of the family of trigonometric functions, is commonly used in scenarios involving waves and circular motion. The basic form is \(y = A \cos(Bx + C) + D\), where each letter represents a transformation or characteristic:- \(A\): Amplitude, changing the height of the wave as previously discussed.- \(B\): Affects the period of the wave.- \(C\): Changes the horizontal phase shift, moving the graph along the x-axis.- \(D\): Adjusts the vertical shift, moving the graph up or down.The function \(y = \frac{5}{2} \cos \frac{x}{4}\) is a transformed version of the basic cosine wave. Here, \(A = \frac{5}{2}\) makes peaks and valleys extend \(\frac{5}{2}\) units vertically from the middle line. \(B = \frac{1}{4}\) increases the period to \(8\pi\), meaning the full wave cycle occurs more slowly.