Graphing a Function and Identifying the Amplitude and Period

Let’s begin by comparing the equation to the form $y=A\sin (Bx).$

• Step 1. We can see from the equation that $A=-2,$ so the amplitude is 2.
  $$\left|A\right|=2$$
• Step 2. The equation shows that $B=\frac{\pi }{2},$ so the period is
  $$\begin{array}{l}P=\frac{2\pi }{\frac{\pi }{2}}\hfill \\ =2\pi \cdot \frac{2}{\pi }\hfill \\ =4\hfill \end{array}$$
• Step 3. Because $A$ is negative, the graph descends as we move to the right of the origin.
• Step 4–7. The x-intercepts are at the beginning of one period, $x=0,$ the horizontal midpoints are at $x=2$ and at the end of one period at $x=4.$

The quarter points include the minimum at $x=1$ and the maximum at $x=3.$ A local minimum will occur 2 units below the midline, at $x=1,$ and a local maximum will occur at 2 units above the midline, at $x=3.$ Figure 19 shows the graph of the function.

Figure 19