write the equation of the trigonometric function shown in the graph.

Explanation:

Step1: Determine the amplitude

The amplitude $A$ is half the distance between the maximum and minimum values. The maximum value is $1$ and the minimum is $ - 3$. So, $A=\frac{1 - (-3)}{2}=\frac{4}{2}=2$.

Step2: Determine the vertical shift

The vertical shift $D$ is the average of the maximum and minimum values. $D=\frac{1+( - 3)}{2}=\frac{-2}{2}=-1$.

Step3: Determine the period

The period $T$ is the distance between two consecutive peaks or troughs. From the graph, $T = 4\pi$. The formula for the period of a trig - function $y = A\sin(Bx - C)+D$ or $y = A\cos(Bx - C)+D$ is $T=\frac{2\pi}{B}$. Since $T = 4\pi$, then $4\pi=\frac{2\pi}{B}$, and solving for $B$ gives $B=\frac{1}{2}$.

Step4: Determine the phase shift

The graph passes through the origin $(0, - 1)$ and has the form of a sine - function (starts at the mid - line value). For a sine function $y = A\sin(Bx - C)+D$, when $x = 0,y=-1$. Substituting $A = 2,B=\frac{1}{2},D=-1$ into $y = A\sin(Bx - C)+D$ gives $-1=2\sin(-C)-1$. Then $\sin(-C)=0$, so $C = 0$.

The general form of a sine function is $y=A\sin(Bx - C)+D$. Substituting $A = 2,B=\frac{1}{2},C = 0,D=-1$ we get $y = 2\sin(\frac{1}{2}x)-1$.

Answer:

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