Question

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 = sin x y = sin(x - π/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:

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