Question

1. find the equation for the graph.

Explanation:

Step1: Identify the general form

The general form of a sine - wave is $y = A\sin(B(x - C))+D$ or $y = A\cos(B(x - C))+D$. From the graph, the amplitude $A$ needs to be determined, and the period is used to find $B$. The vertical shift $D$ and horizontal shift $C$ also need to be identified. Assume the graph is a sine - function of the form $y = A\sin(Bx - C)+D$.

Step2: Calculate the amplitude

The amplitude $A$ is half of the vertical distance between the maximum and minimum values of the function. Observing the graph, if we assume the maximum value is around $y = 4$ and the minimum value is around $y=-4$, then $A=\frac{4 - (- 4)}{2}=4$.

Step3: Calculate the period and $B$

The period $P$ is given as $P = 8\pi$. The formula for the period of a sine - function $y = A\sin(Bx - C)+D$ is $P=\frac{2\pi}{B}$. Since $P = 8\pi$, we can solve for $B$:
\[8\pi=\frac{2\pi}{B}\]
\[B=\frac{2\pi}{8\pi}=\frac{1}{4}\]

Step4: Determine the vertical and horizontal shifts

Assume the graph passes through the origin $(0,0)$ for simplicity (if no other information about phase - shift is given). The vertical shift $D = 0$ (because the graph is centered around the $x$ - axis), and the horizontal shift $C = 0$. So the equation of the sine - function is $y = 4\sin(\frac{1}{4}x)$.

Answer:

$y = 4\sin(\frac{1}{4}x)$