Question

Find an equation for each graph.

Short answer

Equation for the graph is:

$y=4\sin \left(\frac{1}{4}x\right)$

Step-by-step solution

Step 1. Inspecting the y-intercept

Referring to the given graph the type of sinusoid can be determined immediately by inspecting its y- intercept if its at a maximum(highest point) and a minimum(lowest point) or neither

At its y-intercept neither maximum or nor minimum, then the graph is a sine function and hence, $A>0$.

Step 2. Find the amplitude and period

From the graph, $$\begin{gathered} y_{min}=-4 \\ {y}_{max}=4 \end{gathered}$$

$$\begin{gathered} A=\frac{1}{2}(y_{max}-y_{min}) \\ A=\frac{1}{2}(4-(-4\left)\right) \\ A=\frac{1}{2}\left(8\right) \\ A=4 \end{gathered}$$

Using the maximum at$x=2\mathrm{\pi }$ , the next maximum occurs at$x=10\mathrm{\pi }$

Therefore,

$T=8\mathrm{\pi }$

Step 3. Solve for the equation

$$\begin{gathered} \omega =\frac{2\mathrm{\pi }}{T} \\ \omega =\frac{2\mathrm{\pi }}{8\mathrm{\pi }} \\ \omega =\frac{1}{4} \end{gathered}$$

Therefore, the equation of the graph is

$y=4\sin \left(\frac{1}{4}x\right)$