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When we talk about the amplitude of a trigonometric function, we are essentially discussing the height of the wave created by the function. Specifically, in the context of a cosine function, amplitude measures how far the peaks and valleys of the wave are from the centerline, which is typically the x-axis.

Amplitude is always expressed as a positive value, derived from the coefficient in front of the cosine term. For the standard form of a cosine function, given as \( y = a \cos(bx + c) + d \), the amplitude is \( \left| a \right| \). Simply put, just take the absolute value of \( a \).

In the example function \( y = -\frac{1}{3} \cos \frac{1}{3}x \), the coefficient \( a \) is \( -\frac{1}{3} \). So, the amplitude is \( \left| -\frac{1}{3} \right| = \frac{1}{3} \). This tells us that the wave of the cosine function reaches \( \frac{1}{3} \) units above and below the x-axis. The negative sign indicates that the wave's direction is inverted.

The term "period" in trigonometry refers to the distance over which a trigonometric function repeats its values. For periodic functions like the cosine function, the period is the length of one complete cycle of the wave.

In the equation \( y = a \cos(bx + c) + d \), the period can be calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). This formula tells us how long it takes for the wave to start repeating itself.

Let’s apply this to the function \( y = -\frac{1}{3} \cos \frac{1}{3}x \). Here, \( b \) is \( \frac{1}{3} \). Plugging this into the period formula, we get \( \frac{2\pi}{\frac{1}{3}} = 6\pi \). This means that the cosine function takes \( 6\pi \) units along the x-axis to complete one full oscillation and start over.

The cosine function is a fundamental trigonometric function that oscillates between -1 and 1. It describes the cosine of an angle in a right-angled triangle, but in the context of the Cartesian plane, it forms a wave-like pattern.

Written in its standard form as \( y = a \cos(bx + c) + d \), it illustrates several important attributes:

• Amplitude \( a \): How tall or short the wave is.
• Period: Determined by \( b \), as \( \frac{2\pi}{b} \), explaining how long it takes for the cycle to begin again.
• Vertical Shift \( d \): Moves the whole wave up or down.
• Phase Shift \( c \): Shifts the wave left or right along the x-axis.

The function \( y = -\frac{1}{3} \cos \frac{1}{3}x \) exhibits some classic characteristics of the cosine function. It has an amplitude of \( \frac{1}{3} \), indicating heigh variation of the wave peaks from the center. Meanwhile, the negative sign in front of the amplitude flips the graph. The period of \( 6\pi \), deduced from \( \frac{2\pi}{\frac{1}{3}} \), indicates that it requires \( 6\pi \) distance for the cosine wave to fully repeat. These features allow us to graph it accurately over its range.