the black graph is y = sin x. choose the equation for the red graph. y = sin(x + π/2) y = sin(x - π/2) y =…

the black graph is y = sin x. choose the equation for the red graph. y = sin(x + π/2) y = sin(x - π/2) y = sin(x) + 2 y = sin(x) - 2

the black graph is y = sin x. choose the equation for the red graph. y = sin(x + π/2) y = sin(x - π/2) y = sin(x) + 2 y = sin(x) - 2

Answer

Explanation:

Step1: Analyze vertical shift

The general form of a sine function is (y = A\sin(Bx - C)+D), where (D) is the vertical shift. The black graph (y = \sin x) has a mid - line (y = 0). The red graph has a point ((0,-2)). For (y=\sin(x)+k), when (x = 0), (y=\sin(0)+k). Since (\sin(0)=0), if (y=-2) when (x = 0), then (k=-2).

Step2: Check other options

  • For (y=\sin(x+\frac{\pi}{2})=\cos x), when (x = 0), (y=\cos(0) = 1).
  • For (y=\sin(x-\frac{\pi}{2})=-\cos x), when (x = 0), (y=-\cos(0)=-1).
  • For (y=\sin(x)+2), when (x = 0), (y=\sin(0)+2=2).

Answer:

(y=\sin(x)-2)