write the equation of a sine or cosine function to describe the graph

Explanation:

Step1: Determine the amplitude

The amplitude $A$ is half the distance between the maximum and minimum values. The maximum value is $4$ and the minimum is $- 2$. So $A=\frac{4 - (-2)}{2}=\frac{6}{2}=3$.

Step2: Determine the vertical shift

The vertical shift $D$ is the average of the maximum and minimum values. So $D=\frac{4+( - 2)}{2}=\frac{2}{2}=1$.

Step3: Determine the period

The period $T$ is the distance between two consecutive maxima or minima. From the graph, $T = 6$. Using the formula $T=\frac{2\pi}{B}$, we can solve for $B$. Since $T = 6$, then $6=\frac{2\pi}{B}$, and $B=\frac{\pi}{3}$.

Step4: Determine the phase - shift

The graph appears to be a cosine - type graph (starts at a maximum) with no phase - shift, so $C = 0$. The general form of a cosine function is $y=A\cos(Bx - C)+D$. Substituting $A = 3$, $B=\frac{\pi}{3}$, $C = 0$, and $D = 1$ into the formula, we get $y = 3\cos(\frac{\pi}{3}x)+1$.

Answer:

$y = 3\cos(\frac{\pi}{3}x)+1$