Question

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

Explanation:

Step1: Recall the transformation rules for sine functions

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

Step2: Analyze the shift of the red graph compared to \(y=\sin x\)

The black graph \(y = \sin x\) has a point \((0,0)\). The red graph has a point \((-\frac{\pi}{2},0)\). This means the graph of \(y=\sin x\) is shifted to the left by \(\frac{\pi}{2}\) units.
For a function \(y = f(x)\), a shift to the left by \(a\) units gives \(y=f(x + a)\). Here \(a=\frac{\pi}{2}\) and \(f(x)=\sin x\), so the function of the red - graph is \(y=\sin(x+\frac{\pi}{2})\)

Answer:

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