Question

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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Identify the given function: $$y = 2 \cos\left(\frac{1}{3}xight) - 2. $$Notice it is a cosine function with amplitude, horizontal stretch/compression, and vertical shift. 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. Find the period of the function using the formula for cosine period: $$	ext{Period} = \frac{2\pi}{B}$$, where $$B is $$the coefficient of $$x$$ inside the cosine. Here, $$B = \frac{1}{3}$$, so the period is $$\frac{2\pi}{\frac{1}{3}} = 6\pi. $$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. 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.

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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Key Concepts

Vertical Shift in Trigonometric Functions

Period of a Cosine Function

Amplitude of a Cosine Function