the black graph is ( y = sin x ).\nchoose the equation for the red graph.\n( y - 2 = sin x )\n( y + 2 = sin…

Explanation:

Step1: Recall the rules of function translation

For a function (y = f(x)), a horizontal translation is of the form (y=f(x - h)) (shift right by (h) units) or (y = f(x+h)) (shift left by (h) units), and a vertical translation is of the form (y=f(x)+k) (shift up by (k) units) or (y=f(x)-k) (shift down by (k) units).

Step2: Analyze the given graphs

The black graph is (y = \sin x) which has a zero - crossing at ((0,0)). The red graph has a zero - crossing at ((\frac{\pi}{2},0)). Let (y=\sin(x - h)). When (y = 0), (0=\sin(x - h)), then (x - h=n\pi), (n\in\mathbb{Z}). For the basic function (y = \sin x), when (n = 0), (x = 0). For the red graph, when (n = 0), (x=\frac{\pi}{2}). Substituting into (x - h=0) (since (\sin(0)=0)), we get (\frac{\pi}{2}-h = 0), so (h=\frac{\pi}{2}). If we consider the general form of a horizontal shift (y=\sin(x - h)), substituting (h=\frac{\pi}{2}) gives (y=\sin(x-\frac{\pi}{2})). We can also check using the formula (\sin(A - B)=\sin A\cos B-\cos A\sin B). So, (\sin(x-\frac{\pi}{2})=\sin x\cos\frac{\pi}{2}-\cos x\sin\frac{\pi}{2}=-\cos x). But we can also use the transformation rule directly. For vertical - shift equations (y\pm2=\sin x) (i.e., (y=\sin x\mp2)), the mid - line of (y = \sin x) is (y = 0). If we shift (y=\sin x) up by 2 units ((y=\sin x + 2)) or down by 2 units ((y=\sin x-2)), the zero - crossing of (y=\sin x) (where (y = 0)) would not be at (x=\frac{\pi}{2}) for these vertical - shift functions.

Answer:

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