find the equation for the graph. use the red point as your \starting point\.\n$y = ?\\sin(x…

Explanation:

Step1: Determine the amplitude

The amplitude $A$ of a sine - wave $y = A\sin(B(x - C))+D$ is half the vertical distance between the maximum and minimum values. The maximum value is $y = 3$ and the minimum value is $y=-1$. So, $A=\frac{3 - (-1)}{2}=\frac{4}{2}=2$.

Step2: Determine the vertical shift

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

Step3: Analyze the given form

The given form is $y = A\sin(x - C)+D$, where $B = 1$ (since there is no coefficient in front of $x$ inside the sine function). The starting point $(\frac{\pi}{4},1)$ is used to confirm the phase - shift $C=\frac{\pi}{4}$.

Answer:

$y = 2\sin(x-\frac{\pi}{4})+1$