the black graph is y = cos x.\nchoose the equation for the red graph.\n(0,1)\n(π/2,1)\n-2π\n-π\nπ\n2π\n-2π\n…

Explanation:

Step1: Recall the cosine function transformation rules

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

Step2: Analyze the key - point shift

For the black graph (y=\cos x), the key - point ((0,1)) (since (\cos(0)=1)). For the red graph, when (y = 1), (x=\frac{\pi}{2}). We know that if (y=\cos(x - C)), when (y = 1), (x - C=2k\pi) ((k\in\mathbb{Z})). Let (k = 0), if (x=\frac{\pi}{2}) and (y = 1), then (\frac{\pi}{2}-C=0), so (C=\frac{\pi}{2})

Step3: Check the function

Substitute (C=\frac{\pi}{2}) into (y=\cos(x - C)), we get (y=\cos(x-\frac{\pi}{2})). Also, we know the trigonometric identity (\cos(x-\frac{\pi}{2})=\sin x). Another way: The graph of (y = \cos(x)) is shifted to the right by (\frac{\pi}{2}) units. According to the horizontal shift rule for the function (y = f(x)) to (y=f(x - C)) (right - shift by (C) units when (C>0)), for (f(x)=\cos x) and (C = \frac{\pi}{2}), the function of the red graph is (y=\cos(x-\frac{\pi}{2}))

Answer:

(y=\cos(x - \pi/2))