In Exercises 12–13, use a vertical shift to graph one period of the function. y = 2 cos 1/3 x − 2

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1. Identify the given function: \(y = 2 \cos\left(\frac{1}{3}x\right) - 2\). Notice it is a cosine function with amplitude, horizontal stretch/compression, and vertical shift.
2. Determine the amplitude, which is the coefficient in front of the cosine: \(A = 2\). This means the graph will oscillate 2 units above and below its midline.
3. Find the period of the function using the formula for cosine period: \(\text{Period} = \frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the cosine. Here, \(B = \frac{1}{3}\), so the period is \(\frac{2\pi}{\frac{1}{3}} = 6\pi\).
4. Identify the vertical shift, which is the constant added or subtracted outside the cosine function. Here, it is \(-2\), so the midline of the graph is shifted down 2 units from the \(x\)-axis.
5. To graph one period, start at \(x=0\) and plot points through one full period length \(6\pi\), adjusting the cosine values by multiplying by the amplitude and then shifting down by 2. The key points are at \(x=0\), \(x=\frac{6\pi}{4}\), \(x=\frac{6\pi}{2}\), \(x=\frac{3\cdot6\pi}{4}\), and \(x=6\pi\), corresponding to the cosine wave's maximum, zero, minimum, zero, and maximum again, all shifted vertically.

Key Concepts

Vertical Shift in Trigonometric Functions

A vertical shift moves the entire graph of a function up or down without changing its shape. For y = 2 cos(1/3 x) - 2, the '-2' shifts the cosine graph down by 2 units, affecting the midline and range but not the period or amplitude.

Period of a Cosine Function

The period of a cosine function y = cos(bx) is given by 2π divided by the absolute value of b. Here, with b = 1/3, the period is 2π / (1/3) = 6π, meaning one full cycle of the cosine wave spans 6π units along the x-axis.

Amplitude of a Cosine Function

Amplitude is the height from the midline to the peak of the wave and is the absolute value of the coefficient before cosine. In y = 2 cos(1/3 x) - 2, the amplitude is 2, indicating the graph oscillates 2 units above and below its midline.