Finding the Amplitude and Period of a Function

We will solve these problems according to the models.

1. ⓐ $y=2\sin \left(\frac{1}{4}x\right)$ involves sine, so we use the form
   $$y=A\sin \left(Bt+C\right)+D$$

   We know that $\left|A\right|$ is the amplitude, so the amplitude is 2. Period is $\frac{2\pi }{B},$ so the period is

   $$\begin{array}{l}\frac{2\pi }{B}=\frac{2\pi }{\frac{1}{4}}\hfill \\ =8\pi \hfill \end{array}$$

   See the graph in Figure 3.

   

   Figure 3
2. ⓑⓑ $y=\mathrm{-3}\sin \left(2x+\frac{\pi }{2}\right)$ involves sine, so we use the form
   $$y=A\sin \left(Bt-C\right)+D$$

   Amplitude is $\left|A\right|,$ so the amplitude is $\left|-3\right|=3.$ Since $A$ is negative, the graph is reflected over the x-axis. Period is $\frac{2\pi }{B},$ so the period is

   $$\frac{2\pi }{B}=\frac{2\pi }{2}=\pi$$

   The graph is shifted to the left by $\frac{C}{B}=\frac{\frac{\pi }{2}}{2}=\frac{\pi }{4}$ units. See Figure 4.

   

   Figure 4
3. ⓒ $y=\cos x+3$ involves cosine, so we use the form
   $$y=A\cos \left(Bt\pm C\right)+D$$

   Amplitude is $\left|A\right|,$ so the amplitude is 1. The period is $2\pi .$ See Figure 5. This is the standard cosine function shifted up three units.

   

   Figure 5