which of the following is an equation for the graph? a. $2sin(2x+\frac{pi}{2}) + 2$ b. $sin(2x+pi)+2$ c…

Explanation:

Step1: Determine the amplitude

The general form of a sinusoidal function is $y = A\sin(Bx - C)+D$, where $A$ is the amplitude. The distance from the maximum or minimum of the wave to the mid - line is the amplitude. The mid - line of the given graph is $y = 2$, and the maximum value is $y = 4$ and the minimum is $y = 0$. So the amplitude $A=\frac{4 - 0}{2}=2$.

Step2: Determine the period

The period of a sinusoidal function $y = A\sin(Bx - C)+D$ is $T=\frac{2\pi}{B}$. The period of the given graph is $\pi$. Since $T=\frac{2\pi}{B}=\pi$, we can solve for $B$: $B = 2$.

Step3: Determine the phase - shift and vertical shift

The vertical shift $D$ is the value of the mid - line. Here $D = 2$. We know that the standard sine function $y=\sin x$ has a zero - crossing at $x = 0$. For the given function, when $x = 0$, the function value is at the mid - line and decreasing. The sine function $y = 2\sin(2x-\frac{\pi}{2})+2$ has the correct behavior. When $x = 0$, $y=2\sin(-\frac{\pi}{2})+2=2\times(- 1)+2=0$.

Answer:

D. $2\sin(2x-\frac{\pi}{2})+2$