Identifying the Properties of a Sinusoidal Function

Begin by comparing the equation to the general form and use the steps outlined in Example 9.

• Step 1. The function is already written in general form.
• Step 2. Since $A=-2,$ the amplitude is $\left|A\right|=2.$
• Step 3. $\left|B\right|=\frac{\pi }{2},$ so the period is $P=\frac{2\pi }{\left|B\right|}=\frac{2\pi }{\frac{\pi }{2}}=2\pi \cdot \frac{2}{\pi }=4.$ The period is 4.
• Step 4. $C=-\pi ,$ so we calculate the phase shift as $\frac{C}{B}=\frac{-\pi ,}{\frac{\pi }{2}}=-\pi \cdot \frac{2}{\pi }=-2.$ The phase shift is $-2.$
• Step 5. $D=3,$ so the midline is $y=3,\hspace{0.17em}$ and the vertical shift is up 3.

Since $A$ is negative, the graph of the cosine function has been reflected about the x-axis.

Figure 21 shows one cycle of the graph of the function.