Question

the black graph is y = sin x. choose the equation for the red graph. y = sin(x + π) y + 1 = sin x y = sin(x - π) y - 1 = sin x

Explanation:

Step1: Analyze the transformation of the sine function

The general form of a sine function transformation is \(y = A\sin(B(x - C))+D\), where \(D\) is the vertical shift.
For the black graph \(y=\sin x\), when \(x = 0\), \(y = 0\). For the red graph, when \(x = 0\), \(y=1\).

Step2: Check each option

• For \(y=\sin(x+\pi)\), when \(x = 0\), \(y=\sin(\pi)=0\).
• For \(y + 1=\sin x\), then \(y=\sin x- 1\), when \(x = 0\), \(y=-1\).
• For \(y=\sin(x - \pi)\), when \(x = 0\), \(y=\sin(-\pi)=0\).
• For \(y-1=\sin x\), then \(y=\sin x + 1\). When \(x = 0\), \(y=\sin(0)+1=1\).

Answer:

\(y - 1=\sin x\)