Question

the black graph is y = cos x.
choose the equation for the red graph.
(0,1)
(π/2,1)
-2π
-π
π
2π
-2π
-π
π
2π
-2
-2
2
2
y = cos(x + π/2)
y + 1 = cos x
y = cos(x - π/2)
y - 1 = cos x

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:

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