Question

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

Explanation:

Step1: Analyze vertical shift

The general form of a sine function is \(y = A\sin(Bx - C)+D\), where \(D\) is the vertical shift. The black graph \(y = \sin x\) has a mid - line \(y = 0\). The red graph has a point \((0,-2)\).
For \(y=\sin(x)+k\), when \(x = 0\), \(y=\sin(0)+k\). Since \(\sin(0)=0\), if \(y=-2\) when \(x = 0\), then \(k=-2\).

Step2: Check other options

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

Answer:

\(y=\sin(x)-2\)