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 $5$ and the minimum is $ - 1$. So $A=\frac{5 - (-1)}{2}=\frac{6}{2}=3$.

Step2: Determine the vertical - shift

The vertical - shift $D$ is the average of the maximum and minimum values. So $D=\frac{5+( - 1)}{2}=\frac{4}{2}=2$.

Step3: Determine the period

The period $T$ is the horizontal distance between two consecutive maxima or minima. From the graph, $T = 2$. The formula for the period of a sine or cosine function is $T=\frac{2\pi}{B}$, so $2=\frac{2\pi}{B}$, and solving for $B$ gives $B=\pi$.

Step4: Determine the phase - shift

The graph starts at a maximum, so we can use a cosine function with no phase - shift ($C = 0$). The general form of a cosine function is $y=A\cos(Bx - C)+D$.

Answer:

$y = 3\cos(\pi x)+2$