91Ó°ÊÓ

When we talk about the amplitude of a trigonometric function, we're referring to the "height" of the wave, which is essentially the maximum distance it goes from its equilibrium position (the center line of the wave graph).

For the cosine function given in this exercise, the formula is \( y = -2 \cos(2x - \frac{\pi}{6}) \). The amplitude is found by taking the absolute value of the coefficient in front of the cosine function, which is "\( a \)" in the formula \( y = a \cos(bx - c) + d \).

• Here, \( a = -2 \) so the amplitude is \( |-2| = 2 \).
• Amplitude is always positive, as it represents distance.

This means that the graph of our function will oscillate 2 units above and below the central axis (which is \( y = 0 \) since \( d = 0 \)). The negative sign in front of the "2" indicates that our function starts by going downwards rather than upwards.

The period of a trigonometric function tells us how long it takes for the function to complete one full cycle.

In the general cosine function \( y = a \cos(bx - c) \), the period is calculated by the formula \( \frac{2\pi}{b} \).

• For this problem, \( b = 2 \), which makes the period \( \frac{2\pi}{2} = \pi \).
• This indicates that the wave repeats every \( \pi \) units.

Understanding the period is crucial because it tells you how frequently the wave pattern repeats itself over the x-axis. So, for \( y = -2 \cos(2x - \frac{\pi}{6}) \), you can expect one complete cycle of the wave from \( x = 0 \) to \( x = \pi \).

Phase shift describes how the graph of the trigonometric function is slid horizontally from its usual position.

The formula to calculate the phase shift for the function \( y = a \cos(bx-c) \) is \( \frac{c}{b} \).

• In our equation, \( c = \frac{\pi}{6} \) and \( b = 2 \).
• This results in a phase shift of \( \frac{\frac{\pi}{6}}{2} = \frac{\pi}{12} \).

Since the formula \( bx - c \) suggests the shift is to the right, the graph is slid to the right by \( \frac{\pi}{12} \) units. This phase shift adjusts where the cycle of the cosine wave begins on the x-axis.

The cosine function is a fundamental trigonometric function that describes a wave-like pattern. It is periodic, meaning it repeats its values in a regular cycle.

The general form of a cosine function is \( y = a \cos(bx-c) + d \). Here, each variable component influences the graph in different ways:

• "\( a \)" affects the amplitude.
• "\( b \)" determines the period.
• "\( c \)" impacts the phase shift.
• "\( d \)" adjusts the vertical shift, though it is zero in our case.

In the equation \( y = -2 \cos(2x - \frac{\pi}{6}) \), the graph typically starts at the peak, but given a negative sign in front of \( a \), it begins at a trough (negative peak). The cosine function is essential for modeling cyclical patterns and can be found in various scientific contexts, from sound waves to light waves, making it a vital part of trigonometry study.