Question

Find an equation for the graph.

Short answer

The equation of the given graph is $y=-1\cos \left(\frac{4\mathrm{\pi }}{3}x\right)+1$.

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 the minimum is at its y-intercept, then the graph is a cosine function and $A<0$. Then,

$$\begin{gathered} y_{min}=0 \\ {y}_{max}=2 \end{gathered}$$

Step 2. Find the amplitude and period.

Consider the given question,

$$\begin{gathered} A=\frac{1}{2}\left(y_{max}-y_{min}\right) \\ A=-\frac{1}{2}\left(2-0\right) \\ A=-1 \end{gathered}$$

The period is $\frac{3}{2}$. So,

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

Step 3. Form the equation.

Vertical shift,

$$\begin{gathered} b=\frac{1}{2}\left(y_{max}+y_{min}\right) \\ b=\frac{1}{2}\left(2+0\right) \\ b=\frac{1}{2}\left(2\right) \\ b=1 \end{gathered}$$

Consider the form of the function,

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

Substitute the values,

$y=-1\cos \left(\frac{4\mathrm{\pi }}{3}x\right)+1$