write the equation of the trigonometric function shown in the graph. use a positive coefficient on sine for…

Explanation:

Step1: Find the amplitude

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

Step2: Find the period

The period $T$ is the horizontal distance between two consecutive peaks. From the graph, $T=\frac{\pi}{2}$. The formula for the period of a sine - function $y = A\sin(B(x - C))+D$ is $T=\frac{2\pi}{B}$. Since $T=\frac{\pi}{2}$, we have $\frac{\pi}{2}=\frac{2\pi}{B}$, solving for $B$ gives $B = 4$.

Step3: Find the phase - shift

The graph is not shifted horizontally from the standard sine - function position (the mid - line crosses the $y$ - axis at the mid - point of the cycle), so the phase - shift $C = 0$.

Step4: Find the vertical shift

The mid - line of the graph is $y = 1$. So the vertical shift $D = 1$.

Answer:

$y = 4\sin(4(x - 0))+1$ or $y = 4\sin(4x)+1$