16. write the equation of the trig function below

Answer:

$y = 2\sin\left(\frac{\pi}{5}x\right)+1$

Explanation:

Step1: Determine the amplitude

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

Step2: Determine the vertical - shift

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

Step3: Determine the period

The period $T$ is the horizontal distance between two consecutive peaks. From the graph, $T = 10$. The formula for the period of a sine function is $T=\frac{2\pi}{B}$. Since $T = 10$, then $10=\frac{2\pi}{B}$, and solving for $B$ gives $B=\frac{\pi}{5}$.

Step4: Write the equation

The general form of a sine function is $y = A\sin(Bx - C)+D$. Since there is no horizontal shift ($C = 0$), the equation of the function is $y = 2\sin\left(\frac{\pi}{5}x\right)+1$.