Question

Find an equation for the graph.

Short answer

The equation of the given graph is $y=-\frac{1}{2}\sin \left(\frac{3}{2}x\right)-1$.

Step-by-step solution

Step 1. Determine the sinusoidal function.

Consider the given graph,

The type of sinusoidal graph can be determined immediately by inspecting its y-intercept if its at a maximum(highest point) and a minimum(lowest point) or neither.

As there is neither a maximum or minimum at its y-intercept, then the graph is a sine function and the minimum occurs before the maximum and $A<0$. Then,

$$\begin{gathered} y_{min}=-\frac{3}{2} \\ {y}_{max}=-\frac{1}{2} \end{gathered}$$

Step 2. Find the amplitude and period.

Consider the given question,

$$\begin{gathered} A=\frac{1}{2}\left(y_{max}-y_{min}\right) \\ A=-\frac{1}{2}\left[-\frac{1}{2}-\left(-\frac{3}{2}\right)\right] \\ A=-\frac{1}{2}\left(\frac{2}{2}\right) \\ A=-\frac{1}{2} \end{gathered}$$

The period is $\frac{4}{3}\mathrm{\pi }$. So,

$$\begin{gathered} \frac{4\mathrm{\pi }}{3}=\frac{2\mathrm{\pi }}{\omega } \\ \omega =\frac{2\mathrm{\pi }}{4\mathrm{\pi }}\times 3 \\ \omega =\frac{3}{2} \end{gathered}$$

Step 3. Form the equation.

Vertical shift,

$$\begin{gathered} b=\frac{1}{2}\left(y_{max}+y_{min}\right) \\ b=\frac{1}{2}\left[-\frac{1}{2}+\left(-\frac{3}{2}\right)\right] \\ b=\frac{1}{2}\left(-\frac{4}{2}\right) \\ b=-1 \end{gathered}$$

Consider the form of the function,

$y=A\sin \left(\omega x\right)+b$

Substitute the values,

$y=-\frac{1}{2}\sin \left(\frac{3}{2}x\right)-1$