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Understanding the amplitude of trigonometric functions is crucial when graphing them. The amplitude is the distance from the horizontal axis (usually the midline of the graph) to the maximum or minimum point of the wave. It dictates how 'tall' or 'short' the wave of the function appears. In the given exercise, the amplitude is represented by the coefficient in front of the cosine function, and here the amplitude is 3. This means that our graph will oscillate 3 units above and 3 units below the horizontal axis.

Students often mistake amplitude for the vertical shift, but they are distinct; amplitude reflects the vertical stretch or compression of the wave, not its position relative to the axis. So, whether you are graphing a sine, cosine, or any other trigonometric function, remember to view the absolute value of the coefficient to determine the amplitude. This will ensure that you correctly depict the function's peaks and valleys on your graph.

The phase shift is essentially the horizontal movement of a trigonometric function along the x-axis. If you imagine your trig function as a wave moving through time, the phase shift would be how much time the wave needs to start its cycle. It's a key concept to capture the starting point of a trigonometric wave.

In our exercise, the function given was \(g(t)=3 \cos (t+\pi)\). The \(t+\pi\) inside the cosine indicates a horizontal shift. Specifically, the graph of our function is shifted \(\pi\) units to the left from the standard cosine function, which starts at (0,1). If it were minus inside the brackets, \(t-\pi\) for example, the graph would shift to the right. It's a common error to overlook the sign when determining the direction of the shift. Recognizing and accurately plotting the phase shift is essential as it gives the correct starting point for your cycle.

The period of a cosine function, or any trigonometric function, is the length it takes for the function to complete one full cycle before repeating itself. For a standard cosine function, such as \(\cos(t)\), the period is \(2\pi\), which means after every \(2\pi\) radians along the t-axis, the wave pattern repeats.

In the exercise, with our function \(g(t)=3 \cos (t+\pi)\), the period remains unchanged at \(2\pi\). When graphing the function, it's important to capture this repetition accurately by ensuring that every \(2\pi\) distance along the t-axis, the cycle of the wave starts anew. A common error is to change the period when a coefficient is included, but it's worth noting that only coefficients directly multiplying the variable inside the trigonometric function (the frequency) would affect the period. Thus, the step in the provided solution where we graph two full periods from \(-2\pi\) to \(2\pi\) ensures that we see the complete picture of how the function behaves over time.