91Ó°ÊÓ

The period of a function is essentially the length of the smallest interval over which the function repeats itself. For the standard cosine function, the period is

• \(2\pi\)

This means that every \(2\pi\) units along the x-axis, the cosine wave completes one full cycle before starting over again. However, transforming the argument inside the cosine function affects the period. In our exercise, the function is \( \cos\left(\frac{x}{2}\right) \).
This transformation will stretch the period by a factor of 2 because the coefficient \(\frac{1}{2}\) in the argument of the cosine function doubles the length of the x-values needed to complete one full cycle.
Hence, the new period of \( \cos\left(\frac{x}{2}\right) \) is \( 4\pi \). This means from \( x=0 \) to \( x=4\pi \), the function completes exactly one cycle, which fits perfectly into the range specified for graphing in the exercise.

The amplitude of a trigonometric function is a measure of its highest vertical displacement from the horizontal axis. For the basic cosine and sine functions, their amplitudes are typically 1, since they oscillate consistently between -1 and 1. However, transformations such as multiplication change the amplitude.
In our function \( y = 2 \cos^2\left(\frac{x}{2}\right) \), the coefficient 2 affects the range of the function.

• Since \( \cos^2\) term ensures the values are squared, the range prior to scaling is between 0 and 1.
• Multiplying by 2 scales this up to between 0 and 2.

Thus, the amplitude is 2, meaning the highest point of the graph reaches up to 2 from the x-axis, while the lowest point stays at 0 due to squaring preventing negative values.

To accurately sketch a trigonometric graph, identifying key points is crucial. These points offer a framework that the rest of the graph can be drawn through. In our problem, carrying out computations at strategic values of x helps reveal the behavior and structure of the graph:

• At \( x = 0 \), \( y = 2 \cos^2 (0) = 2 \).
• At \( x = \pi \), \( y = 2 \cos^2\left(\frac{\pi}{2}\right) = 0 \).
• At \( x = 2\pi \), \( y = 2 \cos^2 (\pi) = 2 \).
• At \( x = 3\pi \), \( y = 2 \cos^2\left(\frac{3\pi}{2}\right) = 0 \).
• At \( x = 4\pi \), \( y = 2 \cos^2(2\pi) = 2 \).

Plotting these points on a graph provides markers of where the curve reaches its peaks (amplitude), dips (midpoints), and ends one complete cycle (period). Connecting them in a smooth, periodic wave immerses the core nature of the cosine function, completing the graph from 0 to \( 4\pi \).

Trigonometric functions are versatile and can be manipulated to produce a variety of waves by applying transformations. These transformations can involve changes to the amplitude, period, horizontal and vertical shifts, as well as reflections.
In the case of our equation \( y = 2 \cos^2\left(\frac{x}{2}\right) \), several transformations occur:

• The presence of \( \left(\frac{x}{2}\right) \) arguments doubles the period as discussed.
• The coefficient of 2 in front of the function scales up the amplitude by a factor of 2.

Combining these transformations, the graph of \( y = 2 \cos^2\left(\frac{x}{2}\right) \) will oscillate smoothly between 0 and 2, over the expanded duration of \( 4\pi \). The function undergoes its full set of transformations resulting in a taller, horizontally stretched graph compared to the standard cosine wave.