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

Explanation:

Step1: Recall the transformation rules of the sine function

The general form of a sine function transformation is (y = A\sin(B(x - C))+D), where (C) represents the horizontal shift. If (C>0), the graph shifts to the right; if (C < 0), the graph shifts to the left.

Step2: Analyze the horizontal shift

The black graph (y=\sin x) has a key - point at ((0,0)). The red graph has a key - point at ((-\frac{\pi}{2},0)). This means the graph of (y = \sin x) is shifted to the left by (\frac{\pi}{2}) units. For the function (y=\sin(x - C)), when the graph is shifted to the left by (\frac{\pi}{2}) units, we have (C=-\frac{\pi}{2}). Substituting (C =-\frac{\pi}{2}) into (y=\sin(x - C)), we get (y=\sin(x+\frac{\pi}{2})). The equations (y - 2=\sin x) (which can be rewritten as (y=\sin x + 2), representing a vertical shift up by 2 units) and (y + 2=\sin x) (which can be rewritten as (y=\sin x-2), representing a vertical shift down by 2 units) do not match the horizontal shift observed. The equation (y=\sin(x-\frac{\pi}{2})) represents a shift to the right by (\frac{\pi}{2}) units.

Answer:

(y=\sin(x +\frac{\pi}{2}))