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The amplitude of a trigonometric function, like the cosine function in our exercise, gives us important information about the graph's height. It tells us the maximum distance from the baseline or midline to the peak (or trough) of the wave. In simpler terms, it's how tall the wave gets.
For the function given, which is \( y = 10 \cos \left( \frac{\pi}{6} x \right) \), the amplitude is the number in front of the cosine function. Here, the amplitude is 10.

• This means the wave reaches a high point of 10 units above the midline.
• It equally goes 10 units below the midline, forming the lowest points on the graph.

The amplitude is crucial because it dictates the vertical stretch or compression of the graph. A larger amplitude means taller waves, while a smaller one would produce shorter waves.

The period of a trigonometric function tells us how long it takes to complete one full cycle. This is like finding out how long it takes for a pattern to repeat itself on the graph.
To calculate the period of a function like \( y = 10 \cos \left( \frac{\pi}{6} x \right) \), we use the formula: \[\text{Period} = \frac{2\pi}{b}\]where \( b \) is the coefficient of \( x \) inside the cosine. Here, \( b = \frac{\pi}{6} \). So, our calculation will be: \[\text{Period} = \frac{2\pi}{\frac{\pi}{6}} = 12\]This means the wave pattern repeats every 12 units along the x-axis. Knowing the period is significant because it helps in spacing the important points on the graph. Essentially, the graph completes one cycle from start to finish every 12 units.

Phase shift refers to the horizontal movement of the graph along the x-axis. It shows how much the entire wave pattern is shifted sideways.
In the function \( y = 10 \cos \left( \frac{\pi}{6} x \right) \), we look for any additions or subtractions inside the trigonometric part (like slopes of a building) to determine the phase shift. Without any plus or minus sign inside the cosine, the phase shift is zero.

• This means the graph is perfectly aligned with the origin without any left or right movement.
• If, for example, there was \( x - c \) inside the cosine, the graph would shift \( c \) units to the right.
• If it was \( x + c \), the shift would be \( c \) units to the left.

Being aware of phase shifts allows you to understand how the function might start its cycle from a different point, but here, it’s straightforward without any shift.