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The cosine function is a fundamental trigonometric function that plays a crucial role in various mathematical and scientific applications. It's often represented as \(\cos x\). When graphing the cosine function, we observe a distinct wave-like pattern that repeats consistently. This is due to its periodic nature.

Key characteristics of the cosine function include:

• It begins at its maximum value of 1 when the angle \(x\) is 0.
• As \(x\) increases, the value of \(\cos x\) decreases, reaching 0 at \(\pi/2\), then drops to its minimum value of -1 at \(\pi\).
• After \(\pi\), the pattern symmetrically ascends to 0 at \(3\pi/2\) and peaks at 1 again at \(2\pi\).

For the cosine function \(y = -\cos x\), there is a vertical reflection due to the negative sign in front of the cosine. Therefore, all the values of the customary cosine function are mirrored about the x-axis. Instead of starting at a maximum, it starts at a minimum value. This reflection results in the graph starting at \(-1\), reaching 0 at \(\pi/2\), peaking at 1 at \(\pi\), and so forth.

The period of a trigonometric function tells us how long it takes for the function to complete one full cycle of its pattern. For the basic cosine function \(y = \cos x\), this cycle length, known as the period, is \(2\pi\). This means that every \(2\pi\) units along the x-axis, the cosine function repeats its shape.

If we have a modified cosine function, such as \(y = -\cos\frac{1}{2}x\), the period of the function is affected by the coefficient of \(x\). We can calculate this new period by using the formula:

• Period = \(\frac{2\pi}{B}\)
• Where \(B\) is the coefficient of \(x\).

In the example \(y = -\cos\frac{1}{2}x\), the period becomes \(\frac{2\pi}{1/2}\), simplifying to \(4\pi\). Consequently, it takes \(4\pi\) to complete one full cycle, which is double the typical period of a regular cosine function due to adjusting the frequency with \(\frac{1}{2}\). This elongates the wave, creating wider arcs as seen in its graph.

The amplitude of a trigonometric function defines the height of its peaks and depths of its troughs from the function's midline. For a basic cosine function like \(y = \cos x\), the amplitude is 1. This implies the graph oscillates between 1 and -1.

This concept is quite straightforward: the amplitude is the absolute value of the coefficient in front of the cosine term. In a function like \(y = -\cos x\), the amplitude remains positive despite the negative sign because amplitude is always considered as a positive quantity. Thus, the amplitude is:

• Amplitude = \(\left|a\right|\) where \(a\) is the coefficient of the cosine function.

In the example of \(y = -\cos \frac{1}{2}x\), the amplitude is likewise 1. This means the wave's crests and troughs will reach 1 unit above and below its central axis, the x-axis in this instance. The negative sign simply reflects the wave over the x-axis, yet doesn't alter the amplitude. It's essential to remember, amplitude measures magnitude, not direction.