find an equation for the sine graph, f(x): write your answer in the form f(x) = asin (bx + c) + d, where a…

Explanation:

Step1: Determine the amplitude A

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

Step2: Determine the period and B

The period $T$ of a sine function $y = A\sin(Bx + C)+D$ is given by $T=\frac{2\pi}{|B|}$. The graph has a period of $4\pi$. So, $4\pi=\frac{2\pi}{|B|}$, solving for $B$ gives $|B|=\frac{2\pi}{4\pi}=\frac{1}{2}$. Since the graph is not reflected horizontally, $B=\frac{1}{2}$.

Step3: Determine the phase - shift C

The standard sine function $y = \sin x$ passes through the origin $(0,0)$. The given sine - like function passes through the origin, so the phase - shift $C = 0$.

Step4: Determine the vertical shift D

The mid - line of the function is $y = 0$. So, the vertical shift $D = 0$.

Answer:

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