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: Analyze the vertical shift

The general form of a vertical shift of the function \(y = f(x)\) is \(y - k=f(x)\), where \(k\) is the vertical shift. If \(k>0\), the graph shifts up; if \(k < 0\), the graph shifts down.
The black graph \(y=\sin x\) passes through \((0,0)\), and the red graph passes through \((0,- 2)\).
Substitute \(x = 0\) into the equations:

• For \(y+2=\sin x\), when \(x = 0\), \(y+2=\sin(0)=0\), then \(y=-2\).
• For \(y - 2=\sin x\), when \(x = 0\), \(y-2=\sin(0)=0\), then \(y = 2\).
• For \(y=\sin(x+\frac{\pi}{2})\), when \(x = 0\), \(y=\sin(\frac{\pi}{2})=1\).
• For \(y=\sin(x-\frac{\pi}{2})\), when \(x = 0\), \(y=\sin(-\frac{\pi}{2})=-1\).

Answer:

\(y + 2=\sin x\)